MIMF

I also see a lot of lifted bridges. Like Mark, I mostly see cheap plywood guitars where it's obviously due to poor fit and/or poor glue application. But on the few nicely made instruments with lifting bridges, I've seen modified "scalloped" bracing, ones that haven't been cared for properly and have chewed up bridge plates, and in almost every case, a taller than normal bridge with little graduation near the edges.

First, a disclaimer - at the outset, building these models, I didn't figure things were going to get so specific, so they were just rough boards and bits of bone hodge-podged together - I've tried to smooth things up a little, but any objective measurements should be taken with a grain of salt, especially where friction is involved.

David Malicky wrote:Peter, Good to hear, thanks for confirming that! Would you have a rough estimate of how much force it takes to get the bridge to lie down flat?

I lay the (thin top) model on the bathroom scales, and pressed the rear of the bridge down with my finger until it contacted the top - ~15# of pressure. Of course, this is with only 2 strings, and not really tuned to pitch anymore (laziness).

Mario Proulx wrote:Peter, do you have the correct "neck angle" built-in to this test rig?

I'm not sure what is "correct." On the plank model, the string height is 5/8" off the plank top at the saddle, and 5/16" at the nut. This gives an angle of arctan [(5/8-5/16)/25.4] = 0.7 degrees.

Rodger Knox wrote:In the physical model, the bridge's rotation is being limited by the bridge pins. When the clamp is removed, the moment is transfered to the pins.
If that is correct, tighter pins should reduce the rotation.
By the way, has anyone measured the horizontal displacement of the bridge due to string tension? I'd like to know if the estimate I came up with is in the ballpark.

I redrilled and tapered the pin holes for a tight fit on the plank model, and flattened the plank and bridge mating surfaces for a better fit. Despite this, the back of the bridge still lifted, although subjectively slightly less than before, though with the changes from the better fit bridge it is hard to tell.

I removed the bridge pins and tuned it up. The displacement was 3/32 on the treble side and 4/32 on the bass side. Here is where friction plays a large role I expect, and also tightness of the the fit of the windings at the ball ends into the slots in the bridgeplate/top and bridge - mine are quite tight.

And BTW, there was NO lifting of the rear of the bridge with the pins removed.

Jeff Highland wrote:Then from the drag of the strings friction on the top of the hole and the saddle as tensioning?

I think friction at the top of the pin hole would tend to keep the rear of the bridge from lifting, or be neutral. The saddle top is another matter - I guess I could grease it up and see if there is any difference, but I think it's the force of the string pushing the saddle top forward that is the main factor.

Jeff, Thanks for the suggestion, although dimensions don't help as the issues are more fundamental. It helps a lot that you brought up the retaining wall analogue -- now I see where you're coming from. I'm a mechanical engineer, but from what I know of retaining walls, I'd completely agree the method you describe is right for that application and any others that share its valid assumptions. But it's different than a glued bridge in these ways:
- Gravity pulls on the entire planet, so there's no local cancellation of forces like with the bridge's T vs. R, due to the nearby ball-end.
- AFAIK, soil cannot provide tension; glued wood can.
- Since the reaction force of a retaining wall is directed through its base, there's nothing to drive tension even if the soil could provide it. The glued bridge's reaction force may or may not be directed through its base, depending on the magnitude of R.
- Less important: Concrete is much stiffer than soil, so the wall can appropriately be modeled as rigid. Hardwood and softwood don't have that extreme level of difference.
I do think the retaining wall method is good and very insightful for the unglued bridge.

On the experiment, I agree the soundboard is deforming slightly due to the ballend force and making a tiny hump, with the most curvature at the pin holes (as Jeff pointed out earlier). To fit with no gaps, a glued bridge has to push that hump back down. That's another way to understand the origin of the R force (similar to the Superposition method for statically indeterminate problems).

Peter, Thanks for the force estimate! And that's a clever way to measure it. I think your continued experiments are really helpful. On your last post, that's interesting about the bridge not lifting with pins removed. So, in this case, is the bridge basically free to slide forward except for the strings themselves (no nails, glue, or other shear stops, just friction and reorientation of the string in the hole)?
On the deformation of the soundboard/plate from last page, it's actually good if it does a bit, as a real one would, too.

