Plate arching

[J.A.P.]

Here's a question for the experienced builders out there. What is the relationship between arching and the tonal characteristics of an instrument? For example if the overall height of the arch on two mandolins is the same yet the sweep up to top of the arch is different. Naturally this would also mean differences in the recurved area on the hypothetical instruments. Right now I'm thinking mostly about the effect on top plates, but back plates play a significant role too. I'm hoping you all will weigh in on your experiences, preferences and impressions.

Inquiring minds want to know.

Many Thanks,-------------jap

[Michael Lewis]

JAP, there are a lot of variables related to what you ask, but the simple concept is that curves can flex more under compression than straighter sections. Easy flexibility encourages lower frequency, stiffer for high frequency. Everyone has a different touch, so jump in and find what suits your abilities.

[Chris Baird]

I believe it is the other way around, curves flex less for more stiffness.

[J.A.P.]

I'm thinking along the same lines. More arch = stiffer, thus more treble response. Conversely less arch = more flex, thus more bass response. These thoughts are along the line of my question. The instruments I've been building have a standard, (if there is such a thing), arch height. The difference might be in how I get from the recurved area to the top of the arch. I've noticed that I consistently have a larger flatter arch top area and a shorter quicker recurve area. Curiousity has me wondering what would happen to the tone of the instrument if I reversed this situation. Shorter shaper arch top and longer flatter recurve area. These type of building details are what probably distinguishes one builder's sound from another. Anybody have any thoughts along these lines?

Bored in PA-------------jap

[Michael Lewis]

You guys are right in thinking a smaller radius dome is stiffer than a flatter one, but the dome is interrupted with F holes so it isn't really a dome so much. The point about straight sections supporting more than a curved one under compression is easily demonstrated with thin pieces of wood. Apply a compression load to the ends of a thin piece of wood like a yard stick and notice when it starts to collapse, then using the same piece of wood put a slight bend in it and apply a load to it and notice when it starts to collapse. The straight piece will support much more load than the curved piece even though they are the same measurements except for the curve. This translates to a mandolin top in that a lot of dome behind the bridge will pooch up under compression (from the strings tension), where a straighter section from the bridge to the tail piece will support much more load without distorting. This works for me, YMMV

[J.A.P.]

Michael, I think I'm beginning to see the point your you are trying to make. I have see a number of mandolin top that are pouched up between the tailpiece and the bridge. It seem to me that a flatter area behind the bridge can actually act like triangular loading condition. Perhaps, one reason for instruments pooching out behind the bridge is the top of the arch is behind the center of the bridge. This is an interesting thought, but I'm still trying to make a relationship to sound quality.----------jap #

[Dave Cohen]

Despite whatever anyone may tell you, the real answer to J.A.P.'s question is that we don't know the answer to his (her?) question. There is a proliferation of confusion about this and related subjects among luthiers. There is a tendancy to confuse static properties with dynamic ones, which is sorta what Michael did above. There are also popular misconceptions about stiffness and amplitude. On the one hand, you have the misconception about "stiffer is better" among pieces of wood. That one leaves out a lot of physical information which is already in the literature. On the other hand, you see a lot of "If I can make it vibrate more, it will be louder." I'll just leave that at "Total Confusion."

Here are a couple of things to think about. One is that, other things being equal, an arched plate will have higher normal mode frequencies ("eigenfrequencies") than a flat plate will. The catch, of course, is the 'other things being equal' part. We don't necessarily want higher modal frequencies; we want them to be the 'right' frequencies, whatever those are. So we arch the plates, thereby raising the modal frequencies, then thin the recurve, thereby lowering them again. Confused? You should be. The other piece of food for thought is that an arched plate is nonlinear. That is, its' modal frequencies depend on amplitude. You can check this out in a good acoustics text, like Morse, or Fletcher & Rossing. There are some Chinese opera gongs with a domed vibrating area of the plate which illustrate that. When you bang 'em, the amplitude is greatest, and the amplitude gradually decays from that point. What you hear is a change of pitch as the amplitude decays (that is the nonlinear part). For that matter, strings are slightly nonlinear. Ever notice, on your tuner, how the pitch/frequency of the string is higher when you pluck it, then it stabilizes at a lower pitch? Now, are arched mandolin plates nonlinear? I don't know that yet.

[Michael Lewis]

Wow! Thanks for the input Dave. Yeah, I was talking about structure and the question was concerning tone too. Sometimes critical thinking gets messy around here<G>. At any rate, I hope we are all learning to see a bigger picture of what happens as a mandolin does it's job. Anyway, thanks for getting us back on track.