The Displacement D in Millimeters of a Tuning Fork

The Displacement D in Millimeters of a Tuning Fork.

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 people baffled. In this blog post, we explicate the tuning fork mystery using simulation and provide some fun facts almost tuning forks along the fashion.

Explaining the Tuning Fork Mystery

In a contempo video on YouTube from standupmaths, science enthusiasts Matt Parker and Hugh Hunt hash out and demonstrate the “mystery” of a tuning fork. When you strike a tuning fork and hold it against a tabletop, information technology seems to double in frequency. Every bit it turns out, the explanation behind this mystery can exist boiled downwards to nonlinear solid mechanics.

How Does Sound Reach Our Ears?

When you concord a vibrating tuning fork in your manus, the bending motility of the prongs sets the air around them in move. The force per unit area waves in the air propagate as audio. You lot can hear it, but information technology is not a very efficient conversion of the mechanical vibration into audio-visual pressure.

When you agree the stem of the tuning fork to a table, an axial motion in the stem connects to the tabletop. The movement is much smaller than the transverse motility of the prongs, but it has the potential to set the large flat tabletop in motion — a surface that is a far amend emitter of sound than the thin prongs of a tuning fork. The tabletop surface will act equally a big loudspeaker diaphragm.

Our tuning fork.

To investigate this interesting behavior, we created a solid mechanics computational model of a tuning fork. The model is based on a tuning fork that 1 of my colleagues keeps in her pocketbook. The tone of the device is a reference A4 (440 Hz), the textile is stainless steel, and the total length is about 12 cm.

First, permit’s have a look at the displacement every bit the tuning fork is vibrating in its first eigenmode:

The manner shape for the central frequency of the tuning fork.

If we report the displacements in detail, it turns out that fifty-fifty though the overall motion of the prongs is in the transverse management (the
10
management in the movie), there are besides some pocket-sized vertical components (in the
z
management), consisting of ii parts:

1. The angle of the prongs is accompanied with an up-downwardly motion that varies linearly over the prong cantankerous section
2. The stem has an essentially rigid axial move, which is necessary for keeping the eye of mass in a fixed position, as required by Newton’south second constabulary

The displacements are shown in the figures beneath. The mode is normalized so that the maximum full displacement is ane. The top axial displacement is 0.03 and the displacement in the stem is 0.01.

Full displacement vectors in the first eigenmode.

Axial displacements only. Notation that the scales differ betwixt figures. The center of gravity is indicated by the blue sphere.

Now, let’s plough to the sound emission. By adding a boundary element representation of the acoustic field to the model, the audio pressure level in the surrounding air tin be computed. The amplitude of the vibration at the prong tips is set up to one mm. This is approximately the maximum feasible value if the tuning fork is not to exist overloaded from a stress point of view.

Equally can be seen in the figure below, the intensity of the sound decreases rather fast with the altitude from the tuning fork, and also has a big degree of directionality. Actually, if yous turn a tuning fork around its axis beside your ear, the near-silence in the 45-degree directions is hit.

Acoustic level (dB) and radiation pattern (inset) around the tuning fork.

We at present add a 2-cm-thick wooden tabular array surface to the model. It measures ane by 1 grand and is supported at the corners. The stem of the tuning fork is in contact with a bespeak at the center of the table. Equally can be seen below, the sound pressure levels are quite meaning in a large portion of the air domain higher up and outside the table.

Sound force per unit area levels in a higher place the table when the stem of the tuning fork is attached to the table.

For comparison, we plot the acoustic level for the aforementioned air domain when the tuning fork is held upward. The deviation is quite stunning with very low sound force per unit area levels in all parts of the air above the table except for in the vicinity of the tuning fork. This matches our experience with tuning forks as shown in the original YouTube video.

Sound force per unit area levels for the tuning fork when held upwardly.

Is the Double Frequency a Natural Frequency?

So far, nosotros accept not touched on the original question: Why does the frequency double when the tuning fork is placed on the table? One possible explanation could be that there
is
such a natural frequency, which has a motility that is more prominent in the vertical direction. For a vibrating string, for example, the natural frequencies are integer multiples of the central frequency.

This is not the instance for a tuning fork. If the prongs are approximated as cantilever beams in bending, the lowest natural frequency is given past the expression

f_1 = \dfrac{1.875^2}{2 \pi Fifty^2}\sqrt{\dfrac{EI}{\rho A}}

The quantities in this expression are:

• Length of the prong, L
• Young’s modulus, Eastward; ordinarily around 200 GPa for steel
• Mass density, ρ; approximately 7800 kg/m3
• Area moment of inertia of the prong cantankerous section, I
• Cross-sectional area of the prong, A

For our tuning fork, this evaluates to 435 Hz, so the formula provides a adept approximation.

