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Thickness and Stiffness Relationship


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#1 skiingfiddler

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Posted 22 August 2009 - 12:57 PM

I wanted to confirm with you math and science folks the relationship between thickness and stiffness of any given object. I think someone noted some time ago that the stiffness of an object increases/decreases by the cube of the increase/decrease in thickness. So, for example, if you double the thickness of some object, you increase stiffness (resistance to deflection) by a factor of 8 (ie, 2 cubed). Is my understanding generally correct?

If that is generally correct, that obviously has some important implications for violin making, namely that a little bit of thickness change will result in a much larger change in stiffness. For example, a 2.5 mm top increased in thickness to 2.75 mm (a 10% increase) will result in a (1.1 x 1.1 x 1.1 = 1.331) 33% increase in stiffness, which I assume would be quite significant.

Have I gotten this right?

Thanks for any replies.
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#2 Johnmasters

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Posted 22 August 2009 - 02:27 PM

I wanted to confirm with you math and science folks the relationship between thickness and stiffness of any given object. I think someone (John Chee?) noted some time ago that the stiffness of an object increases/decreases by the cube of the increase/decrease in thickness. So, for example, if you double the thickness of some object, you increase stiffness (resistance to deflection) by a factor of 8 (ie, 2 cubed). Is my understanding generally correct?

If that is generally correct, that obviously has some important implications for violin making, namely that a little bit of thickness change will result in a much larger change in stiffness. For example, a 2.5 mm top increased in thickness to 2.75 mm (a 10% increase) will result in a (1.1 x 1.1 x 1.1 = 1.331) 33% increase in stiffness, which I assume would be quite significant.

Have I gotten this right?

Thanks for any replies.

That would be true of a flat plate. This may be a little different with a curved shell as there may be stretching in addition to bending for certain modes. As you thin, weight is proportional to thickness, and this drops slower than stiffness. Some people have used formula used by Harris. I previously gave a simlar one.

In analogy with the Simple Harmonic Oscillator, I mentioned that mode 5 frequency squared times the mass would give an effective stiffness for a plate.
Go down the page to examples, spring/mass.

I have found that this is useful in adjusting for wood properties. Using the formula gives pretty consistent results if you compare various violin parts.

#3 skiingfiddler

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Posted 22 August 2009 - 06:35 PM

John,

Thanks for the confirmation about the flat plate and for pointing out that the situation in a curved plate like a violin top would be more complex. When I think about it, it seems intuitively correct that if I take a flat plate of, say, cardboard or sheet metal and form it into a semi-cylinder (without the sides touching to form a complete cylinder), that semi-cylinder would be more resistant to bending along its length than the flat plate would be, even though thickness has not changed.
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#4 Oded Kishony

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Posted 22 August 2009 - 06:54 PM

I've noticed that stiffness can change in a plate at a different rate than weight, dropping more rapidly at a certain point. Is this an illusion?

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#5 Melvin Goldsmith

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Posted 22 August 2009 - 07:02 PM

I happily bow to greater knowledge here but wood seems not a simple structure....the internal scaffolding and beam structure of reeded fronts makes me wonder if the relationships are not linear?...
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#6 Johnmasters

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Posted 22 August 2009 - 07:48 PM

John,

Thanks for the confirmation about the flat plate and for pointing out that the situation in a curved plate like a violin top would be more complex. When I think about it, it seems intuitively correct that if I take a flat plate of, say, cardboard or sheet metal and form it into a semi-cylinder (without the sides touching to form a complete cylinder), that semi-cylinder would be more resistant to bending along its length than the flat plate would be, even though thickness has not changed.

Yes, it would require some stretching also, I am sure. For "local" bending only, curvature in a shell might involve some stretching over small areas. It is just what you would expect if you visualized it. Flattening half a basketball... this might try to occur if there were certain types of bending modes. Stretching of the perimeter might get involved.

I've noticed that stiffness can change in a plate at a different rate than weight, dropping more rapidly at a certain point. Is this an illusion?
Oded

Do you mean flat plate or violin plate? I suppose it could be complicated for a complicated curvature. But I have no idea how much bending vs stretching there might be. I doubt it is an illusion. I trust your observations.

I happily bow to greater knowledge here but wood seems not a simple structure....the internal scaffolding and beam structure of reeded fronts makes me wonder if the relationships are not linear?...

