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Function of a bridge

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What does a bridge do?

 

Does it move up and down physically drive the top (and body)?

 

Or is it just a medium which waves goes through?

 

Or both? If both, which plays more important part?

 

Does it have its resonance?

 

KYC

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What does a bridge do?

 

Does it move up and down physically drive the top (and body)?

 

Or is it just a medium which waves goes through?

 

Or both? If both, which plays more important part?

 

Does it have its resonance?

 

KYC

 

I remember seeing a thread or two on MN on these issues.

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What does a bridge do?

 

Does it move up and down physically drive the top (and body)?

 

Or is it just a medium which waves goes through?

 

Or both? If both, which plays more important part?

 

Does it have its resonance?

 

KYC

 

It has "two degrees of freedom" for its lowest modes.  Rocking and up/down.   For questions 3 and 4, it is  both because both are the same thing;  don't forget what standing waves are.  There is a free rocking resonance somewhere in the 3k region.  It would have more resonances,  and the first up/down free one whould seem higher because of stiffness. 

 

Question 1:  The two feet both go up and down, and these can be considered the two main motions,  Each motion is a combination of rocking and going up/down.  This breakdown is the same as considering rocking and up/down as the two variables.  (It depends on your choice of coordinates.  Think of the the two channels of a phonograph needle in a stereo record.  Up/down and left/right have the same information as the two coordinates of moving at 45 degrees perpendicular to the left/right edges of the groove.)

 

A mathematical picture is easy if you have had some math.  But otherwise,  it can be very confusing.  For me,  the idea of a bridge is that it has a kind of motion that is intermediate between that of the strings and that of the body.  Ideas of geometric mean come to my mind,  but I have never solved anything with this idea.  The usual trial and error gets the right type of bridge just as it is done with the experience of people who set up violins. (It is a kind of "transformer")

 

But its resonances should not  be thought of without realising that it has partially fixed top and bottom.  I think it is best to think of the entire system with some kind of FEA model.   One could do experiments with the Student Version of some FEA program.  I think the ABACUS one is still about $100 and has 1000 nodes.

 

Still,  knowing the resonances of a free-free bridge can be useful to compare bridges.  That is,  free of the strings and free of the body.

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How much resonance difference in bridges are differences?  What is noticable and what could or what does cause change to sound?   

 

I think that is something you learn by pure experiment.   It is a question about "significance."

 

My GUESS:   The more different the stiffness/mass of the strings is to those of the body,  the more differences would need to be to be significant.

 

The easiest place to get some feeling is in Wikipedia articles about the "Simple harmonic oscillator."  That is the simplest vibrating system.  It is a decent model for exciting any single mode of vibration of a working violin. But one has free oscilations with and without damping.  Also there is the "driven" oscillator which corresponds to the played violin.   If one wants to know the releative amounts of vibration of the different modes of a violin, (corresponding to tone color),  then things get quite hairy.  (The word we used to use for difficult calculations)  In that case,  an FEA model first solves for the normal modes and then is used in a different form for a driving force. 

 

I must admit that I used FEA only for finding resonances and their shapes.  (Distortions of the body due to motion of a single mode.)  The entire program was used in a different aspect for driven systems.  I never learned this,  as I was lazy and that would have been more difficult.

 

But I would suggest that EVERYONE at least look at the article on the SIMPLE HARMONIC OSCILLATOR just to get some feel for it.   There are also treatments for coupled oscillators which require more math but are interesting.  (for the interaction of two or more modes.)  There ought to be a link to this,  or simply google the topic in Wiki.   Even so,  the first and easiest treatment will not include damping (friction).

 

When you add damping,  it gets worse.  There is more than one kind,  but the simplest one is viscous damping.  The differential equations for undamped  systems have both position and its second derivative (acceleration).  Viscous damping adds an extra term times  a constant of some kind multiplying the FIRST derivative (velocity).

 

I would be happy to give a qualitative view of all this if there is an interest.

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Nothing too complicated! The function of the bridge is siply to transfer the vibrations from the strings to the belly of the violin. The longer wavelenghts of the D and G cause the more disturbance at the transfer surface and that is why the left bridge foot needs a bassbar to rest on.

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Its most important function is to keep the strings off the fingerboard.  Everything else just kinda happens.  :ph34r:

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Good humor,   Violadamore.   Wolfjk,  you give a hollow comment as usual.  I hope you do not think your description has any information in it.  Yes,  obviously the bridge moves and that moves the body.