Greg, thanks for the info!

I'm working on an FEA model -- stay tuned!

David Malicky wrote: I'm a mechanical engineer,

Well at least you're not an electrical engineer LOL

I guess I am only suggesting that the unglued bridge FBD is a good start to understanding the forces placed on the glue interface and that you need to quantify those forces and their location before moving on to make presumptions about pressure distributions.
To do that you need to put numbers on the geometry

I'm not sure what is "correct." On the plank model, the string height is 5/8" off the plank top at the saddle, and 5/16" at the nut. This gives an angle of arctan [(5/8-5/16)/25.4] = 0.7 degrees.

<slaps forehead...!>

IF you're going to make a model/test rig, or go through the bother of computer modeling or any other such thing, AT LEAST base it on real-world values.

Sheesh... Go measure a real guitar, and quit guessing!

Have you never built a guitar? Any instrument?

It's interesting to note that up to this point, everyone has implicitly assumed that the tension in the string is PARALLEL to the bottom face of the bridge, at least in the undistorted shape. Due to the arch in most tops, this is not a completely valid assumption, it's probably off a degree or two. Is that enough to have a significant impact on the analysis? I'm guessing it has enough impact so that it shouldn't be neglected.
That's part of the problem here, we're on the edge of chaos theory. The forces are dependant on the exact geometry, and even based on design numbers, they all vary a little from builder to builder. Even what appears to be a simple question may not have a simple answer, because the question assumes a generality that may not be universally true.
I would be interested in comparing the results of a FEA to my envelope estimate. (The bridge moves forward 0.003" as the result of string tension)

up to this point, everyone has implicitly assumed that the tension in the string is PARALLEL to the bottom face of the bridge, at least in the undistorted shape. Due to the arch in most tops, this is not a completely valid assumption, it's probably off a degree or two. Is that enough to have a significant impact on the analysis?

The neck itself will have 1 to 1-1/2° of angle relative to the guitar's top, no matter if it is a true flat top or radiused to some degree. That places the nut-end of the string well -below- the plane of the bridge.

Also, no builder uses a bridge and saddle combination that it 5/8" off the guitar's top.. That's crazy-tall. 3/8" to 1/2" is the norm.

Yes, this will change everything....

Mario Proulx wrote:I'm not sure what is "correct." On the plank model, the string height is 5/8" off the plank top at the saddle, and 5/16" at the nut. This gives an angle of arctan [(5/8-5/16)/25.4] = 0.7 degrees.

<slaps forehead...!>

IF you're going to make a model/test rig, or go through the bother of computer modeling or any other such thing, AT LEAST base it on real-world values.

Sheesh... Go measure a real guitar, and quit guessing!

Have you never built a guitar? Any instrument?

First, I suggest you re-read this part of my last post:

Peter Wilcox wrote:First, a disclaimer - at the outset, building these models, I didn't figure things were going to get so specific, so they were just rough boards and bits of bone hodge-podged together - I've tried to smooth things up a little, but any objective measurements should be taken with a grain of salt, especially where friction is involved.

Second, I am building my first two flat-top acoustic instruments, neither of which is a six string guitar. That's why I ask questions like the topic of this thread.

Third, I believe it is the angle the string makes with the saddle (toward the nut), not the neck angle, that is pertinent to this discussion. I also believe it to have a negligible effect on the torque of the bridge when compared with the break angle. The angle I calculated for this in my model is close to the angles I measured on 3 guitars - 0.3 to 0.6 degrees. This is assuming that the saddle is at a right angle to the top. Additionally, the actual neck angles on these (admittedly cheap) guitars vary from 0.5 to 1 degree, and they all point to an elevation within 1/16" over the top of the bridge, so it appears there may be no "correct" angle, which was my point in the prior post.

Fourth, I agree that the saddle height of 5/8" is too high - it will impart a greater than normal torque to the bridge. Again, I was only interested in qualitative results from my "experiments."

