For Peter KG, Don Noon, and others interested in engineering.

[John Masters]

A flat plate has no bumps including archings.   It is interesting to see what happens in a curved plate,  bumps and all.

Try to flatten a violin plate and you will see stretching of some wood and compression in other places.  If there is excitation at M5,  the outer wood is being stretched back and forth and the central region is compressed.

You can see that the flatter the arching,  the less this happens. The wave equation solutions on a bounded surface are what give the normal modes.  There is obviously a "coupling" of in-plane stretching/compressing to the perpendicular vibrating of the surface bumps.  The curved shell equation is of higher order than that for the flat plate,  and this introduces extra terms.  These terms provide for coupling in "cross terms" which involve the mixing of in-plane stretching and the flattening of bumps. (and vice versa for  comompression/ raising.)

In other words,  the in-plane stresses give restoring forces to bumps.  It is not surprising that string tension and post tension will affect the sound of a violin.

This point has not been mentioned.  It is sound math,  and you can visualize it easily.  The guitar is very much different, by the way. 

 

PS:  Vibration depends on two things..   a mass is moving,  with inertia.  Countering that is a "restoring force" which opposes the motion and brings the situation back to zero distortion.  This happens when the plate passes through sine(frequency) goes through 0, 180, 360 degrees etc.  This is for a single excitation of a normal mode.  Vibrations of other modes can be superimposed,  but they do so in an additive way.  This last point is VERY  important.  It is called the linear superposition of waves.  Radio waves act the same way,  otherwise your radio could not separate different signals.

[Christopher Jacoby]

Are you terming arching distortion as 'bumps'? Thin spots? Warping? The cross arch?

[Don Noon]

We have gone thru this before, and the general conclusion has been that static forces do not have much effect on violin body mode frequencies.

The "restoring forces" you mention due to string tension are constant in direction, and do not depend on the direction of small vibrational deflections... therefore it will not affect the mode frequency. The if the restoring forces changed sign depending on the direction of deflection, yes, then it would make a difference.

[Peter K-G]

Depending on how the arches are made and the neck is set the arch of top or back will be a little higher or lower when strung up. Mostly top arch a little lower and back a little higher. This can decrease B1- and increase B1+

 

0,5 mm arch change can make a difference of 10-15 Hz

[John Masters]

We have gone thru this before, and the general conclusion has been that static forces do not have much effect on violin body mode frequencies.

The "restoring forces" you mention due to string tension are constant in direction, and do not depend on the direction of small vibrational deflections... therefore it will not affect the mode frequency. The if the restoring forces changed sign depending on the direction of deflection, yes, then it would make a difference.

I am not talking only about static loads.  The general motion is what I am talking about.  And you, Don, are pooh-poohing a point about the nature of the wave equation which you have never mentioned.  Please stop being such an expert.  You are an engineer, but how much of a math engineer?   I am not even talking violins.  I am talking curved rigid plates.

 

I did not mention the inflection,  which seems to me to be an important thing to consider,  as the plate curvature goes to zero at this line.  What can you add about the placement of the inflection line ?

[Peter K-G]

John,

 

I follow you a bit by visual imagination (it took awile to find what you are saying), I'm not so good at understanding by reading text. I'm neither a mathematician nor a physician.

[John Masters]

Are you terming arching distortion as 'bumps'? Thin spots? Warping? The cross arch?

Yes,  anything that is added to the dynamics of a flat rigid plate.   And this is not just for violins... 

[John Masters]

John,

 

I follow you a bit by visual imagination (it took awile to find what you are saying), I'm not so good at understanding by reading text. I'm neither a mathematician nor a physician.

As I said above,  this is very general.  It would be part of how violins work.  And has something to do with why archings are done in the first place.  Don's comment is a simple dismissal.  I think he also misses the point.   If this topic cannot be understood outside the consideration of violins,  it is not understood.............

[John Masters]

We have gone thru this before, and the general conclusion has been that static forces do not have much effect on violin body mode frequencies.

The "restoring forces" you mention due to string tension are constant in direction, and do not depend on the direction of small vibrational deflections... therefore it will not affect the mode frequency. The if the restoring forces changed sign depending on the direction of deflection, yes, then it would make a difference.

 

CONSTANT static forces might make a difference between violins with heavy strings and  light stings.  I was talking about  DYNAMIC restoring  forces, so you missed the entire point.  Stop reading so fast..... I think you do it mainly to dismiss ideas.

