When a tuning fork is struck, and held against a tabletop, the peak frequency of the emitted sound doubles — a mysterious behavior that has left many human being baffled. In this blog post, we define the tuning fork an enig using simulation and administer some funny facts about tuning forks along the way.

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### Explaining the Tuning Fork Mystery

In a recent video on YouTube indigenous standupmaths, scientific research enthusiasts Matt Parker and Hugh hunt discuss and also demonstrate the “mystery” the a tuning fork. As soon as you to win a tuning fork and hold it versus a tabletop, it appears to dual in frequency. Together it transforms out, the explanation behind this mystery can be boiled under to nonlinear hard mechanics.

### How does Sound with Our Ears?

When you host a vibrating tuning fork in her hand, the bending motion of the prongs to adjust the air around them in motion. The push waves in the air propagate as sound. You have the right to hear it, however it is not a really efficient switch of the mechanical vibration into acoustic pressure.

When you host the stem the the tuning fork come a table, an axial motion in the stem connects come the tabletop. The movement is much smaller than the transverse activity of the prongs, yet it has the potential to set the huge flat tabletop in motion — a surface that is a far better emitter the sound 보다 the slim prongs the a tuning fork. The tabletop surface ar will act together a large loudspeaker diaphragm.

Our tuning fork.

To investigate this amazing behavior, we produced a solid mechanics computational model of a tuning fork. The design is based on a tuning fork that one of my partner keeps in she handbag. The ton of the device is a reference A4 (440 Hz), the material is stainless steel, and also the full length is around 12 cm.

First, let’s have a look at the displacement together the tuning fork is vibrating in its an initial eigenmode:

The mode shape because that the fundamental frequency the the tuning fork.

If we examine the displacements in detail, it turns out that also though the all at once motion that the prongs is in the transverse direction (the x direction in the picture), there are also some tiny vertical contents (in the z direction), consist of of two parts:

The bending that the prongs is accompanied through an up-down movement that varies linearly over the prong cross sectionThe stem has an basically rigid axial motion, which is vital for maintaining the facility of mass in a solved position, as forced by Newton’s 2nd law

The displacements are displayed in the figures below. The setting is normalized so the the maximum total displacement is 1. The optimal axial displacement is 0.03 and also the displacement in the stem is 0.01.

Total displacement vectors in the very first eigenmode.

Axial displacements only. Keep in mind that the scale differ in between figures. The center of gravity is indicated by the blue sphere.

Now, let’s rotate to the sound emission. By including a boundary aspect representation of the acoustic ar to the model, the sound pressure level in the neighboring air can be computed. The amplitude that the vibration in ~ the prong advice is set to 1 mm. This is approximately the best feasible value if the tuning fork is not to it is in overloaded native a stress allude of view.

As deserve to be seen in the number below, the strongness of the sound decreases rather quick with the street from the tuning fork, and additionally has a big degree of directionality. Actually, if you turn a tuning fork roughly its axis next to your ear, the near-silence in the 45-degree directions is striking.

Sound push level (dB) and radiation pattern (inset) roughly the tuning fork.

We now add a 2-cm-thick wood table surface to the model. It procedures 1 by 1 m and also is sustained at the corners. The stem that the tuning fork is in call with a allude at the center of the table. As have the right to be seen below, the sound pressure levels room quite far-ranging in a large portion of the air domain above and outside the table.

Sound press levels over the table when the stem of the tuning fork is attached come the table.

For comparison, we plot the sound push level because that the very same air domain as soon as the tuning fork is hosted up. The difference is rather stunning with really low sound press levels in all parts of the air over the table except for in the vicinity of the tuning fork. This matches our endure with tuning forks as displayed in the initial YouTube video.

Sound pressure levels because that the tuning fork when held up.

### Is the double Frequency a organic Frequency?

So far, we have not touched on the original question: Why does the frequency twin when the tuning fork is inserted on the table? One possible explanation can be the there is such a natural frequency, which has a motion that is much more prominent in the vertical direction. For a vibrating string, because that example, the herbal frequencies are integer multiples that the fundamental frequency.

This is no the instance for a tuning fork. If the prongs are approximated as cantilever beams in bending, the lowest organic frequency is offered by the expression

The quantities in this expression are:

Length of the prong, LYoung’s modulus, E; usually around 200 GPa for steelMass density, ρ; approximately 7800 kg/m3Area moment of inertia that the prong overcome section, ICross-sectional area the the prong, A

For our tuning fork, this evaluates to 435 Hz, for this reason the formula gives a good approximation.

