How long would a tuning fork vibrate in vacuum?

[I do not know if this is the right subforum]

The answer to the question to the title is: for very long time. However the tuning fork clearly has to stop at some point because some of the energy will turn into heat. However I want to quantify for how long. More specifically I am interested on finding a quantity that determines the dissipation of mechanical vibrations on a bulk piece of metal (in vacuum).

I have found that the best way to simulate this is using viscoelastic equations, which are basically the equations of elasticity but allowing for complex coefficients. The (complex) dynamical modulus has an (imaginary-part) loss modulus that can do the trick but it is getting very hard to find a number for the loss modulus for a given metal.

As it depends on frequency, it would even be better to have a dependence for a large range of frequencies but after looking at a lot of viscoelastic papers I cannot find anything concrete. Does anybody know a good elastic coefficient that can be used to answer to this question in a material specific way?

A tuning fork needs a high Q.
That will be highest for isotropic materials, free of dislocations and free of crystals in a matrix that have mismatched acoustic impedances.

The elastic parameter that you seek, will not be a simple constant for the material, but will also be a function of the production process, heat treatment and time. That is why such a parameter will be difficult or impossible to identify.

Find a metal or alloy that remains temperature stable, without later precipitation changes in crystal structure, or work hardening due to vibration. That immediately rejects alloys of Cu, Al, and probably all the steels, carbon and stainless included. Alloys normally used to attenuate machinery noises, that sound dull when you tap them with a hammer, will make the worst tuning forks.

Look at the phase diagram for possible alloys, then consider drawing impurities from the melt, before refining the pure eutectic alloy.

I would consider high melting point materials, that can be rapidly cooled to a stable glass, but then I would worry that they might creep or crystallise later. Maybe the opposite is safer, single-crystal silicon, grown for the semiconductor industry.

Baluncore said:

A tuning fork needs a high Q.
That will be highest for isotropic materials, free of dislocations and free of crystals in a matrix that have mismatched acoustic impedances.

The elastic parameter that you seek, will not be a simple constant for the material, but will also be a function of the production process, heat treatment and time. That is why such a parameter will be difficult or impossible to identify.

Find a metal or alloy that remains temperature stable, without later precipitation changes in crystal structure, or work hardening due to vibration. That immediately rejects alloys of Cu, Al, and probably all the steels, carbon and stainless included. Alloys normally used to attenuate machinery noises, that sound dull when you tap them with a hammer, will make the worst tuning forks.

Look at the phase diagram for possible alloys, then consider drawing impurities from the melt, before refining the pure eutectic alloy.

I would consider high melting point materials, that can be rapidly cooled to a stable glass, but then I would worry that they might creep or crystallise later. Maybe the opposite is safer, single-crystal silicon, grown for the semiconductor industry.

Aside for the requirements to actually make a good tuning fork, I would like to know at what kind of measurements or coefficients I should be looking for when I have one . I would like to find some reference does not matter the process, of a pure metal. Given the most realistic but ideal material, will it ring forever? And if not for how long?

pines-demon said:

I would like to find some reference does not matter the process, of a pure metal.

Since the crystal structure is critical, that is impossible.

pines-demon said:

Given the most realistic but ideal material, will it ring forever?

No. You will need to sustain the oscillation with some form of amplitude-regulating amplifier, one that does not introduce a phase error. That will apply the energy as a momentary impulse, applied symmetrically, only as the fork crosses zero deflection.

pines-demon said:

And if not for how long?

For a given Q, the oscillation amplitude will attenuate to 1/e = 36.8%, after Q cycles of oscillation.
For a 50 Hz fork, with a Q of 10 million, that will be many multiples of;
1e7 / 50 = 200,000 seconds = 55.6 hours.

After 111.2 hours the amplitude will be 0.368^2 = 0.00135 = 13.5%
After 556 hours the amplitude will be 0.368^10 = 0.000045 = 0.0045%
After n * 55.6 hours the amplitude will be 0.368 ⁿ

There appears to be a limit to the maximum Q of a mechanical resonator.