The second natural frequency of a cantilever beam is

f_2 = \dfrac{4.694^two}{2 \pi Fifty^2}\sqrt{\dfrac{EI}{\rho A}}

This frequency is a factor vi.27 higher than the fundamental frequency. It cannot be involved in the frequency doubling. Notwithstanding, at that place are other mode shapes besides those with symmetric bending. Could one of them be involved in the frequency doubling?

This is unlikely for ii reasons. The beginning reason is that the frequency doubling phenomenon can be observed for tuning forks with different geometries, and it would be besides much of a coincidence if all of them have an eigenmode with exactly twice the fundamental natural frequency. The 2nd reason is that nonsymmetrical eigenmodes accept a meaning transverse displacement at the stem, where the tuning fork is clenched. Such eigenmodes volition thus be strongly damped by your hand, and have an insignificant aamplitude. I such style, with a natural frequency of 1242 Hz, is shown in the animation beneath.

The tuning fork’s starting time eigenmode at 440 Hz, an out-of-plane fashion with an eigenfrequency of 1242 Hz, and the second angle style with an eigenfrequency of 2774 Hz.

The Probable Cause of the Tuning Fork Mystery

Permit’south summarize what we know most the frequency-doubling phenomenon. Since information technology is only experienced when we press the tuning fork to the table, the double frequency vibration has a potent centric motion in the stem. Likewise, we can see from a spectrum analyzer (yous can download such an app on a smartphone) that the level of vibration at the double frequency decays relatively quickly. There is a transition back to the primal frequency as the dominant i.

The dependency on the amplitude suggests a nonlinear miracle. The centric movement of the stalk indicates that the stalk compensates for a modify in the location of the center of mass of the prongs.

Without going into details with the math, information technology can exist shown that for the bending cantilever, the center of mass shifts down by a distance relative to the original length
L, which is

\dfrac{\delta Z}{L} = \beta \left ( \dfrac{a}{L} \correct)^two

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

The important ascertainment is that the vertical motility of the eye of mass is proportional to the square of the vibration amplitude. Also, the center of mass will be at its lowest position twice per cycle (both when the prong bends inward and when it bends outward), thus the double frequency.

With
a
= i mm and a prong length of
50
= 80 mm, the maximum shift in the position of the center of mass of the prongs can be estimated to

\delta Z = 0.2 \left ( \frac{ane}{ 80} \right)^ii \mathrm 80 \, mm = 0.0025 \, mm

The stem has a significantly smaller mass than the prongs, so it has to move fifty-fifty more for the total eye of gravity to maintain its position. The stem displacement amplitude tin thus be estimated to 0.005 mm. This should be seen in relation to what nosotros know from the numerical experiments higher up. The linear (440 Hz) role of the axial motility is of the lodge of
a/100; in this example, 0.01 mm.

In reality, the tuning fork is a more complex arrangement than a pure cantilever beam, and the connection region betwixt the stem and the prongs will impact the results. For the tuning fork analyzed here, the second-order displacements are really less than half of the back-of-the-envelope predicted 0.005 mm.

Even so, the axial displacement acquired by the second-order moving mass effect is meaning. Furthermore, when it comes to emitting sound, it is the velocity, non the deportation, that is important. Then, if deportation amplitudes are equal at 440 Hz and 880 Hz, the velocity at the double frequency is twice that at the primal frequency.

Since the aamplitude of the centric vibration at 440 Hz is proportional to the prong amplitude
a, and the amplitude of the 880-Hz vibration is proportional to
a
2, it is necessary that we strike the tuning fork difficult enough to feel the frequency-doubling effect. As the vibration decays, the relative importance of the nonlinear term decreases. This is clearly seen on the spectrum analyzer.

The beliefs tin can exist investigated in item by performing a geometrically nonlinear transient dynamic assay. The tuning fork is fix in motion by a symmetric impulse applied horizontally on the prongs, and is then left costless to vibrate. It can be seen that the horizontal prong displacement is almost sinusoidal at 440 Hz, while the stem moves upwards and downward in a clearly nonlinear mode. The stem displacement is highly nonsymmetrical, since the 440 Hz contribution is synchronous with the prong deportation, while the 880-Hz term always gives an additional upward displacement.

Due to the nonlinearity of the system, the vibration is non completely periodic. Even the prong displacement aamplitude can vary from i wheel to another.

The blue line shows the transverse displacement at the prong tip, and the green line shows the vertical displacement at the bottom of the stalk.

If the frequency spectrum of the stalk deportation plotted to a higher place is computed using FFT, in that location are two meaning peaks at 440 Hz and 880 Hz. In that location is also a small-scale third peak effectually the 2d bending mode.

Frequency spectrum of the vertical stem displacement.

