# Bass Reflex Enclosure equations of motion

1. Jan 30, 2012

### 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 [Broken]

Here's question 2.

http://dl.dropbox.com/u/11341635/Question2Screenshot.png [Broken]

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 [Broken]

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 [Broken]

The key assumptions from the outcome of question 1,

http://dl.dropbox.com/u/11341635/Assumptions.png [Broken]

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 [Broken]

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. Relevant equations

The method from our supplementary notes is here,

http://dl.dropbox.com/u/11341635/twodegreesfreedom1.jpg [Broken]

http://dl.dropbox.com/u/11341635/twodegreesfreedom2.jpg [Broken]

http://dl.dropbox.com/u/11341635/twodegreesfreedom3.jpg [Broken]

The relevant page from Kinsler and Frey is here,

http://dl.dropbox.com/u/11341635/Kinsler412.jpg [Broken]

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 [Broken]

http://dl.dropbox.com/u/11341635/TwoPort2.png [Broken]

http://dl.dropbox.com/u/11341635/TwoPort3.png [Broken]

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 [Broken]

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 [Broken]

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.

Last edited by a moderator: May 5, 2017