Acoustic-Structural Interaction in a Tuning Fork

Abstract The tuning fork is used to tune musical instruments by sounding a tone at exactly 440 Hz when struck. In this model the goal is to ensure that the prong length of our tuning fork is designed to produce a fundamental eigenfrequency at exactly 440 Hz. A parametric sweep and an eigenfrequency study is first used to calculate the fundamental eigenfrequency as function of the prong length. The length is varied in small increments to find the true prong length needed to produce a fundamental eigenfrequency at 440 Hz. This first study is a pure structural mechanics model. Secondly, the Acoustic-Structure Interaction physics interface is used to model the multiphysics problem of the interaction between the vibrating tuning fork and the surrounding air. In this step the fork is actuated by a harmonic unit force, this corresponds to finding the frequency response of the fork (the Fourier components). A real fork is struck with an impulse force that can be decomposed into Fourier components.

Goal of Eigenfrequency Analysis A theoretical estimation1 of the fundamental eigenfrequency for a tuning fork (cylindrical prongs) is given by where R2 is the radius of the cross section of the prongs, E denotes Young’s modulus, ρ is the density, R1 is the radius of the base, and L is the length of the straight cylindrical part. This estimation is based on the solution for a cantilever beam representing each prong, but this representation is not entirely accurate due to the increased bending stiffness of each prong near the base. This estimation gives us a prong length L of 7.8 cm, but this number cannot be trusted as the exact solution. Therefore, the parametric sweep and an eigenfrequency study will be used to calculate the fundamental eigenfrequency as the prong length is varied in small increments between 7.85 and 7.95 cm to find the true prong length needed to produce a fundamental eigenfrequency at 440 Hz. 𝑓= 1.875 2 𝑅 2 4𝜋 (𝐿+0.5𝜋 𝑅 1 ) 2 𝐸 𝜌 1. Tuning fork, http://en.wikipedia.org/wiki/Tuning_fork

Goal of Acoustic-Solid Interaction Analysis The eigenfrequency analysis shows that the prong length L should be about 7.906 cm, the radiation pattern and magnitude of the sound waves produced when the tuning fork, with this prong length, vibrates around its fundamental resonance of 440 Hz will be calculated. The small but, but insignificant, effect of the added air mass on the tuning fork is also captured in this study. The Acoustic-Solid Interaction physics interface and a Frequency Domain study type are used. The tuning fork is excited by a sinusoidal mechanical boundary load on the top surface of one of the prongs and at a frequency specified in the study: here a range between 435 and 445 Hz. A Perfectly Matched Layer (PML) is used to simulate radiation of the sound waves into infinite space. A Far Field Calculation boundary condition allows for the calculation of the sound pressure at distances outside the modeling domains in postprocessing. In postprocessing we will determine: The Sound Pressure Level (SPL) inside the modeling domain. The Sound Pressure Level (SPL) outside the modeling domain at a radius of 30 cm as a 3D radiation pattern. How the SPL varies at a point based on the frequency of tuning fork excitation. How the instantaneous displacement of the tuning fork is related to the instantaneous sound pressure in the air surrounding the fork.

Parameters Set up Parameters for the tuning fork Geometry and the frequency investigation.

Geometry, Step 1 The base of the tuning fork is formed as a Union between a Cone, Sphere, and Torus.

Geometry, Step 2 The prongs of the tuning fork are formed using two additional Cylinders.

Geometry, Step 3 The air domain is formed using a third Cylinder. Layers are added to the top, bottom, and sides of the air cylinder to create domains to be used as the Perfectly Matched Layer (PML), a layer that models propagation into infinite space.

Materials Add Air as the first material: it will be assigned to all domains by default. Add Steel AISI 4340 as the second material: assign the tuning fork domains to this material to override the Air material assignment there.

Solid Mechanics Eigenfrequency Analysis

Solid Mechanics First set up the Solid Mechanics interface with an Eigenfrequency study is used. Only the tuning fork (solid) domains are included in this analysis.

