Bass Reflex Enclosure equations of motion

[pitchtwit]

1. The problem

This is two questions from my assignment - the second of which I'm stuck on. It's about a loudspeaker. Question one looks at a simple loudspeaker. Question two introduces a bass reflex enclosure to the system.

Here's question 1,

http://dl.dropbox.com/u/11341635/Question%201.png

Here's question 2.

http://dl.dropbox.com/u/11341635/Question2Screenshot.png

Here are my question one key solutions,

The equation of motion for a forced oscillating system,

http://dl.dropbox.com/u/11341635/Equation%20of%20motion%20for%20a%20forced%20oscillating%20system.png

The solution to the equation of motion for a forced oscillating system,

http://dl.dropbox.com/u/11341635/Solution%20to%20the%20equation%20of%20motion%20for%20a%20forced%20oscillating%20system.png

The key assumptions from the outcome of question 1,

http://dl.dropbox.com/u/11341635/Assumptions.png

So I'm supposed to be deriving equations of motion for a loudspeaker that includes the speaker and the air mass of the bass reflex enclosure. Here's a design I did.

http://dl.dropbox.com/u/11341635/Bass%20Reflex%20Design%201.jpg

My problem is understanding two areas.

1. The cone & coil suspension system and the air either side of the the cone & coil suspension system. The air either side has it's own stiffness and damping - so how can I tun this into a series only string of springs - suitable for algebraic manipulation?

Kinsler and Frey (The Fundamentals of Acoustics - scan included below) seem to deal with the problem differently. They consider the electrical circuit equivalent - which is often done in acoustics - which does feature a parallel section - although I don't think that's the same section I'm having problems with. Anyway, that only focuses on the mechanical impedance at the speaker cone"

Another way I thought of tackling the problem was with two-port analysis. I came across this in a previous assignment for another module I do, and I got 10/10 for it, so I'll include it at the bottom as another option.

2. Homework Equations

The method from our supplementary notes is here,

http://dl.dropbox.com/u/11341635/twodegreesfreedom1.jpg

http://dl.dropbox.com/u/11341635/twodegreesfreedom2.jpg

http://dl.dropbox.com/u/11341635/twodegreesfreedom3.jpg

The relevant page from Kinsler and Frey is here,

http://dl.dropbox.com/u/11341635/Kinsler412.jpg

And here is the two-port analysis I did for a previous assignment (this was for calculating the impedance at the front surface with a combination of different surfaces behind.

http://dl.dropbox.com/u/11341635/TwoPort1.png

http://dl.dropbox.com/u/11341635/TwoPort2.png

http://dl.dropbox.com/u/11341635/TwoPort3.png




3. The Attempt at a Solution

Well I haven't really got far. At first I decided to try and adapt a similar model to that given to us in the supplementary notes, and I'll include the diagram here so you can see how that works,

http://dl.dropbox.com/u/11341635/BassReflexDiagram_i.jpg

This would be a drastic simplification, but I don't think that's too much of a problem. This method would involve considering only the radiation coming from the back of the speaker cone. The example in the supplementary notes ends up with two equations for the displacement, which you can see above. The x3 on the right of each of these is puzzling to me. If that's the movement of the ground - wouldn't that just be zero? This would make the whole equation (for both) equal zero, which can't be right. In my case, the ground represents the air in the room - so I guess this would be okay, but how do I get a frequency response curve from that?

Another equation I have from our notes is,

http://dl.dropbox.com/u/11341635/Average%20power.png

What was thinking is that I could get the mechanical impedance at the speaker cone from the Kinsler and Frey equation, and then use that in this average power equation to obtain a frequency response. There are a load of holes in that method though, and I'm pretty sure I wouldn't get a meaningful result from it.

Any help would be great.

 

[KataKoniK]

I would suggest approaching this problem by breaking it down into smaller, more manageable parts. First, focus on understanding the physical systems involved - the loudspeaker, the air mass in the bass reflex enclosure, and the cone and coil suspension system. Then, try to determine the equations of motion for each of these systems separately.

For the cone and coil suspension system, you can use the equation of motion for a forced oscillating system, as you have shown in your key solutions. This will give you the displacement of the cone as a function of frequency.

For the air mass in the bass reflex enclosure, you can use the equation of motion for a mass-spring-damper system, where the air mass is represented by the mass, and the stiffness and damping are determined by the geometry of the enclosure. This will give you the displacement of the air in the enclosure as a function of frequency.

Once you have these two equations, you can then use the principle of superposition to combine them and determine the overall displacement of the loudspeaker cone. From there, you can calculate the frequency response by varying the frequency and solving for the displacement at each frequency.

In terms of the two-port analysis you have done previously, that may also be a useful approach. However, it may be more complicated than necessary for this specific problem. I would suggest starting with the simpler approach outlined above and then, if needed, incorporating the two-port analysis into your solution.

Overall, the key is to break the problem down into smaller, more manageable parts and then combine them to get the overall solution. I hope this helps and good luck with your assignment!