Why do my bass drivers shake violently listening to vinyl

@audionoobie , That should be fine. I do not think the "footers" will add much if anything. Better to have the table on a heavy solid platform. 

@atmasphere , should be and are are two separate issues. I have records the rumble from a poorly maintained lathe is obnoxious. 

The woofer flapping issue is prevalent among people with small woofers that are tuned to go as low as possible. They have to move a lot farther to produce the same results as a large woofer. I have never seen the owner of a 15" woofer complain of this. It is always 10" or below and always with turntables. With many amps going clear down to DC surface irregularities in the vinyl will cause the woofers to flap. They do so in rhythm with the spinning record. This is not necessarily will loud volumes either. But if you do turn it up the problem can become obnoxious. If subwoofers are used and boosted 5 dB like mine you can easily bottom the subwoofer out. Interesting sound for sure. Once digitized it is a simple matter to program in a subsonic filter with a very steep slope sidestepping the issue altogether. If you use subwoofers and play loud using your turntable as the source a subsonic filter is virtually mandatory.

@atmasphere , I was not so fortunate. The 10" drivers in my AR 2ax's use to flop around plenty.

Audionoobie, welcome to vinyl. There is a lot of low frequency rubbish on records in the form of warps and irregularities in the vinyl. It can be made worse if your cartridge and arm are not matched correctly. unfortunately the flapping woofers increase distortion and waste power. Other than making sure your cartridge is appropriately matched the only other way to deal with this is a steep subsonic filter. A two way crossovers used with subwoofers will stop the flapping in your main speakers but will pass it on to the subwoofers so a subsonic filter is still necessary for the best performance. I use a very steep 8th order filter at 18 Hz. It is digital which is the best type of filter to use for this purpose.

Millercarbon, so far you have been totally and completely off base twice in this thread. The OPs problem has nothing to do with the turntable's location. Isolating it will do absolutely nothing. If there is a cartridge mismatch improving that will help otherwise his system is appropriately amplifying the signal that the cartridge is picking up. With smaller woofers this causes excess movement. Next. Stopping that excess movement results in a large improvement in sound quality because of the marked decrease in several forms of distortion. It also results in greater headroom because power is not being wasted. The improvement a subsonic filter will make will be far in excess of any problems it might cause. 
@lewm, subsonic filters are not complicated issues in the digital world, you just program one in. I know this rubs you the wrong way but personally I do not like being stuck in the past. Digital reproduction has advantages that can not be matched in the analog world and the advantages outweigh the disadvantages. I am dead certain you would agree if you could hear it in action. 

I forgot to mention one other major factor that makes this problem worse,
the RIAA curve. When a lacquer is cut the high frequencies are boosted and the low frequencies are cut. On playback the high frequencies are cut which reduces high frequency record noise just like Dolby noise reduction. Unfortunately, low frequencies are boosted making low frequency record noise worse. Only 3 dB means the woofer cones move twice as far. At higher volumes that becomes a huge problem. IMHO subsonic filters are mandatory for people who like to play above 80 dB.

@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.