# String Tension w/ Periodicity + Poisson's ratio =

1. Mar 20, 2015

### Chrono G. Xay

I've been trying to write an equation for the tension of a string where a segment of the string is suspended between two points a set distance apart (the string extends beyond both points in both its initial and final conditions), and involves not only the periodicity of the string's movement when set in motion, but also takes into account the Poisson's ratio of the material.

My understanding is that while it is primarily the change in tension that is responsible for the change in frequency of the string between the two rigid points but also the very slight decrease in mass as more tension is applied and in response the cross-sectional area diminishes.

If I'm understanding correctly this would also involve the equation for Young's Modulus. However, I keep finding going in circles.

Here's what I have:

T = ( m * ( 2 * L * f )^2 ) / l

T - Tension
m - mass
L - total length of string
f - frequency of periodicity
l - 'speaking' length of string (between the two fixed points)

Assuming the speaking length IS the total length, thanks to significant friction, we could ten say that L = l.
T then equals...

T = m * L ( 2 * f )^2
m = V * ρ
V - Volume of string material
ρ - Density of string material

V = A * L
A - Area of string cross-section

T = A * ρ * ( 2 * L * f )^2
A = π * R^2

T = π * ρ * ( 2 * L * r * f )^2
r = D / 2
D - Diameter of string

T = π * ρ * ( L * D * f )^2

D = d0 + Δd
d0 - initial diamter of string
Δd - change in string diamter with tension

Δd = d0 * ν * ( ΔL / L0 )
ν - Poisson's ratio of string material

E = σ / ε = ( F / A0 ) / ( ΔL / L0 )
E - Elastic modulus of string material
σ - Stress on string
ε - Strain of string
F - Force exerted on string
A0 - Initial Area of string cross-section

ΔL / L0 = F / ( E * A0 )
where F = T...

^^^ This is where I get stuck...

Before this I had built up an equation for predicting the needed initial length of string given how long the string needs to be when taught in order to achieve a desired tension and periodicity. [below]

E = ( T / A0 ) / ( ΔL / L0 )
= ( T * L0 ) / ( ΔL * L0 )
ΔL = Lf - L0
Lf = final Length of string (when taught)

E = ( T * L0 ) / ( ( Lf - L0 ) * A0 )

E ( Lf - L0 ) = ( T * L0 ) / A0

Lf - L0 = ( T * L0 ) / ( E * A0 )

Lf = ( ( T * L0 ) / ( E * A0 ) ) + L0
Lf = L0 ( T / ( E * A0 ) + 1 )

L0 = Lf / ( T / ( E * A0 ) + 1 )

It's at this point that I turned to T = ...
...and then got caught in a loop.

Last edited: Mar 20, 2015
2. Mar 25, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Mar 25, 2015

### Chrono G. Xay

That's okay. This equation was one of several projects I'd been working on, and I while this one I hadn't looked at in a while, I knew where I was struggling. The other night I had an epiphany where I concluded that unless there's already tension on the string and you're wanting to do something which would instigate MORE tension you don't need to involve Poisson's ratio.

All the string tension equation wants is the INITIAL--that is: pre-tensioned--dimensions of the string, followed by the length of vibrating string, and the frequency of its motion.

If after the string were tensioned it experienced an excursion, THEN we'd need Poisson's ratio to compute the string's pre-excursion diameter (it's situationally initial dimensions) for the purpose of predicting the tension during excursion.

(If this isn't correct, someone please set me straight.)