mmae 535 report

MMAE 535

Project Report

Evaluation of Noise Cancelling Effort of

Headphones

Haoyang Yan

A20337136

Instructor: Ankit Srivastava

Abstract

This report describes the result of a research project focused on evaluation of noise cancelling effort of two different kinds of material which widely used on two main types of headphones, Circumaural Headphones and In-ear Headphones. Two different kinds of material were considered in this analysis: Polyurethane Foam (PU foam) and Silicone Rubber.

The goal of this project is to check the amount of sound energy left after a given sound pass through the material and evaluate the effort of material in cancelling noise. The analysis consists of a simulation of a given sound pressure applied on the material by using Finite Element Analysis (FEA) method. FEA models were created and analyzed by using ABAQUS (a computer aided engineering program).

Results include the kinetic energy density of given point, stress energy density of given point and total energy of the whole model experienced by the CAD model which represents the material. The results obtained from the analysis were displayed through charts.

Keywords: Headphones, Circumaural Headphones, In-ear headphones, Polyurethane Foam, Silicone Rubber, Finite Element Analysis, ABAQUS

Contents

I. Introduction..........................................................................................3

1.1 Background...................................................................................3

1.2 What to do for this project...........................................................4

II. Method..................................................................................................5

2.1 Geometry Model........................................................................5

2.2 Acoustic Wave Sample..............................................................5

2.3 Fundamental Physical Quantities of Acoustics.......................6

2.4 Viscoelastic.................................................................................9

2.5 FEA Model.................................................................................9

III. FEA Results.....................................................................................15

3.1 Energy Density.........................................................................15

3.2 Total Energy.............................................................................26

3.3 Total Energy Absorption Rate................................................28

IV. Conclusion.......................................................................................30

V. References..........................................................................................31

I. Introduction

Headphones are a pair of portable listening device which are designed to be worn by the users’ ears. They are a kind of converter unit, electric signal could be received from signal source, for example cell phones and music players, and converted to audible acoustic waves by speakers close to the user’s ears.

Headphones are designed to be used for a single person to listening. They were used widely in monitoring at the beginning because private listening environment and higher acoustic quality could be provided for a single user without affecting or being affected by external environment. However, headphones are still widely used in a huge amount of areas nowadays because their high portability and ability to cancel environmental noise. Generally, the form of headphones can be divided into four categories: Circumaural (over-ear), Supra-Aural (on-ear), Earbud and In-Ear. In this project, the ability of noise canceling of Circumaural and In-Ear would be mainly focused on.

1.1 Background

Headphones are used widely for portable media listening in daily life, they would be used with widely environmental noise. They could hardly provide high acoustic quality under a noisy circumstance. In general, the form of headphones could do some help in canceling noise, Circumaural and In-Ear are widely used in noise canceling headphones.

Circumaural headphones (Fig 1.1), also known as full size headphones, have bigger size ear cushions which could contain users’ ears completely. Because this form could completely contain the ear with its big size, circumaural headphones can be designed to fully seal to cut off external noise. In passive reducing noise design, the material of ear cushions plays an important role, Polyurethane Foam (PU foam) are used widely in circumaural headphones design.

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Figure 1.1 Circumaural Headphones Figure 1.2 In-Ear Headphones

In-Ear headphones (Fig 1.2) is another widely used form in passive reducing noise design. They are small headphones with much better portability than circumaural headphones designed to wear by inserting into the user’s ear cannels. With the unique way of wearing, In- Ear headphones is also a good form to block environmental noise. Silicone Rubber is the material used mostly in designing In-Ear headphones’ cushion.

1.2 What to do for this project

This project aims to analyze the characteristics of acoustic wave propagation in headphones cushion with two kinds of material, and tries to find out which one would be better in cancelling noise by analyzing acoustic wave energy. The main method for this project is Finite Element Analysis.

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II. Method

The Finite Element Analysis (FEA) method is mainly used in this project. It is a method to approximately predict the reaction of a product in real world with applied boundary conditions.

