plate shape

[John Bertotti]

I think I have the answer to this question but want to double check. My presumption is if the total square inches of the sound holes is the important part for a traditional mando with the modes not being affected by the shape of the holes then can I assume the same for the plates. If total square inches of the plate remains constant can the shape be seriously modified without affecting the modes? I realize this opens up a whole case of worms with graduation and arching, which will affect the modes, but I'll worry about that after I get the shape I want. Thanks all John http://www.mandolincafe.net/iB_html/non-cgi/emoticons/rock.gif

[Chris Baird]

The only mode the hole has any real effect on is the body air resonance(helmholtz). The mechanics that dictate that the shape isn't significant for holes and air resonances is different than that which dictates the importance of the shape of plates. #However, experiments have been done with violins and I have read reports that a great many shapes can be used and give good tone. #What alterations had to be made for compensation I do not know. I imagine a different arching and graduation would be needed for each different shape.

[Dale Ludewig]

This question might well be answered by Dave (Dr.) Cohen. He's an expert an analyzing this stuff, particularly the hole size impact on the sound of the instrument. If I understand him correctly, the Helmholz frequency (resonant frequency of the body cavity)is immune to changes to Fhole (or any other shape) size unless it becomes extreme. On the other hand, I agree that plate shape and size and graduation become major issues. I'm in a sense working on the same problem because I've started two mandola's (Emory Lester style- see my website for photos- not trying shameless self promotion) that I hope to have to show at IBMA. Plates are larger but the same thickness total. The graduations will have to be different and I imagine I must call on intuition from my little brain. I don't think I can get the same break angle over the bridge because the body is longer but I can't make the bridge higher. Is the string tension higher or lower? I haven't figured that out yet and may call some other builders of note that I am glad are friends of mine. I guess (this is my idea and partly how I work), you press down on the top with what you think the final pressure of the strings will be and see what the top does as far as depression and tonal response when tapped upon. As with normal mandolins, I believe it is largely intuition. You can just about hear the final product. (just about!)
Just my .02 worth.
Dale Ludewig
www.ludewigmandolins.com

[craigw]

Dale, I was schlepping through the Loar Picture of the day thread, under "Post a picture of your Mando" and came across this picture of a vintage H-5: www.f5journal.com/pic_day/76967/76967_1.jpg
It appears that there is more arch in the top plate giving it the added height proportionately that in turn would give a greater break angle. I have never seriously looked a H-5 mandolas until now but intend to build an H-5 soon. My first thought was it looked like a pregnant guppy and "what's up with that" but with your question re: the break angle I wonder if this the reason for a seeming exagerated arching. I would be interested to hear from those more experienced with vintage H-5s regarding engineering of the plates.

[Dave Cohen]

There seem to be several questions here.

Regarding plate shape: normal modes are global, i.e., they encompass the entire plate. So the fundamental or (0,0)plate mode for an oval plate will be oval shaped, while the same mode for a circular plate will be circular in shape. they will still be (0,0) modes; that is, they will have no nodes except at the perimeter. The higher modes will be global as well. F'rinstance, the oval plate will have a sideways rocking or (1,0) mode that is oval shaped, while the circular plate will have a (1,0) mode that is circular in shape. The dimensions will affect the frequencies of the modes. In an oval plate, one would expect the longitudinal rocking mode to have a lower eigenfrequency than the sideways rocking mode. All of this assumes isotropic plates, i.e., stiffness the same in all directions. Wood is not isotropic, but the textbook principles still apply within limits.

In my last round of holography experiments, I looked at several different mandolin body shapes: a Lyon & Healy A, a Vega 205 Cylinderback, a couple of Neapolitans, several old Gibson oval hole mandolins and a mandola, and a Loar F5. I can't divulge too much of this before the manuscript is submitted and reviewed, but the differences between the Neapolitans and the Vega were interesting. Both are essentially ladder-braced. The very narrow Neapolitan bodies had no top plate modes lower than 500 Hz, while in the much wider Vega, the lowest member of the (0,0) multiplet occured lower than 400 Hz. So there you have some hints about the effects of body shape.

