The Cantenary Arch

The main arching of a violin, viola, cello, and this Archtop guitar is formed on the inside with catenary arches.

A catenary arch is what you get if you stretch a chain out, and let it dangle:

That is what it is, but what we are really thinking about is the other way around; like this:

You see it all the time on concrete overpasses.  I see it on the steel truck beds.  They are almost flat when loaded down, but rise up like this photo above when they are empty.

A catenary curve is very strong.  We don't want violins and guitars to be VERY strong.  They have to vibrate.  But the stronger we design them, the thinner the wood can be.  This makes the response faster.

Of course, wood choice has a lot to do with this as well.  The bellies usually bring much more to the table sound wise than the backs do.  The Euro Sycamore, Acer, that I'm using for the back is just starting to have some sound to it.  It is still quite thick, especially around the edge, but now I can get a series of ring tones that were suppressed by the wood itself before.  The curly redwood belly is as thick as can be right now, and it shakes with resonance.

Everything on the inside is checked with the chain.  Just a 2 foot chain from Walmart.  The ends are set in from the edges.  Further in on the lower bouts, and closer at the middle.  There is more recurve on the lower bout than the middle.  It helps to draw it out first.  With violin posters it is easy to do.   This one is another story.  I carved inside and out until I had an idea, and then I went with it.

There is a full diagonal cross arch from the bass top to the treble bottom.  The diagonal cross arch from the little wing to the bass bottom is broken up.  They meet at the point where the first diagonal arch went.  The top end is deeper, giving a very curvy, stiff area.  The other end is more like the arch on the other side.

There is also the same kind of arch down the centerline.

The cross arches go through the diagonals, and end at the centerline.  But at the top things are different.  The cross arches there start out perpendicular to the diagonal arch.  They go from edge to edge, and through the diagonal.  The small wing is almost centered on the diagonal, and the diagonal is the deep point.  The bass side isn't, and the deep point is further towards the neck.  When the cross arches get past the area where they go from end to end, they gradually change to cross arches from the side to the middle.

It reminds me of the line pictures you see of finite element analysis.  Each design constraint pulls on the other, and the final shape is the smallest way that the surface can be draped around the form.

On the outside I mark the point of inflection.  The outside can be roughed really close to size across the middle, and fairly close up to the point of inflection.  I have this down to about 4.5 mm or so at most of the inflection line; except for the ends.  From there the thicknesses taper to 6 -7.5 mm in the middle.  That's about as far as you can go until you define the finished outline, and cut the recurve on the inside in.

The part at the top will have to wait until the recurve is cut in.  It seems to be a lot more complicated.

Yes, I am going to have a tapered Guadagnini button.

The Curate Cycloid

I use a curate cycloid for most of my cross arches.  Some models are slight variations of it.  This guitar arch is very wide.  To get the rise off the top that I want, the cycloid can't be the full width.  I'm doing it in the Guadagnini style, so I'm bringing the arch low point to 7-12 mm from the outside edge,  and that will be perfectly flat.  There will be no dip.  The arch will just rise.

4 mm thick at the edge.  This is for the back.

The most important point to know in carving a cycloid is the inflection point.  This is where it changes from concave at the end, to convex across the middle.  This point is at a different place on every cycloid.

If the arch is narrow and tall, the inflection point will be pushed out more to the edge. It will also be lower relative to the height of the arch. If the arch is low and wide, the inflection point will get closer to the midpoint in height, and will be further in from the edge.

This will show you how it works:

70 is the angle of rotation.  You will need to check every degree and see the difference in angles.  If the point is before the inflection, the angle is continually increasing.  After the inflection the angle between the points decreases.  You have to use all the numbers to the right of the point.  The angle changes are very slight, and rounding off will mess you up.  Here's what this one looked like.  I write down some things, and the difference between the lengths in width I put in the memory:

You can see that the angle doesn't change that much, and each degree of rotation makes very slight movement.  Armed with that information it is easy to draw a cycloid:

I can readily see where the arch needs to come down, and about how much.  I also see that the middle is still far too thick, since the outside arch in the middle is where I want it.   I wanted it high.  Guadagnini was not Strad, and he certainly wasn't del Gesu.  It is just about 29 mm from the bottom.  I mark up the outside of the back, and then can trim it down some.

