Resonant frequency calculations for 'solid object' systems

[HalcyonicBlues]

Hi, thanks for stopping by :) This is for a big assignment (Yr 12). I asked my physics teacher this, but she couldn't give me a definite answer, and neither could my music teacher who previously studied physics, and then he messaged an engineer he knew, who hasn't replied yet...

~

Given that the equation for estimating the fundamental frequency of a string (like for a guitar or piano) is along the lines of:

f= [√(T/[m/L])]/2L

So it involves tension, mass and length.

But how about the resonant frequency calculation for a solid that isn't stretched over anything and so...has 'no' tension? (In this instance, an object like a ruler 'twanged' on the edge of a desk).

The Attempt at a Solution

I thought of comparing it to a spring, because although the wave motion is often demonstrated as longitudinal and not transverse, a spring doesn't necessarily have any initial tension applied. Then that would involve spring constants and angular frequency... Is such a comparison appropriate at all?

Some other places I read suggested using stiffness (ie. Young's modulus):

f = √([stiffness/m]/2∏)

The problem I have with this is when I have other equations I want to use (to calculate inharmonicty - but that's a different story) that include both stiffness and tension as variables. And I don't think that just sticking '0' into the tension field would really work...?


Hannah x

 

[Simon Bridge]

It is no accident that musical instruments tend to be long and narrow - this makes for a single dominant resonance so you can get a note. However, the other dimensions contribute, and so does the material, so different instruments sound different even when playing the same note.

You probably want to look at a rectangular plate, like a Xylophone key.
It will have vibration modes, usually different, along each axis.
It can also vibrate logitudinally (strike on the end instead of a face).

It can get quite complicated:
http://en.wikipedia.org/wiki/Vibration_of_plates

 

[AlephZero]

Some other places I read suggested using stiffness (ie. Young's modulus):

f = √([stiffness/m]/2∏)

That is the right idea, but it leaves the qiestion of what exactly you mean by "stiffness" and "mass" in the formula.

There is nothing really special about the vibration of "solid musical instruments", compared with solid objects in general.

You didn't say what level of knowledge you have. If you haven't done calculus, you won't be able to understand how the formulas are derived. In any case, there are only "formulas" for objects with simple shapes. Try a google search for vibrations of a cantilever beam, for example.

For objects with more complcated shapes, you would need to do computer simulation to find the frequencies.

To calculate inharmonicity of a string, you can use the idea of the f = √([stiffness/m]/2∏) formula. You can think of the tension in the string as generating a "stiffness" when the string moves laterally. You can feel the force created by that "stiffness" as you displace the string sideways when you pluck it. In mechanical analysis this effect is called the "load stiffness" or "stress stiffness" because it is proportional to the forces or stresses acting on the material, not in the properties of the material itself. You can add together the load stiffness and the "elastic stiffness" created by Young's modulus for the material.