The Science of bridge break angles and why they don't matter.
By Richard Jernigan
I finally got around to doing some calculating. I find that the change in string tension never comes near overcoming the friction. Here's what I did. I invite anyone so inclined to check my work. Pushing the string down to pluck it increases its length. The increased length stretches the string, increasing the tension. As the string vibrates, the tension decreases as it passes back through its rest position and increases again as it moves transversely. Al Carruth's String Theory" post gives a good picture of this: http://tinyurl.com/4xneskf
The maximum change in tension occurs just before the string is released. If it's ever going to slide across the saddle, that's when it's going to happen. If you pluck the string 1/5 of the way from the bridge to the nut, on a 650mm scale you can only push it about 8mm before it will rattle against the frets.
I used the following properties for nylon:
String diameter = .04 inch--D'Addario EJ 45 g-string
http://tinyurl.com/3wn273x
Modulus of elasticity = 400,000 pounds/square inch (PSI)
Here's the calculation:
additional string length due to pluck = 0.31 mm
strain = 0.31/650 = 0.00047 (dimensionless)
stress = strain*modulus = 0.00047*400,000 = 189 PSI
additional tension = stress*cross sectional area of string = 0.24 lb
This is a small fraction of the 12 lb static string tension.
As a sanity check, it takes about 0.92 Ib of downward force to displace the string by 8mm, 1/5 of the way from the bridge to the nut. Pushing on my kitchen scale, this feels about right.
For the string to slide across the saddle, the 0.24 Ib additional tension must overcome the frictional force. The frictional force is the product of the force with which the string is held against the saddle and the coefficient of friction. The force with which the string pushes against the saddle is twice the tension (24.Ib) times the cosine of half the break angle. The "break angle" here means the angle between the long part of the string and the part behind the saddle.
The smallest coefficient of friction I could find for nylon was 0.18, from the same source as the modulus above.
Most values I found are nearly twice this. Even using this small value for the coefficient of friction, a break angle of 173.7 degrees suffices to resist the increased tension due to stretching the string.
Put another way, sloping the string down only 6.3 degrees from the horizontal behind the bridge
produces enough friction to counterbalance the additional tension of the vibrating string. I have never seen such a shallow break angle, even with the looped tie of a 6-hole block.
It seems reasonable that generations of luthiers would have found a break angle that results in efficient transmission of string energy to the saddle, without slippage. RNJ