[Peter K-G]

I have no doubt, you will outclass me anytime with this. Me I'm just imagine/visualize things, Businss, emotions, violins, sailingboats, yachts..(I have constructed sailing boats and yachts)

My wife call me a Life artist

[John Masters]

I have no doubt, you will outclass me anytime with this. Me I'm just imagine/visualize things, Businss, emotions, violins, sailingboats, yachts..(I have constructed sailing boats and yachts)

My wife call me a Life artist

 

Then visualize this:  It is not difficult.  Cut a slice off a basketball and try to flatten it........... it is not a Euclidean surface,  it is not a plane. Imagine the stretching of the perimeter needed to flatten it.  That is ALL I am talking about.  A violin plate is not a Euclidean surface either.  But there is an important line of points at the inflection.  A small sample of wood here is curved in only one direction,  like a cylinder. It could be flattened here.

 

The inflection has never been addressed here in any kind of sensible way.  I think that one could learn a lot by learning to place it in a certain region.   The curtate cycloid does this.  But there is more than one way to use CC.  One can make them transverse the entire length,  or possibly radially from ywo points in the upper and lower bouts.  This will make a different arch.  I believe Michael Darnton does this.

[Don Noon]

I mistakenly thought we were discussing violins.

And this appeared to be a reference to static forces:

"It is not surprising that string tension and post tension will affect the sound of a violin."

Whatever. I'm not a math engineer, have no interest in becoming one, and therefore can not contribute anything useful to this topic. Carry on.

[Carl Stross]

A flat plate has no bumps including archings.   It is interesting to see what happens in a curved plate,  bumps and all.

Try to flatten a violin plate and you will see stretching of some wood and compression in other places.  If there is excitation at M5,  the outer wood is being stretched back and forth and the central region is compressed.

You can see that the flatter the arching,  the less this happens. The wave equation solutions on a bounded surface are what give the normal modes.  There is obviously a "coupling" of in-plane stretching/compressing to the perpendicular vibrating of the surface bumps.  The curved shell equation is of higher order than that for the flat plate,  and this introduces extra terms.  These terms provide for coupling in "cross terms" which involve the mixing of in-plane stretching and the flattening of bumps. (and vice versa for  comompression/ raising.)

In other words,  the in-plane stresses give restoring forces to bumps.  It is not surprising that string tension and post tension will affect the sound of a violin.

This point has not been mentioned.  It is sound math,  and you can visualize it easily.  The guitar is very much different, by the way. 

 

PS:  Vibration depends on two things..   a mass is moving,  with inertia.  Countering that is a "restoring force" which opposes the motion and brings the situation back to zero distortion.  This happens when the plate passes through sine(frequency) goes through 0, 180, 360 degrees etc.  This is for a single excitation of a normal mode.  Vibrations of other modes can be superimposed,  but they do so in an additive way.  This last point is VERY  important.  It is called the linear superposition of waves.  Radio waves act the same way,  otherwise your radio could not separate different signals.

 

Your point being ...?

[Peter K-G]

Then visualize this:  It is not difficult.  Cut a slice off a basketball and try to flatten it........... it is not a Euclidean surface,  it is not a plane. Imagine the stretching of the perimeter needed to flatten it.  That is ALL I am talking about.  A violin plate is not a Euclidean surface either.  But there is an important line of points at the inflection.  A small sample of wood here is curved in only one direction,  like a cylinder. It could be flattened here.

 

The inflection has never been addressed here in any kind of sensible way.  I think that one could learn a lot by learning to place it in a certain region.   The curtate cycloid does this.  But there is more than one way to use CC.  One can make them transverse the entire length,  or possibly radially from ywo points in the upper and lower bouts.  This will make a different arch.  I believe Michael Darnton does this.

 

Maybe this can add something, balance between the place where you get straight lines:

 

[Peter K-G]

Good night guys

[curious1]

Then visualize this:  It is not difficult.  Cut a slice off a basketball and try to flatten it........... it is not a Euclidean surface,  it is not a plane. Imagine the stretching of the perimeter needed to flatten it.  That is ALL I am talking about.  A violin plate is not a Euclidean surface either.  But there is an important line of points at the inflection.  A small sample of wood here is curved in only one direction,  like a cylinder. It could be flattened here.