To actually see the 2d-order term at 880 Hz in action, we tin subtract the part of the stalk vibration that is in stage with the prong bending from the full stem displacement. This displacement deviation is seen in the graph below as the red curve.

The total centric stem displacement (blue), the prong bending proportional stem displacement (dashed green), and the remaining second-order displacement (reddish).

How did we perform this calculation? Well, we know from the eigenfrequency analysis that the aamplitude of the axial stem vibration is nigh 1% of the transverse prong deportation (really 0.92%). In the graph higher up, the dashed green curve is 0.0092 times the electric current displacement of the prong tip (not shown in the graph). This bend can be considered equally showing the linear 440 Hz term — a more than or less pure sine moving ridge. That value is and then subtracted from the total stem displacement, and what is left is the red curve. The second-club displacement is zero when the prong is straight, and peaks both when the prong has its maximum inwards bending and when information technology has its maximum outward bending.

Actually, the scarlet curve looks very much like it is having a time variation proportional to sin2(ωt). It should, since that displacement, according to the analysis above, is proportional to the foursquare of the prong displacement. Using a well-known trigonometric identity,
\sin^2(\omega t) = \dfrac{1-\cos(2 \omega t)}{ii}. Enter the double frequency!

Dissimilar Tuning Forks

Commenters on the original video from standupmaths have noticed that some tuning forks work ameliorate than others, and with some tuning forks, it is difficult to see the frequency doubling at all. Every bit discussed above, the first criterion is that you hitting it hard plenty in order to get into the nonlinear regime. But there are also geometrical differences influencing the ratio betwixt the amplitude of the two types of vibration.

For instance, prongs that are heavy relative to the stalk will cause large double-frequency displacements, since the stem must move more in order to maintain the center of gravity. Slender prongs tin can have a larger amplitude–length (a/L) ratio, thus increasing the nonlinear term.

The blueprint of the region where the prongs run across the stem is important. If information technology is potent, so the amplitude of the cardinal frequency vibration in the stalk will exist reduced, and the relative importance of the double-frequency vibration is larger.

The cross section of the prongs will likewise accept an influence. If we return to the expression for the natural frequency

f_1 = \dfrac{i.875^2}{ii \pi L^2}\sqrt{\dfrac{EI}{\rho A}}

information technology tin be seen that the moment of inertia of the cross department plays a role. A prong with a square cantankerous section with side
d
has

I = \dfrac{d^4}{12}

while a prong with a circular cantankerous department with diameter
d
has

I = \dfrac{\pi d^4}{64}

Thus, for two tuning forks that wait the aforementioned when viewed from the side, the i with a square contour must have prongs that are a gene one.14 longer to give the aforementioned fundamental frequency. If we assume the same maximum stress due to bending in the two tuning forks, the i with the square contour tin can have a transverse deportation amplitude, which is 1.142
larger than the circular ane considering of its higher load-conveying capacity. In improver, if the stem is kept at a fixed size, and then it will get proportionally lighter when compared to the longer prongs. All these contributions end up in a seventy% increment in vertical stem vibration amplitude when moving from a circular profile to a square profile.

In addition, tuning forks with a circular cross section unremarkably have a blueprint that is more flexible at the connection between the prongs and the stalk, and thus a college level of vibration at the cardinal frequency.

The conclusion is that a tuning fork with a foursquare cross department is more than likely to exhibit the frequency-doubling behavior than 1 with a circular cantankerous section.

Do We Hear the Frequency Doubling?

In most cases, the reply is “no.” The central frequency is still at that place, even though it may have a lower amplitude than the 1 with the double frequency. Only the way our senses work, we hear the fundamental frequency, although with a different timbre. Information technology is hard, simply non impossible, to strike the tuning fork and then hard that the sound level of the double frequency is significantly dominant.

Conclusions

The frequency doubling occurs due to a nonlinear phenomenon, where the stem of the tuning fork must move upward, in order to compensate for the small lowering of the heart of mass of the prongs as they approach the outermost positions of their bending move.

Note that it is not the fact that the tuning fork is connected to the table that causes the frequency doubling. The reason that we measure it in that case is that the sound emitted by the resonating table surface is caused past the axial stalk motion, whereas the sound we hear from the tuning fork that is held upwards is dominated by the prong bending. The motion is the same in both cases, as long as the impedance of the tabular array is ignored. In fact, you can measure the doubled frequency with a tuning fork when held up too, just information technology is 30 dB or and so below the fundamental frequency.

Next Steps

• Watch the original videos from standupmaths on YouTube:
• The Tuning Fork Mystery: unexpected vibrations
• The Tuning Fork Mystery: an unexpected update
• Read more about the intersection of tuning forks and simulation on the COMSOL Weblog:
• The Tuning Fork Application
• How to Perform Multimaterial Optimization in COMSOL Multiphysics