Apologies for name dropping, but Dr. Woodhouse seems to feel that most of the violin itself acts in a pretty linear way for most reasonable considerations. The bow-action, however, he feels has non-linear aspects that are important. (or may be, whatever turns up)

#7 Oded Kishony

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Posted 22 August 2009 - 08:07 PM

Also from Woodhouse (since you've already conveniently dropped his name) mode 5 has quite a bit of stretching which may be why your stiffness formula works so well.

I've only observed this non linear drop of stiffness once so far but I'll be on the look out for it.

(on arched plate, btw)

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#8 Johnmasters

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Posted 22 August 2009 - 08:41 PM

Also from Woodhouse (since you've already conveniently dropped his name) mode 5 has quite a bit of stretching which may be why your stiffness formula works so well.
I've only observed this non linear drop of stiffness once so far but I'll be on the look out for it.
(on arched plate, btw)
Oded

Yes, it takes account of a combination of bending and stiffness. Also, it considers both crossgrain and end grain stiffnesses. It is just a measure of some gross characteristic.

#9 Wm. Johnston

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Posted 24 August 2009 - 02:11 PM

If that is generally correct, that obviously has some important implications for violin making, namely that a little bit of thickness change will result in a much larger change in stiffness. For example, a 2.5 mm top increased in thickness to 2.75 mm (a 10% increase) will result in a (1.1 x 1.1 x 1.1 = 1.331) 33% increase in stiffness, which I assume would be quite significant.

Have I gotten this right?

Thanks for any replies.

This would be pretty well correct for a flat plate or one without too much curvature.

The question of how does a small increase in thickness affect the stiffness of the completed violin is different though. The result that you have described is only true for a beam bending about it's center of mass. If the beam is not bending about it's center of mass then you have to include a second term to determine the stiffness of the beam, this comes from the parallel axis theorem. The additional stiffness added to the beam is something like A*d where A is the cross sectional area of the beam and d is the distance from the centroid to the bending axis. This second term is important in many structures which are designed to be both stiff and lightwieght, skiis, I-beams...

I think but don't know for sure, that this additional stiffness term is going to be important for violin vibrations. It may be that at low frequencies where the violin vibrates in a beamlike way that this additional stiffness term is very important but for higher frequencies where the violin divides up into smaller vibrating areas that this addition term might not be important. I just don't know at this point but it is something that I spend a lot of time pondering over.
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#10 Wm. Johnston

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Posted 25 August 2009 - 01:02 PM

Sorry, I made a mistake in what I posted yesterday.

The extra term added to the stiffness by bending along an axis different than the center of mass is A*d^2, I forgot the distance squared portion yesterday when I posted this.
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#11 Don Noon

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Posted 25 August 2009 - 01:53 PM

Just to gather together the flat-plate relations...

Stiffness => t^3

Mass => t

Frequency => sqrt (stiffness/mass) => t

where t is thickness

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#12 Anders Buen

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Posted 25 August 2009 - 04:39 PM

Just to gather together the flat-plate relations...

Stiffness => t^3

Mass => t

Frequency => sqrt (stiffness/mass) => t

where t is thickness

There was an article in the CASJ (Catgut Acoustical Societys Journal) by Hajo G Mayer on a regression model he had extracted from a set of violin and viola top plates he had made. It uses the sound speed of the wood, the mass and the average thickness as input. I think he based the model search on a curved plate model by Cremer. There the mode 2 and mode 5 frequencies are calculated from a t^2 dependancy.

I think is is not easy to assess how the stiffness of an arched plate is dependant on the arch height and arch shape. But I am sure it is. One way to measure the stiffness is to look at how much the plate deflect under a certain pressure from a force. David Meyer has made a device for Doug Cox to do such measurements on violin plates. A similar technique is used by archtop guitar builders and in violin and hardanger making. Makers in my family has used such a device too. I have ordered one of these with an ability to measure the deflections for a given force on a stringed up instrument as well.

If one is measureing the mode frequencies through the thinning process one should be able to get the changes in Hz per mm thinning. My experience is that mode 5 drops about 60Hz per mm thinning including some bass bar trimming and length shortening. But also this relation is somewhat dependant on where the plate is thinned. I recently saw data from a 250step thinning experiment conducted by Marty Kasprzyk where he had managed to keep the tap tones for mode 2 and mode 5 constant (the modes were even a bit higher after the last step) even after thinning off the plate 14 gram wich was a mass reduction of about 19%.
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