In practical engineering filed, FEA method could be widely used in many areas. This method is usually performed by a number of FEA software on the market, such as HyperMesh, ABAQUS, LS-DYNA, and so on. ABAQUS, which is a widely used universal FEA software, is the software used for this project. The main process of ABAQUS begins with importing or creating CAD model, creating material and section properties, defining assemble part, setting up analysis steps and variable output, generating mesh, running whole analysis by submitting job, and finally post processing.

II.1 Geometry Model

The geometry CAD model is shown below as Fig 2.1, it is mainly like a block. The dimension of the CAD model is 40mm×40mm×50mm. In this simulation, a certain sound pressure parallels to the Z axle and points to the positive direction of Z axle would be applied on one of 40mm×40mm surfaces, so that the sound wave would pass through the CAD model which represent the material block.

Figure 2.1 Geometry CAD Model

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II.2 Acoustic Wave Sample

For making the simulation become much closer to daily life, an audio sample of major road noise has been collected in this project. Analysis for the collected audio sample has been done by using MATLAB.

The analysis results are shown below:

Fig 2.2 shows the detail of oscillography of the audio sample. It could be seen that the wave is basically a harmonic wave because any complex acoustic wave could be regarded as the combination of sine wave, and sine wave is the wave of pure tone. The time of the audio sample is about 60 second, and the amplitude range, which describes the loudness in acoustics, is mostly between +0.5 to -0.5.

Figure 2.2 Oscillography of Audio Sample

Fig 2.3 shows the detail of frequency spectrum of the audio sample. This plot describes the amplitude of different frequency. It could be seen that the amplitude of frequency between 0 Hz to 5000Hz is much higher than others, which means the loudness of this audio sample is mainly in the range of 0 Hz to 5000Hz.

Figure 2.3 Frequency Spectrum of Audio Sample

II.3 Fundamental Physical Quantities of Acoustics

In this project, some basic physical quantities would be used for simulation.

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Acoustic wave is the propagating form of sound, it could be seen as the pressure vibration propagation among the elastic medium. Acoustic wave is caused by the vibration of sounding objects. The propagation of acoustic wave needs medium. Acoustic wave is a kind of longitudinal wave, it propagates along the same axle as the direction of particle vibration, but it sometimes could also be mixed with transverse wave when it propagates in solid.

Acoustic wave is usually classified through its frequency. Based on the frequency range for human ears to capture is from 20 Hz to 20000 Hz, acoustic waves could be classified into three main categories, infrasonic wave (whose maximum frequency is 20 Hz), audible wave (whose frequency range from 20 Hz to 20000 Hz), and ultrasonic wave (whose minimum frequency is 20000 Hz). As it shown in Fig 2.3, the collected audio sample could be seen as infrasonic wave and audible wave because its loudness is mainly between 0 Hz and 5000Hz. As infrasonic wave could not be captured by human ears, this project would only focus on audible wave.

2.3.1 Speed of Acoustic WaveThe speed of acoustic waves has the same characteristic as other kind of waves, it

is affected by the medium properties it propagating. Essentially, speed of sound is the speed of pressure perturbation propagating in medium.

c=√ Kρ

(2.1)

Equation 2.1 describes how to calculate the speed of sound, where c is the speed of sound, ρ is the density of medium, K is the bulk modulus, it could be calculated from equation 2.2,

K= dPdPdρ

(2.2)

where dP and dρare small variations of pressure and density. In liquid and solid, variations of K and ρcaused by temperature and pressure would be too tiny to be considered, so that the speed of sound could be considered as a constant in one medium. For solid medium, K could be regarded as shear modulus E for longitudinal wave and elastic modulus G for transverse wave. In this project, longitudinal wave would be considered, so that equation 2.1 would become

c=√ Eρ

(2.3)

2.3.2 Sound PressureSound pressure, also could be called acoustic pressure is a physical quantity

widely used for describing acoustic. It is the variation of atmospheric pressure

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perturbed by a sound wave. Sound pressure could be used to calculate other physical quantity, for example particle velocity.

p=−k ∂ y∂ x

(2.4)

Equation 2.4 shows the equation of sound pressure based on its definition, where p is the sound pressure, k is called volumetric modulus of elasticity, x is the position of an element, and y is the average displacement of the molecules in the x direction. In practical, it is easy to get the data of sound pressure by measuring directly, Fig 2.4 shows some data of sound pressure in daily life. As discussed before, the audio sample is collected for major road noise, the relevant data has also been highlighted on Fig 2.4.

