Adding tension to a model of a Timoshenko beam?

Question

I am interested in modeling a Timoshenko beam.

However, I would also like to add tension to the beam so that it can vibrate like a string as well as a beam.

The Timoshenko beam is described at that link by two coupled equations, where subscripts are derivatives of that variable:

$ρA w_{tt} = κAG(w_{xx}- φ_x) + q\tag{1}$

$ρIφ_{tt} = EIφ_{xx} + κAG(w_x- φ)\tag{2}$

Where $ρ$ is density, $I$ is second moment of area, $E$ is Young's Modulus, $G$ is the shear modulus, $κ$ is the Timoshenko shear coefficient, $A$ is cross sectional area, $w$ is transverse displacement, $φ$ is rotation of a segment, and $q$ is external force/load.

To solve a unified equation of motion, they isolate $φ_x$ from equation (1), take the $x$ derivative of equation (2), and then substitute in various derivatives of the $φ_x$ from equation (1) into (2).

This also helps to eliminate $φ$ which would be hard or impossible to solve for in a finite difference approach.

This gives a final equation of motion of:

$EIw_{xxxx} + ρA w_{tt} - (ρI + \frac{ρAEI}{κAG})w_{xxtt} + \frac{ρAρI}{κAG}w_{tttt} = q + \frac{ρI}{κAG}q_{tt} - \frac{EI}{κAG}q_{xx}\tag{3}$

Now let's imagine a very simple model for a vibrating string:

$ρA w_{tt} = (T + σA)w_{xx}\tag{4}$

where $T$ is basic tension and $σ$ is the extra stress on a given segment from lengthening (strain).

How could I combine these two systems? My idea is to just add these terms into equation (1) so I'd have:

$ρA w_{tt} = κAG(w_{xx}- φ_x) + q + (T + σA)w_{xx}\tag{5}$

Then I could just combine (5) with (2) to create a new version of (3).

But I'm wondering if this makes sense. Wouldn't these tension terms also affect equation (2)? I.e. Wouldn't I have to add something to equation (2) also?

Answer 1

I guess that you are right. In the case of a stiff string, using Euler-Bernoulli assumptions (see the reference), you end up with the following:

$$ (T + \sigma A)\frac{\partial^2 w}{\partial x^2} - EI \frac{\partial^4 w}{\partial x^4} = \rho A \frac{\partial^2 w}{\partial t^2}\, .$$

You arrive to this equation after doing a balance of normal and shear forces, and moments, using the Euler-Bernoulli assumptions. I think that you could do a similar derivation replacing the hypotheses for the Timoshenko theory ones.

Reference

• Fletcher, N. H., & Rossing, T. D. (2012). The physics of musical instruments. Springer Science & Business Media.