The second natural frequency the a cantilever beam is

This frequency is a variable 6.27 higher than the basic frequency. It cannot be involved in the frequency doubling. However, there space other mode shapes besides those v symmetric bending. Might one of castle be involved in the frequency doubling?

This is unlikely for 2 reasons. The very first reason is that the frequency doubling phenomenon can be observed for tuning forks with different geometries, and also it would certainly be too much of a simultaneously if every one of them have actually an eigenmode with precisely twice the basic natural frequency. The second reason is the nonsymmetrical eigenmodes have actually a far-reaching transverse displacement in ~ the stem, wherein the tuning fork is clenched. Such eigenmodes will hence be strong damped by her hand, and also have an trivial amplitude. One such mode, v a organic frequency that 1242 Hz, is displayed in the computer animation below.

The tuning fork’s first eigenmode in ~ 440 Hz, one out-of-plane mode with one eigenfrequency that 1242 Hz, and also the second bending mode with one eigenfrequency the 2774 Hz.

### The Probable cause of the Tuning Fork Mystery

Let’s summary what us know around the frequency-doubling phenomenon. Due to the fact that it is just experienced as soon as we push the tuning fork come the table, the dual frequency vibration has a solid axial motion in the stem. Also, we deserve to see from a spectrum analyzer (you have the right to download such an app on a smartphone) the the level of vibration in ~ the double frequency decays relatively quickly. Over there is a transition back to the fundamental frequency together the leading one.

The suspended on the amplitude argues a nonlinear phenomenon. The axial activity of the stem indicates that the stem compensates because that a change in the ar of the facility of fixed of the prongs.

Without going into details through the math, it deserve to be shown that because that the bending cantilever, the center of fixed shifts down by a distance relative to the original size L, i beg your pardon is

Here, a is the transverse activity at the tip and the coefficient β ≈ 0.2.

The important observation is the the vertical movement of the center of fixed is proportional come the square that the vibration amplitude. Also, the center of mass will certainly be at its lowest place twice per cycle (both as soon as the prong bends inward and when it bends outward), thus the double frequency.

With a = 1 mm and also a prong length of L = 80 mm, the maximum transition in the position of the facility of mass of the prongs can be estimated to

The stem has a substantially smaller mass 보다 the prongs, so it needs to move even an ext for the total center of gravity to preserve its position. The stem displacement amplitude can thus be estimated to 0.005 mm. This should be seen in relation to what we understand from the number experiments above. The straight (440 Hz) part of the axial movement is that the stimulate of a/100; in this example, 0.01 mm.

In reality, the tuning fork is a more facility system 보다 a pure cantilever beam, and the connection region between the stem and the prongs will affect the results. Because that the tuning fork analyzed here, the second-order displacements space actually less than half of the back-of-the-envelope guess 0.005 mm.

Still, the axial displacement brought about by the second-order relocating mass result is significant. Furthermore, as soon as it comes to emitting sound, the is the velocity, no the displacement, the is important. So, if displacement amplitudes are equal at 440 Hz and 880 Hz, the velocity in ~ the twin frequency is twice that in ~ the fundamental frequency.

Since the amplitude the the axial vibration at 440 Hz is proportional to the prong amplitude a, and also the amplitude the the 880-Hz vibration is proportional come a2, it is essential that we strike the tuning fork hard sufficient to endure the frequency-doubling effect. Together the vibration decays, the relative prominence of the nonlinear hatchet decreases. This is clearly seen on the spectrum analyzer.

The behavior can be investigated in information by performing a geometrically nonlinear transient dynamic analysis. The tuning fork is set in movement by a symmetric impulse used horizontally top top the prongs, and also is then left cost-free to vibrate. It deserve to be watched that the horizontal prong displacement is almost sinusoidal in ~ 440 Hz, if the stem move up and down in a plainly nonlinear manner. The stem displacement is highly nonsymmetrical, since the 440 Hz contribution is synchronous with the prong displacement, if the 880-Hz term constantly gives an additional upward displacement.

Due to the nonlinearity the the system, the vibration is not fully periodic. Also the prong displacement amplitude have the right to vary native one cycle come another.

The blue line reflects the transverse displacement in ~ the prong tip, and also the green line mirrors the upright displacement at the bottom the the stem.

If the frequency spectrum that the stem displacement plotted above is computed utilizing FFT, there are two far-reaching peaks in ~ 440 Hz and also 880 Hz. Over there is also a little third peak approximately the second bending mode.