Figure 2.4 Data of Sound Pressure

2.3.3 Bulk Modulus

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Bulk modulus, also known as incompressibility modulus, is a physical quantity describing the measurement of material deformation to uniform pressure. It could be defined by equation 2.5

K=−V ∂ p∂ V

(2.5)

where K is the bulk modulus, V is the volume, p is the pressure.

II.4 Viscoelastic

Viscoelasticity is a combination of both viscous characteristic of liquid and elastic characteristic of solid. In general, polymer would experience the process of fusion and flow (transfer to liquid from solid), and then the process of cooling and hardening (transfer to solid from liquid) during the manufacturing process. This would lead to both liquid and solid properties of polymer, which would be viscous and elastic respectively, performing under different circumstance.

Material like PU foam and silicon rubber used in this project has viscoelastic properties, this kind of material could be called viscoelastic material. With its viscoelastic properties, viscoelastic material is widely used in damping structural vibration and noise. In this project, viscoelastic properties would be included and mainly focused on damping noise.

II.5 FEA Model

The FEA model is built based on the CAD model discussed in 2.1. The detail of the FEA model would be discussed in this section.

2.5.1 Materials PropertiesThere are two important material behaviors need to be defined when using

ABAQUS to do an acoustic simulation, which is called Density and Acoustic Medium, bulk modulus is required in Acoustic Medium.

As discussed above, materials used in this project are PU foam and Silicon Rubber, which are used widely in noise cancelling headphones design.

Polyurethane (PU) is a polymer composed with carbamate units in main chain structure. There are two main categories of polyurethanes, some are thermoplastic polyurethanes, but most of polyurethanes are thermosetting polymers, which means the material would not melt when being heated. Polyurethane is mainly used to make different kinds of foam and plastic sponge, for example high-resilience foams, rigid foams and microcellular foams.

Silicone rubber is a rubber-like material, its main chain structure is alternatively composed with silicone atom and oxygen atom, while organic units would connected

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with silicone atom. Silicone rubbers are widely used in industry, and there are multiple formulations.

Rubber like material could be seen as incompressible material. Incompressible means the total volume would not change after the material being compressed. Poisson’s Ratio of ideal incompressible material is 0.5, while in practical it would be less than and close to this value. This kind of material has is an ideal material which does not exist, however, rubber like material is close to incompressible.

Properties of these two kinds of materials could be in a wide range, because of the ways to manufacture them and other polymers or fillers contained. Table 2.1 shows some physical properties data range of PU foam and silicone rubber,

Polyurethane Foam Silicone Rubber

Density 0.0250 - 1.39 g/cc 0.700 - 3.80 g/cc

Modulus of Elasticity 0.000138 - 3.45 GPa 0.000517 - 1.90 GPa

Poisson’s Ratio 0.300 - 0.750 0.470 – 0.499

Shear Modulus 0.00122 - 0.870 GPa 0.0000689 - 1.83 GPa

Table 2.1 Physical Properties Data of PU foam and Silicone Rubber

PU Foam PropertiesIn this project, the density of PU foam would be 4.07E-10 tone/mm^3, Poisson’s

Ratio would be 0.38. Hyperfoam is the material behavior used to describe PU foam properties, relevant data of PU foam is provided by the test data (provided by ABAQUS manual) shown below in Table 2.2 and Table 2.3, test data would be imported in ABAQUS directly:

Nominal Stress

(MPa)

Nominal Strain Nominal Transverse Stress

(MPa)

