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Is deflection a linear equation?
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Author:  meddlingfool [ Mon Oct 06, 2014 8:58 am ]
Post subject:  Is deflection a linear equation?
This is another question geared towards my first 12 string.

Here is my thought process.

Once I know what percentage of extra tension a 12 string set will give vs a six string set, I will know how much extra stiffness the soundboard and bracing will require.

As an example, if 12 strings give 2x the tension (I know it doesn't but for simplicity's sake), I could simply make the top 2x stiffness.

So, to use round numbers...

If my tops normally deflect to .100 under a given weight, if I make a top deflect .050, have I made that top twice as stiff?

Does it track linearly like that, or is there an exponential thing going on like there is with the cube stiffness rule?

Thanks...
Author:  Jfurry [ Mon Oct 06, 2014 9:26 am ]
Post subject:  Re: Is deflection a linear equation?
Ed I've been following your threads,my curiosity has gotten the best of me. Please bear with my stupidity.
Your question. If my top delflect to.1 under a given weight . Deflects what vibration ,sound wave, ? How do you change deflection ? Why do you change deflection?Howdo you measure deflection ?
Author:  meddlingfool [ Mon Oct 06, 2014 9:52 am ]
Post subject:  Re: Is deflection a linear equation?
Hi Jeff,

Deflection is how much your top bends (deflects) under a load.

In my jig, the two points the top rests on, and the weight placed on the top remain constant. Underneath between the two points is a micrometer. This will tell you how much your top deflects when you put the weight on it. This will let you sand your tops to a consistent stiffness.

How? To change deflection, change the thickness of the top.

As to why, to change the tone. If your guitars are always coming out too tight, boxy, and trebly, your tops are too thick. So you want to make them thinner, but by how much? Well, you could use this to measure what you do now. Say it deflects to .100, but you don't like the sound. Next one, thin it til it deflects to .150. You like it more, but think you can do better. Next one, thin til it deflects .250. Oops, too much, now it sounds floppy. But at least you have data and a way to measure the stiffness of your tops. (The numbers there are just examples and have no bearing on what your real life metrics might be)

Measured with the dial caliper underneath. Simple but effective...

Image
Author:  Goodin [ Mon Oct 06, 2014 12:04 pm ]
Post subject:  Re: Is deflection a linear equation?
Thank you for that advice and pic Ed. Great idea with the dial caliper. What do you use for a weight?
Author:  meddlingfool [ Mon Oct 06, 2014 12:54 pm ]
Post subject:  Re: Is deflection a linear equation?
An empty Glengoyne tube filled with 5lb worth of pennies. 18" between rests...
Author:  Jim Watts [ Mon Oct 06, 2014 1:01 pm ]
Post subject:  Re: Is deflection a linear equation?
Ed, yes it's linear assuming you only change the load. Using the set up you show in your photo, deflect the top with one can of beans then set another can on top of the first one and you'll notice twice the deflection.
Hope that helps.
Author:  J De Rocher [ Mon Oct 06, 2014 1:29 pm ]
Post subject:  Re: Is deflection a linear equation?
Jim Watts wrote:
Ed, yes it's linear assuming you only change the load. Using the set up you show in your photo, deflect the top with one can of beans then set another can on top of the first one and you'll notice twice the deflection.
Hope that helps.


I would have guessed to opposite. If you take Ed's setup but without a weight on the wood and just press down in the center with your hand, the initial deflection would take very little pressure. I think if you keep adding pressure with your hand, the amount of additional deflection for each added increment of pressure would get smaller until at the extreme, the wood barely deflects at all and then suddenly breaks. Same as what happens when you bend a stick until it breaks. I think the deflection would be close to linear at small deflections but become distinctly non-linear as the deflection increases. This is an untested guess on my part. I'll be interested to see what Ed finds if he tries your suggestion. If his initial weight gives a 0.100 deflection, I would bet a home brew that doubling the weight would give less than 0.200 deflection.
Author:  meddlingfool [ Mon Oct 06, 2014 1:56 pm ]
Post subject:  Re: Is deflection a linear equation?
Actually that's a good point, I can experiment myself if I can find another five pounds somewhere...
Or even one, really. One more pound should yield 20% more deflection if it's linear...
Author:  J De Rocher [ Mon Oct 06, 2014 1:59 pm ]
Post subject:  Re: Is deflection a linear equation?
I was curious enough about this that I set up a deflection rig like Ed's and did the test using an unsanded spruce plate (0.175 thick), a dial gauge, and two weights labeled as being 3 lbs each but were actually 3.6 and 3.7 lbs.

