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4 weeks ago
#1
Gear Maniac

calculate stiffness of air "spring" between 2 masses (M-A-M resonance)

Hi community
Since some days i try find some research of how to calculate the dynamic stiffness of the air-spring between a normal double Stud wall.
We all know the typical formula, where the dynamic stiffness is given by only the distance between the 2 mass layers. But if the air cavity is open at the sides, the stiffness will change and the formula we have see everywhere are useless, because they assume a closed cavity with a given distance.
I would be extremely thankful if somebody of you guys know some research papers where the topic seen in the attached picture is described a little more into depth. Thx a lot for your help.

Last edited by mikahanau; 3 weeks ago at 11:19 PM..
3 weeks ago
#2
Lives for gear

I might be incorrect, but i have never heard of a calculation available for this. If the is no seal, there is no spring.

If i'm wrong, I really look forward to hearing others comments on this.
3 weeks ago
#3
Lives for gear

Stiffness = Force / Deflection

Dynamic Stiffness (Kd) = Force / Varying Deflection

Air? hmmmm
3 weeks ago
#4
Gear Maniac

Quote:
Originally Posted by JayPee

Stiffness = Force / Deflection

Dynamic Stiffness (Kd) = Force / Varying Deflection

Air? hmmmm
In German it is called "dynnamische Steifigkeit", and this term is correctly used for describing the properties of the air between 2 masses in an mass-air-mass system, which acts as a spring (s` in MN/m³). If the terms are different in english physics, please let me know.
thx
3 weeks ago
#5
Lives for gear
3 weeks ago
#6
Gear Maniac

Quote:
Originally Posted by JayPee
I guess you know the formula to calculate the mass air mass resonance between a normal single studwall, so hopefully you can tell me the real name of what is red part in the formula. When i find out the right egnlish name, i hope i can get more usefull search results.
My only english book for acoustics is the "Master Handbook of acoustics" and in this book there is not much usefull infoarmation about this topic at all, so i hope you will understand what i am lookig for, by just seeing the formula, which should be the same all over the world.

3 weeks ago
#7
Lives for gear
What's is this book?
3 weeks ago
#8
Gear Maniac

Ok i will ask in a more simple way, that no special terms are needed to understand the question: Will the green version have the same Mass air mass resonance frequency like the orange version in the picture? If "NO" then why? and how can i calculate it ?

I will call it now just "stifness" of the airlayer.... So if the there is more airvolume , the stifness should get lower, and the resonance should get lower too. The question is, if the big coupled airvolume left of the room, will act the same if i just make the distance of the 2 masses bigger, or if there is % of how much of the big orange area left of the room will be used. I hope like this it is more clear what i am looking for to understand in a physical way.

Last edited by mikahanau; 3 weeks ago at 12:46 AM..
3 weeks ago
#9
Gear Maniac

Quote:
Originally Posted by JayPee
What's is this book?
It is called " Schallschutz und Raumakustik in der Praxis" from W. Fasol and E.Veres, and is a satndard lecture for study Architecture and Acoustics in Universitys in Germany.
3 weeks ago
#10
Gear Maniac

Quote:
Originally Posted by JayPee
What's is this book?
Ahh and dont get confused. In this chapter its more about how different coupling materials affect the resonance between the the mass layers. Can be air, insulation, Styrofoam, and what ever other materials. Out of this basic formula, you get the other formulas you will find all over this forum, where you have fixed number for insulated walls and just the depth of the air cavity
3 weeks ago
#11

I don't have a math formula for it, but my Fingerspitzgefühl says the spring should be softer (lower) in the narrow red section than in the green one, as it is in direct connection with the larger red volume with deeper depth. Why? The pressure inside a, say pear shaped ballon, is the same in all its volume. The "counter force" (spring) if I put a pressure with my finger on the ballon shouldn't change no matter where I put my finger on its shape. That's whats feels intuitively right, but acoustics can often fool you.
3 weeks ago
#12

Quote:
Originally Posted by Jason Foi
I might be incorrect, but i have never heard of a calculation available for this. If the is no seal, there is no spring.