 

The inflection has never been addressed here in any kind of sensible way.  I think that one could learn a lot by learning to place it in a certain region.   The curtate cycloid does this.  But there is more than one way to use CC.  One can make them transverse the entire length,  or possibly radially from ywo points in the upper and lower bouts.  This will make a different arch.  I believe Michael Darnton does this.

Hi John,

My background is decidedly not in engineering or mathematics. But I get your picture, I think. When you speak of inflection poitnts do you mean where the arch changes direction?

[curious1]

Maybe this can add something, balance between the place where you get straight lines:

Back_arch_tuned_StraightTangentLine_Correct.jpg

I guess I do something like Peter. The points in green are all the same height with the arch passing through a straight line represented by the green line. The blue line is arched area in the center.

I'm not sure if this what the thread is about but if it is.... Hurray!

If not, never mind.

[Carl Stross]

Then visualize this:  It is not difficult.  Cut a slice off a basketball and try to flatten it........... it is not a Euclidean surface,  it is not a plane.

 

 

But it is.

I see we're diving head on into Hausdorff Topological Spaces. A meaty subject, worth at least 20 pages.

 

By the way : ever heard of developable surfaces ?

[Oded Kishony]

I had a discussion about this with Jim Woodhouse some time ago. The subject was the use of what you refer to as 'efective stiffness' . Jim thought that the use of mode 5 in the calculation was a good choice because that mode includes stretching whereas other modes don't. You'll have to ask Jim for details.

oded

[John Masters]

I mistakenly thought we were discussing violins.

And this appeared to be a reference to static forces:

"It is not surprising that string tension and post tension will affect the sound of a violin."

Whatever. I'm not a math engineer, have no interest in becoming one, and therefore can not contribute anything useful to this topic. Carry on.

That was a kind of aside.  A preload to the dynamic forces. These could alter the shapes of the zero-position of the bumps.

[John Masters]

Your point being ...?

My point being (and to Don also)  Get your minds out of the gutter.  Talk vibrations,  forget violins.  To discuss varnish generally, perhaps forget violins.   Don't go to a Violin University.   Go to a university.

 

The point is this:  vibrations in curved shells have extra stuff over vibrations in rigid flat plates.  You should know this.  And what is the need for 8th order differential wave equations?  Well,  there are those coupling cross terms in the solutions.

 

The violin is a good application,  but not the only one............ BUT  think about the inflection,  that certainly should be of interest.   Isn't that about the place where makers graduate the thinnest ?  Sounds reasonable to me.

[John Masters]

Hi John,

My background is decidedly not in engineering or mathematics. But I get your picture, I think. When you speak of inflection poitnts do you mean where the arch changes direction?

Yes,  thank you,  you get A+.  A curtate cycloid has a slope that goes to zero at each end.  But the curvature goes to zero somewhere toward the end.   In other words, even though the slope is not zero at the inflection,  the rate of change of the slope DOES go to zero.   This is called in calculus the second derivative.

[John Masters]

But it is.

I see we're diving head on into Hausdorff Topological Spaces. A meaty subject, worth at least 20 pages.

 

By the way : ever heard of developable surfaces ?

I am not a topologist,  and know little about it.  But I think one can make a statement about the need for an 8th order equation because of coupling of the bending stresses on a bump and the in-plane stresses.  We could perhaps write Gregory Perelman for help.

 

Please write me at  jmluthier10@yahoo.com.   I really would like to know a bit about these things.  I am fond of math and physics generally.

[John Masters]

I had a discussion about this with Jim Woodhouse some time ago. The subject was the use of what you refer to as 'efective stiffness' . Jim thought that the use of mode 5 in the calculation was a good choice because that mode includes stretching whereas other modes don't. You'll have to ask Jim for details.

oded

Exactly.............. what I said about the nodal line being totally inside the plate was the same statement as Jim's

[John Masters]

image.jpg

I guess I do something like Peter. The points in green are all the same height with the arch passing through a straight line represented by the green line. The blue line is arched area in the center.

I'm not sure if this what the thread is about but if it is.... Hurray!

If not, never mind.

The line of points where the inflection is zero depends on where you draw the CC.  If they go the entire distance from one end to the other,   the line is guitar-shaped and goes to the endpoints.  (there is no recurve at the ends of the plates.   However, there is enough wood to make a small cosmetic one.)

 

If there is a large longitudinal inflection,  here the end compressions cause a dip to be pushed in near the ends.  The area is small but I did not like it..  I found that in FEA.   I decided to not use the arching where a point is chosen in the bouts and the CC are drawn radially from here to the edges.