calculus project, need help with equations

[roaldd]

Hi,

I am doing a calculus project in which I will build a subwoofer (already done), and show the math involved with tuning the box based on the theil small parameters. the box is sealed with a db500 12"

12" dB-500 SVC

Qts 0.447
Qes 0.481
Qms 6.347
Fs 24Hz
Res 3.25Ω
Ls 2.36
Lp 3.51
Rp 2.91Ω
Dia 241mm
Vas 71.5l
mms 169g
cms 244um/M
bl 13.3Tm
Spl 86dB

I'm kind of in a bind here and need some quick replies. thanks a lot in advance.

Roald

[joecarrow]

This looks to me like it's a bit beyond what I did in Calculus- more like a Modeling and Control of Dynamic Systems, which involved some differential equations at times. Although some of the equations can be fairly simple, their derivation is less so.

What you've basically got is a resonant system (the woofer) driving another resonant system, the ported box. The ported box is a helmholtz resonator; there are many websites out there describing these and their math in great detail. Here's the first entry from Google:

http://www.phys.unsw.edu.au/jw/Helmholtz.html

Definitely an interesting school project. I think you'd need to come up with an signal block diagram using most of the terms involved in order to come up with the system transfer function. Once you have the transfer function, you would use that to construct a bode plot, which would (if you set it up right) predict the system response in relation to an input.

Maybe someone else will correct me if I'm off track, but that's the way I understand it.

[Dennis H]

This should get you started. The pic is from Siegfried Linkwitz, showing the electrical equivalents of a sealed woofer. That electro-mechanical conversion is what the whole T/S model is based on.

http://www.linkwitzlab.com/thor-design.htm

[Paul H]

Every once in a while it occurs to me that there just might be a couple of engineers and/or geeks hanging out in this forum :B

[JoshK]

joe is right, this is a Diff Eq (and by the looks of it a Partial Diff EQ) problem, beyond what you'd learn in Calc I-IV in college. I majored in applied math (even got a MS in it). But I didn't know how to parameterize the problem because I haven't studied this area enough.

This isn't terribly complicated, but isn't lower classmen level IMHO. It is easier to use some empirical heuristics to simplify the problem and then to use some numerical analysis to solve a specific problem using computers. This is the kind of stuff I did in my masters.

If you tell me what the problem is suppose to involve and at what level, I might help you come up with a reachable problem.

[Dennis H]

Looks more like algebra then diffeq to me but then it's been decades since I studied this stuff so I probably don't have a clue. I guess it gets into diffeq if you want to study what the system is doing at any point in time rather than the overall SPL response but the overall response looks pretty straight forword with SL's formulas -- basically what WinISD, etc do.

[Feyz]

In addition to what is mentioned already:

Since the electro-mechanical circuit in the end uses capacitors and inductors to model, the solution is a linear differential solution. But because SPL/impedance response is for steady state sinusodial response, using AC phasor analysis method the differential equation turns into a linear non-differential equation and is solved that way easily.

One quick online reference I could find (Look at the whole Section 4):

http://en.wikibooks.org/wiki/Circuit_Theory#Section_4:_AC_Analysis

For the electro mechanical model of a loudspeaker in a box, the course book is:

http://users.ece.gatech.edu/~mleach/audiotext/

One article from the author that may give you some ideas, though it shows how to use SPICE to solve the circuits:

http://users.ece.gatech.edu/~mleach/papers/spice_electro.pdf

[joecarrow]

Looks more like algebra then diffeq to me but then it's been decades since I studied this stuff so I probably don't have a clue. I guess it gets into diffeq if you want to study what the system is doing at any point in time rather than the overall SPL response but the overall response looks pretty straight forword with SL's formulas -- basically what WinISD, etc do.

It's funny- it really does boil down to some algebra, but only once you do the differential equations to come up with this shorthand method. Feyz says basically the same thing, but sounds smarter saying it.

[Feyz]

Well, I wasn't trying to sound smarter
The thing is phasor method needs to be used (or some similar one) to solve the equations, and I felt like that needed to be explicitly mentioned for the help of the original poster.

[JoshK]

I think I remember some of that from school...

We used to call this a linear system of differential equations. I don't recall anything named phasors, but I do remember transformations that took diff eq's to algebra and then back.

[Tommythecat]

I think some of us are confused by your statement of how you'd like to show "how to tune" the box. In fact, joecarrow's first post deals with Helmholtz resonators and talks about ported enclosure analysis - when the poster said it would be modeling a sealed box.

Are you looking to optimize a sealed enclosure to a specific "alignment" given driver 'x'? or simply analyze the result of putting 'x' driver in 'y' sized sealed enclosure?


Beyond that, this is a very complex and intricate problem that uses several topics from different fields of math (algebra, linear algebra, differential equations). I started trying to explain some things(typing them out), but it would really take something like 25+ solid pages of work to quickly explain the motivation, derivation, and execution of creating a mathematical speaker model. I don't take this subject lightly, partially because I haven't yet perfected the model I use to model a driver in a sealed enclosure.

[joecarrow]

Hey, great catch! I thought I double-checked to make sure that he was really talking about a ported box. It said "tuning", so I guess that's what tripped me up.

Honestly, I've been stretched thin lately with my new job and helping my partner prepare for a cross country trip.