The break angle is another consideration. There are so many factors here that it is difficult to answer in a forum like this. You can change your break angle with either bridge height or plate arch height. The very tall bridges in violas and 'cellos can contribute to nonlinearities resulting in wolf notes, but that kind of wolf note is not a factor in plucked string instruments as far as I know. Still, steep neck angles make for an instrument which is awkward to play. Another consideration is that doubling the break angle will not double the down force from the strings on the top plate. There was an article by a German fellow in American Lutherie some issues back with the formula for the down force. You will also find it in Fletcher & Rossing's text. The problem with depending on the static down force as a measure of how hard the top plate is driven is that the plate is driven by motions of the bridge in response to motions of the strings, i.e., the driving force is a dynamic one, not static. The magnitude of the dynamic forces are small compared to the static down force. The bridge undergoes rocking motion as well as up-and-down motion, and more complex motion as well. When the strings are struck by a pick dragged across them (instead of pulled straight up and released downward toward the plate), the string motion is almost certainly (circularly) polarized. Plus, with double strings, there is the problem of a changing phase relationship during the string motion decay as well. Only with the up-and-down bridge motion can the dynamic force be related to the magnitude of the static down force, and the dynamic force will be a very small fraction of the static down force. Shoot, I would just try for a comfortable combination of bridge height and neck angle. You will be far better off trying to affect the stiffness of the plates by thoughtfully removing wood, as luthiers usually do. That will hopefull make for the strongest possible coupling between plate modes and air modes, which is the best shot that a luthier has at making an instrument which efficiently radiates sound, imo.

[Chris Baird]

Dave,
Thanks for the post. It was well written and understandable for us non-physicist(at least this one). I've hinted before about static force on the top plate as possibly altering the modes. Have you done any experimentation to determine just how much if any change occurs with a practical change in static force. i.e. observe the modes then raise the bridge a good deal and observe again. It would be interesting to find an instrument that showed a marked change in tone to the ear with a bridge height adjustment and then find out exactly what was happening with the change using holography.

[Dave Cohen]

Not much physical likelihood or plausibility in that. Normal mode shapes are dominated by geometry, i.e., mandolin-shaped plates give mandolin-shaped modes. Having looked at most of the different types of mandolins, I can say with a little more confidence than before that ALL of the different types of mandolins share the same basic mode shapes. The frequencies and the ordering of the modes are affected by shape, bracing pattern, etc. But most mandolins have a doublet (0,0) or "trampoline" mode, a sideways rocking or (1,0) mode, a longitudinal rocking or (0,1) mode, a twisting or (1,1) mode, a (2,0) mode (which Australian luthier/physicist Graham Caldersmith calls a "long tripole mode"), and so on. Owing to differences in shape, bracing pattern, etc., one or a few of the modes might be missing in some mandolins.

A flat isotropic plate with a constant down force aplied silultaneously with a dynamic (i.e., time varying) driving force would not show much, if any, difference in mode shapes. The modal frequencies might be altered, but I'm not even sure of that. There is something potentially more interesting with arched (i.e., mandolin & archtop guitar) plates. Certain Chinese opera gongs, like guitar strings only more so, are nonlinear. That is, the frequency ("pitch") changes with amplitude. So with flat ones, you hear a falling pitch as the amplitude decays, whereas with the arched ones, you hear a rising of the pitch as the amplitude decays. Are mandolin plates like that? I don't know. I haven't seen any evidence for that yet. I have significantly varied the current to the coil in driving the plates for holography experiments, but haven't noticed a change in modal frequencies as a result. So the early answer is a NO - qualified, but with a few data points. To know for sure, I would have to design some experiments to look for such an effect. A possibility, but I have a few things on my plate ahead of that.

[Chris Baird]

Thanks, So what do you think the role of a static load on the top plate plays in effective sound radiation? Why is it part of the design and why does it work to enable the dynamic couplings(if it does)?

[Dale Ludewig]

Craig, I went and looked at that picture but couldn't see the back. I agree that a higher arch in the top would increase the break angle, although the neck joint angle would have to be changed accordingly. I do put a slightly higher arch in my mandolins than the 5/8" standard.
I made a mandolin a few years ago with the thought that I'd change the neck angle (top arch remaining the same) such that you'd use a taller bridge and still have good action. I figured that this would give a greater break angle and drive the top harder. It is a sweet sounding instrument but I think I realized that you can go too far on the break angle and then the static downward force on the top will act as a "brake" (pun intended) on the top and impede its vibration, loosing volume in the process. I guess the normal break angles that we see and use came about after much experimentation. I should have realized that.

Dale Ludewig

[John Bertotti]

Excellent information! Thanks Dave. And Thanks everyone else for the additional questions I also wanted answers to but didn't know to ask.
One last question for clarification. Dave, plate modes are global round for round oval for oval. Is that actual body shape or is that relative to the shape of the recurve carved in the plate? Like a round carved area on an oval plate. John http://www.mandolincafe.net/iB_html/non-cgi/emoticons/biggrin.gif

[Dave Cohen]

Geez, you guys really wanta know this stuff. That's a good thing. I could only wish that my students would have been so interested. At the end of a lecture, I would ask "Any questions?" Usually no response, so I would reply with "OK, then I guess that you are ready for a quiz tomorrow!"