Frequency spectrum of the upright stem displacement.

To actually check out the second-order term at 880 Hz in action, we can subtract the component of the stem vibration the is in phase through the prong bending native the complete stem displacement. This displacement difference is checked out in the graph below as the red curve.

The total axial stem displacement (blue), the prong bending proportional stem displacement (dashed green), and the continuing to be second-order displacement (red).

How walk we carry out this calculation? Well, we know from the eigenfrequency evaluation that the amplitude the the axial stem vibration is around 1% that the transverse prong displacement (actually 0.92%). In the graph above, the dashed environment-friendly curve is 0.0092 times the existing displacement that the prong guideline (not shown in the graph). This curve deserve to be thought about as showing the direct 440 Hz hatchet — a an ext or less pure sine wave. That value is climate subtracted from the total stem displacement, and what is left is the red curve. The second-order displacement is zero once the prong is straight, and also peaks both as soon as the prong has its preferably inward bending and also when it has actually its maximum external bending.

Actually, the red curve looks very much like it is having actually a time variation proportional to sin2(ωt). It should, since that displacement, according to the evaluation above, is proportional come the square the the prong displacement. Using a well-known trigonometric identity, sin^2(omega t) = dfrac1-cos(2 omega t)2. Get in the twin frequency!

### Different Tuning Forks

Commenters top top the original video from standupmaths have actually noticed that some tuning forks work better than others, and also with some tuning forks, it is challenging to view the frequency doubling at all. As debated above, the very first criterion is the you hit the hard sufficient in bespeak to gain into the nonlinear regime. However there are also geometrical distinctions influencing the ratio between the amplitude that the two varieties of vibration.

For instance, prongs the are heavy relative come the stem will cause large double-frequency displacements, due to the fact that the stem must move an ext in bespeak to keep the center of gravity. Slender prongs have the right to have a bigger amplitude–length (a/L) ratio, thus increasing the nonlinear term.

The design of the an ar where the prongs satisfy the stem is important. If that is stiff, then the amplitude that the fundamental frequency vibration in the stem will be reduced, and the relative importance of the double-frequency vibration is larger.

The cross section of the prongs will additionally have an influence. If we go back to the expression because that the organic frequency

it have the right to be checked out that the minute of inertia that the cross section plays a role. A prong through a square cross section with next d has

Thus, for 2 tuning forks that look the same when viewed native the side, the one with a square profile must have actually prongs that space a factor 1.14 much longer to give the same fundamental frequency. If we assume the exact same maximum stress because of bending in the 2 tuning forks, the one through the square profile deserve to have a transverse displacement amplitude, i m sorry is 1.142 larger than the circular one due to the fact that of its greater load-carrying capacity. In addition, if the stem is kept at a fixed size, then it will become proportionally lighter when contrasted to the longer prongs. Every these contributions end up in a 70% increase in upright stem vibration amplitude when moving from a circular profile to a square profile.

In addition, tuning forks with a one cross section usually have a style that is more flexible at the connection between the prongs and also the stem, and thus a higher level that vibration in ~ the an essential frequency.

The conclusion is that a tuning fork through a square cross section is much more likely to exhibit the frequency-doubling habits than one v a circular cross section.

### Do us Hear the Frequency Doubling?

In most cases, the price is “no.” The an essential frequency is tho there, also though the may have a reduced amplitude than the one with the dual frequency. But the means our senses work, we hear the fundamental frequency, although with a different timbre. It is difficult, yet not impossible, to strike the tuning fork so difficult that the sound level of the double frequency is substantially dominant.

### Conclusions

The frequency copy occurs because of a nonlinear phenomenon, wherein the stem the the tuning fork must relocate upward, in order to compensate for the tiny lowering the the center of mass of the prongs as they strategy the outermost location of their bending motion.

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Note that it is no the fact that the tuning fork is linked to the table that reasons the frequency doubling. The factor that us measure that in that case is the the sound emitted by the resonating table surface ar is caused by the axial stem motion, vice versa, the sound we hear from the tuning fork that is organized up is conquered by the prong bending. The movement is the exact same in both cases, as long as the impedance the the table is ignored. In fact, you deserve to measure the doubled frequency through a tuning fork when hosted up together well, but it is 30 dB or so below the fundamental frequency.

### Next Steps

Watch the original videos native standupmaths ~ above YouTube:Read much more about the intersection of tuning forks and also simulation ~ above the stclairdrake.net Blog: