Action of inflection for a plate under compression.

[Johnmasters]

I modeled a dome in FEA. The radial cross section is a curtate cycloid. The edges were fixed and a downward force was applied in at a radial distance indicated by the central dark blue ring.

The colored second picture shows the deformation on the undeformed surface. One can see that the maximum bending occurs at the inflection. I believe that this is because there is no "curvature"* at the inflection and therefore requires no stretching of the plate.

Bending takes less energy than stretching the plate. I don't know if this is of interst to you, but it says at least something. Other models show that the inflection moves outward somewhat when a compression like this is applied. It may be a good idea to string up a white model, let it set and perhaps creep for a week or so, and then rescraped the edges to push the inflection back to whereever you may want it.

* By curvature, I mean the product of the two curvatures in the principal directions. Inside the inflection, the curvature is positive, and it is negative outside the inflection.

[Johnmasters]

Isn't it odd that the thinnest graduation is usually near the inflection? I wonder why this is? Also, isn't it curious that modern makers do not usually (except for Oded) touch up graduations from the outside, perhaps after stringing up.

This seems an important point if you are interested in more than discussing an ancient spirit varnish recipe or figuring out standard construction techiques.

[Oded Kishony]

Hi John,

If I understand this model correctly then the area of greatest inflection ( same as stress? most change of shape?) is at the perimeter (red color)?

This would correspond to the purfling area on an instrument?

How does the area of greatest inflection correspond to areas of vibration?

This is a very interesting model but one needs to keep in mind how much vastly more complex an actual violin is.

Oded

[Adam Edwards]

Also oded , the thick lip ( edgework ) on the other side of the fluting fits in so well as the mass that moves in harmony with this process that we have seen

thanks for your efforts john

[GMM22]

Hi John,

... one needs to keep in mind how much vastly more complex an actual violin is.

Oded

I am positive Mr. Masters realizes the violin is a complex thing. The fundamental principle of his model is to reduce parameters to a more reasonable number for primary investigation. This is the only way to start to make headway in understanding any complex system. One begins with first principles of one aspect, and modifies the hypotheses as new information presents itself.

[Don Noon]

... one needs to keep in mind how much vastly more complex an actual violin is.

Oded

Yep, just watch the Strad3D clips and see the plates flapping like a triple-jointed gooneybird.

It is no surprise that maximum bending occurs at the inflection point. By definition, the curvature of the inflection point is zero, and it is the curvature that creates stiffness.

[Darren Molnar]

edit-answered my own question

[Oded Kishony]

George Stoppani has a method of identifying 'hot spots' (max bending) in his modal analysis-your inflections reminds me of that.

I can also identify 'hot spots' for any mode at which a string vibrates using reciprocity, simply by listening for the area with the greatest amplitude.

I wonder if in this model you could apply a lateral compression on opposite sides to simulate string forces? That might be interesting.

Oded

[Johnmasters]

Hi John,

If I understand this model correctly then the area of greatest inflection ( same as stress? most change of shape?) is at the perimeter (red color)?

This would correspond to the purfling area on an instrument?

How does the area of greatest inflection correspond to areas of vibration?

This is a very interesting model but one needs to keep in mind how much vastly more complex an actual violin is.

Oded

I don't know about creep and I don't know about vibration. I will keep thinking about it. But, the old model of a bellows at the edge seems valid.

This view is only of the displacement in Z, vertical to the plane of the disk. There are now stresses shown. The perimeter is red, the least displacement. You can see the chart to the left.

The inflection on the model is about 2/3 out from the middle. The inflectoin has moved out, it seems, for the colored photo. It is hard to see, but you can compare these in the pictures.

Half of the circle would be like the outer bouts. I think the violin is different in the bouts, but qualitatively perhaps not extreamly. The only things I pointed out would be the things I mentioned. I will later do it with anisotropic wood. I am keeping it in mind...

The important point to me is that benidng is easier than stretching. To flatten the center or edge, things must be stretched. At the inflection, this is not true, maximal bending occurs here, and I am sure that is true of a violin with a somewhat different shape.

Also oded , the thick lip ( edgework ) on the other side of the fluting fits in so well as the mass that moves in harmony with this process that we have seen

thanks for your efforts john

Thanks for reading.. I expect the purfling and extream edge pretty much moves with the ribs. This should not affect the main observation about the bending vs stretching of the plate.

I left the edge free to rotate (hinged edge) as you might want in an actual violin. It is only fixed in position, for possible hinged motion.

I am positive Mr. Masters realizes the violin is a complex thing. The fundamental principle of his model is to reduce parameters to a more reasonable number for primary investigation. This is the only way to start to make headway in understanding any complex system. One begins with first principles of one aspect, and modifies the hypotheses as new information presents itself.

Well thanks to you, but for the others. This is what I intended... The violin is more oddly shaped, but the point about the inflection being "flat" is the important point. It bends more easily here. I will be making a finer-mesh violin plate later in another program with more points available.

It is no surprise that maximum bending occurs at the inflection point. By definition, the curvature of the inflection point is zero, and it is the curvature that creates stiffness.

Not surprising at all, why do you think I set the excercise for myself in the first place? For the others, a thin band of the model containing the inflection would be "flat" in the sense that it would be like cutting a thin circumferential slice from a cone. This could be cut radially, unbent, and lain flat on a surface. It can bend in any direction without stretching. This is not true of the spherical-like central portion or the inside-of-a-donut type outer rib which has a negative curvature.

I was looking for more exotic topological features about the three types of curvatures there.. (3) I found only what you mention, but I am not giving up. I did see the inflection move outwards...... for a maker, this might be nice to keep in mind. Perhaps one should check this for magnitude by stringing up a white violin. Possibly one might wish to re-scrape the surface after it settles down. And as you say, the more complicated shape may be an issue...... where to re-scrape and where not to ??

George Stoppani has a method of identifying 'hot spots' (max bending) in his modal analysis-your inflections reminds me of that.

I can also identify 'hot spots' for any mode at which a string vibrates using reciprocity, simply by listening for the area with the greatest amplitude.

I wonder if in this model you could apply a lateral compression on opposite sides to simulate string forces? That might be interesting.

Oded

And what do you do with the "hot spots"? That would be nice to know.

Modeling local loads can be done and also use [isotropic] wood instead of steel. I intend to do this. The problem with fine-mesh violin plates is that it takes a lot of spreadsheet calculations to find the elevation of the arch to be curtate cycloids. Also, one must first decide on a longitudinal arch. I have done this for a 450-point plate, but the number of calculations go up as the square of the fine-ness.

Another interesting thing: Does the static loading correspond to a zero-frequency mode in some sense? It does in buckling of a column. This also interests me, even though it is not like a straight column.

[Johnmasters]

see above

[Don Noon]

Another interesting thing: Does the static loading correspond to a zero-frequency mode in some sense?

"Mode" means a natural frequency, which is not what is going on with static loading. However, if you are interested in frequencies well below a real "mode", then static forces have some use. But the static force has to be in the same direction as the low-frequency vibration of interest.

For example: static string load is along the string length, and causes neck bending, body bending, and other similar things. In PLAYING an open G, the sound-producing action is lateral on the bridge, and well below any structure mode of the body. So you'd have to put a lateral static load at the bridge in the model to see the important flexing to produce the low G note. This could also be important for Helmholtz pumping action, and other notes below the lowest (~400Hz) structure mode.

[Oded Kishony]

John Masters writes:

And what do you do with the "hot spots"? That would be nice to know.

I'm still discovering. What I (think I) know is that not all hot spots are equal. How 'active' a hot spot will be is determined by the overall shape of the mode in question, ie how much phase canceling is happening. Second, where in the overall spectrum the peak is located.

Finally how well you can hear that frequency

I don't mean to give you homework assignments but if it isn't too much trouble how about modeling a higher and lower 'arch'? I suspect the bands will change in proportions.

Oded

[Johnmasters]

"Mode" means a natural frequency, which is not what is going on with static loading. However, if you are interested in frequencies well below a real "mode", then static forces have some use. But the static force has to be in the same direction as the low-frequency vibration of interest.

For example: static string load is along the string length, and causes neck bending, body bending, and other similar things. In PLAYING an open G, the sound-producing action is lateral on the bridge, and well below any structure mode of the body. So you'd have to put a lateral static load at the bridge in the model to see the important flexing to produce the low G note. This could also be important for Helmholtz pumping action, and other notes below the lowest (~400Hz) structure mode.

I know what a mode is. If I supply radial loads at the edge, the shape changes very much like the first mode. The V for static loading is minimized (as you know) and the V in the Lagrangian presumably is also. I will look at the shape of the deformation under radial loading and compare it with the first mode with no loading.

I am not interested in complicated violins at this point. I have thought about the whole thing for a long time. You always reply with "I am a professional engineer and have a better picture of things than you do." I resent this, and wish you would stop doing it.

John Masters writes:

I'm still discovering. What I (think I) know is that not all hot spots are equal. How 'active' a hot spot will be is determined by the overall shape of the mode in question, ie how much phase canceling is happening. Second, where in the overall spectrum the peak is located.

Finally how well you can hear that frequency

I don't mean to give you homework assignments but if it isn't too much trouble how about modeling a higher and lower 'arch'? I suspect the bands will change in proportions.

Oded

Not at all, I love suggestions for different models. My disk has a high arch, and a lower one would move the inflection inboard in proportion to the diameter. I showed this two or three years agod with a little graph which compared a true curtate cycloid of a given height, one with half the height, and one that was simply the taller one with the y axis rescaled to give a lower curve. This last one was not a true curtate cycloid, and I called it a "pseudo curtate cycloid." So I think that says something about higher and lower arches.

In other words, you can't simply scale up the height in a taller arch. You also have to move the inflection outboard to get a true curtate cycloid (for the transverse cross sections).

A flatter dome might show a larger motion of the inflection, in proportion to the model I have done so far.

As for the "hot spots" I wonder if you want the plates to be "as flat as possible". The main question is if this is consistent with maximum static strength......... I do know that the graduation can be thinned maximally at the inflection. And this is where you normally see it, or thereabouts.

If you want to try an experiment yourself, deform a white violin by stringing up and try to find any change in position of the flat places. They may have a very small change, I need to use actual wood constants and dimensions to look at this in the next model. Perhaps a viable approach would be to string up and the scrape the outer portions. This is what THEY did, but I don't know if they did it with the violin strung up. But that would be a natural conclusion for them to make, wouldn't it? After a preliminary shaping, let it is sit a few weeks and scrape it again. Maybe do this several times. The final inflection would be where it wants to be stable.

[Oded Kishony]

My sense is that they weren't very concerned about static movement except perhaps at the F holes where it would be more obvious. I'm working under the assumption that they did what they could to alter and adjust the sound of the instrument.

In the higher and lower arch does the area of inflection change? Also in a curtate cycloid one can alter the 'recurve' placing it further in or out how does this change the inflections?

Oded

[GMM22]

Exactly how does your FEA program define bending?

[Johnmasters]

My sense is that they weren't very concerned about static movement except perhaps at the F holes where it would be more obvious. I'm working under the assumption that they did what they could to alter and adjust the sound of the instrument.

In the higher and lower arch does the area of inflection change? Also in a curtate cycloid one can alter the 'recurve' placing it further in or out how does this change the inflections?

Oded

I have the opposite view. The most obvious things they would have seen in earlier examples is warping. At some point they had to decide how thin and light they could make things. Maybe after developing a design, they could modify it for tone, knowing that it would be pretty stable.

The inflection is a line. I am assuming that the edges are pushed out against ribs that resist expansion. I modeled this by constraining the edge. Thehere is some downward force from the bridge, I don't know the role of the post and bar for an example like mine; how much support they give. At present, I know that an inward force from the edge (neck or saddle) would give an outward drift of the inflection.

I used curtate cycloids. Any recurve would seem qualitatively similar. Perhaps it is not good to have an inflection immediately in front of the neck or saddle. I draw my longitudinal arch to no have and inflection, although the wood is thick there.

The point is how does the position of the inflection change with loads. At least a curtate cycloid of a given height and length has a unique position for the inflection. Moving it in or out gives a kind of pseudo curtate cycloid. I think that the loading shifts the inflection so that the loaded curve is no longer an exact curtate cycloid although the difference may be too small to see. I need to find this out, about how much it should move using actual violin materials and dimensions.

Exactly how does your FEA program define bending?

You can see a graphic of various displacements and rotations. I did not look at rotations. I inferred the bending from an exagerated picture of the deformation. I will include this. This is the deformation exagerated on the model itself. The true shape is not the dip in the middle, but the middle does go down slightly.

Here you can see where the maximal bending is.

I think that static loading is the zero-frequency eigenmode in the sense that it is the solution to the relavant DE with w^2 = 0. By relevant DE I mean the one for the system at hand whether you can write it down or not. It seems to work for beams and plates so I think it would work for other similar DE's.

Thanks for that, I wll do some thinking about it.

Now I can't get this thing to go away............ This picture is for a dome compressed inward around the circumference, no loads on the central part.

[Wm. Johnston]

Another interesting thing: Does the static loading correspond to a zero-frequency mode in some sense? It does in buckling of a column. This also interests me, even though it is not like a straight column.

