Which equation?

Construction and repair of Classical Guitar and related instruments
vesa
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Which equation?

Post by vesa » Sun Mar 10, 2019 1:47 pm

Which equation to use to calculate the strut bending stength.
Eg. strut shape triangular, height 5 mm, width 4 mm.
I will try new strut thicknesses (the shape will be triangular)
in my next build and I would like to compare a standard Torres 3X7
(which I have used until now) with new ideas.


Vesa
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Antonio Marin nr. 813 1995 (Bouchet)
Vesa Kuokkanen 2016

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geoff-bristol
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Re: Which equation?

Post by geoff-bristol » Mon Mar 11, 2019 12:23 am

I would say the dynamics of a wedge shaped beam is pretty complex !

I have been thinking the same thing of late - I would be tempted just to make up two shapes from the same piece of spruce and test them. I would think in quartered spruce - the higher narrower beam would be stiffest ? However - the smaller footprint for width can lead to more distortion of the top ( if using a domed top ) than the 7mm width.

I have been thinking fans could be wider and flatter - ie 12mm wide x 1.5 ? .......just a thought

vesa
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Re: Which equation?

Post by vesa » Mon Mar 11, 2019 7:15 am

The problem is that I would like to if I get enough rigidity
when going from wide and low to thin and high (no sagging).
The area is very easy to calculate, just google ¨isosceles triangle area¨
and you get a calculator but the strength..??..
Maybe in Trevors books?
Of course every piece of wood is different but some maths would
be helpful to get me in the ballpark.
It is interesting that Romanillos and HyA have used very thin struts
combined with top thicknesses on the thin side (without the top sagging).
Marin has also thin and high (and only 5) in his Bouchet inspired,
but the top is a bit thicker (have not a Hacklinger, so I can not measure it).

Vesa
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60moo
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Re: Which equation?

Post by 60moo » Mon Mar 11, 2019 1:10 pm

The bending strength of any material of any cross section at any location along its length is proportional to the material's modulus of elasticity multiplied by its second moment of area, or E x I.

"E" depends on the wood being used.

"I" (if you want exactitude) is trickier to calculate, because a guitar strut - if glued along its entire length - acts structurally in composite with the soundboard, and the answer would therefore also depend on the spacing and orientation of the struts.

(But for this exercise, you're only wanting to compare relative stiffnesses of different cross sections, so none of the above need enter into the equations. For relative strengths, we can even assume that the struts are simply supported - it doesn't really matter. And the area of the cross section does not directly come into the calculations.)

So, for any triangle of base length b and height h, the second moment of area I(t) = 1/36 (b) x (h) x (h) x (h)
i.e. base times the height cubed divided by 36.

For any rectangle of base length b and height h, the second moment of area I(r) = 1/12 (b) x (h) x (h) x (h)

For any semicircle of radius r, the second moment of area I(s) = 1/4 Pi x (r) x (r) x (r) x (r) or one quarter Pi times radius to the fourth.

[P.S. What I'd like to know is: Why do they call them "struts"? In engineering, a strut is a structural member principally designed to take axial (i.e. end) compression; a "beam" is the term used for a member that takes bending, viz. tension and compression, which is what's really going on with guitar bracing.]

Alan Carruth
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Re: Which equation?

Post by Alan Carruth » Mon Mar 11, 2019 3:28 pm

If you weigh the top for a classical guitar before putting on the bracing and after, you'll find that the bracing is not all that much of the total mass. I'm not as good about keeping track of that as I should be, but that is, in part, because I've come to the conclusion that the bracing is really the place to try to save weight; it's the to. I did quickly pull records of a few guitars I've made over the past several years (there's nothing so useful as a file cabinet!), and although I kept the records in different ways, there's some useful information.

The most useful numbers for this discussion come from my #90. pretty much a 'straight up' classical in Indian rosewood and European spruce. The overall weight of the completed top was 174 grams. Of that, the upper transverse brace, and associated bracing (such as the sound hole patch) weighed 20 grams, and the fans in the lower bout weighed 11.5 grams.