 

But how,  why,   how much,  under what conditions,  and how does everything happen anyway ?  And you still are not talking in terms of standing waves.  Learn that if nothing else.

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I was following and comprehending everything up to the damping.  Viscous damping makes me think that something is being muffled soundwise or vibration wise.  Or is viscous damping totally opposite?  I'm understanding damping {friction} as a hampering of something. 

 

I understand Wolfjk and V's explainations.  I can cut the chamfers, remove wood from bridges,  plane and sand bridges to thicknesses that are given to me but how do I know I have the right blank to start with at the beginning? 

 

You know there's always an interest in a qualitative view Johnmasters- go ahead.  I can't even imagine how that would go.

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I was following and comprehending everything up to the damping.  Viscous damping makes me think that something is being muffled soundwise or vibration wise.  Or is viscous damping totally opposite?  I'm understanding damping {friction} as a hampering of something. 

 

I understand Wolfjk and V's explainations.  I can cut the chamfers, remove wood from bridges,  plane and sand bridges to thicknesses that are given to me but how do I know I have the right blank to start with at the beginning? 

 

You know there's always an interest in a qualitative view Johnmasters- go ahead.  I can't even imagine how that would go.

 

I thank you for asking a serious question,  Uncle Duke.  Yes, exactly.  the damping is a muffling.  How muffled depends on how much damping.  But damping is still high even in a full-tone violin.

 

A violin is not very efficient.  Consider how easy it would be to put one watt of power into a bow stroke.  That would be one meter per second of bow speed times about four ounces of tug on your hand (from the resistance of the string......... you are putting in energy after all.)

 

One watt of acoustic power is a lot,  and the violin would be damned loud.

 

Way back at the beginning of the Catgut Acoustical Society publications (CAS)  John Schelling estimated the efficiency of the violin to be about 4%.  This sounds OK to me.  Most of the bowing energy goes into heating up the body.  What is left is about 4% acoustic energy.  Damping is the loss due to (useful) pumping of the air plus all the loss from bending and stretching.  Take a tree twig and bend it rapidly and touch it to your upper lip.  You will see that it has warmed up.  This heat energy is a total loss of mechanical work you have done.

 

Flexing the wooden parts and the strings causes most of the losses.  Think of tuning forks,  they ring a lot and are made of spring steel.  A wooden tuning fork would damp out much more quickly. 

 

I used the term "viscous" damping because it is proportional to the amount of movement of the materials.  That is the common term.  It is proportional to the speed of a motion.  It "acts" like a viscosity.  That is the only reason.  The term multiplying a first derivative (velocity) of the position of a point in the violin gives this.  A stationary piece of wood will not have losses.  But you get the losses by trying to make it bend and stretch. 

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There are scientific studies and analyses out there.....

Somebody might locate a useful one in non-specialist friendly language.

The movement/oscillation of the bridge is relatively complex I would imagine

and obviously only part of the overall interaction of bow on stringed instrument.

 

This short description gives some idea of the motion of the bridge and the sound box.

 

 http://acoustics.phas.ubc.ca/node/4

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Thanks for the picture,  Omobono.  That is what it looks like.  Other things of interest is how the feet drive the body and how the strings move the bridge.  Also,  the excitation of various modes of a violin vibration with perhaps a different amount of damping for each mode.  There is no mention of the all-important relative dampings in this article.

 

Driving damped oscillators (or normal modes of a violin)  is where you find all of the tone aspects.  It is not an easy problem to illustrate.  It requires FEA and numerical analyses.

 

I think that FEA simulations would be the only way to analyse an ACTUAL violin being played.  Otherwise the math is too complicated.   FEA solves thousands of equations in thousands of unknowns in a matter of seconds.

 

If you took Algebra II in high school,  likely you solved three equations in three unknowns.  That involved using determinants and was a somewhat messy.  Solving 4 or 5 simultaneious equations in the same number of unknowns would be a time-consuming effort. So you can see that an FEA model with thousands of points would need a high-speed computer.  The stick model given by the German guy (I forgot his name) is a rough picture.  My own FEA models had 1000 points available.  A powerful FEA  software package is going to COST you.  Somebody stole the calculator section of ABAQUS and made a version with unlimited points,  but it is fussy and hard to use.  I can go find the name if you are interested.