Damping Add Damping to Linear Elastic Material 1; specify the Damping type as Isotropic loss factor. Specify the Isotropic structural loss factor for Steel AISI 4340 to be 0.001 (under the Materials node), that is 0.1 %. This value depends highly on the type of steel and the quality of the cast as well as other factors. In a real application it should be measured.

Parametric Sweep Add a Parametric Sweep to Study 1. Sweep the parameter L (prong length) over range(0.0785,1e-4,0.0795).

Eigenfrequency Study In the Eigenfrequency study step, specify the Desired number of eigenfrequencies as 1 and Search for eigenfrequencies around 440 Hz. Solve only for the Solid Mechanics physics in this Study.

Default Plot: Mode Shape The default plot will show the mode shape and displacement of the tuning fork at its fundamental eigenfrequency of about 437 Hz.

User-Defined Plot: Eigenfrequencies Add a 1D Plot Group with a Global plot. Use Solution 2 for the Data set and plot the eigenfrequency (freq) on the y-Axis and the Outer solutions (L) on the x-Axis. This shows how the fundamental eigenfrequency for the tuning fork decreases with increasing prong length. correct length

Calculate Eigenfrequencies Add a Global Evaluation under Derived Values. Evaluate solid.freq on Solution 2 (Parametric Solution) and see the results in Table 1.

Acoustic-Solid Interaction Frequency Domain Analysis

Perfectly Matched Layer Add a Perfectly Matched Layer to the domains in the outer cylindrical layer. Choose the Geometry Type as Cylindrical, the Center Coordinate as (R1,0,0), and choose the Coordinate stretching type as Rational.

Linear Elastic Material 1 Add the tuning fork (solid) domains to Linear Elastic Material 1.

Damping Add Damping to Linear Elastic Material 1.

Boundary Load Add a Boundary load of 1 N/m2 to the top surface of one of the tuning fork prongs in the z direction. This is a harmonic actuation of the fork.

Far Field Calculation Add a Far Field Calculation to the boundaries of the inner cylinder of air. Specify the Type of integral as Full integral.

Frequency Domain Study In the Frequency Domain study step, specify the range of frequencies using the parameters defined earlier: range(f0-df,2*df/9,f0+df). Solve only the Acoustic-Solid Interaction physics in this study.

Solution 4, Selection Duplicate Solution 3 to create a Solution 4. Add to Solution 4 a Selection and include all the non-PML domains.

User-Defined Plot: Acoustic Pressure, Slice Add a 3D Plot Group with a Slice Plot using Solution 4 as the Data set. Plot the expression acsl.p_t on the Zx-plane.

User-Defined Plot: SPL, Multislice Add a 3D Plot Group with a Multislice Plot using Solution 3 as the Data set. Plot the expression acsl.Lp. Add an Arrow Surface plot of the normal vectors.

Default Plot: Far-Field SPL (3D) This default plot gives a 3D view of the Far Field Sound Pressure Level (SPL). Specify the center of the Sphere to evaluate on. Note that you can adjust the angular resolution (for better graphics) and the distance from the center the SPL is evaluated at (this can be outside the modeling domain).

User-Defined Plot: Response (SPL) Add a 1D Plot Group with a Point Graph and Global plot. In the Point Graph, select point 38 and plot the expression: acsl.Lp In the Global plot, plot the expression: subst(acsl.ffc1.Lp_pfar,x,R1,y,0,z,100[cm]) The resonance is a bit off the desired 440 Hz and can be finetuned with a finer sweep over the length L or using an optimization. The response is here for a unit load and only the relative amplitude is of interest.

User-Defined Plot: Acoustic Slice + Structural Volume Add a 3D Plot Group with a Slice plot and a Volume plot; add a Deformation to the Volume Plot. In the Slice plot, plot the instantaneous pressure acsl.p_t in the ZX-plane. In the Volume plot, plot the structural displacement, acsl.disp.