Why the woofer moves badly when playing certain LPs

You need a subsonic filter in your phono stage like this one that has a RUMBLE setting  Gain-DIP switch settings on/off.
3.18 us off / 7950 us on DIP switches 7 & 8
http://www.wyetechlabs.com/preamps/phono/phono.html

@atmasphere Thanks for your comment. I enjoy reading your posts on various forums and consider you an expert on things audio. So it's with respect that I disagree with your comment: 

This statement is incorrect. If the mechanical resonance is corrected the amplitude of the woofer movement will be decreased.

The resonance mechanism you and others propose involves forced vibration in which some source of periodic excitation acts close enough to the resonant frequency of the tonearm/cartridge to excite resonant response. This is a plausible explanation, although it doesn't explain the noticeable discontinuity in the woofer oscillations, roughly (exactly?) once per revolution, in the video. In other words, I see two distinct periods in the video. The shorter period corresponds to the woofer oscillations, the longer period corresponds to the very regular disruption of the woofer oscillations at about 0.5 Hz.

The mechanism I'm proposing doesn't involve resonance. The warp acts like a mild ski jump that imparts a vertical acceleration to the cartridge/tonearm once per revolution --- enough to flex the cantilever but not enough for the stylus to lose contact. If the tracking force is too light, the cartridge will "bounce" repeatedly in the groove with the cantilever acting as a spring. The bounces generate several cycles of woofer oscillations until they are disrupted by the next pass over the warp and the process repeats.

I'll confess to having had exactly this problem early on with my current turntable. The cartridge and tonearm were carefully matched, I used a microscope to align the cartridge, tracking force was carefully set, and the table was well isolated and featured very low rumble. Nonetheless, the sound wasn't all that good. Then, when playing a particular LP, I noticed my subwoofer undergoing oscillations very similar to those in the OP's video. I at first assumed some form of resonance was causing the problem, but after several days, I couldn't find a cause. Then I noticed a relation between a modest warp in the record and a repeating pattern in the oscillations, very similar to the OP's video. But then why was my expensive tonearm and cartridge tracking so poorly?

The answer had to do with a shortcut I developed in operating the cueing lever that worked fine on my old table but didn't fully release the cueing support on my new table. That last bit of support reduced the effective tracking force. All it took to fix the problem was to fully depress the cueing lever. Subwoofer oscillations gone and sound quality vastly improved.

Sorry for the long post. Can anyone else see the periodic disturbances of the oscillations in the video?

I would check external things like what the table is sitting on and how far it is away from the speakers and don’t forget the floor. Tonearms just don’t  play the record. They can pickup the room environment. The floor can be a big culprit sending the bass waves straight through the tonearm causing the woofers to move.  

Have you checked to see how low your woofer is supposed to go before your excursion is too much?

The resonance mechanism you and others propose involves forced vibration in which some source of periodic excitation acts close enough to the resonant frequency of the tonearm/cartridge to excite resonant response. This is a plausible explanation, although it doesn't explain the noticeable discontinuity in the woofer oscillations, roughly (exactly?) once per revolution, in the video. In other words, I see two distinct periods in the video. The shorter period corresponds to the woofer oscillations, the longer period corresponds to the very regular disruption of the woofer oscillations at about 0.5 Hz.

The mechanism I'm proposing doesn't involve resonance. The warp acts like a mild ski jump that imparts a vertical acceleration to the cartridge/tonearm once per revolution --- enough to flex the cantilever but not enough for the stylus to lose contact.

If the cartridge has too high a compliance (too low a mechanical resonance) then it will behave as you describe. If the cantilever is stiffer (less compliance) the mechanical resonant frequency will be higher and thus the amplitude of the warp energy will be reduced. Or- if the mass (not the tracking pressure) near the stylus is reduced (by changing out the hardware and getting rid of the stylus guard if there is one) the effect is the same.