The top for my #102, in walnut and redwood, weighed 142 grams when it was taken to thickness and trimmed to shape, and 174 grams with all the bracing. #118, cedar and Morado, had a 'bare' top that weighed 131 grams, and the final weight was 167 grams.

Bridges tend to come in a little above 20 grams, so the bridge weighs almost as much as all of the bracing put together. You can't do much to reduce the mass of the upper transverse brace, and it's not really worth trying anyway: that's structural, and on most classical guitars the soundboard below the waist bar is what's producing most of the sound.

Mote that I 'tune' my tops off the guitar, and try hard to avoid removing any material once I've assembled it. These weights are within a few grams of 'actual'. In the tuning process I sometimes (not often enough...) keep track of how much mass I've taken off the braces. It's usually not much more than about 5-10 grams. start to finish. This make a big difference in the 'tap tones' of the plate, and (I hoe) accounts for some part of whatever success I've enjoyed.

Wright's computer model found that reducing the mass of the top by 30% would produce a useful increase in sound output; something people can hear. If you used a normal top, and left off all of the bracing, you'd get a useful reduction in mass. How long would it hold up under string tension? OTOH, a 'sandwich' top can weigh 40% less than a 'normal' one, and have the same overall stiffness. That's why so many people pursuing 'loud' guitars are going to lattice and sandwich tops, to reduce the mass of the plate.

The bottom line is that however you try to optimize brace profiles, it won't make a lot of difference in the weight. I would think offhand that by careful design you could reduce the mass of the bracing by, say, 5%. That's 5% of the (upper limit) total brace weight of about 30% of the top, so 1.5 grams? You can do better by reducing the mass of the bridge, which typically can weigh almost as much as all of the top bracing together.

vesa
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Re: Which equation?

Post by vesa » Mon Mar 11, 2019 3:53 pm

Thanks a lot 60moo
60moo wrote:
So, for any triangle of base length b and height h, the second moment of area I(t) = 1/36 (b) x (h) x (h) x (h) i.e. base times the height cubed divided by 36.
Yes it is the relative strength that interests me
Sorry my maths are a bit rusty
But did you mean:
if eg. height is 5 and base 4:
1) 1/36 x 4 x 5 x 5 x 5 = 13.88
or 2) 1/36 x 4 x 5 x 5 = 2.77

compared with 3 and 7 (Torres):
1) 1/36 x 7 x 3 x 3 x 3 = 5.25
or 2) 1/36 x 7 x 3 x 3 = 1.75

But in either case 4 x 5 has much higher bending strength than 3 x 5.

Vesa
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geoff-bristol
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Re: Which equation?

Post by geoff-bristol » Mon Mar 11, 2019 8:00 pm

Somehow - when the parts are 3mm x 7mm - beam and strut seem meaningless ? ...but I take your point :?

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Trevor Gore
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Re: Which equation?

Post by Trevor Gore » Mon Mar 11, 2019 9:42 pm

vesa wrote:
Mon Mar 11, 2019 7:15 am
The problem is that I would like to if I get enough rigidity
when going from wide and low to thin and high (no sagging).
The area is very easy to calculate, just google ¨isosceles triangle area¨
and you get a calculator but the strength..??..
Maybe in Trevors books?
Of course every piece of wood is different but some maths would
be helpful to get me in the ballpark.
It is interesting that Romanillos and HyA have used very thin struts
combined with top thicknesses on the thin side (without the top sagging).
Marin has also thin and high (and only 5) in his Bouchet inspired,
but the top is a bit thicker (have not a Hacklinger, so I can not measure it).

Vesa
Yes. Section 4.4 in the design book (Design of braces and bracing systems, 23 pages) covers flexural rigidity (E * I) for different sections, different materials, composite braces and how to measure and calculate all you need to know, including how to accommodate the specific properties (density and elastic moduli) of the wood you are about to use. Also includes comparative charts of EI for a variety of historic and contemporary guitars.
Trevor Gore: Classical Guitar Design and Build

printer2
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Re: Which equation?