If i'm wrong, I really look forward to hearing others comments on this.
If there is air, there is a spring.
The resonance frequency of the spring drops when the cavity gets larger.
If you use speaker simulation software you van easily see what happens with the resonance frequency when you increase the cavity/loudspeaker enclosure.
I'm not Stuart so I would have to dive into it to give the equations :-).
3 weeks ago
#13
Lives for gear

Quote:
Originally Posted by bert stoltenborg
If there is air, there is a spring.
The resonance frequency of the spring drops when the cavity gets larger.
If you use speaker simulation software you van easily see what happens with the resonance frequency when you increase the cavity/loudspeaker enclosure.
I'm not Stuart so I would have to dive into it to give the equations :-).
Interesting...thanks.. i get that there would be resonance, because there is a cavity, but i just dont see how its still a MSM isolation system if its not sealed. Sound would leak out like crazy.
3 weeks ago
#14
Lives for gear

What Bert said.

I did a test many years ago in an Acoustic Laboratory with a double stud wall. tested the TL of the wall. Then cut a hole in one leaf of the wall, to ventilate the air cavity. Attached an acoustic duct to the hole so that TL would not go down due to leakage.

It was a test of how much the stiffness of the air cavity might be reduced since the wall cavity volume was now connected via an air path to the whole room (on one side of the wall).

Perhaps the hole wasn't big enough but there was no effective change to the M-A-M and TL performance.

M-A-M is a low-ish frequency, usually. Large wavelength. Probably fairly stiff locally.

May have been better off connecting a vacuum pump to reduce the pressure inside the wall cavity...
3 weeks ago
#15
Gear Maniac

Quote:
Originally Posted by Sebg
What Bert said.

I did a test many years ago in an Acoustic Laboratory with a double stud wall. tested the TL of the wall. Then cut a hole in one leaf of the wall, to ventilate the air cavity. Attached an acoustic duct to the hole so that TL would not go down due to leakage.

It was a test of how much the stiffness of the air cavity might be reduced since the wall cavity volume was now connected via an air path to the whole room (on one side of the wall).

Perhaps the hole wasn't big enough but there was no effective change to the M-A-M and TL performance.

M-A-M is a low-ish frequency, usually. Large wavelength. Probably fairly stiff locally.

May have been better off connecting a vacuum pump to reduce the pressure inside the wall cavity...
Intressting. How big was the hole compared to the over all surface of the wall?
The question is aslo if the shape of the aircavity is relevant, or just the volume itself.

For me it would make sense, that the left big orange part of my sketch, would be in some way coupled to the orange area just between the leafs.
I can see this realy good in floating floors, which realy breath and you can feel like a wind at the sides of them, when your outside.
The question would be how much a mostly open airlayer affect the resoance of the MAM compared to a seald airlayer?
3 weeks ago
#16
Lives for gear

An interesting addition to what Bert already explained: Not really the same, but related... - If you look at the equations for calculating the resonance of slot walls and perforate panel devices, the only parameter related to the cavity, is the depth... there is no width or height parameter... Only the depth matters ... In other words, it doesn't matter if your slot wall is 20 meters wide, or 20 cm wide, nor does it matter if it is 8 feet high or 8 inches high, you will get the same resonant frequency in all of those cases, (all other factors being equal)... Only the depth matters.

(OK, so this isn't really the same thing as what the OP asked, because slot walls and perf panel devices are not MSM walls, but it is interesting, and somewhat related...)

It seems to me that what the OP is looking for, is the spring constant of air. Also related to Hooke's law. Or perhaps Bulk Modulus... but that's getting a bit outside my pay grade...

- Stuart -
3 weeks ago
#17
Gear Maniac

Quote:
Originally Posted by Soundman2020
An interesting addition to what Bert already explained: Not really the same, but related... - If you look at the equations for calculating the resonance of slot walls and perforate panel devices, the only parameter related to the cavity, is the depth... there is no width or height parameter... Only the depth matters ... In other words, it doesn't matter if your slot wall is 20 meters wide, or 20 cm wide, nor does it matter if it is 8 feet high or 8 inches high, you will get the same resonant frequency in all of those cases, (all other factors being equal)... Only the depth matters.

(OK, so this isn't really the same thing as what the OP asked, because slot walls and perf panel devices are not MSM walls, but it is interesting, and somewhat related...)

It seems to me that what the OP is looking for, is the spring constant of air. Also related to Hooke's law. Or perhaps Bulk Modulus... but that's getting a bit outside my pay grade...