1 0.0140 0.08 0.0046

2 0.0334 0.16 0.0166

3 0.0533 0.24 0.0366

4 0.0853 0.32 0.0573

5 0.1280 0.40 0.081710

6 0.1653 0.48 0.1098

7 0.2080 0.56 0.1394

8 0.2560 0.64 0.1666

9 0.2987 0.72 0.1904

Table 2.2 Simple Shear Test Data of PU Foam

Nominal Stress

(MPa)

Nominal Strain

1 -0.0217 -0.05

2 -0.0317 -0.10

3 -0.0367 -0.15

4 -0.0402 -0.20

5 -0.0433 -0.25

6 -0.0467 -0.30

7 -0.0504 -0.35

8 -0.0542 -0.40

9 -0.0604 -0.45

10 -0.0668 -0.50

11 -0.0759 -0.55

12 -0.0909 -0.60

13 -0.1083 -0.65

14 -0.1410 -0.70

15 -0.1933 -0.75

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16 -0.2896 -0.80

Table 2.3 Uniaxial Test Data of PU Foam

Viscoelasticity of PU foam would also be included in this project, relevant data is provided by test data shown in Table 2.4 below.

Value of the long-term, normalized shear modulus

0.5000

gR Time

1 1.0000 0.0001

2 0.9695 0.001

3 0.9417 0.002

4 0.8722 0.005

5 0.7913 0.010

6 0.7043 0.020

7 0.6233 0.050

8 0.5736 0.100

9 0.5271 0.200

10 0.5013 0.500

11 0.5000 1.000

Table 2.4 Relaxation Test Data of PU Foam

Where gR is normalized shear relaxation modulus, its value is in the range of 0 ≤ gR (t)≤1.

Silicon Rubber Properties

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In this project, the density of PU foam would be 1.31E-9 tone/mm^3. Hyperelastic is the material behavior used to describe silicon rubber properties, relevant data of silicon rubber is provided by the test data (provided by ABAQUS manual) shown below in Table 2.5 and Table 2.7, test data would be imported in ABAQUS directly:

C10 (MPa) C01 (MPa) D1

3.2 0.8 0.0125

Table 2.5 Mooney-Rivlin Coefficients of Silicon Rubber

Where C10, C01 and D1 are defined as temperature-dependent material parameters in ABAQUS. C10 and C01 could be calculated from the experimental data. D1 is the coefficient would describe compressibility, the material would be incompressible when D1=0.

Based on C10, C01 and D1, the initial Young’s modulus (E), the bulk modulus (K) and Poisson’s ratio (ν) could be calculated based on the following equations.

E=6(C 10+C 01) (2.6)

K= 2D 1

(2.7)

K could also be calculated from:

K= E3(1−2ν )

(2.8)

Substituting equation 2.6 into equation 2.8, the following equation could be got:

K=6 (C 10+C 01)3 (1−2 ν )

=2(C 10+C 01)1−2ν

(2.9)

Substituting equation 2.9 into equation 2.7,

D 1= 1−2νC 10+C 01

(2.7)

Hence, some other physical quantities, like Young’s modulus and Poisson’s ratio, of the silicon rubber used for this test could be calculated. Those data would be shown in Table 2.6 below

Silicone Rubber

Density 1.31E-9 tone/mm^3

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Modulus of Elasticity 24 MPa

Poisson’s Ratio 0.475

Table 2.6 More data for Silicone Rubber

Viscoelasticity of silicon rubber would also be included in this project, relevant data is provided by test data shown in Table 2.7 below.

g_i Prony k_i Prony tau_i Prony

0.3 0.1 0.1

Table 2.7 Frequency-Dependent Prony Parameters of Silicon Rubber

Where g_i Prony represents the shear relaxation or shear traction relaxation modulus ratio, k_i Prony represents the bulk relaxation or normal traction relaxation modulus ratio, tau_i Prony represents the relaxation time.

2.5.2 Solver TypeSolver type could be set up in Step section in ABAQUS. As discussed in 2.2 and

2.3 above, this simulation would contain frequency response analysis, and the frequency would be considered in this simulation is from 20Hz to 5000Hz.

