Break Angle at the Saddle

[TallDad71]

I am looking for some guidance on Break Angle at the Saddle but can't find any in the forum. I wonder if anyone has given any serious thought to the mechanics of this.

What I think to be true so far is:

On a standard steel string guitar different strings have different tensions when tuned to EADGBE, typically middle strings have more tension.
A luthiers dream is to make each string have equal presence on the guitar, if a top E string is exerting less pressure on the saddle than a D string then it is going to have a lesser effect on the soundboard.

I have done some mechanics sums and discovered the following:

The string tension exerts and downwards (and forwards) pressure on the saddle, I will call this the Resultant! The bigger the Resultant the more energy that gets driven into the saddle and then the soundboard.

The height of the saddle is not a determining factor on how big that Resultant force is. This is great as the design of a saddle is fixed to the players comfort.

The Break Angle is a determining factor on how big the Resultant force is, the shallower the angle the lesser the Resultant. If the break angle is 1 degree very little energy will get transferred. If the angle is 89 degrees lots of energy will transfer but the string will distort and the saddle will break.

In order to ensure that each string exerts the same Resultant force on a guitar then the break angle must be adjusted in line with the tension that the string exerts.

My sums suggest that if you have a Break Angle of around 30 degrees to the horizontal then the D string may have 30% more Resultant force on the saddle than the top E string with the same Break Angle.

In order to even out the Resultant forces across each string the Break Angles will have to differ by around 7 degrees! The only way to make the break angle different is to change the position where the string enters the bridge block, either by cutting a channel for strings with lower tension to increase their Break Angle or by moving the peg backwards for string with higher tension to reduce their Break Angle. If you were to move the peg then it would have to go back around 2mm!

My gut feeling is that the sums aren't hard, and that string makers will have already done the work on making sure each string has equal presence, based on the idea that break angles are near enough the same across the six strings. Perhaps different tensions help to even out presence, I don't know


Has anyone tackled this issue and come to a satisfactory conclusion?

Would anyone like to go through my sums and see if I am thinking correctly?

[jeffhigh]

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The string tension exerts and downwards (and forwards) pressure on the saddle, I will call this the Resultant! The bigger the Resultant the more energy that gets driven into the saddle and then the soundboard.



Would anyone like to go through my sums and see if I am thinking correctly?

This is the major problem with your reasoning.
It is a common but mistaken belief that the downforce of the string on the saddle (what you call the resultant) is proportional or otherwise related to the output of the guitar.
The transverse signal from the string, which is the major driver of the soundboard, sits on top of the static string downforce.
Whilst you need a certain amount of downforce (created by the break angle) to keep the string in contact with the saddle and define the end of the speaking length of the string, more does not equate to increased transfer of string energy.

[TallDad71]

Ok thanks for that.

So if you need a certain amount of downforce to promote the forward backward motion of the saddle the break angle is largely irrelevant. However there would still be an optimum level of downforce, albeit within much wider limits, how would you determine this?

If that's the case then the height of the saddle is relevant. At 12mm high for example it will rock along the transverse direction of the string. At 17mm high the saddle is a longer lever and will rock more freely. At 7mm high it would be a short lever and rock less freely.

How would you determine then the optimum saddle height for a guitar I wonder. If it were too high the bridge would struggle to stay attached and in one piece and the soundboard might just be overpowered.

Again I think I am soon to be reassured that there is no formula. Build it, make copious notes, review the finished guitar to try and discern which of the thirty tweeks between builds which effect on the tone.

A luthiers life!

[jeffhigh]

The vibration from the string is transmitted to the soundboard in two ways

Firstly, and the major sound producer is transverse to the string, which drives the soundboard in and out.

Secondly and with much less energy and at twice the frequency is the tension change signal. This is the one which rocks the bridge. Some writers like Siminoff mistakenly assign this as the major sound producer.

You can produce sound without much elevation from the soundboard via the first mechanism (eg the flamenco guitar with low bridge) you just don't get much from the tension change signal.

For a steel string guitar 12-14mm above the soundboard seems to be the range which works well without having to overbrace.

Acceptable break angle IMHO for a SS acoustic is somewhere between about 10 and 45 degrees. Less than that the string starts moving about when plucked, too high and the force pushes the saddle forward, possibly breaking out the front of the bridge. The steep angles you get on the bass strings on a Martin are too high as far as I am concerned.

[TallDad71]

Again thanks for the help.

I think we agree! I have produced a couple of diagrams which show the maths for different break angles 20 and 45 degrees.

Is there any chance you could have a look and explain your thinking in the context of these diagrams please?

