Tractrix horn calculators

[capo72]

Yes jc I did it the way you explain. Knowing the height of the cross section at any given point allows you to compensate the width (curve) to get the correct area. I'll dig out my paper work tonight and see exactly what I did. It's been well over a year ago since I did the layout. I'm going to try to plot some curves from the calculator via Autocad today and then i'll compare them to what I did by hand. But I agree with you that the curve (width) should be plotted based on the area with a known height and constantly adjusted to remain tractrix from the throat to the mouth.

Jeremy

[jwc]

Here is an example of the image2 or blue.

There is no change in the vertical height of the horn. So the "flare" you see on the top and bottom would be the same as what greg is calling the "center plane" of the horn. When calculating this flare....I knew the internal height was 25" throughout the length of the horn. Since I knew the area at each length (tractrix formula), then I can calculate the width of the flare by dividing the area by the 25"Hieght.

With these midrange horns we've been talking about lately. we make the "flares" differently to account for the vertical expansion. So they do not represent the center plane of the horn. They also don't represent a flare that lays "flat". They only represent a tractrix flare when they are "tilted" at the angle of vertical expansion.

jc

[greg928gts]

Edgar - YOU GOT IT! Great diagrams that explain what I noticed when I started building a set of horns and thus brought up this question.

I ran this whole thing by my foreman Derek today. Right away he said "if nothing else, just look at what will happen to the length of the horn as you tilt the template up at the angle", and he's absolutely right and this is shown in Edgar's diagrams. Not even thinking about the curve, the length of the template is different at the center plane than it is at the angle of the top piece.

It is in fact a different curve all-together, the top of the horn and the center plane.

Greg

[greg928gts]

I think that because we sand the edges of the top and bottom back square to the sides we have the same curve although it may be shifted back toward the mouth.

No, sanding the edges back to square is not correcting the curve difference.

Greg

[greg928gts]

Now if you have a "known" vertical expansion. Lets say at 4" along the center of the horn....you know a line from the center of the throat to the center of the mouth. Yes 4" from the throat.....Let's say the calculator for tractrix (not all the extra stuff that "calculators" give you) wants the area at 4" to be 35 sq in. Well.....if you know exactly what the hieght of the horn is at 4" from the throat.....then can't you calculate the width of the horn at 4". For instance.......if you know the horn will be say 5" tall at 4" from the throat. You know the tractrix formula dicates a 35 sq in area at that location......then won't the width of the horn at that 4" location be 7"?

Yes. But if you cut the top and bottoms of the horn using the calculated curve, the end result when measured with a tape measure will be different at some points of the horn, mostly different towards the mouth. In other words, it may not be 7 after it's built.

In order to get 7 at that point, the curve of the top and bottom pieces of the horn must be adjusted slightly to account for the angle that those pieces are sitting at, and to put the side pieces in the right place so that the measurement will be 7.

Greg

[greg928gts]

Greg. The flares you showed us. Did you say the one on the left "lines up" with the one on the right when you tilt up the mouth edge to the appropriate height of the horn mouth? Is this what I understand you to say?

jc

Yes, more or less. In other words I had to come up with the "new" curve by trial and error and it may not be perfect. But if I held that template that is on the left in the picture up at the angle it will be at when built, and then dropped a plumb pencil down from the edge of it and plotted the curve on graph paper, it would be very close to the calculated curve that the tractrix calculator gave me. BTW, this is basically how I fudged it.

Greg

[tigerwoodKhorns]

Great 3D figures.

I may have missed this but is the equation for the center of the horn (and thus wrong if used for the top and bottom) or is the equation tailored so that it is correct for the top and bottom pieces of the horn?

If it is incorrect, the correct points can be determined without too much math in AutoCAD if someone here has 3D capabilities to draw out the horn using the equation as the center (I don't know if AutoCAD was used for the earlier figures) and can "grab" either the top or bottom piece and put it in the X-Y plane and then grab the coordinates for the flare.

I am guessing that you want the outter coordinates (need to know the thickness of the wood - probably 3/4") because when you sand it to match the flare, you do not take any material off of the outter portion, just the inner portion, right???