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After looking at the bridge simulation I can vision a bowed e string fingered note causing the g side to raise or gain tension.  I can't see it the other way around yet-  bow the g string and causing e string side to raise.  I need to think about how things work.  Maybe one time i'll start with a fully thinned down bridge and switch them out with incrementally thicker ones.  A lot of unneccessary cutting but I may learn something.

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Good humor,   Violadamore.   Wolfjk,  you give a hollow comment as usual.  I hope you do not think your description has any information in it.  Yes,  obviously the bridge moves and that moves the body.

 

But how,  why,   how much,  under what conditions,  and how does everything happen anyway ?  And you still are not talking in terms of standing waves.  Learn that if nothing else.

Hi John Masters,

I'm no scientist! but i'm interested. I watched and listened to talks given Woodhouse on bridge movements, animations by Shcleske, Sygmuntowitz, and George Stoppani and learned something! When a wave exits a body, it leaves some energy behind that moves the object it originated from. It is rocket science. When you look at various body violin body animations, you can observe that the movements are caused by the sound waves exiting the reeds in the belly wood and the cell cavities in the back, ribs and neck/scroll.

I think a simple experiment should confirm this! If you lick your finger and rub it on the edge of the violin it should alter the movements.

I also think that most of the exagerated and irregular movements in animations are caused by soundwaves exiting the damaged ends of the reeding in the belly wood. (The photos of the DR Clare Barlow talk at Newark clearly show it)

I also think that there is something more to waves than we know of.

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After looking at the bridge simulation I can vision a bowed e string fingered note causing the g side to raise or gain tension.  I can't see it the other way around yet-  bow the g string and causing e string side to raise.  I need to think about how things work.  Maybe one time i'll start with a fully thinned down bridge and switch them out with incrementally thicker ones.  A lot of unneccessary cutting but I may learn something.

 

That is a small effect peculiar to a violin.  My suggestion is to get some good idea foundation about vibrations in general.  It will help your intuition when thinking of a violin.   Also,  try to keep different effects in order as to likely size.  It is a mistake to go straight to a violin even though that is your interest.  A violin is a total mess when you try to look at all possible details.  The details go in after the basic vibration model.

 

To understand how things work,  look at the gross aspects first.  Then introduce a small perturbation and determine for yourself if it is likely to be important.  The problem is not the violin,  it is how all bodies can vibrate.  Along with vibrations go waves.  In this case,  standing waves.  (steady-state vibrations)  Read the wiki article and google various topics in "vibration".  I don't know what you will find,  maybe a lot.

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Hi John Masters,

I'm no scientist! but i'm interested. I watched and listened to talks given Woodhouse on bridge movements, animations by Shcleske, Sygmuntowitz, and George Stoppani and learned something! When a wave exits a body, it leaves some energy behind that moves the object it originated from. It is rocket science. When you look at various body violin body animations, you can observe that the movements are caused by the sound waves exiting the reeds in the belly wood and the cell cavities in the back, ribs and neck/scroll.

 

This is not right.  You have a steady state vibration.  A small amount of this pumps air.  On the whole,  a free vibration is damped by internal friction.  Just understand steady-state vibrations and assume that they will pump a little air.   Woodhouse and the others certainly understand this stuff,  and I think they would criticise you just as I am doing.

 

It seems that you still are thinking of vibrations or waves "flowing" from one place to another.  Again,  read the Wiki article on the SHO and learn about standing waves.

 

I think a simple experiment should confirm this! If you lick your finger and rub it on the edge of the violin it should alter the movements.

I also think that most of the exagerated and irregular movements in animations are caused by soundwaves exiting the damaged ends of the reeding in the belly wood. (The photos of the DR Clare Barlow talk at Newark clearly show it)

I also think that there is something more to waves than we know of.

 

Nonsense ..... You may detect something,  but you are misinterpreting what it is.  The animations do not introduce wave radiation in the first place. They are FEA calculations from the normal modes and no radiation is part of the model.  (Unless it is a very large and complicated mode)  What if the violin is made of plastic,  then where is the "reed" ?  You would get a similar model.

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What does a bridge do?

 

Does it move up and down physically drive the top (and body)?

 

Or is it just a medium which waves goes through?

 

Or both? If both, which plays more important part?

 

Does it have its resonance?

 

KYC

 

This seems very similar to your other threads about what we know, and impedance, etc.     Again, in honesty probably no one truly knows.  And the highly developed practices of good setup people are more meaningful than any theoretic understanding.

