[CAUT] A different Tension Formula

[RicB]

Hi folks... just got done decifering McFerrins approach to Tension in 
base strings... and it seems a bit more accurate then what I've seen on 
other spreadsheets.

Essentially... one starts by observing that the winding material is 
actually a circle... and that the space it takes up is mixed with the 
space inbetween windings... air if you will.  Given the diameter of the 
winding material the ratio of  air space in a square whos sides are 
equal to the diameter of the wire in lenght is of course

D^2- PI(D/2)^2 : PI(D/2)^2.

If D is 1 mm then PI(D/2)^2 is 0,785 mm and D^2- PI(D/2)^2 us 0,215 mm

In simplest form this simple means that 78.5% of the the total space 
taken up of the square mention above is by the winding itself and 21.5 % 
of this same is taken up by air.  Since the density of copper is 8.94 
grams per cm^3 and that of air is negligible you can multiply the 
density of copper by how much per 1 mm copper there actually is in a 
winding... ergo 78.5 % of 8.94 densities.  This equals 7.02 grams per 
cm^3 and is quite acuratly the effective density of the copper wound 
around the string. 

 From this point you only need to know the ratio of copper winding to 
steel core to figure out the total effective density of the string per 
cm ^3. Of course a string is not a cube of some length... but then we 
have already figured out the air portion so this works just dandy.

Figuring the ratio of string to winding is easy.... Total diameter 
divided by core diameter.

When you have that figure you simply run an easy formula to find how 
many parts of winding there is for 1 part of wire core.  Say you have a 
1:4 ratio copper to steel.  This works out to 0.25:1.  Since the winding 
is both above and below the core it is taken twice... so you have 0.5 : 
1.  1.5 parts in all.  So you simply put the 1.5 in the following 
equation... 

(PI*(1.5/2)^2 - PI*(1/2)^2) / PI*(1/2)^2) to get the total parts of 
copper densities to core densities. 

You then add up the total densities of each and divide by the total 
resulting parts to get the average density per cm ^3 of the string as 
follows.

The above example works out to 1.25 : 1.  So

(1.25 * 7.02 + 1 * 7.85) / 2.25 = 7,38888~ = s (weight-density)

You then just plug that into the standard formula for Tension  T = 
(f*L*d)^2 * s * PI/g  where g = 981 cm per sec^2.... a familiar quantity 
me thinks.

For a 5:1 ratio string you would get
(PI*(5/2)^2 - PI*(1/2)^2) / PI*(1/2)^2) = 7.05 = s (weight-density)... 
and plug that into the same formula for T.

In other words... if you know the core diameter... and the total 
diameter... you can use the former 2 equations to find a very exacting 
value for the wound strings tension.

Cute eh... ?   Now for how to figure out this Stiffness / Inharmonicity 
bit McFerrins way....

Cheers
RicB