Post by printer2 » Mon Mar 11, 2019 11:55 pm

Fred

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60moo
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Re: Which equation?

Post by 60moo » Tue Mar 12, 2019 4:07 am

vesa wrote:
Mon Mar 11, 2019 3:53 pm
Thanks a lot 60moo
60moo wrote:
So, for any triangle of base length b and height h, the second moment of area I(t) = 1/36 (b) x (h) x (h) x (h) i.e. base times the height cubed divided by 36.
Yes it is the relative strength that interests me
Sorry my maths are a bit rusty
But did you mean:
if eg. height is 5 and base 4:
1) 1/36 x 4 x 5 x 5 x 5 = 13.88
or 2) 1/36 x 4 x 5 x 5 = 2.77

compared with 3 and 7 (Torres):
1) 1/36 x 7 x 3 x 3 x 3 = 5.25
or 2) 1/36 x 7 x 3 x 3 = 1.75

But in either case 4 x 5 has much higher bending strength than 3 x 5.

Vesa
Vesa - in both those examples the correct answer is 1) i.e. the height is cubed (not squared).

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60moo
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Re: Which equation?

Post by 60moo » Tue Mar 12, 2019 4:26 am

This paper is a combination of 1st year engineering structural analysis and 2nd year stress analysis. The beauty of the classical guitar is that a proper analysis is far more complex than that shown, as we're dealing with a plate/beam set up, with special boundary conditions. I'm sure someone's done a finite element analysis for their PhD somewhere on this topic. Would make a very interesting read - but my bet is that the human ear would still leave it for dead if we're looking for optimal design outcomes.

printer2
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Re: Which equation?

Post by printer2 » Tue Mar 12, 2019 11:14 am

60moo wrote:
Tue Mar 12, 2019 4:26 am
This paper is a combination of 1st year engineering structural analysis and 2nd year stress analysis. The beauty of the classical guitar is that a proper analysis is far more complex than that shown, as we're dealing with a plate/beam set up, with special boundary conditions. I'm sure someone's done a finite element analysis for their PhD somewhere on this topic. Would make a very interesting read - but my bet is that the human ear would still leave it for dead if we're looking for optimal design outcomes.
A beam was all that was asked for, too busy these days to go further.
Fred

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James Lister
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Re: Which equation?

Post by James Lister » Tue Mar 12, 2019 11:27 am

I think for what you're asking all you need to know is that the stiffness is proportional to the width of the strut, and proportional to the height cubed, so a tall, narrow strut is more efficient than a low, wide one. As you go narrower, the size of the gluing surface becomes a problem, which is where a triangular strut is advantageous.

As Alan points out, the mass of the struts is small compared to that of the soundboard itself, so if you're really interested in producing a lighter top with the same stiffness, you need to reduce the soundboard place thickness and increase the size of the struts. Note however that the "greatest guitar of our epoch" (according to Segovia) had a relatively thick top and small braces, so efficiency isn't everything.

James
James Lister, luthier, Sheffield UK

TomBeltran
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Re: Which equation?

Post by TomBeltran » Tue Mar 12, 2019 3:40 pm

James Lister wrote:
Tue Mar 12, 2019 11:27 am
I think for what you're asking all you need to know is that the stiffness is proportional to the width of the strut, and proportional to the height cubed, so a tall, narrow strut is more efficient than a low, wide one. As you go narrower, the size of the gluing surface becomes a problem, which is where a triangular strut is advantageous.

James
John Gilbert may have had this in mind, in constructing his fan braces, which looked like a half an I-beam. His bridges may have been light given the way he scooped out the center, but to me, they had a solid, heavy look. My first thought was that for some reason, he was trying to strengthen the outside edge. I typically use rather narrow (3mm) high (5-6 mm) fan braces, which I first saw in a 1964 or 1965 Ramirez 1a that was a prototype for the Western Cedar top. It seems to me, that using a wide, narrow, fan might have the effect of simply being a thicker top. And I appreciate the input of the mathematically endowed among us. Who knew there could be a practical use for math. :)

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