Calculating bends in a tapered TL

[bwbass]

I think I'm finally ready to start making sawdust on my first TL design, but I wanted to get some opinions on my method of calculating the line length through the bends.

What I'm building is a small sub using the Adire Koda 8. I've come up with a 67" line with a 2.5 x Sd : .3 x Sd taper, with the driver mounted at the closed end of the line. Modeled in MJK's MathCAD worksheets, I get an F3 around 31Hz or so, and with some heavy stuffing in the first couple of segments, response looks pretty smooth out to 120Hz.

OK, the issue is, how to best calculate the line length through the bend while maintaining a constant taper. After thinking about it for a while, I decided that the best solution would be to treat the path of the line around the corners as a section of an equiangular (logarithmic) spiral, since the width of the cabinet needs to increase as a linear function of the length of the line as it does in the straight segments. I built up a spreadsheet to calculate the total line length this way from the start and end cross sections and the length of the first straight segment, and then fiddled with the input till the line came out to the desired 67".

I drew the whole mess up in cad, and threw in a couple of 45 degree corner reflectors at points tangent to the spiral bends.

It seems to me that this would be the best way to maintain a constant taper, but it's based on the assumption that the sound waves will follow the path of "best fit" as the line narrows, rather than making a 90 degree turn in the corners. It seems logical to me, but I'm interested in what others have to say about this.

[MJK]

Hi bwbass,

Looks good to me. I have found that for a single fold TL that the geometry of the bend is not so important for the first few standing waves. Bob Brines has also demonstarted that the 45 degree reflectors have very little impact at the lower frequencies of a TL. Since you are building a sub, this should make the bend a fairly insensitive variable in your enclosure. I think your corner geometry will work well and I would not spend too much more time refining the design of the fold.

Hope that helps,

[bwbass]

Thanks, Martin. I looked at Bob Brines' data on the effects of corner reflectors, but it seems to me all that this shows is that the discontinuity in cross section caused by a square corner does not affect the line length "seen" by the driver. Therefore there must be a "correct" line path through a rectangular bend, regardless of corner reflectors.

If I had chosen to calculate my line with a square-cornered line path through the bends, the length through the bend would have been about 27% longer (4/pi), and therefore I would have made the width of the line on the other end of the bend larger, since it's further down the line. I would think that the discontinuity in the taper before and after the bend would have a noticeable effect, moreso than the presence or absence of corner reflectors. If there's one correct way that the line length is "seen" through the bend, the other methods must cause a deviation from the modeled response.

So which way is right?

[MJK]

bwbass,

If I take the sketch you have provided, assume it is dimensionally accurate for your design, eliminate the corner reflectors, then calculate a new path that is based on a square corner, how much does the total calculated length change? My guess, less than 10%. If your "physics" line length is within 10% of what your design drawings show, you are probably more accurate then the other approximations you are making in the design that you may not even realize you are making. Please don't overthink a small detail when other variables in the design produce a much larger error.

[bwbass]

Martin, thank you for all you help and advice. I certainly don't want to start a fight over making more work for myself! I just thought the topic was worthy of discussion... of course there are greater variables involved that are harder to quantify or control - my thinking is that since this is a purely geometric problem, once a methodology is established, this one element can be quantified and accounted for.

You are correct that the change in the length of the line through the bend is around 10 - 11% between my spiral and a rectangular model. Overall change amounts to about 3% over the total line. Doesn't seem like much, but another way of looking at it is that if the rectangular model is correct, then the spiral model is two inches too short and has an abrupt .25" stairstep in the taper halfway down the line. Or vise-versa if the spiral model is correct.

My copy of MathCAD explorer has mysteriously died, so I can't model a line with this discontinuity at present. I wonder if anyone would like to try a model comparison using tl sections and post results. (a generic driver would be fine, as long as the taper ratio is similar.) MJK, I know you're too busy to do this yourself, I'm hoping someone else will jump in.

Anyone?

[MJK]

You should be able to download and install a new copy of MathCad Explorer from my site. I guess the link went dead a while back. Once the dead link was identified, I fixed it so it is working again.

[bwbass]

Thanks for the link - I did a system restore and re-installed it and it works fine.

OK, I've now officially spent too much time thinking about this!

Below is my design modeled as if the spiral path is correct...

[bwbass]

.... and then as if a rectangular-cornered path is correct (the line is about 2 inches longer through the bend, but cross sectional areas stay the same.)

[bwbass]

The difference is small, but noticeable, especially in the higher frequency range (not important for this project, though.) Nothing ghastly, though, and I suppose the final result will undoubtably be influenced more by how evenly I fluff the stuffing in the line than by +/- 2 inches of line length....

Even so, if a tapered line had 3 or 4 bends in it the compounded discrepancy could be quite noticeable. So what is the most accurate method of determining the line length through the bends, disregarding the somewhat minor practical value of using one method over another?

[MJK]

bwbass,

The difference is small, but noticeable, especially in the higher frequency range (not important for this project, though.)

Agreed, but the assumption in the worksheet that the driver and open end are coincident is a much bigger source of error then the small inaccuracy in length. Think about how the high frequency resposne might change if the driver and open end were modeled 30 inches apart. This is the kind of thing I was emphasizing yesterday with respect ot other sources of errors.

Even so, if a tapered line had 3 or 4 bends in it the compounded discrepancy could be quite noticeable. So what is the most accurate method of determining the line length through the bends, disregarding the somewhat minor practical value of using one method over another?

Agreed again, this is why I do have not designed many multi-fold TL's. At some point modeling the fold will become more important. When I am worried that my MathCad model may not be accurately predicting the first mode I double check it with an ANSYS acoustic FEM model of the air in the enclosure. After calculating the natural frequencies and mode shapes in ANSYS, I revisit how I modeled the line length in MathCad and make adjustments to reflect the ANSYS results. If ANSYS shows a corner is being "shorted", then I remodel in MathCad to match the FEM path result. Typically I have found my MathCda models to be very close to the ANSYS predictions up to the point where modes across the line area start to occur (~ 500 Hz for the lines I have modeled and built).