[colterphoto1]

" It is in fact a different curve all-together, the top of the horn and the center plane. "

YES

[jwc]

greg. I still don't think you and I are seeing eye to eye on this.

You realize that everything edgar has pointed out is just what we were saying in the other thread. What you are discovering is the same fault in Bruce Edgars article on how to build a midrange horn. YES....the length of the "Flare" will change as you stand it up on an angle.

Please tell me what is wrong with the way I just described how to calculate the width based on a known height? What do you mean you had to fudge the numbers? You shouldn't have to fudge anything.

jc

[jwc]

Greg. The flares you showed us. Did you say the one on the left "lines up" with the one on the right when you tilt up the mouth edge to the appropriate height of the horn mouth? Is this what I understand you to say?

jc

Yes, more or less. In other words I had to come up with the "new" curve by trial and error and it may not be perfect. But if I held that template that is on the left in the picture up at the angle it will be at when built, and then dropped a plumb pencil down from the edge of it and plotted the curve on graph paper, it would be very close to the calculated curve that the tractrix calculator gave me. BTW, this is basically how I fudged it.

Greg

Greg...what exactly are you letting the calculator do for you? Don't let that program calculate anything other than the tractrix area at a certain horn length.

jc

[jwc]

Now if you have a "known" vertical expansion. Lets say at 4" along the center of the horn....you know a line from the center of the throat to the center of the mouth. Yes 4" from the throat.....Let's say the calculator for tractrix (not all the extra stuff that "calculators" give you) wants the area at 4" to be 35 sq in. Well.....if you know exactly what the hieght of the horn is at 4" from the throat.....then can't you calculate the width of the horn at 4". For instance.......if you know the horn will be say 5" tall at 4" from the throat. You know the tractrix formula dicates a 35 sq in area at that location......then won't the width of the horn at that 4" location be 7"?

Yes. But if you cut the top and bottoms of the horn using the calculated curve, the end result when measured with a tape measure will be different at some points of the horn, mostly different towards the mouth. In other words, it may not be 7 after it's built.

In order to get 7 at that point, the curve of the top and bottom pieces of the horn must be adjusted slightly to account for the angle that those pieces are sitting at, and to put the side pieces in the right place so that the measurement will be 7.

Greg

Are you referring to the manual way of calculating or what the program tells you what the "curved" sidewalls will be? Yes...if you calculate a "flare" based on what you are calling the "center plane", then YES...the error will be more toward the mouth as the "mouth" part of the flare is more "off the center plane" that the throat portion of the flare. Again....if you account for the height differences from throat to mouth...then the flare you make will not be a curve of the center plane but a flare appropriate for the vertical expansion as edgar pointed out in his red image.

I disagree with the second statement. If you know that the horn will be 7", then once your flare is tilted up in it's appropriate postion (angle), then it shoud be dead on.

jc

[greg928gts]

Greg.....I think I know now what you have been calling the "center plane". The flare that I have drawn out isn't the "center plane". That would be the "blue" in edgars diagram. What I have drawn out...the flare.....is the red...as I know what the height of the 223Hz tractric horn is from thoat to mouth. So I basis my calculations are from the "center axis length of the horn". The flare doesn't represent the "center plane".

I see no information at all that indicates that the tractrix calculator is giving us this curve. I now have three reasons why I believe the tractrix calculator is giving us the center plane curve.

1. Because the side plane of the horn is the same up and down the side (as you said JC with a horn with no vertical expansion). This is the curve we want the side of the horn to be, that is the curve of the plane (as shown in diagram #2 by Edgar).

2. Because we are plotting the angle for the vertical expansion on flat graph paper in two dimensions, and because we are plotting the curve of the horizontal expansion on flat graph paper in two dimensions. This IS the curve that is desired for the sides of the horn. I believe this is the center plane curve. For the tractrix calculator to be giving us anything but this center plane curve, drawn in two dimensions, they would HAVE to clarify that in the instructions for using the calculator.