This image shows a break angle of 20 degrees. The blue line is the saddle the brown the string and the red is my Resultant. T is the tension on the string when in equilibrium
It shows that the horizontal force is nearly SIX times the vertical force at the top of the saddle, 0.98T versus 0.17T

Screen Shot 2017-02-27 at 07.12.22.png (88.78 KiB) Viewed 14052 times

This image shows a break angle of 45 degrees.
It shows that the horizontal force is less than THREE times the vertical force at the top of the saddle. 0.92T versus 0.38T

Screen Shot 2017-02-27 at 07.12.33.png (92.35 KiB) Viewed 14052 times

Now comparing the two

The vertical forces aren't radically different (0.98T versus 0.92T) but the horizontal forces are twice as different (0.17T versus 0.38T). The diagonal force on the guitar is also increased (0.34T versus 0.77T) though I am not too sure what effect that might have on the sound.


You said that the vertical forces were the major sound producer, this might strongly influence the Monopole response.
The horizontal forces were less of a sound producer, this might strongly influence the Long Dipole response.

So perhaps the break angle influences the MIX between long dipole and monopole response?

Whatever MIX a luthier chooses, wouldn't it be beneficial equalise the vertical and horizontal pressure across all strings?

These calculations don't take account of the height of the saddle. I agree that the height of the saddle will make a difference to the sound and that perhaps there is even an ideal height for each set of strings. However for any set height of saddle the resultant forces will be the same as above.

[jeffhigh]

I'm not sure why you have halved the angle, and I think you have transposed sine and cos, but it could be me being rusty on this.
There are a few other problems with the calcs too

But the major problem is that these are static forces not the signal forces

The key issues to understand are that

-these are STATIC forces from string tension at the point of contact with the top of the saddle.

-The variation in break angle (assuming overall height of saddle above soundboard is constant) does not affect the overall forces applied by the bridge to the soundboard on a fixed bridge (pinned or tied)

-Vibrational forces from the string are NOT proportional to the static forces produced by string tension, nor is their transmission to the saddle or bridge changed by different Static forces produced by break angle as long as there is sufficient to maintain contact.

[TallDad71]

Think I'm finally starting to get my thick head over this.

I want to do some calcs on the torque that is applied on the system in terms of bridge rocking.

Thanks for challenging me, I will overcome this affliction.

[jeffhigh]

Torque is pretty simple
For static torque of the bridge on the soundboard it is just saddle height above the soundboard in metres by string tension in Newtons.
As far as the tension change signal it is just a much smaller plus and minus force superimposed on the static string tension

[TallDad71]

I want to explore this thread to a half decent conclusion in order to help out people who believe as I did.

But is this right?

In a guitar with an external point of contact such as an archtop the main producer of the sound is the vibrational energy of the strings, I am thinking lots of sine and cosine curves all jumbled up. The saddle transfers those vibrations into the body which excites the air and hey presto!

Screen Shot 2017-02-28 at 08.12.46.png (7.33 KiB) Viewed 13988 times

The break angle need only be the minimum sufficient to prevent those vibrations from passing behind the saddle to the dead strings.
_________________________________________
A steel string guitar is different as the strings are attached to it. When the string vibrates it has two sources of energy. The first is the same as above and the second is related to the shortening and relaxation of the string.

Now You and Siminoff disagree as to which is the major sound producer so for purposes of my understanding I will call it 50% vibrational and 50% longitudinal pull and relaxation.

The saddle acts as a lever and the entire bridge and soundboard underneath the bridge area are rotated forwards. The rocking of this soundboard moves great amounts of air inside the chamber. This creates a much louder guitar than an archtop. I will assume that the pivot for the rocking is near or under the saddle, in reality it could be anywhere depending on the structure of the soundboard.

Screen Shot 2017-02-28 at 07.42.35.png (10.52 KiB) Viewed 13988 times

The law of moments tells us that the higher the saddle from the pivot the easier it is for the for the string to rock the bridge block. The easier it is, the easier it can move the bridge block and the air inside the chamber, this will make for a louder guitar. A 1mm change in the height of the saddle will have near a 10% change in the Moment and have a noticeable effect on the guitars sound.

So what effect does break angle have? Vibrational energy accounts for 50%, longitudinal energy is influenced heavily by saddle height, what is left for break angle, perhaps not too much?

The D and G strings are higher on the saddle and have more tension in them. Surely they should be louder then? But they aren't any louder than the other strings on my guitars which puzzles me.

I am a stubborn sod and I still think it has a mechanical effect! But I now believe that the effect is small and extraordinarily difficult to calculate.

Here is the effect on 20 degree

Here is the effect on a 45 degree break angle

The blue lines show the effect on the bridge of the longitudinal energy, same as previous posts. It seems to me that a steeper break angle will have more effect at rocking the bridge on the pivot (wherever that might be!) and a narrower angle will have less effect.

At this point my maths and physics fails me and I can't calculate whether the effect counts for 10%, 1% or even 0.1% of the overall sound. Anecdotal evidence suggests its much closer to the 1% than the 10% though.