Lastly, to give you some feedback, I found your post to be offensively patronizing. I may be ignorant, but I'm not stupid. I have found the answer to my original question, and will not be posting any more of my data as they are not up to your standards.

Everybody take a breath.

Peter Wilcox wrote:
"Third, I believe it is the angle the string makes with the saddle (toward the nut), not the neck angle, that is pertinent to this discussion. I also believe it to have a negligible effect on the torque of the bridge when compared with the break angle. "

Measurements I made on a classical guitar a while back showed that, at least for that case, the break angle did not change the amount the bridge rotated under tension, but did alter the location of the axis of rotation. The lower the break angle, the further back from the saddle the axis was. For those tests, I used an 18-hole tie block, which allowed me to either bring the strings off the saddle and through the holes in the front of the block, to give a 'normal' break angle of about 25 degrees, or to pass them string over the back edge of the tie block, which gave a break angle of about 6 degrees with a 'normal' string height off the top of 11mm. Putting in a much taller saddle (strings 18mm off the top: don't try this at home!) restored the break angle to about 25 degrees. The bridge rotation was measured by taking the displacement 50mm in front of and behind the saddle. For a given string height above the top the total displacement was the same, but a greater break angle gave a little more dip in front, and less rise behind, the bridge. Raising the string height off the top gave more displacement, of course, but the stationary point was the same as it was with the low saddle and normal tie. It's possible that a tie bridge works differently from a pin bridge in this respect, but I can't see why it would, and will await the advent of some data.

IMO. the tiny angle the string makes with respect to the plane of the top should be negligible: .7 degrees out of parallel won't change the numbers to speak of.

Alan Carruth / Luthier

If you think that only the break angle counts...

Try this:

Tie or otherwise affix a rope to the end of a 2x4 or something similar. Butt the free end of the 2x4 against something immovable, like an open door's threshold, and standing upright, pull on the rope. Notice how easily the end of the 2x4 comes up? Right! Now, kneel-down behind the threshold, and pulling the rope at 12 inches off the floor, try to raise the 2x4. Pretty damn hard to do, isn't it? Right! But given enough strength, you can still get the 2x4 to rise.... Now, place the rope as close to the floor as you can, and pull once more. Go ahead, pull like your life depends on it. Can't get the 2x4 to rise, can you? Of course not!

And that is why any model or test rig that doesn't make use of the proper angles is at best of little use, and at worse, gives very incorrect and misleading notions of what the forces at work here actually do.

Pull on the 2x4 with a 5 ft rope and feel the force needed to raise it. Then drop the end of the rope 0.7 degrees (~3/4") and pull again. Can you feel the difference?

I'm sure we could, and the difference between a correct model and the one used will be more than 0.7 degrees.

I'm not sure if I agree with Mario, or if he agrees with me, but we do agree!
In my analysis of the compression displacement, I had to make half a dozen basic assumptions about the strusture. Material and thickness of the top, Young's modulus, stress distribution, string gauge, and a couple of others I don't remember offhand. This is a much simpler analysis. I have a high degree of confidence that my results are within a factor of 3 of the correct answer. That's why I'm interested if anyone has measured that displacement on an actual guitar. Mathematical models frequently need to be calibrated to obtain reliable results. I'm afraid there are too many variables with too high an uncertainty factor on the values of those factors (remember, we're talking about wood here, nonhomogeneous, anisotropic wood) for a mathematical model to produce any meaningful results.
A man has got to know his limitations...

Below are some initial results from a simplified 2.5D FEA model of the bridge, top, and plate. In FEA, it's best to start simple and then add complexity in stages -- that helps understand incremental effects and catch errors. Here's the basic model:

Bridge: 1.5" x .35", integral saddle (no slot), string break angle over saddle = 30 deg.

Top: 0.110" thick.

Bridgeplate: 0.100" x 1.7", extends 0.1" beyond bridge fore and aft.

Pins: Omitted and assumed inactive, as with unslotted pins in a slotted bridge.

