Tuning forks, and symmetric structures

[GregLocock]

A few questions, some of which I know the answer to, others I don't.

1) Why does a tuning fork have two prongs rather than one (or I suppose three)?

2) Is the dominant resonant mode anti symmetric (cantilevers swaying in phase) or symmetric (prongs clapping)?

3) How does the tuning fork 'select' the mode from (2) as the dominant mode?

4) Does anyone have a classical tuning fork? If so could they post a good quality wav file of the complete excitation/decay cycle?

http://www.tms.org/pubs/journals/JOM/0511/Burleigh-0511.html

is somewhat relevant, I know. I don't like that wav file!

5) if we were to strike just one prong why does the other prong not behave as a harmonic absorber for the first prong (this is really a more general question about symmetrical modes)?










Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

Typically we hold a tuning fork loosely. Or if we sit it flat on a surface, it might be something like a simply supported boundary, but definitly not a clamped boundary.

Looking at the simpler case of holding it in our hands, it is a very loose support. It cannot provide a reaction force at the high natural frequency of the tuning force.

So the tuning fork cannot vibrate in the in-phase mode (unless perhaps you clamped the stem in a vise). It vibrates in the out-of-phase mode.


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[electricpete]

Another thing we note is that touching the vibrating blades quickly dampens the vibraiton. But touching the stem does not dampen the vibration. Once again it tells us the stem is not vibrating. It is just transmitting force between those two prongs which are vibrating out of phase.

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[electricpete]

And another obvious question: why do we build tuning forks based on the out-of-phase mode?

If we used an in-phase mode, we would of course have to support the tuning fork rigidly. That in itself is not much of a problem. But more improtantly, the resonant frequency would be very sensitive to the stiffness of the tuning fork support... not what you want for a fork that is supposed to create a repeatable frequency.

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[GregLocock]

Yes, I agree that in practice the mode is clapping, so the handle is nodal.

I'm not really sure I agree with "So the tuning fork cannot vibrate in the in-phase mode (unless perhaps you clamped the stem in a vise). It vibrates in the out-of-phase mode.
"

Consider a simple beam, freely suspended. I'm fairly certainly it does vibrate in the lowest frequency mode where the wavelength is twice the beam's length.



Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

The in-phase mode I was discussing was the one you could get if you clamped the stem in a vice. You can't get that modewithout clamping the stem.

You are right there certainly are other possible in-phase modes to consider - including the one you describe which is free/free condition of a beam which is not symmetric end to end. But that would not work when you put the end down on a table and prevent the end from moving. And it would also be stopped by holding the stem.

It brings to mind a question, though. Why is it that the tone gets louder when you put the stem on a table?

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[GregLocock]

Ah, that one is easy.

A typical tuning fork has a size of about 0.1 m, and a radiating area of say 0.002 m^2. The wavelength of 440 Hz is around 0.7 m, so an affective radiator at that wavelength will need to have an area of about 0.7^2/10 = 0.05 m^2. If the area of tha rdaiator is less than that too much of the air (basically) slips around from the positive pressure side to the negative pressure side rather than radiating outwards.

There is another property generally called the radiation efficiency (a silly word since it can be greater than 1). This is the efficiency with which the vibrating structure couples into the air, at a given frequency. It is basically the relationship between the wavelength of the bending waves in the structure, compared with the wavelength in air at the same frequency. When they match perfectly you get a 3 dB bump in the SPL produced. Steel and aluminium in typical automotive thicknesses have a maximum radiation efficiency around 1 kHz or so - that's why engines sound thrashy, and whines are so common, even though there is far more energy available at lower frequencies.





Cheers

Greg Locock

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[MikeHalloran]

The .wav file appears to demonstrate a beat among several competing modes... which results from hitting the fork to excite it.

Our high school physics teacher (brilliantly, imho) arranged to have our introduction to sound taught by the highest ranking music teacher. He demonstrated the discordant note produced by striking a tuning fork, and then demonstrated the pure tone produced by tapping it gently (almost rubbing it) against something softer, like a Pink Pearl.




Mike Halloran
Pembroke Pines, FL, USA

 

[electricpete]

Greg - It makes good sense as you say that a wooden table would be good radiator due to the large area and the "efficiency" of wood emitting sound (like violins and guitars). It seems a little bit at odds with the picture of the two prongs moving out of phase and the stem not moving... if stem didn't move at all, then it wouldn't excite the table. I guess it must be a smaller vibration of the stem (maybe axially?) that allows the vibration to couple to the wooden tabletop. (?)

