Turntable Causes Speaker Cones To Excessively Move Rapidly

Yogiboy appropriately mentioned tonearm / cartridge mismatch as the most likely culprit. I agree. Your response assumes the vendor would not sell you a mismatched combination, but that may not be true.

Look up the effective mass of your tonearm as well as the rated compliance of your cartridge. Go to vinylengine.com, open a free account, and surf to their cartridge resonance evaluator. This will calculate your resonant frequency. The higher the compliance, the lower the resonant frequency on any given tonearm.

As an owner of a Technics TT with an aluminum arm, the arm mass can be pretty high. I had the same problem running a Shure V15-III. The problem resolved when I installed an Ortofon Concord, which has a lower compliance AND lowered the effective tonearm mass (no headshell or fastener mass).

All "rumble" or "subsonic" filters are not alike.  AND, both your subwoofer and speakers are vented...which makes them susceptible to noise generated by normal record playback at frequencies below their "cutoff point"...tuned resonance frequency,  For the REL this is probably around 20Hz, and for the Sopra 1s around 50Hz...so both are wide open to oscilate when fed  0-12Hz.  The Lux filter is probably 6dB/oct below 20Hz.  You need a steeper filter.  This can be obtained with an outboard phono preamp whose design inludes a sharp filter.  The other solution is to use sealed box speakers and subwoofers that are "loaded" to roll off below their resonance frequency.  

@otimmons I am using a Ortofon 2M black lvb and it still does it. Not sure if that would be close enough to yours. 

The Ortofon has a compliance of 22 compared with 22.5 for the Shure. The resonant frequency in a 22g tonearm is 6 Hz. Anything less than 8 can give the unwanted resonance. you may have a mismatch.

look for a cartridge with a dynamic compliance less than 16, preferably less than 13 x 10^-6 cm/dyne

buy a KAB Rumble filter and call it a day.  It took away the oscillating with my former VPI Prime.  I no longer have the problem with my Technics SL1200G and no longer use the filter.