Propagation velocity of transverse wave on a bar

[luckydog]

I need to know the propagation velocity of a transverse wave on a long thin bar or rod. In terms of material properties, such as E and density, and in terms of geometry such as I (2nd moment of area).

I'm a physics grad, so reasonably versed in such things. But can neither find nor derive an expression. So help appreciated.

Either an expression for propagation velocity, or some help in finding an expression for compliance per unit length, from which I can probably derive it.

Thx.

LD

 

[Studiot]

The elastic equation for a shear (transverse) wave in a solid medium is

$$\frac{1}{{{\beta ^2}}}\frac{{{\partial ^2}{\theta _x}}}{{\partial {t^2}}} = {\nabla ^2}{\theta _x}$$

where

$$\beta = \frac{\mu }{\rho }$$

and

$${\theta _x}$$ is the one dimensional displacement

$$\mu$$ is the shear modulus = modulus of rigidity

$$\rho$$ is the density

t is, of course, time

This is independent of the shape of the object.

Hope this helps

 

[luckydog]

Thanks studiot. Appreciated.

However, the application of that isotropic equation doesn't seem to fit well. In the case of a long thin rod, from published solutions for beam modal self resonance, transverse wave velocity appears to be dependent upon profile shape, ie second moment of area I. For example, two beams of equivalent mass/length but differing in I (say one is hollow but with larger radius), have different transverse modal self resonant fs for the same length. Therefore differ in propagation velocity. Presumably the compliance/length differs with I, and therefore profile ?

I was hoping for a solution that readily embraces differing area profiles. Presumably the isotropic equation can be adapted or applied, but I can't see how.

 

[Studiot]

I'm not sure what you are looking for.

The velocity I offered is a material property. It is for the propagation of waves through the medium.

There are oscillatory modes available to rods, beams and other structural elements, by virtue of their shape, but this is a different thing. This is about vibration of the element as a whole.

The equation I posted is also a simple approximation. Depending upon your application there are effects noted which depend upon the application. For example concrete piles are long thin rods and commonly tested by shear wave pulses. There is some reference to this NDT method on the web.

 

[luckydog]

Hi Studiot. Yes, thanks for posting the equation, it's correct and I understand it for isotropic solids. But the application I have requires to know the transverse pulse response propagation time in a long thin aluminium rod, with a circular cross section that could be hollow with a relatively thin wall. I wish to control the propagation time for a fixed length, by selecting material and radial geometry such as outer and inner radius.

The reason I mention vibration modal frequencies of beams is that i believe it illustrates how propagation velocity of a rod is also a function of radial profile (for a constant mass/length) as well as a function of material properties. Presumably, stiffness/length varies with I...

However the isotropic solid propagation equation seems not to readily accommodate variation due to various I values associated with different radial profiles. Or I can't see how to apply it so that it does.