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[electricpete]

Mike - I hadn't heard that. Was there an explanation? Striking it harder excites different mode shape? Is there some non-linear effect?

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[MikeHalloran]

Different mode shapes is how it was explained. Gentler excitation lets it assume its own dominant mode.



Mike Halloran
Pembroke Pines, FL, USA

 

[MikeyP]

Is this a homework question? - Just kidding.

Here's my theory:

Let's assume we are talking about "normal" operation of a tuning fork - Holding the stem about 2/3 of the way up and striking one of the tines on something.

Assume that the fingers have the potential to add damping but negligible stiffness.

The lowest mode (symmetric) has the tines bending to an extent and to compensate for the displacement of mass of the tines, the stem moves up and down. However, it moves up and down *as a rigid body* due to symmetry. Therefore there is no strain in it so our fingers cannot add damping.

With the "in phase" mode, as the two tines bend to the right then the stem must also bend to the right. This now means that the stem is no longer a rigid body component and it has bending strain, hence the fingers will tend to damp out this mode. Hence the symmetric mode dominates.

This agrees with my experience of using tuning forks (unfortunately I don't actually own one). When you first strike it on something hard, there is a short-lived "clang" of lots of high freq modes which is audible from some distance away. These modes decay very quickly leaving just the symmetric mode which can only be heard by holding the fork to the ear. This mode persists for a long time due to lack of damping.

If you want to use a tuning fork quietly (in a performance situation), then you pinch the tines towards each other and then release. Obviously the symmetric mode will dominate and you don't get the initial clang. Another technique is to strike it on the rubber heel of your shoe. Those of you who use hammer testing for modal analysis will know that this concentrates all the energy at the low frequencies.

The fact that the symmetric mode involves the stem going up and down also explains how it transfers energy to a table when touch the stem to it.

M

--
Dr Michael F Platten

 

[MikeyP]

BTW if anyone has the inclination to do an FE model, I would be interested to see at what frequency the "in phase" mode occurs I suspect that the second bending of the tines may actually be the 2nd freq - not the in phase mode.

M

--
Dr Michael F Platten

 

[GregLocock]

FEA on the way.

Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[GregLocock]

http://www.geocities.com/greglocock/gallery/gallery.htm

Top 3 links. Mesh quality is (a) disgusting (b) important (notice the change in frequencies from refining the mesh around the shouldeers) (c) as good as it is going to get. The model is free free so I've ignored the first 6 modes.

The single leg has a first mode at around 1560 Hz, the tuning fork proper has a first mode of interest at 498 Hz, symmetric, and an antisymmetric mode at 1621 Hz.

I'm a bit surprised by the (lack of) relationship between those numbers.

There's a detail you can't see from the screenshots - the first mode causes the handle to pump up and down axially, ideal for exciting a sounding board.



Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[MikeyP]

The links don't work for me, Greg.

How do I claim my prize? ;-)

M

--
Dr Michael F Platten

 

[electricpete]

The FEA link works for me. The handle moving axially as Greg and Mike said seems like the explanation for why it works with a sounding board.

I'm not sure what the one-prong simulation is supposed to show. I notice it still has 524 hz mode, so I assume there was some symmetry condition imposed to make it act like a 2-prong. But the figure doesn't show the symmetry I expect for a 2-prong out-of-phase.... symmetry is not there. Is this just a simulation of a tuning fork with one prong completely removed... and just a coincidence that the frequency comes out the same?

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[GregLocock]

For your (and my) further confusion I've added two more models, this time clamped at the base.

Yes, I just cut one of the prongs off. I don't really understand why you don't like that as an approach.

Incidentally when you read the frequencies it is the data in the lower left corner of each window that is definitive. I haven't updated the window titles.

Mikey, is it that you can't read .png files?



Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[MikeyP]

Ah, When I use Firefox it redirects to a geocities error page. Works on IE. Could be my security settings.

M

--
Dr Michael F Platten

 

[GregLocock]

I'll add a table of the results of primary interest, I am now reasonably confused.

So electricpete, what would you like to see as a comparison for the single prong?



Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.