- Stuart -
I have to disagree with you over here. When you watch the formula of a perforated wall, you will see that the depth is just used to determine the airvolume behind the perforation, since the volume changes automatical when you change the area of the slotted wall, without changing the % of perforation, so the relation between this parameters stay the same. The formula for perforated panels is based on the helmholtz formula.

Do we agree that all of this resonators, with the same volume and opening area, will have the same resoance freq. ? (i just show it 2 d for better understanding)

I think it is an intressting topic, also for construction of wall assemblys which would be seen as "bad" triple leafs, while in reality maybe the coupled airspace apart from the narrow area would take down the resonance so low, that the triple leaf effect probably won`t matter anymore, and even give better soundisolation than avoid the triple leaf in this special case.
I hope there will be more input on this, and hopefully some research of this topic can be found somewhere.
cheers
3 weeks ago
#18

Quote:
Originally Posted by mikahanau
I have to disagree with you over here. When you watch the formula of a perforated wall, you will see that the depth is just used to determine the airvolume behind the perforation, since the volume changes automatical when you change the area of the slotted wall, without changing the % of perforation, so the relation between this parameters stay the same. The formula for perforated panels is based on the helmholtz formula.

Do we agree that all of this resonators, with the same volume and opening area, will have the same resoance freq. ? (i just show it 2 d for better understanding)

I think it is an intressting topic, also for construction of wall assemblys which would be seen as "bad" triple leafs, while in reality maybe the coupled airspace apart from the narrow area would take down the resonance so low, that the triple leaf effect probably won`t matter anymore, and even give better soundisolation than avoid the triple leaf in this special case.
I hope there will be more input on this, and hopefully some research of this topic can be found somewhere.
cheers
If the opening is small compared to the volume of the enclosure you have a tuned device. in that case the trick doesn't work.
Philio de Haan did a test with an ampeg in a car on studiotits that illustrates this. http://forum.studiotips.com/viewtopi...cedes&start=20
3 weeks ago
#19

My speculation: To calculate the total volume of the L-shaped red section is easy. If one add together total outer length with the total inner length, take the average of that and divide the volume with that average one should get the average depth. Maybe not totally correct as from the 1 st picture the smaller section should have smaller distance between studs and therefore a stiffer wall area but it may give a ballpark figure. For the second picture with 3 versions; if those where loudspeakers with identical internal volume, the speakers' tuning / resonance should be the same. Resonances within / for each wall would be different though if measured with an accelerometer or a laser because of their varying length, width and stiffness.
3 weeks ago
#20
Gear Maniac

Quote:
My speculation: To calculate the total volume of the L-shaped red section is easy. If one add together total outer length with the total inner length, take the average of that and divide the volume with that average one should get the average depth. Maybe not totally correct as from the 1 st picture the smaller section should have smaller distance between studs and therefore a stiffer wall area but it may give a ballpark figure. For the second picture with 3 versions; if those where loudspeakers with identical internal volume, the speakers' tuning / resonance should be the same. Resonances within / for each wall would be different though if measured with an accelerometer or a laser because of their varying length, width and stiffness.
Thats is more or less what i think, but just as you, only with my "Fingerspitzengefühl"
I guess there will be kind of a coupling factor between the large orange area,a nd the slim orange area, which just let a part of the large area be effectly used as part of the air-spring.

Too bad i live in Mexico, where we usaly don`t have Drywalls in our houses, just brickwalls. Otherwsie it would be easy to make some tests with just a isobox with a speaker inside, infront of a normal studwall

cheers
3 weeks ago
#21

Quote:
Originally Posted by mikahanau
Thats is more or less what i think, but just as you, only with my "Fingerspitzengefühl"
I guess there will be kind of a coupling factor between the large orange area,a nd the slim orange area, which just let a part of the large area be effectly used as part of the air-spring.

Too bad i live in Mexico, where we usaly don`t have Drywalls in our houses, just brickwalls. Otherwsie it would be easy to make some tests with just a isobox with a speaker inside, infront of a normal studwall

cheers
If you have two chambers coupled by a hole you have a bandpass system, as in the link is explained by Philip.