3. The most important and hardest to argue with, because the length of the horn from throat to mouth is part of the horn calculation, and yet if you build the horn by cutting the tops and bottoms according to the calculated curve, the horn WILL NOT be the length figured into the tractrix calculation. See diagram #5 by Edgar. If you build the horn so that the curve of the sides ends up being the calculated curve, then the length of the horn will be what is part of the calculation. The only way to do this is to cut the tops and bottoms with a different curve.

Unless I see some good hard evidence to the contrary, I'm going to assume that the tractrix calculator curve is the center plane of the horn, and adjust the curve cuts of the tops and bottoms of my horns so the resulting side plane will be the curve that is given by the tractrix calculator. Just makes sense to me now.

Greg

[greg928gts]

JC - I didn't have to fudge numbers, I had to fudge the curve of the cuts I was making to the top and bottom pieces so that the curve of the sides of the horn matched the calculated curve.

Regarding the side curves, the only thing I'm getting from the tractrix calculator is a point by point reference that I'm plotting on graph paper. This gives me a curve. I now believe almost certainly that this curve is the curve at the center plane of the horn or the side plane of the horn and that no adjustments have been made in the calculations to give me the "flare" curve. The builder of the horn has to make those adjustments to the cuts of the top and bottom pieces. If not, the length of the horn will change from the dimension entered into the original calculation.

Greg

[jwc]

Greg...the technique I showed you will give you the EXACT length the horn should be. If it doesn't...you are not doing it the way I do. I can tell you that the 223 Hx horn I came up with has the exact length it supposed to. Yes...If I were to calculate manually the center plane with the tractrix formula...yes my horn would be shortened. BUT it isn't.

The thing is...when you make the appropriate flare to account for the variation in height as you get to the mouth...this flare cutout is longer than the horn length when you lay it flat on the table...but once you prop up the mouth to the appropriate angle...the length is perfect.

Like I said before....if you and I were sittin in the same room.....you would see what I'm saying.

What you are argue....I am agreeing with. BUT...what I'm saying to you is there is a "trick" to make that flare approriate when you bring it to an angle. There should be no fudging. This Flare that I create is NOT the center plane. I only use tractric calculators to give me the area at a certain horn length. That info by itself is clueless of anything to do with the center plane.

jc

jc

[tigerwoodKhorns]

JC,

I see what you are saying and it seems easy enough to do as opposed to my AutoCAD solution. Do you have a spreadsheet that calculates the proper points using the known area and height (derived from the angle of the top and bottom pieces)?

What Fc do you use for a K Horn?

[Edgar]

Don't let that program calculate anything other than the tractrix area at a certain horn length.

Yes and no.

Remember, the tractrix curve is a line in two dimensions. The tractrix horn has a circular cross section, and the radius of the cross section at any point along its length is equal to the distance from the tractrix curve to the centerline at that same length. Most tractrix calculators compute the area of the tractrix horn at any point along its length as the area of the circular cross section at that length. While easy to compute, it does not accurately represent how the wave front propagates down the horn. The wave front is curved outward -- Roy has described this as a "bubble". The bubble is NOT spherical, but it's pretty close. So a better approximation of the area expansion can be obtained by computing the area of a spherical section at each point along the length of the circular tractrix horn.

However, when translating the area expansion of the circular tractrix horn into any other cross section (square, rectangular, elliptical, etc.), the shape of the "bubble" appropriate for THAT cross section also needs to be taken into account. For a rectangular tractrix horn that only expands in one dimension (the blue figure in my Image2), the bubble will be approximately cylindrical. For other shapes, it will be different and generally much more complex. In fact, the best way to determine the actual bubble shape is finite element analysis.

I guess the point of all of this is to tell you not to get too worried about matching the dimensions exactly. Even if you do, it'll still be wrong. So all that you can do is design it, build it, and see if it works.

Edgar (Greg)

[jwc]

Edgar. I agree with everything you say except for two things.

You can make the horn with the correct area at a certain length. Round or not.

