Natural frequencies in solids.

Factual Questions
1

Well, maybe metals rather than the generic “solids”.

Watching Mythbusters right now, they’re working on a theory of Tesla’s. It involves the “natural frequency” of metal in this case. My two nephews are watching it with me and are trying to understand what they mean by “natural frequency”.

I understand it in an abstract sense. I know what it means and how it’s supposed to work, but I’m at a complete loss to describe/explain it.

Can anyone break this down and explain in English what it is and how it’s supposed to work?

Thanks.

2

Say you’ve got a flexible rod, and you hold it in the middle. If you try and move the middle up and down the ends are doing to flex. If you move it up and down too fast you’ll move the middle back down before the ends have finished moving all the way up. If you move it too slow then the ends will go all the way up and will start bouncing back down before you start moving down. But, if you move the middle up and down at exactly the right frequency, you can time it so that you pull down at exactly the same time as the ends want to go down on their own. This will make the ends come down with much more force than they would have if you had been moving too fast or too slow. The exact frequency where this happens is the natural frequency of the rod.

It’s fairly easy to determine the natural frequency of most things. Just whack it. When you whack soemething, you hit it with an impulse which has a whole bunch of frequencies all in it at once. Anything that’s not a frequency that the material naturally wants to vibrate at will dampen out pretty quick, so any frequency that lasts longer will be the natural frequency of the object. A wine glass is a good example. If you hit a wine glass with a spoon, it tends to “ding” at the same frequency every time. That’s the natural frequency of the wine glass. Just like the ends of the flexible rod will move down with more force (and possibly break the rod) if you hit the rod’s natural frequency, wine glasses can shatter if you hit their natural frequency as well, which was something else that was demonstrated on another episode of mythbusters.

Object made of a single material, like a flexible rod, wine glass, or the mythbusters bridge (made of steel) will often resonate very dramatically if you vibrate them at or very close to their natural frequency. Objects made out of many different materials like a brick wall tend not to resonate so much.

3

And just in case there was any confusion, the natural frequencies are properties of an object, not just a material. The shape and especially size of the object are important as well. For instance, a six-foot steel rod will have a natural frequency twice as high as a twelve-foot steel rod.

4

You might start out with a flexible metal rod and point out that metal flexes. Then go on to say that even as a solid block, metal still flexes. Then show them how Jello shakes and say that metal shakes like that only not so much. The frequency with which is shakes depends upon the shape and the material the block is made of.

5

Every physical thing has an infinite number of natural frequencies. As others have said, this depends not only on the material (specifically, the mass and the stiffness of the material) but also the physical shape of the object.

You can show the effect of shape by holding a stiff metal ruler so it overhangs the edge of a desk. Twang it, and it will make a noise. Now move it so it doesn’t overhang so much (the “shape” changed–it’s shorter) and twang it again. Higher pitch, higher frequency.

But why do they have natural frequencies? Imagine a brick bouncing up and down on a spring (Java applet simulation). It takes time for the spring to stretch and slow down the brick–time for the energy transfer, in other words. The brick is mass, the spring is stiffness, and time is frequency. Change the mass or stiffness, and the frequency (time) changes.

Fairly simple objects–a string, a rod, a bowl-- have relatively simple natural freqencies that can be calculated. These frequencies are related to each other, and get progressively higher. Here’s a graphical picture of the first few frequencies of a vibrating string. When someone talks about “the” natural frequency, they’re typically talking about the first, or fundamental natural frequency (remember the ruler over the edge of a desk?). This is physically menaingful because the fundamental frequency is usually the “strongest”–exhibiting the most energy and highest amplitudes (cool video [scroll down] of the fundamental frequency of a wine glass).

If you want to demonstrate fundamental frequencies and overtones to your nephews, head to the back yard and grab a garden hose. With one person on each end, swing it back-and-forth to make a single loop. That’s the fundamental. Swing it faster, and you can make two loops, with a stationary point in the center. Faster yet, three loops. Or, if you’ve got a Slinky, stretch that out on the floor between two people and shuck it back-and-forth.

More complicated, three-dimensional, objects will have mode shapes that are very different-- swinging back-and-forth in all directions. Here’s a set of animations of a manifold system. In these complicated systems, saying “the” natural frequency is less meaningful, because there may be little relationship between frequencies with different mode shapes. Very large, complex structures will have multiple natural frequncies, and it’s hard to point to which one might be the “worst”–the infamous Tacoma Narrows Bridge being an example (download a a movie thereof here).

And one last thing–materials also have some internal damping that tends to reduce the amplitudes of resonance. Materials like glass and steel have low internal damping, so they will “ring,” or continue to oscillate well after you tump them. Other materials will have a higher internal damping, and will tend to “thud” when you thump them. Two items made out of materials with the same mass and stiffness but different damping might have very similar natural frequencies (not exactly the same, but close enough), but you would perceive them as different because one would ring longer than the other.

-zut, whose PhD thesis included a fair amount of calculation of natural frequencies.

6

Also how the thing is mounted in whatever it is mounted in.

Piezo electric crystals for electrical oscillators have several different natural frequencies depending on the mode of oscillation. Sometimes these various frequencies are mechanicall coupled together and can cause a circuit to have two resonant frequencies, jumping from one to the other at random. Supression of unwanted modes can be a problem in some crystal configurations.

7

Atoms and groups of atoms that are bound to one another have natural frequencies representing the springiness of the bond and the mass of the atoms or groups. This is the basis of infrared spectroscopy, and it IS a property of the material rather than the object. More specifically, there are several frequencies associated with each atom bond combination, because the bond can flex like a lever, or extend and retract, or twist - generally the extension-retraction is the highest frequency and the twist is the lowest (IIRC).

8

Here’s a nice google video demonstrating the various resonance frequencies of a metal sheet.

9

Just to make something clear (which I think that chronos’ contribution might, most uncharacteristically, caused confusion with), the natural frequencies of the sort being discussed in this threadare properties only of the gross structure of the object, and depend upon the material only to the extent that the density and elastic modulus are given by the peoperties of the material.

There are, however, frequencies of vibration that are properties only of the material and its phase, and don’t depend on its structure. These are the phonon frequencies of the material, and they vary from material to material. But they won’t show up in experimentrs of the sort the mythbusters were doing, and for bulk ophysical situations, can be ignored.

10

Hi guys,
I learnt from you that the natural frequency depends on the material, its shape and its condition of mounting the object. Now a question has come to my mind. Assume we have two rods with the same shape and different materials. We fix one of the ends of these rods and experimentally find the natural frequency of the rods. Assume we find that the natural or let’s say the fundamental frequency of the rod 1 is higher than the rod 2. Now the only thing we change is the condition of mounting, let’s say the boundary conditions. For example now we fix both ends of the rods. If we can find the fundamental frequency under this condition, can we be sure that natural frequency of rod 1 will be again higher than rod 2?
By this question, I’d like to have sense about the effect of boundary conditions on natural frequency.
Thank you so much

11

Barring extraordinary circumstances, yes.

12

Two other terms would be better … Harmonic frequency or resonant frequency .
The cause is the internal reflection of the wave… it reflects around, echos… resonates.
This is often due to the distance of travel being a half wave length.

There may be more of them frequencies … so the terms help make it clear … the lowest frequency shows the effect greater, for various reasons, so the lowest frequency which causes resonance is then the resonant frequency and then the other higher frequencies which are due to mathematical progress of the frequency, as in they are frequencies where a wavelength is added ( Wavelength = speed of wave/frequency), these frequencies are the harmonics.