# Acoustics of metal alloys question

I design percussion instruments and I have been researching for some time now the acoustic properties of different common alloys such as phosphor bronze, brass, copper, nickel silver (german silver) and so on.... Specifically I want to test the amplitude and frequency modes of each. This would be in several different forms though; I need to test the alloys in a simple bar (8"-12") form, pipe/tube form (2" OD x 4' L), and most importantly with a drum cylinder with and without drum heads attached.

I am an audio engineer so the recording and testing is not a problem. I just need someone to point me in the right direction as far as how to test and measure. What would be awesome is if I could find a program that would accurately simulate the acoustic principals that way I could test anything in any configuration without trying to find all these metals myself.

Thanks! - (this is my first post)

Donnie

PS... if anyone has any audio measurements they would like tested let me know and I might be able to help you out.

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With accurate material properties you can come up with simple analytical models that can predict some of these for you. For simple beam testing it's pretty simple to treat a single degree of freedom beam as follows.

You could derive the equations yourself from the fourth order differential equation for bending, but a good engineering reference book (Roark) will give you expressions for natural frequencies for beams with varying end conditions to save you the trouble. For instance, the nth natural frequency of a beam is:

$$\omega_{n}=\alpha^{2}_{n}\sqrt{\frac{EI}{m L^{4} }}$$

where $$\alpha_{n}$$ is a constant depending on the clamping conditions and the modal number, I is the moment of inertia, m the mass, L the length and E the Young's Modulus of your material. Note you can break m down into $$\rho A$$, the density multiplied by the cross sectional area. There are entire data tables published elsewhere for the clamping constants for each modal number.

There is another expression (which will be given in the reference either as a combination of hyperbolic and trigonometric sines and cosines, or as a combination of powers of e) to give you the modal shapes. As for cylinders, that would require a little more thought but I'm willing to bet there are solutions out there for them. Bear in mind that both for real beams and cylinders there will be more than just the one degree of freedom, therefore other modes and frequencies will be present.

In terms of measuring excitation, you could mount accelerometers at various points on the body and then excite your system either through an instrumented impact hammer or a shaker. Either way you can test at various frequencies and measure the response at various points to show the mode shapes and amplitudes. It's often worth trying to predict the modal shapes for simple excitations before setting up sensors in order to ensure you don't mount a transducer at a nodal point.

Have a look at the Bruel and Kjaer website for some decent primers on vibration testing.

Hope this helps.

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Wow! Thank you so much for all the great information! This gives my mind a lot of numbers to crunch. Would MatLab, Mathematica, Comsol etc. be a reliable program for this kind of analysis?

Thanks again,

Donnie

PS... I actually have a pair of Bruel and Kjaer 4006's

You could do it Excel if you really wanted to! Matlab is perfect (Octave for open source if you're that way inclined), and I've carried out similar analyses in MathCAD, LabVIEW and even in C++.

This is actually a typical exercise (for the beams anyway) for mechanical engineering students in a stress analysis, vibrations or finite element analysis course. You can model the system in an FE package, and verify your answers approximately against reference analyses (such as Roark) or by actually deriving the solutions yourself.