Why the woofer moves badly when playing certain LPs

Put the speaker grilles back on. If you can't see them pumping it can't bother you. (Only partly tongue in cheek).

I remember when I was young, but I never faced this phenomenon when I played pirated LPs on a cheap turntable. Even if this is the case with certain LPs, I don't think this is the only cause of LPs pressing.
A lot of people have given me good advise, and especially I'm going to focus on the rumble and try to solve this problem.

I'm sorry for not being able to thank everyone who gave me good advice
Thank you very much.

I experienced exactly the same thing through my ProAc D38's, the woofers would flap about all over the place without producing anything audible. My phone preamp has a subsonic filter, engaging that sorted it out. I was reluctant about the idea of a filter at first, but I couldn't detect any audible change to the sound quality so was more than happy to leave it on. 

The idea of going to lower mass mounting screws is a good one (worth a try- its cheap; maybe also remove the stylus guard), better yet a cartridge with slightly lower compliance.

A cartridge with lower compliance (15cu) that i mentioned also has light mass itself (5.5 - 6.5g instead of 8g AT). Could be an ideal solution if the OP's problem is cart/tonearm resonance.

@starbusters look at Pickering 3000/SP:

Stylus Type: Nude Stereohedron
ORIGINAL STYLUS: D3000SP
Compliance: 15cu @10hz
Contact Radii: .0028 (71u)
Scanning Radii: .0003 (8u)
Stylus Tracking Force: 1 gram (+1/2g) (-1/4g)
Setting with Brush: 2 gram resulting operation tracking force 1 gram. Optional range is (+1/2g) (-1/4g)
Frequency Response: 10 Hz to 30 kHz +
Output: 5.0 mV
Channel Balance: Within 1 dB @ 1kHz
Channel Separation: 35 dB @ 1kHz
Cartridge DC Resistance: 600 ohms
Cartridge Inductance: 350 mH
Cartridge Color: Brown
Cartridge weight: 5.5g (6.5g with brush)
Load Resistance: 47-100k Ohms
Load Capacitance: 275 pF
SP version was made in 1989, discontinued in 1994

Cleeds, since when is a resonance point a "brick wall" situation. The resonance peaks then rolls off. If your woofers do not move at all visibly and you are playing vinyl either your woofers do not go down very low or your cartridge is too stiff for your tonearm and your bass is rolling off prematurely. I suppose you could also be listening at the volume of a mouse squeak. I would never use an analog filter. It would damage the bass too much but digital is a whole other world. I could survive without it but volumes with certain records would have to be limited and It might rob enough power to affect the room control function. There are no downsides to a digital filter up to 80 dB/oct. With a 3 dB down point at 18 Hz the effect is totally inaudible. 
The problem Cleeds is that the source is vinyl, a medium that is imperfect at best. So, treating it at it's source means not playing records.