When a loudspeaker driver moves back and forth, the enclosure in which in sits will also vibrate to some extend. We will try to investigate how much it vibrates, how much this will affect the total sound field, and what to do about if, if anything.

Imagine a loudspeaker driver positioned in an enclosure as in the crude figure below with rigid masses, a force generator, and single spring. The green diaphragm will move one way for an given applied force, whereas the yellow-ish enclosure will move in the opposite direction. Part of the applied force also goes into compressing the softer parts of the driver, but at frequencies above the driver's resonance, most of the force will be related to the masses moving. The whole system is put on rollers, so it is free to move horizontally.

Now, from the above we can set up a force equilibrium equation, or alternatively do an analogy circuit. I tend to do the latter, but feel free to do either. From this, we can plot both the velocity of the driver, and the velocity of the enclosure, for given masses of either. We can for example notice, how the movement of the enclosure actually affects the total electrical impedance of the system more and more, as the mass of the enclosure is decreased.

The sound output from the enclosure is related to an oscillating motion, and so intuition could be built from looking at the analytical solution for an oscillating sphere and an oscillating piston, whereas the sound output from the driver is related to the sound output from a piston in a baffle (see also the previous blog post). With this in your arsenal, you should have a fairly good sense of at least the orders of magnitude between the output from driver alone, the enclosure on its own, and their combined effect, without necessarily having been given a good description of the overall geometry. You will then get an idea as to the ratio of acceleration*area for the driver, and for the enclosure as well, and will be somewhat informed as to their individual importance. As always, try and gather information as early on in the project as possible, if nothing else to get your brain in gear.

Of course, no speaker is ever placed on a frictionless floor, and so one has to expand on the model, to get a more realistic model. You can for instance still assume a rigid enclosure, but now positioned on two supports, so that the enclosure has 2 DOFs on its own. But now you also have to take into account where the driver(s) sits relative to the enclosure geometry.

I have not seen much work done in the way of analytically characterizing loudspeaker vibration like this, but I could be wrong, so please put a comment below with any references, you may know of. You should be able to come a long way, before actually having to switch to numerical methods. You could also look into claims regarding rubber feet vs. spikes, for your particular loudspeaker at least.

As a next step we can assume that the enclosure walls are in fact not rigid. The internal pressure of the enclosure then affects the sound output from the enclosure, as this pressure exert force on the walls, causing them to vibrate and emit sound. For this, it is probably a good idea to do an eigenmode analysis, as the one I did some time back, when I was thinking about a similar topic. Two trapezoid shapes; a square one and a more general isosceles one, were examined (extrude in a certain height to get the full 3D enclosure) in this plane only. I was interested in the distribution of eigenmodes to see if this could explain why loudspeaker engineers always go for certain shapes; this will maybe be addressed at a later point.

The eigenfrequency analysis can give you some insight as to where you may experience excessive vibrations, but it will depend on the actual excitation from the driver(s).

At some point you will likely have to do a full Finite Element Analysis, when you know your geometry in more detail. This has been e.g. been done by KEF in their nice white paper on the LS50, from which the figure below was taken.

From https://www.kefdirect.com/media/wysiwyg/documents/ls50/ls50_white_paper.pdf

Here, the diapragm displacement is kept in the figure to illustrate the scale between it and the enclosure displacement. Again, keep in mind that the surface area of the enclosure is much larger than the diapraghm, but also that the enclosure has its own characteristic radiation patterns.

Great work has also been done by Cobianchi and Rousseau from Bowers And Wilkins with their COMSOL paper+presentation and AES paper. Their way of work is very appealing to me, and I would like to see more like this in the future for other aspects of loudspeaker design.

If you have access to FE software, you can experiment away yourself. Plastic vs wood, bracings or not, and whatever else you want to try out. I have show an example below; the simulation was done in COMSOL Multiphysics, and so was the geometry.

Done with COMSOL Multiphysics, www.comsol.com

I think many loudspeakers for home use are actually quite sturdy, once you get past the very cheap ones. But there are other loudspeaker applications, where the construction is quite different such as conference speaker, and bluetooth home speakers, where you are not exactly looking for the ultimate HiFi experience, but where you do indeed sometimes have fairly light enclosures, where perhaps the individual parts are joined with clips, making some more intricate contact problems compared to standard home speakers. For such applications, it can be of great value to do Finite Element Analyses (FEA), to establish vibration patterns and sound radiation characteristics.

With car speakers you have similar challenges, as weight is an issue, and oftentimes drivers have to be put in very challenging structural and acoustics environments. You can visit e.g. MVOID's homepage, and see how some of these issues are tackled.

Finally, even though you might think that there is a consensus that enclosure vibrations must be killed at all cost, some loudspeaker manufactures actually have a different philosophy, where the sound from the total system is tuned via having fairly thin-walled enclosured. Examples are Harbeth and Spendor, and I think this is generally a British thing(?). When the end result is a subjetive experience, as with listening, there are bound to be different camps. Many other manufactures proudly show off their excessive internal bracings in their sales material, and such stiffening will certainly affect the radiated sound, though it is not often clear how much.

What do you think? Is enclosure vibration an important issue, or should you first prioritize other areas of the design?

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