Thanks Jeff.

[jeffhigh]

When analysing the torque on the bridge, you can basically assume that the string is pulling to the left at the top of the saddle and the rest of the string, saddle and bridge acts as one with the connection to the soundboard providing resistance to the bridge rotation hence T x saddle height is the Static torque and tension change x saddle height is the signal torque.
So break angle is not really influencing eithe static or signal torque.

I can't remember the exact numbers but from memory, I think Trevor Gore has calculated the signal change contribution as around 25% for a steel string and half that for a nylon string

[Trevor Gore]

Would anyone like to go through my sums and see if I am thinking correctly?

You need to clearly separate the two issues you are discussing here: the static torque load on the bridge (which drives the structural design of the top and its bracing) and the dynamic loading due to the string pluck which drives the sound production. The challenge is to make the top stiff enough to survive the static loading over the long term whilst being flexible enough to be easily driven by the comparatively miniscule dynamic forces from the vibrating string.

That static problem is simple, as Jeff has indicated. The problem can be regarded as a rigid body mechanics problem (rigid in that the bridge is very stiff compared to the top plate). Consequently, the strings can be assumed to be "welded" to the top of the saddle (for a "flat top" guitar) and the torque on the bridge is the total string tension multiplied by the height of the string off the guitar top (as Jeff pointed out). Notice that the break angle doesn't come into consideration - nor should it. Al Curruth (a respected luthier who does a lot of experimentation) has examined the break angle problem experimentally (although he hasn't formally written up his results yet) and finds (as expected) that sound production is independent of break angle (but dependent on saddle height). There are a few minor discussion points around those results, but I'll leave those for now. The magnitude of the torque has a major influence on the design of the top and it's subsequent longevity.

How the string drives the top is another matter altogether. There are two components to the drive force. One that pumps the top in and out, usually called the perpendicular (to the top plane) transverse (to the string's axis) string force and the tension change force which acts in-line with the string and rocks the saddle. The transverse force is about 4 times the magnitude of the tension change force for a steel string guitar and acts in the direction where the top is least stiff and drives the monopole mode of vibration, which is the most efficient sound radiating mode by far. Thus, the transverse force pumping the top is, by orders of magnitude, the largest sound producer.

As a string vibrates, the angle it makes at the ends (e.g. the bridge) changes a small amount; let's call this α (alpha). The whole string tension multiplied by sin α is the transverse component of the string force acting perpendicular to the plane of the soundboard, and this alternating force drives the soundboard and produces the majority of the sound. As is happens, (in theory, at least, but borne out by experimentation) the angle of the string at the ends only has two values; +α and -α. So the alternating force driving the soundboard is actually a square wave for a central pluck of the string, with a magnitude of ~4 newtons per string for a 3mm pluck amplitude (steel strings).

As the string vibrates, it changes length (due to the non-straight path it takes between the ends). The length change can be calculated from the one-dimensional wave equation and knowing the elasticity (stiffness) of the string material, a tension change can be computed. This tension change "tugs" on the top of the saddle to produce a rocking motion. However, as the string goes through two extremes of motion per cycle, there are two "tugs" per cycle, so the tension change force drives the rocking motion (the long dipole mode) at twice the fundamental frequency of the string's transverse vibrations. As the soundboard is stiff in this direction (to resist the static string load) not much motion is produced and so not much sound, though it is an audible component. As the string vibrates, its path length varies linearly, with the result that the tension change force is a triangular wave at twice the fundamental string frequency, as previously mentioned.

Whilst all this stuff is standard engineering and has been known for hundreds of years by those active in the field, it is far from common knowledge, but is fundamental to the function of a guitar and hence a guitar's design, if best advantage is to be made of the meager drive forces that are available.

If you want to know the full story in all its glorious detail and lots more besides, you probably know where to find it by now!

[TallDad71]

Now I get it.

I have been struggling to make my maths fit the realities of the situation. I read Carruth a few days ago and noted that the break angle is irrelevant in his experiments.

Your sentence "Consequently, the strings can be assumed to be "welded" to the top of the saddle" solved the mathematical problem for me.

In a static state it can be claimed that there is equal tension in the string on both sides of the saddle, before and after the break angle. This allowed for all manner of Resultant forces to be calculated.

In a vibrating string the tensions either side are no longer equal, so any maths that I chose to do suddenly became a lot less relevant. The maths involved when the string is "Welded" to the saddle is a lot easier and confirms the points both of you have made.

Thanks for your help chaps.

Trevor, your book is very well respected. Brexit has screwed the value of Stirling for imports and as soon as it increases again I shall get a copy sent over.

[Dave M]

Do it now, Tall Dad do it now. Don't reckon the pound is going to be going up any time soon.