Why do my bass drivers shake violently listening to vinyl

Different signal from each. Vinyl putting out more bass than digital. The question is how does it sound? Is it distorted? If so, then you may have a problem. 

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

You likely have a mismatch between cartridge compliance and cartridge/arm mass.  Do a search.

@audionoobie - Some questions:

• Do the woofers visibly pulse in and out i.e. you can see the movement
• Is the "shaking" consistent for every album
• are the albums warped, so the pulsing matches the album warps 
• is the platter free from warps
• is the bearing in good shape
• is the stylus clean and in good condition
• what kind of stylus are you using
• does your phono stage have a sub-sonic filter

Take a read of this thread
Phono-pre: subsonic filter on or off ? | Audiogon Discussion Forum  

It might give you a clue as to what is going on

Good Luck

@williewonka interesting thread before it went off topic. The KAB rumble filter was brought up. Here is an excerpt from their site which accurately describes what I am seeing:

Getting into LP's and startled at the big speaker cone movements that you're seeing? Thinking a better turntable will help? It may not! This very low frequency energy is called rumble. Much of it is actually cut into the record groove when the master disc was made. But the main source is feedback exciting the natural stylus/arm resonance between 7 and 12 hz. This produces a slow easily visible in and out motion of your woofers. If you cannot see this motion, the RF-1 will not solve your feedback problem. If you can see this motion, the RF-1 will solve your feedback problem.

• Do the woofers visibly pulse in and out i.e. you can see the movement - Yes
• Is the "shaking" consistent for every album - Yes
• are the albums warped, so the pulsing matches the album warps - No
• is the platter free from warps - Yes
• is the bearing in good shape - Brand new Technics 1210GR
• is the stylus clean and in good condition - Yes brand new AT VM760SLC
• what kind of stylus are you using
• does your phono stage have a sub-sonic filter - Not sure. It is a Herron VTPH-1. I'll have to research that.

@melm It seems my cart is a very popular choice according to the Technics SL1200 FAcebook group. Many users using this cart.