Reference;
Proceedings of the Eurosensors XXIII conference.
Experimental evidence of thermoelastic damping in silicon tuning fork.
C. Muller*, J. Baborowski, A. Pezous, M.-A. Dubois
RF and Piezo Components, CSEM, 2002 Neuchâtel, Switzerland

Short extract;
"
2. Thermoelastic damping theory.
In the 1930‟s Zener predicted that thermoelastic losses may be a limitation to the maximum Q-factor of resonators and developed a model to quantify the phenomenon [7-9]. Basically, the principle of thermoelastic damping is the following: When a mechanical structure vibrates, there are regions where compressive stress occurs and others where tensile stress occurs, in a cyclic way given by the vibration frequency. Accordingly, compressed regions heat up and stretched regions cool down. Hence a temperature gradient is established between different regions of the system. However, to set the mechanical system in vibration, energy must be provided, leading to a non-equilibrium state having an excess of energy. Disregarding thermoelastic damping, the vibration could persist indefinitely in an elastic body that is perfectly isolated from its environment. However, local temperature gradients lead to irreversible flows of heat, which are a dissipation mechanism that attenuates the vibration until complete rest is achieved.
"

pines-demon said:

Given the most realistic but ideal material, will it ring forever? And if not for how long?

My intuition tells me that even an ideal crystal without any impurities and dislocations cannot ring forever, but the timescales are hard to quantify. Assuming the phonons are initially all in one mode, this situation cannot persist because of nonlinearities. The modes are not perfect harmonic oscillators. (It's the nonlinearity of the intermolecular potential for example that causes crystals to expand with temperature.) Even if there isn't scattering by dislocations, there are interactions between the phonons that lead to a spread of the initial energy into many other modes, and the number of modes for a macrocopic object is of course huge. I admit that I'm at a loss at quantifying this, but the timescale will surely depend also on the size of the object. (Assuming it floats in free space.)

WernerQH said:

My intuition tells me that even an ideal crystal without any impurities and dislocations cannot ring forever, but the timescales are hard to quantify. Assuming the phonons are initially all in one mode, this situation cannot persist because of nonlinearities. The modes are not perfect harmonic oscillators. (It's the nonlinearity of the intermolecular potential for example that causes crystals to expand with temperature.) Even if there isn't scattering by dislocations, there are interactions between the phonons that lead to a spread of the initial energy into many other modes, and the number of modes for a macrocopic object is of course huge. I admit that I'm at a loss at quantifying this, but the timescale will surely depend also on the size of the object. (Assuming it floats in free space.)

What about Umklapp processes?

pines-demon said:

What about Umklapp processes?

No, I was thinking of a phonon decaying into two phonons of lower frequency. Nonlinearities can produce phonons with the sum or difference frequencies. Eventually there would arise a thermal distribution.

Quartz crystals make good, but not perfect clocks.

The physically largest tuning fork, made from the heaviest material, with the thinnest springy restoring force arms, will have the lowest frequency. So for the same Q, the vibration amplitude will decay at the lowest rate per time, so it will keep vibrating for the longest time, but not forever.

Baluncore said:

The physically largest tuning fork, made from the heaviest material, with the thinnest springy restoring force arms, will have the lowest frequency. So for the same Q, the vibration amplitude will decay at the lowest rate per time, so it will keep vibrating for the longest time, but not forever.

The whole question is what is that "not forever". A week? a human lifetime? the age of Earth? of the Universe?

pines-demon said:

The whole question is what is that "not forever"

If the tuning fork was the size of a galaxy, it would still be vibrating in a billion years.

Quartz tuning forks (of the type that are used in watches) are frequently used as sensors in vacuum in a range of applications; for example in AFM. The "output signal" from these sensors is either a change in frequency of a change of Q (they are often operated in a phase locked loop)
Their Q does go up in vacuum; but only maybe a factor of 10 or so.

Just google "tuning fork sensor" and you will get a range of hits; including many that describes operation in vacuum

So the general answer to the question is "longer than in air", but not by maybe more than 1-2 orders of magnitude before some other effect source of loss starts to dominate.

Baluncore said:

If the tuning fork was the size of a galaxy, it would still be vibrating in a billion years.

But when will it stop?

pines-demon said:

But when will it stop?

The amplitude of oscillation will fall towards zero, but will never get there.
At some point, thermal noise will be greater than the amplitude of oscillation.

pines-demon said:

But when will it stop?

How long is a piece of string?

Your question is essentially "How long will it take for an oscillator of unspecified Q to reach an unspecified amplitude so small I will call it zero?"

Vanadium 50 said:

Your question is essentially "How long will it take for an oscillator of unspecified Q to reach an unspecified amplitude so small I will call it zero?"

More like specifying what elastic parameter we should look at to characterize it, providing a realistic value for it. If specifying a Q helps, be free to detail an example.

pines-demon said:

More like specifying what elastic parameter we should look at to characterize it, providing a realistic value for it. If specifying a Q helps, be free to detail an example.

“The only interesting answers are those which destroy the question”. —Susan Sontag

It would help us find the right answer, if you could ask the right question.
Why do you need to know ?
What are you trying to achieve ?

pines-demon said:

But when will it stop?

The decay of the oscillation will (ideally) be exponential so the simple answer is Never.