 

Still, it's fun to take a stab at these ambitious questions.    So, IMHO.....

 

What does a bridge do?

 

1) Stop one end of the playing string length, and one end of the tailpiece side effective string length.

2) dampen, reflect, modify, and pass through vibrations along the strings

3) establish a string height above the belly, and a string angle across the bridge, which govern down force onto the belly.

4) modify and communicate vibrations from strings into violin body

 

Does it move up and down physically drive the top (and body)?

 

Yes. The bridge will move in all available modes, but up and down pulse through the feet probably dominates motion at the feet.  However, the top of the bridge is more directly connected to the strings and probably moves in a more complex combination of up and down, back and forth(along string length direction), side to side, and twisting.  But given the narrow waist and general configuration, much of these motions probably communicate back into the string motion, with up and down feet motion probably being the dominant communication into the violin body.

 

Or is it just a medium which waves goes through?

 

No.  The effective component length of the bridge will be too short for it to look like a wave transmission medium for any but exceedingly high frequencies.  For most frequency relevant to music making, I belief the bridge will look like a lump component linking the strings to the violin body.  However, the elasticity, compressibility, and dampening of the bridge will surely impact this behavior

 

Or both? If both, which plays more important part?

 

Perhaps for high enough frequencies there might be transmissive behavior, but I suspect that would be a very very high frequency.  I believe wave medium behavior is either completely irrelevant, or extremely secondary.

 

Does it have its resonance?

 

Of course it has resonances, throw it on a glass table and hear the 'tink' a bridge makes.  But are they relevant?  I doubt it very much.  Instead, I believe the bridge is best seen as a lump component linkage allowing the strings to drive the violin body.  However, this link is neither absolutely ridge, elastic, or massless.  It also has the difficult job of connecting between the different impedances of the strings and body.  The evolved standard shape gives the right combination of stiffness and flexibility in the right places, and in the right directions, combined with a minimal wood mass, and an appropriate dampening, neither too little nor too much.  

 

The bridge shape is vastly stiffer in the plane of the bridge than at right angles to that plane.   The dominating 'lump component' linkage behavior will be in the plane of the bridge.  The resonances for 'in plane' compressions of the bridge are probably very very high frequency, and therefore not greatly relevant.

 

But twisting and back and forth motions out of plane will be much less stiff, and will therefore have the lower resonances that will be in a more relevant frequency range.  These are probably the resonances we hear when stimulating a bridge as a free component outside a violin system.  But in place within a violin system, these out of plane resonances of the bridge are probably not relevant to driving the violin.  I believe these lower impedance vibrations are going to tend to go back into the string motion much more than traveling into the violin body.

 

 

Those are my 'best guess' ideas.  But these are just wild wild guesses.  In truth, I've no idea. 

 

Do such musings help in the business of cutting a good bridge?  Maybe they fuel the intuition a little?

 

Some people believe they are adjust the bridge resonance when they trim the bridge.  I prefer to think we are adjusting the strength, mass, and dampening of the bridge, which effect the tone.  And, that we are adjusting the connection and separation of components of the bridge (arms, top arc, feet, etc), which effects how the strings talk to each other and talk to the violin body.

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This seems very similar to your other threads about what we know, and impedance, etc.     Again, in honesty probably no one truly knows.  And the highly developed practices of good setup people are more meaningful than any theoretic understanding.

 

Still, it's fun to take a stab at these ambitious questions.    So, IMHO.....

 

What does a bridge do?

 

1) Stop one end of the playing string length, and one end of the tailpiece side effective string length.

2) dampen, reflect, modify, and pass through vibrations along the strings

3) establish a string height above the belly, and a string angle across the bridge, which govern down force onto the belly.

4) modify and communicate vibrations from strings into violin body

 

Does it move up and down physically drive the top (and body)?

 

Yes. The bridge will move in all available modes, but up and down pulse through the feet probably dominates motion at the feet.  However, the top of the bridge is more directly connected to the strings and probably moves in a more complex combination of up and down, back and forth(along string length direction), side to side, and twisting.  But given the narrow waist and general configuration, much of these motions probably communicate back into the string motion, with up and down feet motion probably being the dominant communication into the violin body.

 

Or is it just a medium which waves goes through?