And......I tell people not to rely on the calculations provided by calculators. You can make a midhorn with vertical expansion just like your red image. You don't have to accept being off a little. There is a trick. Only way I know to do it is do it on paper.

[greg928gts]

Just got off the phone with JC.

Maybe I'm wrong. I still don't know.

JC explained to me that we were accounting for the height of the horn in our calculations of the side curve, so therefore the curve really is the "flare" of the horn. And as I'm talking to him on the phone that all makes sense and I'm feeling rather foolish about making such a stink. But I no sooner got off the phone and started looking at my plots and now I'm not sure again. The plotted curve ends up at 14.33", which is the centerline length of the horn, so it still looks like the center plane curve to me. UGGGGGH!! I don't want to figure this, I just want someone to tell me how to build it. I'm an idea person and a builder. The stuff in the middle, you know the stuff engineers do, that's just not my gig.

JC, I think I'd like to go over these numbers and the resulting plots again with you, if you don't mind. Off to bed right now, busy work day tomorrow, guests tomorrow night, so I don't know when right now. But if you have time later this week I'd really appreciate it.

Greg

[WMcD]

I looked at the problem while making exponential horns with tilted but flat tops and bottoms.

The drawing here is exaggerated in that it uses a 45 degree angle. You have to imagine this is a side view of the horn to be built. The major center axis (x) is at the center of the horn, and H is the top plate.

I'll assume the goal is to have a traxtrix (or anything else) in two dimensions at the center axis, so it is at right angles to this view, say, coming toward us.

The divisions showed can be one inch down the center of the horn.. But to make things show up, I've used five divisions to equal one with curved brackets, then it is "projected" it on to the top plate.

This is easy math because we know that the 45 degree triangle shown has sides of 1, 1 and square root of 2. So the H is 1.414 (I just wrote 1.4).

Therefore, if you want to make things more precise, using this as an example, you'd redraw the tractrix for the top and bottom plates. The "toward us"
numbers are the same. But you have to stretch the x axis by a factor of 1.414 at every point.

Probably you're using some other angle and we have to calculate a different stretch factor. If we know the angle (that is a lower case alpha I've used) then it is pretty simple trig.

- - - -

This does not solve three issues.

1) As Bruce Edgar pointed out, you have to use a band saw and tilt to top and bottom plates when cut to get the side of the plate come out correctly. But I sanded in one case, and filled in in another with putty. I think that if you draw the stretched curve on the bottom of the top plate, you sand away the excess and things are right. This issue is most pronounced at the mouth.

Not having a band saw, I tilted the plates and then used a hand-held drill with a very small bit. I tried to keep the bit plumb (this would be easy with one of the drills with a bubble level built in) to mark off the points. Then I used a hand held jig saw to make the cut. Somewhat slopply.

2) The pure tractrix is really a horn of circular cross-section. A horn with a square cross section with curved tops/bottoms and left/right sides can honor the curve too, by compromise. Once we depart from a cross section of regular polygon (?) we have to make some decisions. (a) We can use an equivalent cross section. I believe this is what Bruce Edgar and probably Al K did. Or ( we can use the flat top and bottom plates, use tractrix at the sides, and call it close enough.

3) We really have not been calculating the lenght of the side walls as we bend the wood. There is probably a calculus solution. I've used a spreadsheet to calculate use a running total with a Pathagorian (sp) calculation. It is quite un-necessary because the side wall is maybe 130 percent of the center axis and were just bending thin ply.and that sort of overage doesn't cost much. You can just use a piece of cardboard once the top and bottom plates are in a jig to find the needed lenght.

Best,

Wm McD

[Edgar]

You can make the horn with the correct area at a certain length. Round or not.

You can certainly make a horn of any shape have the same cross sectional area as the circular cross section of the tractrix horn. But my point is that, even if you do THAT perfectly, you still haven't correctly translated the area expansion of the actual wave front because the wave front doesn't propagate as a plane.

And......I tell people not to rely on the calculations provided by calculators.

When I design tractrix horns I use my own tractrix calculator, because that way I know exactly what assumptions it uses.