3.6 lb, deflection = 0.107
7.3 lb, deflection = 0.185
Author:  meddlingfool [ Mon Oct 06, 2014 2:14 pm ]
Post subject:  Re: Is deflection a linear equation?
Ok, so...

.107x 2 is .214

.185 is 86% of .214, so it's very close, but not quite double...
Author:  J De Rocher [ Mon Oct 06, 2014 2:58 pm ]
Post subject:  Re: Is deflection a linear equation?
As a practical matter, assuming double when measuring small deflections seems like it would put you in the ball park.
Author:  Jfurry [ Mon Oct 06, 2014 2:59 pm ]
Post subject:  Re: Is deflection a linear equation?
Thanks for taking the time to explain. I do really appreciate it.
Author:  David Malicky [ Mon Oct 06, 2014 3:10 pm ]
Post subject:  Re: Is deflection a linear equation?
There are two factors: the material and the geometry.

Aside from viscoelastic effects, dried wood is a linear material (below its yield stress):
Stress = Modulus * Strain
Force = Stiffness * Deflection
See Fig 3: http://www.fhwa.dot.gov/publications/re ... 7/sec1.cfm

For small deflections, a beam also shows a linear response. At high deflections, the beam's curvature will skew the response non-linearly. I don't recall a rule of thumb for how much deflection is not small, but I'd guess 0.25" deflection over 15" is still very linear.

There is one more important geometry factor, though, if doing thin flat plates: Any warp or distortion in the plate can dramatically affect the linearity of the response. Two causes:
- If the plate's warp prevents it from contacting the supports evenly, then the initial load will only load a portion of the plate. With increasing load and deflection, more of the support starts to engage the plate, and so the plate appears to stiffen (but it's just an artifact of the way it is being tested).
- If the plate's warp has small domes and dishes in it, these will open and close with loading, also skewing response non-linearly.
A prior post on this issue: viewtopic.php?f=10101&t=26060
Author:  meddlingfool [ Mon Oct 06, 2014 3:39 pm ]
Post subject:  Re: Is deflection a linear equation?
I think there's an easy way out of this, now that I've had a second cup of coffee.

If I know that the 12 string set adds 62% more tension, could I not simply add 62% more weight than usual and then thickness to the same deflection?
Author:  J De Rocher [ Mon Oct 06, 2014 3:47 pm ]
Post subject:  Re: Is deflection a linear equation?
That sounds like a good idea. The wonders a good stimulant can work!
Author:  James Orr [ Mon Oct 06, 2014 4:53 pm ]
Post subject:  Re: Is deflection a linear equation?
meddlingfool wrote:
I think there's an easy way out of this, now that I've had a second cup of coffee.

If I know that the 12 string set adds 62% more tension, could I not simply add 62% more weight than usual and then thickness to the same deflection?


That sure makes sense to me. But so does:
Quote:
If my tops normally deflect to .100 under a given weight, if I make a top deflect .050, have I made that top twice as stiff?

That said, math is not my gift, so don't put any reliance whatsoever in this post ;)
Author:  Tom West [ Mon Oct 06, 2014 5:02 pm ]
Post subject:  Re: Is deflection a linear equation?
Interesting, hope T.G. shows up to add to the data.
Tom
Author:  meddlingfool [ Mon Oct 06, 2014 5:22 pm ]
Post subject:  Re: Is deflection a linear equation?
That makes sense too James, for 2x, but...