Steady-state dynamics, Direct with Linear perturbation is the solver type used in this project, and the frequency range would be 20Hz to 5000Hz. Number of points, which means the number of point needed to be solved among the whole frequency range, needed to be set up here.

2.5.3 Boundary ConditionFor solving problems in ABAQUS, the boundary conditions are usually set up in

BCs. For preventing the CAD model move after the load being applied, boundary condition should also fix the position of the model. In this project, boundary conditions are defined on surfaces except surfaces perpendicular to Z axles, just as shown in red shaded surfaces in Fig 2.5 below, those surface would be fixed in 6 directions:

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Figure 2.5 Boundary Condition

2.5.4 Load and Sound SourceFor solving problems in ABAQUS, the load could be set up in Loads section. In

this simulation, the load is set up by choosing Pressure as the load type. As discussed in 2.3.2, the acoustic pressure data would be used as the pressure in this simulation, the value would be 6E-007 MPa. The acoustic pressure would be applied on the surface shown in Fig 2.6 below, and it would be point along the positive direction of Z axle.

Figure 2.6 Acoustic Pressure Load

2.5.5 Mesh DetailsThis simulation is made based on ABAQUS, considering viscoelastic properties

need to be included, solid (continuum) elements should be used for solving hyperplastic material. Table 2.3 shows the detail of the mesh detail.

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Element Size 1mm

Element Type C3D8R

Element Count 80000

Table 2.8 Mesh Detail

III. FEA Results

As discussed above, the element type is a solid (continuum) element and the dimension of the CAD model is 40mm×40mm×50mm. For balancing the calculation time and result, 10 is the number of point would be picked up among the chosen frequency range.

III.1 Energy Density

Energy density is a physical quantity to measure the amount of energy stored in a certain system per unit volume or mass. In this simulation, kinetic energy density and strain energy density would be considered to get the total energy in the material.

III.1.1 FEA modelKinetic Energy Density

The detail of kinetic energy density in the FEA model for both materials would be shown below from Fig 3.1 to Fig 3.22. In ABAQUS, kinetic energy density is represented by EKEDEN. For observing the detail inside the model, view cut of the ZY plane would be used.

Figure 3.1 PU foam base state Figure 3.2 Silicon rubber base state

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Figure 3.3 PU foam 20Hz Figure 3.4 Silicon rubber 20Hz

Figure 3.5 PU foam 573.3Hz Figure 3.6 Silicon rubber 573.3Hz




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Strain Energy DensityThe detail of strain energy density in the FEA model for both materials would be

shown below from Fig 3.23 to Fig 3.44. In ABAQUS, strain energy density is

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represented by SENER. For observing the detail inside the model, view cut of the ZY plane would be used.

Figure 3.23 PU foam base state Figure 3.24 Silicon rubber base state


Figure 3.27 PU foam 573.3Hz Figure 3.28 Silicon rubber 573.3Hz



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3.1.2 Data analysisFor comparing the sound wave energy in the model for both material, three points

would be picked up as datum points in the model for getting data. These three points would be on one of ZY surfaces which could divide the model in half, and be 0mm, 25mm and 50mm apart from the surface respectively where the sound load would be applied. The detail for the three points is shown below in Fig 3.45 in red

Figure 3.45 Detail for the Three Points Picked

Kinetic Energy DensityThe detail of kinetic energy density data picked at each point for both materials

would be shown below.

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Figure 3.46 PU foam 0mm Figure 3.47 Silicon rubber 0mm

Fig 3.46 and Fig 3.47 shows the detail of kinetic energy density data at 0mm of PU foam and silicon rubber respectively, that’s the position where acoustic pressure applied. In Fig 3.46, the data varies from 0 to 0.5e-009 in mJ/mm^3 (because the CAD model was built in millimeter, 1 mJ/mm^3=10E-06 J/m^3), while in Fig 3.47, the data varies from 0 to 0.15e-012 in mJ/mm^3, it is clearly to be noticed that the data of silicon rubber is much smaller than PU foam’s. It could be easily found that the kinetic energy density of PU foam increasing and decreasing rapidly at the beginning then becoming steady in Fig 3.46, while the trend of silicon rubber is steady at the beginning but increasing rapidly at the end in Fig 3.46. The detail data of Fig 3.46 and Fig 3.47 would be shown in Table 3.1 below:

