I wrote out a whole thing that was wrong but I found this guys math that seems correct to me.
Credit to
Len Zenith Jr
of ProsoundWeb
"Newtons 3rd law states that for every action there is an equal and opposite reaction;
force=mass x acceleration or f=ma
so using newtons 3rd law;
mass(cabinet) x acceleration (cabinet) = mass(cones) x acceleration (cones)
Assuming the mass of the cones of a double 18" cabinet is ~210 grams each so mass(cones) = 420 grams for the both of them or mass(cones) = 0.42 kg
and acceleration = 2 x distance / time squared
lets find acceleration of the cones at 60 hz:
time @ 60 hz = 1/60 = 0.0167 seconds
distance of cone travel (x-max) lets assume full bore is 18mm = 0.018 meters
acceleration (cones) = 2 x 0.018 m / (0.0167 s)squared
acceleration(cones) = 129 m/s2
assume the mass(cabinet) = 112 kg
back to newtons 3rd law:
mass(cabinet) x acceleration (cabinet) = mass(cones) x acceleration (cones)
112 kg x acceleration (cabinet) = 0.42 kg x 129 m/s2
gives us acceleration (cabinet) = 0.484 m/s2
now distance the cabinet moves d = 1/2at2
d= 1/2 x 0.484 m/s2 x 0.0167 s x 0.0167 s
d= 0.0000674 m
d = 0.067 mm
So there you have it, hanging in free space with a 60 hz tone playing at full balls to the wall 18mm x-max the cabinets are vibrating back and forth 0.067 mm or
less than 3 thousands of an inch.
If you think about it, a driver sitting on the workbench playing at x-max doesn't even move the driver, nevermind the whole subwoofer cabinet. "
I cant see the force of the drivers playing any roll in the force transmitted to the floor. Also by this math spikes and anything "coupling the speaker to the floor" is unnecessary because even teflon coated surfaces will have enough static friction to hold the cabs in space.
The real question would be cabinet vibrations I suppose but I would surmise spending more money damping your speaker would be a better use of time.
Interesting approach to calculate cabinet vibration. In case of a fully "floating", i.e. a decoupled enclosure from the floor, only the effective mass of the cabinet at the moving cones will matter.
In case of sine waves causing the vibration, one can use the 2nd derivative of the speakers cone excursion to obtain the acceleration. When speaker enclosure and cone stimulation are sinusoidal, the 2nd derivative will also be sinusoidal. Therefore my conclusion is that the excursion of the cone multiplied by its mass is equal to the excursion of the cabinet times its effective mass where the woofers are mounted.
Regarding the mentioned parameters I applied the ones from my modified Dynaudio Focus 220 speakers which are slim towers of about 1m height and behave completely different than one may assume.
From older Dynaudio Esotec 17W LQ Thiele-Small speaker data the woofers seem to have an effective cone mass of 15g x2, since there are two stacked. Their linear excursion is rated at 6.5mm peak-to-peak as a practical orientation.
Now here comes the surprise: The cabinets weigh about 20Kg, but the effective mass at the woofers is much lower. I used a kitchen scale to "tilt" the enclosure from the top back. The force, when the rear spikes went off the ground, was about 2Kg, app. a 1/10 of the total mass. This means that 6.5mm of cone excursion translate into app. 100um cabinet movement at the top of the tower. This magnitude seems reasonable, as touching the top front or rear of the enclosure with a fingernail feels accordingly.
This amplitude might not be much, but the effect on the tweeter above the woofers is inferior.
From Dynaudio Esotec D-260 Thiele-Small tweeter paramters their linear excursion is about 300um pp (with comparable sensivity as the woofer).
What happens now is that the tweeter output will suffer from Doppler-effect modulation, with comparable excursion of the foreign modulation signal as its own.
Since I have been suffering from annoying distortions, when listening to these speakers, I went a little deeper into music theory and human sound perception.
For instance, when string based instrumentation with vocal company occured, I could hear these distortions.
What is happening here could be abstracted by some (violin) signal around 400Hz modulating the (singers) formants around 2.7KHz. Doing some phase modulation math yield sideband-tones at around -30dB. If they were inside the same hearing band (so called Critical Bands, which have around 400Hz bandwidth at 2.7KHz), they would probably not be audible, but outside the same Critical Band they are probably audible (as distortion). That could be as well the explanation why double bass and drums do not cause that much audible distortion in comparison.
Now what has this to do with the coupling and decoupling of speakers?
In this case, the annoying distortions may be lowered a lot by "nailing" the enclosures as much as possible to solid ground to virtually increase the enclosure´s effective mass. I my case I had these speakers dropped into a wooden floor with their spikes first, when the distortions were worst. Large steel pads (10cm dia. 1cm thick) between the the wooden floor helped a lot, but hollow-drilling the wooden floor down to the cast-plaster, gluing small semi-hard copper shims onto the plaster surface (for protection and ease of alignment) and carefully adjusting the speakers on them appears as the best possible solution. I would have gone also through the plaster layer, down to the concrete floor, but hurting heating pipes close by was too risky.
In parallel I also mounted 3Kg flatsteel bars onto the upper rear inside of the enclosures to balance the enclosure and moved the (anyway rattling) crossover boards away from there, to an iron-free nonmagnetic external cabinet.
The steel-mass also improved the bass attack and precision a lot.
(during mod phase)
(final setup with lowered bass roll-off to separate bass-reflex range extension and room resonance)
So from my perspective there is a trade-off between coupling and decloupling speakers.
- When the enclosure is vibrating too much (lightweight tower enclosures, as described above), coupling is the option.
- When the speakers induce audible resonances into the structure (as into bookshelves), decoupling is optimal.
Some late remarks concerning the above estimation:
At 400hz, the cone movement will be much smaller than the rated 6.5mm and my assumption regarding the effective mass needs correction as well. The mentioned 2Kg "lift-off" (mass)force is as well lower than the effective mass, which the membrane movement drives (the speakers themselves are about 3Kg plus part of the enclosure). Anyway the "lift-off" force is definitely reached when sweeping at decent levels and low frequencies. The enclosure starts to shake on the spikes then...I leave the above for discussion anyway, since the observed distortion effects are still valid (for me).