In practice:
if you have a nice decoupled double brickwall and sound is pushing at one of the wall leafs the leaf acts (in theory) as a piston on the air in the gap between the leafs. That's why only the depth is imortant.
If we put windows in the leafs the window will normally have less mass than the leaf.
Therefore the sound can move the window more easely, the excursion of the window is larger than that of the leaf and the air between the windows is compressed more than in the cavity between the stone leafs.
If we don't seal the window gap the over pressure will have the air leak to the gap between the stone leafs reducing the spring tension between the window planes and thus reducing resonance and increasing sound reduction.
3 weeks ago
#22

Quote:
Originally Posted by mikahanau

I think it is an intressting topic, also for construction of wall assemblys which would be seen as "bad" triple leafs, while in reality maybe the coupled airspace apart from the narrow area would take down the resonance so low, that the triple leaf effect probably won`t matter anymore, and even give better soundisolation than avoid the triple leaf in this special case.

cheers
I don't see how this would work.
3 weeks ago
#23
Lives for gear

I suspect "dynamic stiffness" in this case refers to the adiabatic stiffness of air. (As opposed to air's stiffness under isothermal conditions.) This would correspond to the compression of air at audible frequencies without insulation. If insulation is added to the cavity, the sound waves transfer heat to the insulation and the process becomes isothermal. The stiffness under isothermal conditions is approximately .7 times the stiffness under adiabatic conditions.

I'm not entirely sure I understand the rest of the question but basically if the cavity is completely "open" then the stiffness of the air within the cavity is zero. The stiffness (restoring force) of the air is due to the difference in pressure between the air within the cavity and the air outside of it, and if the cavity is completely open then there is no pressure difference and therefore no restoring force.

If the cavity is only partially open, such that the effective openings are small then you will have a "leaky" resonator with lowered Q and multiple tuning frequencies. (Like a Helmholtz resonator with varying neck sizes.).

One thing to keep in mind is that all of these functions are continuous, so there is no sudden switch from a Helmholtz/membrane resonator to a free standing panel the moment there is some air leakage -- it's actually a smooth transition. There is always some amount of air leakage in any assembly.

As for the picture you posted with three cavities of the same internal volume but different shapes, it must be kept in mind that mass-spring models are "lumped parameters" models. (You frequently see this phrase in the world of speaker design or in electronics.) Lumped parameter models are only good approximations at "low enough" frequencies. (What exactly "low enough" might mean depends on the context and the accuracy you require.)

In the case of modeling an enclosed volume of air as a spring, a lumped parameter model requires that the pressure within the cavity is equal at every point within the cavity. In other words, there cannot be local pressure variations within the cavity. To satisfy this condition, the wavelength at the relevant frequencies must be large in relation to the dimensions of the boundaries. If the wavelength is quite large relative to internal boundaries (greater than 4 times at least) then there won't be significant local variations in pressure within the cavity. Also, small areas within the geometry will need to be large enough for air to freely move, so that there is no turbulent air flow. (As an extreme example, we cannot have a long, 1" thick corridor with a 90 degree bend. The sharp bend and small size creates air turbulence, much like in a badly designed ported speaker.)

Turbulent air flow and variations in pressure within the cavity violate the assumptions of the lumped parameter model.

In general there is no analytic solution for arbitrary shapes. The formulas you find online or in textbooks are based on specific analytic shapes (cylinders, rectangles, spheres, etc.). They are frequently "close enough" to give you a good guide to the real world. But solving these sorts of problem for arbitrary geometries with higher precision will require some form of numerical technique such as finite element method. Coupled oscillator problems are quite difficult and if you have ever attempted to solve a coupled second-order differential equation then you know what I mean. FEM breaks these geometries up into many tiny pieces and solves a matrix equation that represents coupled oscillators with many (thousands or tens of thousands or more) of degrees of freedom. This is impractical for a human to do by hand but can be done relatively simply by a computer.

Unfortunately FEM packages are usually many thousands of dollars, and most don't have implementations that work well for vibrations in air. (They mostly do structural vibrations.) The loudspeaker design software SoundEasy (approximately \$250) actually has a very simple FEM implementation that can find room modes for relatively simple shapes. The elements are "brick" shaped and fairly large, so it can't conform well to detailed geometries but it can give good results for spaces like L-shaped rooms and cavities.

Sorry there isn't a simpler answer but hopefully this gives some food for thought.

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