 

No.  The effective component length of the bridge will be too short for it to look like a wave transmission medium for any but exceedingly high frequencies.  For most frequency relevant to music making, I belief the bridge will look like a lump component linking the strings to the violin body.  However, the elasticity, compressibility, and dampening of the bridge will surely impact this behavior

 

Or both? If both, which plays more important part?

 

Perhaps for high enough frequencies there might be transmissive behavior, but I suspect that would be a very very high frequency.  I believe wave medium behavior is either completely irrelevant, or extremely secondary.

 

Does it have its resonance?

 

Of course it has resonances, throw it on a glass table and hear the 'tink' a bridge makes.  But are they relevant?  I doubt it very much.  Instead, I believe the bridge is best seen as a lump component linkage allowing the strings to drive the violin body.  However, this link is neither absolutely ridge, elastic, or massless.  It also has the difficult job of connecting between the different impedances of the strings and body.  The evolved standard shape gives the right combination of stiffness and flexibility in the right places, and in the right directions, combined with a minimal wood mass, and an appropriate dampening, neither too little nor too much.  

 

The bridge shape is vastly stiffer in the plane of the bridge than at right angles to that plane.   The dominating 'lump component' linkage behavior will be in the plane of the bridge.  The resonances for 'in plane' compressions of the bridge are probably very very high frequency, and therefore not greatly relevant.

 

But twisting and back and forth motions out of plane will be much less stiff, and will therefore have the lower resonances that will be in a more relevant frequency range.  These are probably the resonances we hear when stimulating a bridge as a free component outside a violin system.  But in place within a violin system, these out of plane resonances of the bridge are probably not relevant to driving the violin.  I believe these lower impedance vibrations are going to tend to go back into the string motion much more than traveling into the violin body.

 

 

Those are my 'best guess' ideas.  But these are just wild wild guesses.  In truth, I've no idea. 

 

Do such musings help in the business of cutting a good bridge?  Maybe they fuel the intuition a little?

 

Some people believe they are adjust the bridge resonance when they trim the bridge.  I prefer to think we are adjusting the strength, mass, and dampening of the bridge, which effect the tone.  And, that we are adjusting the connection and separation of components of the bridge (arms, top arc, feet, etc), which effects how the strings talk to each other and talk to the violin body.

The bridge does determine certain boundary conditions,  as you say.  These become restaints on the wave equations.  I was interested in the solutions of various equations given boundary conditions.  But you need to understand the generalities to say anything useful about all those things.

 

Anyone who tries to look at the motions of the bridge while connected to strings and bridge will get nothing of interest except by experimentation.  That is my final opinion.

 

I can't say anything more to people who do not understand basic waves and vibrations.

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Hi John Masters,

I'm no scientist! but i'm interested. I watched and listened to talks given Woodhouse on bridge movements, animations by Shcleske, Sygmuntowitz, and George Stoppani and learned something! When a wave exits a body, it leaves some energy behind that moves the object it originated from. It is rocket science. When you look at various body violin body animations, you can observe that the movements are caused by the sound waves exiting the reeds in the belly wood and the cell cavities in the back, ribs and neck/scroll.

 

This is not right.  You have a steady state vibration.  A small amount of this pumps air.  On the whole,  a free vibration is damped by internal friction.  Just understand steady-state vibrations and assume that they will pump a little air.   Woodhouse and the others certainly understand this stuff,  and I think they would criticise you just as I am doing.

 

It seems that you still are thinking of vibrations or waves "flowing" from one place to another.  Again,  read the Wiki article on the SHO and learn about standing waves.

 

I think a simple experiment should confirm this! If you lick your finger and rub it on the edge of the violin it should alter the movements.

I also think that most of the exagerated and irregular movements in animations are caused by soundwaves exiting the damaged ends of the reeding in the belly wood. (The photos of the DR Clare Barlow talk at Newark clearly show it)

I also think that there is something more to waves than we know of.

 

Nonsense ..... You may detect something,  but you are misinterpreting what it is.  The animations do not introduce wave radiation in the first place. They are FEA calculations from the normal modes and no radiation is part of the model.  (Unless it is a very large and complicated mode)  What if the violin is made of plastic,  then where is the "reed" ?  You would get a similar model.

 

Hi John Masters,

The Woodhouse calculations and geometry are very pleasing and interesting! Simplified, they show the movements of the brige caused by the transfer of sound by the shorter waves of the high pitch and the longer ones of the D and G string.