The area in the middle is how much the cycloid is extended in the middle.  It just blends in with the rest.  The numbers show the thicknesses.  It is still very thick.  I didn't know what I needed.  The inside needs to go up about 2 mm on the long arch.  I made a gauge for measuring.  I found a cheap TTC dial gage that measures in TENTHS  of a mm.  It was on sale for a crazy low price.  Maybe it's overkill; but it works great, and is just deep enough for the arch top.

It comes in handy for fine finishing.  Especially on the back where the thickness can vary by 100% or more from the edge to the middle.  The belly is not as difficult as far as thicknessing goes.  It doesn't have a ton of travel, and maxes out at 11 mm.  No problem; anything that thick would be useless for sound anyway.  I was thinking more in the way of 3 mm to 7.5 mm.   We'll see.

I roughed some off the outside, and wanted to see what digging the inside arches deeper would do.  It is working out nicely.  The long arch is a straight catenary, and the diagonals are straight catenaries now too!

Next time we'll go over catenary curves.  It is the essence of the instrument.

The Back Arching Started

Ok. Let's talk about the arching on the back. First things first. I've never held, or looked at an archtop. So I don't have any idea how it really looks! Why am I making one without seeing one? Having faith that it will work out is a very valuable thing. It is essential in life. In making an arch top you have to have an idea as well.

 Here's the premise: GB Guadagnini's grandson comes over, and shows him his latest guitar. The old man looks at it, admires its outline, strums the strings a little, and notices how it sounds. He wonders about the flat plates, and looks inside. Seeing all the braces he says to his grandson, "Whatsamatter you guy? You forgot everything I taught you? Get me some wood! I'll show you how it's done."

 Yeah old man Guadagnini had no more idea about archtop guitars than I do. But he was still going to make one. How would he approach it? Probably like building a cello. They have just about the same range. But the guitar has a much smaller footprint. More strings. More pressure wanting to budge the belly up, but less pressure pushing it down. How do you bring out the low notes with such a small area, and with such a puny volume to support them? The archtop addresses the pressure problem somewhat. The larger string angle brings more downforce on the bridge, and reduces the squishing force of the strings on the body and neck. A carved belly will counter the downforce, and TWO large 'bass' bars will take up the contracting force of the strings on the belly.

 He was intrigued by the laminated neck, and thought that would work just fine. We'll add a truss rod, because everyone wants one. So let's see what I've come up with so far. I used a 1100 mm radius for the outside long arch. Why? I have a gauge for one, and it gives an arching of the right height. Maybe GB had one? Two diagonals on the inside generate the arch on the inside, but they are not solid arches. They rise up quickly from both ends, but only go 3/4 of the way or so. It leaves it thicker in the middle. The cross arches are the same way. I have the central area on the outside pretty much the way I want it, except for the upper area. the recurve area of the bottom isn't there yet. But NONE of the recurve area is close, Especially at each number 3 in the middle.

 Yesterday I took over the dining room table:

The area in the middle is the area that is not part of the cycloid that forms the outside cross arch. The arch rises too quickly on the edges to make that work. So the central area continues the arch, matching the width of the outline. The line outside of that marks the area that is pretty much, "to print." This is basically how I do violins as well. The central area is defined first.



To determine what I had, and see what I needed, I had to use my plastic profile gauge.  But it was too short.  So I went to Rockler Hardware, and picked up another to make it 20" long.  It was more of an ordeal than I expected.  They are identical, so to put them together, one of them has to have the hundred or more red plastic pieces pulled out, and reversed.  If you have one, you'll know what I mean.  They are angled on the back and the angles will not match.  It's all set now, and I'm not taking it apart.

Taking patterns inside and out 60 mm apart, I drew them on an 18 X 24 sheet of drawing paper, and examined them to see what I needed to do.  I drew cycloids on the ends of the outside, and determined how much of the center would just be extension.  Turned out to be 50-80 mm.   Then I turned the paper upside down, and figured out where the ends of the cross arches should be, and how deep they have to be at the center.  Then I had to check and make sure that the diagonal arches still fit with the deeper, wider cross arches.

I did the long arch as well, and I saw that the gauge does its job.  Now I have carving to do.