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Apr 22 2014

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Tractrix Ribbon Horn …

{{{{ Update 2014-04-25 …

Things are looking good. I have just purchased some additional 20cm raster boards, and sliced them perpendicularly into narrow, 3 cm wide pieces.  A lot of pieces.

I also purchased a run of tape, a polyester fabric, that is normally used for hemming and / or , finishings of the underside / bottom side of curtains. The tape is 10 cm wide, so two runs shall fit nicely the whole “height” of the narrow wooden sticks.   The idea is so as to glue the tape, single sided, on top of the neatly arranged, narrow pieces of wood.  Basically, after cutting the board into narrow pieces, narrow sticks, I then “recreate” the shape of the original uncut board by arranging the sticks next to each other, and then by gluing the fabric on top on them.

In this manner,  I restore the “original shape” of the board. I bind the board together with the poliester fabric tape, running along the whole length of the board, The fabric now performs the role of “hinges” between the individual 3 cm wide wooden segments.

Essentially, this shall allow, after drying up, to make a curved wall out of this “shutter style” recreated wooden board.  Obviously, the curved walls will be arranged along the trace of a tractrix curve, and then firmly glued to the floor and the ceiling planes of each individual horn.

As the tractrix horns are convex from the inside of the horn, and concave from the outer side of the horn walls, this implies that the glued fabric shall be from the outer side of the horn and “to be seen” by bystanders.  This is the reason why I made myself the extra effort of finding a special fabric, thin, even, without any external texture, but at the same time, very strong. Provided that we perform the gluing task orderly and evenly, the fabric shall essentially disappear under a layer of mahogany wood stain varnish, which is foreseen as finishing touches.

}}}

 

As the saying goes … “I Told you So” …

Indeed, they told me.  The word of the month is “TRACTRIX”.

So here I am, back to square one, or almost square one, thinking about the final shape of my ribbon horns.

The previous concept was a lazy concept. A doctrine stating: “as little tools and wood handcraft work as possible” concept.   An ideal idea for the two-left-handed, the carpentry-skills-decapitated, or the DIY-lazy folks in general.

But was that REALLY what I wanted ?   If you want to have a really good horn, you simply can not “cheat” … or “cut corners”  (literally: making curves actually *does* involve some filing down, or “cutting” of corners, so as to make them smooth … ) ….

Well, as demonstrated in some of those earlier texts of mine (photos included), and by listening to what I am currently listening to now, I have actually proven that yes, you ** CAN ** “cut corners” by not smoothing the corners.

But is that what is really what I want and with what I shall be really happy in the long term ?

Not necessarily.

So now, I have a slightly modified concept, a slightly new horn philosophy. Let us now try to take a somewhat different stance. We shall still, albeit only partially, stick to the concept of “as little tools and woodwork as possible”, but now we shall relax this rule a bit.  Now I think that I found a reasonable compromise, a solution as to how to obtain the best of both worlds:

a). I want to have ***Curves***  !!!!  AND:
b). A fairly small amount of woodwork, that goes along with it.

Having said the above, you might wish to ask:  What type of Curves ? What are we talking about here ?  Are we talking about a milling / grinding / rolling / cutting a full blown radial horn ?

Along it’s rotational axis ?

No, not exactly.

For me, that would be too complicated, for starters, at least.  I actually do want to keep this fairly uncomplicated …

Therefore, I want to “stick” to my initial concept of the standardized “height”, to the constant vertical distance between the “floor” and the “ceiling” of the horn.

I want to stick to my fixed vertical height of 20 cm.  But the side panes of the horn – ah, this a totally different story. Now I want to “model” so as to follow a specific curve.

How do I model a curve, using pre-cut pieces of wood ?   Well, I have this idea. Cut a whole lot of small wooden sections, say of a width of 2cm wide and the (standard) 20 cm height. Now put them all adjacent to one another and then, from the top side, glue of them a piece of strong linen cloth, precut to size. At best – of a uniform color, as it will later be “seen” from the outside.

When the contraption, the wooden sections with cloth, dries out, you will actually be able to create a “curve” out of the resulting planar piece. The cloth shall work as “hinges”, so you will be able to force those 2 cm wide sections of wood to closely follow a curved shape, by slightly tilting each next section by a tiny angle, using the supporting cloth as a hinge of sorts. Essentially, you can now allow the wooden wall to follow any concave section of curve that you may imagine, and this with quite a good approximate tolerance. The narrower the cutting raster of the pieces of wood, the better the approximation.