But, in practical terms, we can compare the accuracy of a watch based on a metal tuning fork (Bullova Accutron: 1 minute per month) and a quartz watch (Tag Heuer: 1 second per month). The accuracy relates to the Q factor of the oscillator (yes- there will be other details) so perhaps we could expect a factor of 1:60 for quartz and steel.
A high quality mechanical (tic-tok) chronometer would have a drift of several minutes per month (very much dependent on how it's used). But the escape wheel is rotating in air so that could account for some of the drift. This was tried in the Vacuum Chronometer. As its name suggests, it achieved the chronometer standard but not significantly better so maybe the other losses dominate in a tic-tok watch.

Baluncore said:

“The only interesting answers are those which destroy the question”. —Susan Sontag

It would help us find the right answer, if you could ask the right question.
Why do you need to know ?
What are you trying to achieve ?

Characterize the dissipation of mechanical waves between metals, with numbers.

Numbers? 6! 11! 19! Are we done?

If not, what is missing? It's been pointed out multiple times that the decay falls toward zero but will never reach it. You can say that answer isn't good enough, but then you need to tell us what good enough is.

@pines-demon
What are we supporting here? What is the application? Why do you need to "Characterize the dissipation of mechanical waves between metals, with numbers"?

The application is still so poorly defined, that no one answer is worth the investment.

Baluncore said:

@pines-demon
What are we supporting here? What is the application? Why do you need to "Characterize the dissipation of mechanical waves between metals, with numbers"?

The application is still so poorly defined, that no one answer is worth the investment.

I did a very general post and have had some interesting answer so far. I am still exploring what would be needed for such characterization.

Going back to the top of the post I suggested to find some kind of loss modulus for metals. This loss modulus provides the decay of mechanical waves in viscoelastic materials. The problem is that this is not usually tabulated for metals, so I am trying to find an alternative material-dependent parameter that has well tabulated values. It could be some relaxation time that provides some insight on the dissipation of heat of mechanical waves in a metal. Or some complex speed of sound.

Vanadium 50 said:

Numbers? 6! 11! 19! Are we done?

If not, what is missing? It's been pointed out multiple times that the decay falls toward zero but will never reach it. You can say that answer isn't good enough, but then you need to tell us what good enough is.

I agree it never attenuates completely. But suppose that you have three tuning forks made from different metals, with the same purity, shape, mass etc. what kind of material dependent parameter do you need to characterize the attenuation? If you need a number, let's say to 90% (or ##1/e## if you prefer simple exponentials).

pines-demon said:

I agree it never attenuates completely. But suppose that you have three tuning forks made from different metals, with the same purity, shape, mass etc. what kind of material dependent parameter do you need to characterize the attenuation? If you need a number, let's say to 90% (or ##1/e## if you prefer simple exponentials).

Why not look at some actual results from the literature?
For example, consider the 2020 article A 5.86 Million Quality Factor Cylindrical Resonator with Improved Structural Design Based on Thermoelastic Dissipation Analysis (https://www.mdpi.com/1424-8220/20/21/6003). The authors model and fabricate cylindrical resonators of fused silica, operate them in a vacuum, and investigate the effects of annealing and chemical etching on their performance. They calculate resonance quality from ##Q =\pi f\tau## by measuring the frequency ##f## and the ring-down time ##\tau## (i.e., where the initial amplitude falls to ##1/e##) and find ##e##-folding times in the hundreds of seconds. Quoting from the paper:
"The results of the chemically etched resonator at ##0.01\text{ Pa}## are depicted in Figure 11. The resonant frequency of the low-frequency axis decreases to ##5939.7\text{ Hz}## while the decay time increases to ##314.9\text{ s}##, which gives a ##Q ##factor of ##5.86×10^6##. The resonant frequency of the high-frequency axis is ##5941.0\text{ Hz}##, and the decay time is ##293.6\text{ s}##, which gives a ##Q## factor of ##5.48×10^6##. To our knowledge, this is the highest ##Q## factor reported in cylindrical resonators to date."

pines-demon said:

I did a very general post and have had some interesting answer so far. I am still exploring what would be needed for such characterization.

There are too many confounding parameters. Without a scale and an application, this discussion is pointless.

Baluncore said:

his discussion is pointless.

I think the point was "Hay, let's stump PF!"

Vanadium 50 said:

I think the point was "Hay, let's stump PF!"

So far PF seems to be less into discussion and more into sarcastic comebacks.

pines-demon said:

So far PF seems to be less into discussion and more into sarcastic comebacks.

Very few newbies here at PF get this feeling. There's a reason for that.

This thread is done. Please try to do better in the future here. Thank you.