On a different note the present wave theory is incomplete. The doppler effect and the red shift implies that there is a continuous line between the observer and the source. Therefore the speed of sound and the speed of the the wave is different. There should be (or rather there IS, but not yet found) a mathematical formula to exactly calculate the speed of the wave that causes the sound.

Regarding "wickipedia" some of the facts are way way out. Take it with a pinch of salt!

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Thank you David and others for interesting answers.

 

Slightly different subject, I noticed that there are trends that makers put extremely thin bridges.

 

People believe the thinner the better, but I don't agree with that.

 

Thin bridge last about a month or two before it bends terribly, anyway thin bridge doesn't make the violin or cello sounds better or

respond faster, gross misunderstanding.

 

KYC

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People believe the thinner the better, but I don't agree with that.

 

I don't know who these "people" are, but they apparently don't understand that instruments are inherently different, and what may be better for one might be worse for another.

 

But on the topic of thin (I see the word "light" there) bridges...

 

post-25192-0-35772200-1432392216_thumb.jpg

 

This is a little test I did recently.  The difference in response of this lightweight bridge compared to a moderately light (1.77g) normal one looked like:

post-25192-0-98364100-1432392218_thumb.jpg

 

The higher frequencies gained significantly, as should happen according to theory.  However, it wasn't better.  Gaining power in the 2 - 4 kHz range I'd say was good... but far overshadowed by even stronger gains in the frequencies above that, making the thing unbearably gritty and harsh.

 

I had seen the Woodhouse paper on bridge effects, and tested them previously without being able to "tune" a bridge rocking frequency to get the predicted desirable shaping of the bridge hill frequencies.  This recent bridge test was another attempt, and another unsatisfactory result.  I did carve away the waist to get the stiffness down, which succeeded in flattening out the response... but not shaping it in any desirable way. 

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I don't know who these "people" are, but they apparently don't understand that instruments are inherently different, and what may be better for one might be worse for another.

 

But on the topic of thin (I see the word "light" there) bridges...

 

attachicon.gifLight Bridge.JPG

 

This is a little test I did recently.  The difference in response of this lightweight bridge compared to a moderately light (1.77g) normal one looked like:

attachicon.gifLight Bridge response.jpg

 

The higher frequencies gained significantly, as should happen according to theory.  However, it wasn't better.  Gaining power in the 2 - 4 kHz range I'd say was good... but far overshadowed by even stronger gains in the frequencies above that, making the thing unbearably gritty and harsh.

 

I had seen the Woodhouse paper on bridge effects, and tested them previously without being able to "tune" a bridge rocking frequency to get the predicted desirable shaping of the bridge hill frequencies.  This recent bridge test was another attempt, and another unsatisfactory result.  I did carve away the waist to get the stiffness down, which succeeded in flattening out the response... but not shaping it in any desirable way. 

Are you now going to thin the top some more for your light bridge?

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Hi John Masters,

The Woodhouse calculations and geometry are very pleasing and interesting! Simplified, they show the movements of the brige caused by the transfer of sound by the shorter waves of the high pitch and the longer ones of the D and G string.

On a different note the present wave theory is incomplete. The doppler effect and the red shift implies that there is a continuous line between the observer and the source. Therefore the speed of sound and the speed of the the wave is different. There should be (or rather there IS, but not yet found) a mathematical formula to exactly calculate the speed of the wave that causes the sound.

Regarding "wickipedia" some of the facts are way way out. Take it with a pinch of salt!

 

Yes, I like Woodhouse too.  I have not seen his illustrations.  "Movements  caused by the transfer of sound."  OK,,,,  Consider that there are reflections.  For a steady-state condition,  you have reflected waves.  These and the "outgoing" waves add to make a standing wave.  So just call it a standing wave. 

 

You can get this picture from ordinary FEA with no driving force or damping.  You need only get normal modes of an entire violin and select an image of the bridge.  If the pictures are of different frequencies, they could be various modes of the entire violin.  Low ones show most movement in the bass feet,  the high ones show most motion in the treble foot.  It is not a mystery.

 

If you actually wanted to do this,  you would soon see how it works.  I will bet Woodhouse does not mention red shift and doppler shift.

 

Red Shift and Doppler effect?  We regard the experimental violin as stationary.  Speed of wave?  It is zero on the surface of the violin;  they are standing waves.   The radiated waves (small) are ordinary acoustic waves.

 

Wikipedia has good math and science.  People who post know what they are doing.  At least the ones that have written the article on the SHO which you obviously have not read.

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