I think that static loading is the zero-frequency eigenmode in the sense that it is the solution to the relavant DE with w^2 = 0. By relevant DE I mean the one for the system at hand whether you can write it down or not. It seems to work for beams and plates so I think it would work for other similar DE's.

[Johnmasters]

I think that static loading is the zero-frequency eigenmode in the sense that it is the solution to the relavant DE with w^2 = 0. By relevant DE I mean the one for the system at hand whether you can write it down or not. It seems to work for beams and plates so I think it would work for other similar DE's.

see above

[Wm. Johnston]

Half of the circle would be like the outer bouts. I think the violin is different in the bouts, but qualitatively perhaps not extreamly. The only things I pointed out would be the things I mentioned. I will later do it with anisotropic wood. I am keeping it in mind...

Once you take into account the anisotropy of the MOE and the non-circular shape of the violin I think that they will balance each other out and behave similar to what you have shown here. Wood is more flexible across the grain than along it but the violin is narrower across. I can't remember where I saw it but I remember seeing a plot of the fourth root of the MOE of spruce as a funtion of angle (or something like this, it was probably in a VSA journal) and this worked out to be an "almost violinshape". Violinmakers would've arrived at this by trial and error but I think that there is probably something to this.

[GMM22]

There has been several references here to the inflection point being the zone of maximum bend. If I read your explanation correctly John, under deformation, the various colours more correctly correspond to a quantity of displacement from an initial set of points in a reference frame.

Is this not a different phenomenon from the term bending as it might normally be understood, i.e., if we first define a suitably short chord length, and then examine a section of your shape for maximum bending, that being defined as the greatest change in ratio in the perpendicular distance of the midpoint of a chord of a curve, to a point on the curve?

[Johnmasters]

There has been several references here to the inflection point being the zone of maximum bend. If I read your explanation correctly John, under deformation, the various colours more correctly correspond to a quantity of displacement from an initial set of points in a reference frame.

Is this not a different phenomenon from the term bending as it might normally be understood, i.e., if we first define a suitably short chord length, and then examine a section of your shape for maximum bending, that being defined as the greatest change in ratio in the perpendicular distance of the midpoint of a chord of a curve, to a point on the curve?

I am sorry, I had the wrong picture. Above is the correct one for a loaded dome. I placed a circle of downward forces on the nodes 4 from the center. You can see the blue ring that corresponds to the depression from this.

Both the edge and dome are pushed down, but the RATE of compressing downward is maximum around the inflection. The picture is the actual deformation blown up in scale and added to the original. The table shows the amounts of the deformation. The elevations of like colors are not necessarily the same, they are added to the deformed shape itself.

Your statement is correct, the colors are the deformation. The colors are not the bending, but you can see rate of change of the deformation to be the bending.

[lvlagneto]

Have you ever tested or built a plate model from scan data? Can you test Maya models... or objs?

[Don Noon]

I am not interested in complicated violins at this point. I have thought about the whole thing for a long time. You always reply with "I am a professional engineer and have a better picture of things than you do." I resent this, and wish you would stop doing it.

John,

Many of my comments (and especially those in this thread) are directed toward the wider viewership, who might not have thought about all this stuff for a long time, and might get some benefit from my viewpoints written at a level well below what would be appropriate for a direct response to you. If I was writing directly to you, I could see that it might be condescending... that is not my intent.

[Johnmasters]

John,

Many of my comments (and especially those in this thread) are directed toward the wider viewership, who might not have thought about all this stuff for a long time, and might get some benefit from my viewpoints written at a level well below what would be appropriate for a direct response to you. If I was writing directly to you, I could see that it might be condescending... that is not my intent.

I am sorry that I overreacted. We have corresponded before, and I know that I should not have.

[Johnmasters]

Have you ever tested or built a plate model from scan data? Can you test Maya models... or objs?

I don't know what maya models are, or objs. I have not built a model from scan data, scan data of what? At least in this case, there is likely no scan data. My present experiments with violins are with Chinese whites. They seem to have copied a good arch and have a close quality control. The use a CNC router, I think.

This experiment is a lot more clear on its meaning. The dome had no load on the center, just radial forces all around the edge. Pointed inward. The plate buckles upward and the edge moves in. You can now see how the inflection is the region of most bending. This is because the curved portions inside and outside are stiffer by having double curvature.

This one has the radial forces outward. The stretching flattens the outer portion and also pull down the central dome.