Materials: Isotropic elastic, Poisson's ratio = 0.35 for all. Bridge and plate are the same material, E=1000 ksi. Top has an E of 500 ksi. Of course, those are just ballpark numbers -- it's usually not meaningful to guess a precise isotropic E for an extremely orthotropic material such as wood. In FEA, I always start with isotropic for the usual reasons. Using the Wood Handbook and Trevor's book, I have some estimates for the orthotropic properties (9 constants) for Rosewood and Spruce -- I've done some runs and will show those results shortly (initial distributions appear similar to iso).

Loads: 25 lb simulated string tension, parallel to the top, using geometry to apply resultants on saddle, ramp, and ballend areas. I'll look at the incremental effect of the slightly angled downward component next.

Constraints: Front end of top fully fixed. Rear of top fixed laterally and vertically, free in fore-aft direction. (I also tried the rear constraint fully fixed; it had a small effect on the bridge reactions.) The sides of the model have symmetry/mirror constraints, which simulates other loaded bridge slices on both sides of this one (i.e., continuity -- neither air, walls, nor unloaded material). So this is most like the D or G string slice.
Here's the model with loads and constraints (click to enlarge; the ramp loads are a bit buried, sorry):
Mesh and Solver: the mesh has controls on element size and quality and refinements in important corners (as small as 0.001"). I used Creo Simulate (Pro/Engineer Mechanica), which uses higher-order "p-elements". Here's a mesh of 45k tetrahedral elements:
Limitations: As Trevor mentioned a bit ago, FEA is not the real world, although it is a pretty sophisticated model of it. Some things are hard to simulate, such as the stresses in an inside sharp corner. Mathematically, there is infinite stress (a "singularity") at the very corner edge. Real materials experience microscopic yielding there. So, FEA stresses are increasingly inaccurate as the corner is approached. There are a number of standard techniques to handle this, though:
1) Make the corner elements really small to confine the singularity to a tiny region. Ignore the stresses in those elements. I used 0.001" elements in the bridge-top corners.
2) For stresses very near the corner, never rely on the absolute stress value; only use for comparison.
Other limitations above.
Next post...

Resulting deflections (sanity check... not very meaningful with ballpark iso Es): Below are the stresses in the vertical direction (Syy), sliced through centerline. The front and rear corners show very large stress concentrations: compresion (blue) and tension (red). The middle shows the propogation of the ball-end load through the thickness, meeting with the string ramp load -- a self-equilibrating "stress column". Here are stresses in the vertical direction applied to the top surface (same as the pressure distribution on the bridge bottom). That ball-end stress column shows up as the "R" distribution.

The shear stresses on top surface are similar to the vertical stresses, except for sign: Finally, here are the Von-Mises stresses on the centerline and top surface (V-M combines shear, normal, and all other stresses in 1 number. These plots have a different stress scale.) So, there are huge stress concentrations at the front and rear of the bridge. That can be interpreted in a few ways, I think equally applicable:
- The bridge mostly behaves as a rigid body for these torque reactions.
- As the bridge's rear is pulled away from the top, the tension has to negotiate the sharp corner. This is similar to the classic stepped-bar in tension stress concentration. Bad sharp corner, bad.

Feedback, concerns, and suggestions on the model are welcome! I'm currently working on material and geometry variations to see relative effects. Also a model of the unglued bridge (hard to do... non-linear contact).

Love the color!

(and the analysis!)

Maybe that sharp corner is what drives the top. May be a good thing!

Another option is we are seeing impedance mismatch. Just throwing out ideas here.

I like the Von-Mises plot. That's getting saved.

Interesting, and still digesting the findings....

Can you please repeat the tests, with a 15/16" wide bridge with a bridgeplate at exatly one inch wide, centered under the bridge? And perhaps increase the string tension to 28lbs.

I ask because personally, I've seen more problems on heavily built guitars than on very lightly ones, and I've long suspected stress risers in the heavily built ones as the likely culprit(poor workmanship would be my second suspect...). I'm interested in seeing what the modeling will show!