PU Ki net i c 0mm Rubber Ki net i c 0mmFrequency [Hz] Energy densi ty [mJ / mm̂3] Energy densi ty [mJ / mm̂3]

1 20 2. 76149E- 15 3. 67681E- 192 573. 333 4. 88052E- 10 4. 27298E- 163 1126. 67 6. 85018E- 13 1. 15601E- 144 1680 5. 60073E- 12 2. 04387E- 155 2233. 33 5. 69166E- 11 9. 38234E- 166 2786. 67 2. 36500E- 12 4. 09297E- 147 3340 2. 02230E- 11 6. 64413E- 158 3893. 33 3. 52199E- 11 2. 30442E- 149 4446. 67 1. 34163E- 10 5. 00068E- 15

10 5000 1. 08685E- 10 1. 52559E- 13

Table 3.1 Data of Fig 3.46 & Fig 3.47


Fig 3.48 and Fig 3.49 shows the detail of kinetic energy density data at 25mm of PU foam and silicon rubber respectively, that’s the position at the geometric center of the CAD model. In Fig 3.48, the data varies from 0 to 0.45e-09 in mJ/mm^3, while in Fig 3.49, the data varies from 0 to 0.27e-012 in mJ/mm^3, the data of silicon rubber is still much smaller than PU foam’s. It could be easily found that the kinetic energy density of PU foam at this position has a similar trend as shown in Fig 3.46, while the trend of silicon rubber is also similar to the one shown in Fig 3.46. The detail data of Fig 3.48 and Fig 3.49 would be shown in Table 3.2 below:

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10 5000 5. 38114E- 11 5. 07216E- 14



Fig 3.50 and Fig 3.51 shows the detail of kinetic energy density data at 50mm of PU foam and silicon rubber respectively, that’s the position assumed to be around human ears in this simulation. In Fig 3.50, the data varies from 0 to 0.5e-09 in mJ/mm^3, while in Fig 3.51, the data varies from 0 to 0.13e-012 in mJ/mm^3, the data of PU foam has almost the same trend as shown above in Fig 3.46 and Fig 3.48, while the data of silicon rubber increased and decreased rapidly among the range of 3400Hz to 4500Hz. The data of silicon rubber is still much smaller than PU foam’s. The detail data of Fig 3.50 and Fig 3.51 would be shown in Table 3.3 below:



10 5000 9. 60439E- 11 1. 26402E- 13


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Figure 3.52 PU foam Kinetic Total Figure 3.53 Silicon rubber Kinetic Total

Fig 3.52 and Fig 3.53 shows the detail of kinetic energy density data at 0mm, 25mm and 50mm of PU foam and silicon rubber respectively, which means all data are shown together.

Strain Energy DensityThe detail of strain energy density data picked at each point for both materials

would be shown below.


Fig 3.52 and Fig 3.53 shows the detail of strain energy density data at 0mm of PU foam and silicon rubber respectively, that’s the position where acoustic pressure applied. In Fig 3.52, the data varies from 0 to 0.35e-009 in mJ/mm^3, while in Fig 3.53, the data varies from 0 to 0.16e-012 in mJ/mm^3, it is clearly to be noticed that the data of silicon rubber is much smaller than PU foam’s. It could be easily found that the strain energy density of both PU foam and silicon rubber could not maintain a steady trend. The detail data of Fig 3.52 and Fig 3.53 would be shown in Table 3.5 below:

PU St rai n 0mm Rubber St rai n 0mmFrequency [Hz] Energy densi ty [mJ / mm̂3] Energy densi ty [mJ / mm̂3]