It now suffice to “draw” the curve on the floor and on the ceiling of my wooden support box frame, and simply glue in this “curved” plane into the box, so as to form a curved “left” vertical wall, and a curved “right” vertical wall, defining the curved shapes of the sides of the new horn. Obviously, the driver unit to be used is still a ribbon driver, longitudal in shape. Therefore the above described method of “shaping” the throat of the horn (the narrow side) seems a reasonable compromise.

Both the throat, as well as the mouth of the horn, will be rectangular, and not oval. The side walls of the horn will be curved.  On the other hand, the top and bottom of the horn will be flat planes, set a constant 20 cm apart from each other.

OK. So now we have some curves. But are we talking about parabolic curves ?
Hyperbolic curves ?
Logarithmic curves ?

Well, no.   I am going for a very special curve:

The TRACTRIX.

It is not a well known curve, and indeed, a bit of a pain, a bit of a headache in terms of the maths. It has very good sound wave propagation properties, providing with the least amount of distortion to the sound.

The tractrix curve can be conceptually “visualized” by a tractor pulling a piece of wooden timber, a wooden log of a fixed length.  The other end of the log draws our tractrix curve.

You may say, “No big deal” … Well, actually, it is a big deal.  Mathematically, at least.

The thing is, the movement of the tractor is PERPENDICULAR to the initial positioning of the wooden log.

Or, yet a different visualization: long vehicle, TIR truck trying to make a right turn in a narrow street, pulling a long load … look at the path as covered by the wheel on the back axis of the truck below:

traxtrix 1

OR: a dog on a leash concept …

dog leash

or a watch pulled across a table …

TRCTX watch

or a child with a toy being pulled by a string … or  .. etc, etc….

TRCTXD

A good visualization of this concept is to be found here:  http://mathworld.wolfram.com/Tractrix.html

“…

The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1999, pp. 250-251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog’s master, the curve has the descriptive name hundkurve (dog curve) in German. Leibniz found the curve using the fact that the axis is an asymptote to the tractrix (MacTutor Archive).

From its definition, the tractrix is precisely the catenary involute described by a point initially on the vertex (so the catenary is the tractrix evolute). The tractrix is sometimes called the tractory or equitangential curve. The tractrix was first studied by Huygens in 1692, who gave it the name “tractrix.” Later, Leibniz, Johann Bernoulli, and others studied the curve.

In Cartesian coordinates, the tractrix has equation

 x=asech^(-1)(y/a)-sqrt(a^2-y^2).

…”

The maths of the tractrix seem to look forbidding, but the sech(-1) function can be written out in a more “human” readable form, as follows:

x = a * ln( ( a + Sqrt( a * a - y * y ) ) / y ) – Sqrt( a * a – y * y )

The formula contains symbols with following meaning:  

x – Assuming that the X axis goes axially inside into the horn, then the value “x” is the distance from the “mouth” of the horn.
a – is the Radius at the mouth (in other words: the big “sound output” opening. This can also be visualized as the “ROD LENGTH”, or the “fixed length” of the leash (in the example of the dog on a leash), or the Trailer Length, in the example of the long truck making a right turn).
r  - is the radius, or the distance x between the side wall of the horn and its central axis, at a distance of “x” centimeters deep into the horn, counted from the mouth opening.
ln(z)  - is a function: the natural logarithm of “z”.
Sqrt(z) – is a function: the square root of “z”.

 But there is a problem associated with this formula. The issue is that it presents the “depth into the horn” – the value “x”, as a function of the distance “r” (or: radius) of the horn walls from the central axis at this point “x”.  So, simply speaking, it presents the function: x = F(r). 

Anybody, who wants to build such a tractrix horn, actually shall need the opposite: r = F(x).

Tough luck, Guys. There is no simple way to find the reverse function to the one presented above. So what it comes down to is to make a table in Excel, with a pairing of “x” and “r” values, and simply work off such a table. Give it a few round of try and see … you will get there, eventually.

This is a very new concept, only but an idea.  Only but the beginning of a new (horny) road.

I have yet to purchase the linen cloth :)

I have yet to cut the wooden sticks for the “curve” sections.

I will keep you posted on my progress on this one   { if at all any :)  }

 Cheers, 

Ziggy.

P.S.:   Here are some source links, if you find this “Tractrix” subject interesting:

http://www.ugr.es/~fmartin/gi/curves.pdf

http://mathworld.wolfram.com/Tractrix.html

http://www.reocities.com/agalavotti/tractrix.htm

http://fullrangedriver.com/singledriver/horndesign.html

http://users.etown.edu/s/sanchisgr/HistoryOfMathematics/Calculus2/Worksheets/W2.pdf

http://rfdz.ph-noe.ac.at/fileadmin/Mathematik_Uploads/ACDCA/TIME2008/TIME08_contribs/Steyn/Steyn.pdf

 

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