Tuning Fork

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This example simulates a tuning fork for tuning musical instruments and, if correctly design, should sound the note of A, 440 Hz. It computes the fundamental eigenfrequency and eigenmode for the tuning fork. Although the example seems to be somewhat academic in nature, the eigenfrequencies and eigenmodes of microscopic tuning forks are also used in quartz watches and other electronic devices.

Model Definition

The model geometry is shown in Figure 1. The fundamental frequency of the fork is determined by the length of the prongs, the cross-section geometry of the prongs, and the material properties of the fork.

Figure 1: Tuning fork geometry.

The following formula gives a theoretical estimation for the fundamental frequency of a tuning fork with cylindrical cross section of the prong (Ref. 1):

(1)

where R2 is the radius of the cross section of the prongs, E denotes Young’s modulus, and ρ is the density. The length of the prong can be estimated as

(2)

where R1 the radius of the base, and L is the length of the straight cylindrical part, see Figure 1.

In the fundamental eigenmode, the prongs move according to the figure below. Thus, the eigenmode is symmetric with a symmetry plane placed between the prongs.

The advantage with the shape of the fundamental eigenmode is that the relative displacements in the handle are very small, which makes it possible to hold the fork without damping the vibration. This also allows to make use of the theoretical estimation for the frequency Equation 1 which is based on the solution for a cantilever beam representing each prong.

The parameters used in the model are: R1 = 7.5 mm and R2 = 2.5 mm. The fork material is Steel AISI 4340, for which E = 205 GPa and ρ = 7850 kg/m3.

For the frequency f = 440 Hz, Equation 1 and Equation 2 give the length of the prong cylindrical part as L = 7.8 cm. This presents an underestimation because the part of the prong near the base has larger bending stiffness compared to that for a straight cantilever beam.

To fine-tune the fork, you will use parametrized geometry and gradually increase the cylinder length starting from the above given estimation. To achieve this, you set up a parametric sweep with respect to parameter L.

1. Tuning fork, http://en.wikipedia.org/wiki/Tuning_fork

COMSOL_Multiphysics/Structural_Mechanics/tuning_fork

Modeling Instructions

From the menu, choose .

In the window, click .

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In the window, click .

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In the tree, select .

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In the tree, select .

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
L	7.8[cm]	0.078 m	Cylinder length
R1	7.5[mm]	0.0075 m	Base radius
R2	2.5[mm]	0.0025 m	Prong radius

You can build up the fork geometry efficiently using predefined geometry primitives.

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In the toolbar, click .

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In the window for , locate the section.

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In the text field, type R2.

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In the text field, type 2e-2.

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From the list, choose .

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In the text field, type 2.

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Locate the section. In the text field, type R1.

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In the text field, type -R1.

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Locate the section. From the list, choose .

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In the text field, type -1.

Sphere 1 (sph1)

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In the toolbar, click .

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In the window for , locate the section.

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In the text field, type 4e-3.

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Locate the section. In the text field, type R1.

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In the text field, type -(R1+2.25e-2).

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In the toolbar, click .

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In the window for , locate the section.

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In the text field, type R1.

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In the text field, type R2.

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In the text field, type 180.

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Locate the section. In the text field, type R1.

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Locate the section. From the list, choose .

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In the text field, type 0.

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In the text field, type 1.

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Locate the section. In the text field, type -90.

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In the toolbar, click and choose .

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In the window for , locate the section.

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Clear the check box.

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Click in the window and then press Ctrl+A to select all objects.

This completes the handle and base of the fork.

Add two cylinders to represent the prongs.

Cylinder 1 (cyl1)

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In the toolbar, click .

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In the window for , locate the section.

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In the text field, type R2.

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In the text field, type L.

Cylinder 2 (cyl2)

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In the toolbar, click .

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In the window for , locate the section.

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In the text field, type R2.

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In the text field, type L.

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Locate the section. In the text field, type 2*R1.

Use virtual geometry operations to avoid short edges and narrow regions. This will improve the mesh generation.

Ignore Edges 1 (ige1)

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In the toolbar, click and choose .

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On the object , select Edges 22, 23, 29, 32, 33, 39, 42, and 43 only.

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In the toolbar, click .

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Click the button in the toolbar.

The completed geometry should look as shown in the following figure:

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In the toolbar, click to open the window.

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Go to the window.

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In the tree, select .

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Click in the window toolbar.

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In the toolbar, click to close the window.

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In the window for , locate the section.

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From the list, choose .

Free Triangular 1

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Right-click and choose .

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Select Boundaries 6 and 24 only.

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In the window for , click .

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In the window, right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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Select Domains 1 and 3 only.

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Right-click and choose .

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In the window for , locate the section.

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In the text field, type 50.

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In the window, under click .

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In the window, right-click and choose .

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In the window for , click .

The meshed geometry should look like that in the figure below.

Set up a parametric sweep with respect to the cylinder length L and search for an eigenfrequency in the vicinity of 440 Hz.

Parametric Sweep

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In the toolbar, click .

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In the window for , locate the section.

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In the table, enter the following settings:


Parameter name	Parameter value list
L (Cylinder length)	range(0.078,1e-4,0.0795)

Step 1: Eigenfrequency

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In the window, under click .

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In the window for , locate the section.

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Select the check box.

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In the associated text field, type 1.

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In the text field, type 440.

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In the toolbar, click .

Solution 1 (sol1)

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type 1e-3.

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In the toolbar, click .

To see all computed eigenfrequencies as a table, follow these steps:

Global Evaluation 1

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In the toolbar, click .

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In the window for , locate the section.

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From the list, choose .

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From the list, choose .

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Click in the upper-right corner of the section. From the menu, choose .

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Go to the window.

You can see that the eigenfrequency closest to 440 Hz occurs for the cylinder length of 0.0791 m. Further fine-tuning can be performed if necessary.

To see the eigenmode that corresponds to this frequency, do the following:

Mode Shape (solid)

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In the window, under click .

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In the window for , locate the section.

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From the list, choose .

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In the toolbar, click .

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Click the button in the toolbar.

In this figure, you can clearly see that mode is symmetric, and the displacements at the handle are very small compared to those of the prongs. This means that holding the tuning fork at the handle will dampen the vibrations negligibly.