If I want to make it 62% stiffer, what do I take it to? Pretty sure it would not be .062. You'd need to work backwards from .100.

Like, 100 -25 % is 75, but 75 + 25% is 93.75. Not sure how to make that calculation, particularly adding for a 15% variance...scritch. Scritch scritch.
Author:  meddlingfool [ Mon Oct 06, 2014 5:36 pm ]
Post subject:  Re: Is deflection a linear equation?
And I guess the braces need to be 62% stiffer as well...
Author:  Trevor Gore [ Mon Oct 06, 2014 5:45 pm ]
Post subject:  Re: Is deflection a linear equation?
I must encourage you to look in that big black book more often, Ed!

1) Yes, deflection is linear under load for "small deflections". (But it's a cube rule for the central deflection if you change the span)
2) You need to take the "slop" out of the system first (as Dave Malicky says). So, for example, pre-load with 1 lb, zero the dial gauge, then load with 5lb.
3) Your wood needs to start out flat. If it's dished or domed you have a structure which adds stiffness and will also likely change geometry as it deflects
4) Planed wood is stiffer than sanded wood at the same thickness. Coarse thickness sanding mashes the surface, where most of the stress is, making it unable to sustain a load. So you have material on the surface that adds thickness but not stiffness

Look into the parallel axis theorem. If you double the top stiffness and double the bracing stiffness and then glue them together, you will likely get quite a bit more than twice the stiffness of the reference structure.
Author:  meddlingfool [ Mon Oct 06, 2014 7:32 pm ]
Post subject:  Re: Is deflection a linear equation?
Thanks Trevor,

I do need to do a bit more studying. I haven't actually re looked at it for some time now. Ever since it answered my major questions, it's been full steam ahead without much chance for review...
Author:  Tom West [ Tue Oct 07, 2014 8:45 am ]
Post subject:  Re: Is deflection a linear equation?
Trevor Gore wrote:

Planed wood is stiffer than sanded wood at the same thickness. Coarse thickness sanding mashes the surface, where most of the stress is, making it unable to sustain a load. " So you have material on the surface that adds thickness but not stiffness."


Trevor : Thanks for another gem!!! Thinkers know this stuff, quite obviously I have to do some of that thinking stuff.
Tom
Author:  Alan Carruth [ Tue Oct 07, 2014 10:31 am ]
Post subject:  Re: Is deflection a linear equation?
I got thinking about the 'cube rule' a few years back when I got a 12-string order not making one in some time. I'd started to measure wood properties, and had come up with a simple way to get close to the correct thickness given the Young's modulus of the wood. It's a bit different from your deflection testing, but not so much in the end. Anyway, it occurred to me that if the stiffness along the grain, which is the most important thing structurally, varies as the cube of the thickness, then you just scale the thickness and brace height by the cube root of the tension difference. In other words, if the 12 string carries 68% more tension then you just increase the thickness and brace height by the cube root of 1.68, or about 20%. In the event, I assumed double the load, and increased everything by 25%, and ended up with a cannon. After all, you'll have 68% more energy at a given string amplitude, driving a top that's only 20% heavier.

If you don't calculate the Young's modulus, why not just increase the load on your deflection setup by the 68% tension increase factor? Assuming the bridge geometry is the same it should work, since the static load is linear.
Author:  David Malicky [ Tue Oct 07, 2014 12:47 pm ]
Post subject:  Re: Is deflection a linear equation?
Alan, yes, that method gets the longitudinal stiffness right, both for the individual components and for the "T" cross-section formed by the brace and top.

The stresses in the braces are a bit higher since stress goes by the 'square rule'. But stress is more sensitive to brace cross-section shape (rectangular vs triangular vs...).
Author:  meddlingfool [ Tue Oct 07, 2014 2:30 pm ]
Post subject:  Re: Is deflection a linear equation?
That's the plan at this point. Might also use 5/16" bracing instead of the usual 1/4"...
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