For the questions, let's see now. Chris, the short answer is that I don't know exactly what the contribution of the static down force is. Luthier Doug Woodley had an article in Mandolin Quarterly a while back in which he speculated that too much down force from a too tall bridge "swamped" the response of the mandolin to the dynamic force. I think that he was intuitively thinking along the same lines that I am. Because of my accursed temperement, I will have to arrive at the conclusion analytically before I will be able to say anything with any confidence.

As for efficiency of sound radiation, that comes from the frequencies of the (0,0) multiplet bracketing the frequency of the Helmholtz air resonance. That's why I like Don McRostie's "deflection tuning" method, b/c whether he knows it or not, he is using the method to bring the plate mode frequencies down to where they will couple w/ the Helmholtz mode.

John, the actual body or plate shape. Look at some holograms, either from our mandolin papers, or from some of the numerous papers on classical guitars, and you will see what I mean.

[Chris Baird]

Dave, Thanks again. I too believe that just as too little static force is conducive to weak tone so is too much static force. I would certainly like to know should you ever find out just what is happening when changes in static load take place. It seems to me that a plate would act like a string in that increasing the tension applied raises the modes. I envision the bridge like a finger bending a string. Only the bridge,under sting tension, is applying tension to a plate/shell.

[Michael Lewis]

Good discussion folks, and thanks Dave! You know we want more. http://www.mandolincafe.net/iB_html/non-cgi/emoticons/wink.gif

[sunburst]

Dave,
The plate modes pretty much ignore the minimun area or recurve area and involve the whole plate.
Do you think, as I've begun to suspect, that the graduations, particularly in the top, are more for support of static forces than for any affect on the dynamic? In other words, the top works like a leaf spring with more stiffness at the area of static load, and therefore can be of less mass than an evenly carved plate.
I've read that violin makers are more concerned with arch than graduation, and have read of Strads with nearly even thicknesses in the tops.

Funny how the more I learn the less I know. I'm carving plates for two mandolins now, and I'm wondering why I'm doing what I'm doing anyway.

[Dave Cohen]

John, I'm not sure what you mean by 'ignore the recurve area'. In doing the holography, I deliberately didn't push the plate(s) too hard by applying too much current to the coil for two reasons. One was that I didn't want to fry the coil, and the other was that you get a clearer holographic image if you don't have too many fringes. If I had pushed the coil harder with more current, the image quality would not have been as good, but the fringes would clearly have encompassed the entire plate. So, in a few words, the modes DO extend into the recurve area, but the amplitude is lower there, dropping to zero only at the very edge of the plate.

I too have seen the CASJ articles, particularly by Jeff Loen, but also earlier ones, which described not only uniformly graduated plates, but also some plates from old Italian violins which were REVERSE graduated, i.e., thinner in the center. I'm not sure what to make of that. You cetainly have my blessings if you want to try something like that. I am all for experiment. But I think that you have to remember that a plate is modeled well by a mass on a spring. That is because is IS a mass, and it is also elastic, i.e., "springy" Recall from Physics 101 that, other things being equal, a greater mass will oscillate at a lower frequency. Also, if one mass is oscillating connected to a stiffer spring and an equal mass is oscillating connected to a softer spring, the mass connected to a softer spring will oscillate at a lower frequency. What does this have to do with graduations? Plenty. When you remove wood from a plate, you are both softening the spring AND removing mass. So, depending on where you remove the wood from, you could be raising the modal frequencies, lowering them, or even leaving them alone! Now, this is not Cohen & Rossing mandolin stuff; it goes way back to the early '80s & the Journal of Guitar Acoustics. This is where the Rossing articles (which I mentioned in an earlier post in this thread) in The Big Red Books of [/I]American Lutherie[I] come in. No point in me repeating the stuff over the internet, just taking up bandwith. Rossing, as usual, did a great job explaining the concepts. In particular, look at the two-mass model and the three-mass model.

[sunburst]

I guess "ignore the recurve area" was a poor choice of words, I meant that the recurve doesn't seem to act as a hinge as some people speculate, but instead is involved in the modes along with the rest of the plate.
I'm still trying to recover from various misinformation that I've received over the years and trying to develope some understanding of how the plate dynamics relate to the carving that I am doing. I'll have to see if my library can get me in touch with the Big Red Book.
Thanks.