kitarist -> String Tension formulae (RE: Solera Flamenca Strings) (Feb. 3 2021 21:03:13)
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ORIGINAL: kitarist quote:
the tension is a theoretical number based on a calculation of variables. [..] The formula itself is an accurate calculation of the tension (as long as we use the proper stretched-and-stabilized-string diameter): if we somehow measure that real tension directly with some contraption, it would be the same as what we calculated with the formula. [..] I'll post about the formula in a day or two; I want to first figure out how the different formulations floating around are related to one another so I can explain it clearly. Sorry about the delay. (In the equations below, the first number is just an equation number and not part of the formula; also, please ignore the over-expaining in places as I wrote this up first to reply in another forum). Density is mass per unit volume (so 'density' is typically short for 'volume density'; in physics usually denoted by Greek letter ρ (rho)). UW in D'Addario's formula, what they call 'Unit Weight', is a mass per unit length, i.e. linear density. In physics it would be typically denoted by the Greek letter μ (mu). The two are related - and in the case of a string with diameter d, the relationship looks like this: (1) μ = (π/4) * (d^2) * ρ i.e. just considering the string as a cylinder with cross-sectional area A = (π/4) * (d^2). It just so happens that the squared velocity v^2 of a wave on a string of diameter d, density ρ, and thus linear density μ = (π/4) * (d^2) * ρ, with the string under tension T, is (2) v^2 = T/μ On the other hand, any wave velocity is related to its frequency f and wave period λ (Greek lambda) by v = f * λ. For a wave on a guitar string the wave period is twice the guitar scale length L, so (3) v = 2 * f * L From eqs. (1), (2) and (3) it follows that the tension T can be expressed as: (4) T = 4 * μ * (f^2) * (L^2) , and also as: (5) T = π * ρ * (d^2) * (f^2) * (L^2) The tension formulae (4) and (5) would give you tension force in Newtons, i.e. [ kg*m/(s^2) ] (kilograms*meters/seconds_squared), and expect to be fed L and d in meters, and densities in kg/(m^3) for ρ or kg/m for μ. That's it. From here on all the apparent differences in formulae seen floating on the internets are due to using different units for tension, length (affects L and d) and densities (people use different units for convenience or out of previous practices), and to absorbing the π or 4 into the resultant combined unit-conversion factor. Case A: Mimmo's (Aquilla) equation for string tension gives tension T in kg force instead of Newtons; expects diameter to be entered in mm (not meters), and expects density to be entered in grams/(cm^3), not kg/(m^3). We will use the tension equation in its form (5) as the starting point. Accounting for these changes in units and absorbing the π means that the right-hand side in eq. (5) (without showing π) has to be divided by (1000/π) * 9.80665 = 3121.6 approximately, where 9.80665 is an average Earth acceleration due to gravity. Thus equation (5) becomes: (5.1) T = ρ * (d^2) * (f^2) * (L^2) / 3121.6 which is essentially Mimmo's equation except for rounding the unit-conversion factor to the nearest integer. Case B: In contrast, D'Addario's equation gives you tension T in lbf (pounds force) and expects linear density entered in lb/in and scale length in inches. We will use the tension equation in its form (4) as the starting point. This little table shows you why the change in tension units from N to lbf means dividing eq. (4) by approx. 4.45 and the change in length units from meters to inches means dividing (4) by approx. 39.37; the two combined mean dividing eq. (4) by 386.1: Thus equation (4) becomes: (4.1) T = 4 * μ * (f^2) * (L^2) / 386.1 which is essentially D'Addario's equation except for a small correction of the actual factor - it is 386.1; not 386.4.
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