10 5000 3. 43341E- 10 5. 93037E- 15

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Fig 3.54 and Fig 3.55 shows the detail of strain energy density data at 25mm of PU foam and silicon rubber respectively, that’s the position at the geometric center of the CAD model. In Fig 3.54, the data varies from 0 to 0.75e-012 in mJ/mm^3, while in Fig 3.55, the data varies from 0 to 0.27e-012 in mJ/mm^3, the data of silicon rubber is almost the same with PU foam’s. The trend of strain energy density of PU foam is steady at the beginning but increase and decrease rapidly among the range of 2200Hz to 3200Hz as shown in Fig 3.54, while the trend of silicon rubber is also steady at the beginning but increase rapidly at about 4500 Hz shown in Fig 3.46. The detail data of Fig 3.54 and Fig 3.55 would be shown in Table 3.6 below:



10 5000 3. 61215E- 11 2. 69081E- 13



Fig 3.56 and Fig 3.57 shows the detail of strain energy density data at 50mm of PU foam and silicon rubber respectively, that’s the position assumed to be around human ears in this simulation. In Fig 3.56, the data varies from 0 to 0.2e-09 in mJ/mm^3, while in Fig 3.57, the data varies from 0 to 0.27e-017 in mJ/mm^3, the data of silicon rubber is much smaller than PU foam’s. The trend of PU foam has a

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similar trend with the kinetic energy density trend of PU foam at the beginning, but increases rapidly at about 3400Hz. The detail data of Fig 3.56 and Fig 3.57 would be shown in Table 3.7 below:



10 5000 2. 03310E- 10 3. 45601E- 15


Figure 3.58 PU foam Strain Total Figure 3.59 Silicon rubber Strain Total

Fig 3.58 and Fig 3.59 shows the detail of strain energy density data at 0mm, 25mm and 50mm of PU foam and silicon rubber respectively, which means all data are shown together.

III.2 Total Energy

As mentioned above in 2.5.5, the volume of each element would be 1mm^3, so the data of energy density could represent the relevant energy of the given nodal in this simulation.

As discussed above in 3.1, it is not easy to describe the trend in some plot. Based on the element volume mentioned above, the sum of kinetic energy density data and strain energy density data would be the total energy, which would present a more obvious trend, experienced by datum picked. Table 3.8 and 3.9 would show the total data of both PU foam and silicon rubber at each datum picked:

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PU Total 0mm PU Total 25mm PU Total 50mmFrequency [Hz] Energy [mJ ] Energy [mJ ] Energy [mJ ]

1 20 7. 59818E- 13 4. 81971E- 14 4. 54638E- 152 573. 333 6. 38107E- 10 4. 24146E- 10 6. 93887E- 103 1126. 67 1. 38333E- 11 3. 33354E- 12 3. 06946E- 124 1680 6. 65082E- 12 1. 41280E- 12 6. 42819E- 125 2233. 33 1. 52816E- 10 1. 37058E- 11 6. 66158E- 116 2786. 67 6. 11561E- 12 8. 43671E- 11 6. 21428E- 117 3340 2. 64514E- 11 1. 29656E- 11 2. 34789E- 118 3893. 33 5. 08858E- 11 8. 14739E- 12 5. 55177E- 119 4446. 67 2. 28289E- 10 1. 27046E- 10 1. 93333E- 10

10 5000 4. 52026E- 10 8. 99329E- 11 3. 11995E- 10

Table 3.7 Total Energy of PU foam

Rubber Total 0mm Rubber Total 25mm Rubber Total 50mmFrequency [Hz] Energy [mJ ] Energy [mJ ] Energy [mJ ]

1 20 1. 54312E- 15 2. 41945E- 16 2. 45607E- 162 573. 333 2. 31370E- 15 4. 86329E- 16 6. 03939E- 163 1126. 67 2. 25111E- 14 9. 54773E- 15 1. 55188E- 144 1680 5. 41007E- 15 5. 89034E- 15 1. 22372E- 145 2233. 33 2. 09802E- 15 5. 57068E- 15 1. 77430E- 146 2786. 67 9. 54246E- 14 1. 10843E- 14 3. 66607E- 167 3340 1. 58870E- 14 7. 10118E- 15 3. 88697E- 148 3893. 33 1. 84749E- 13 1. 65340E- 14 1. 32869E- 139 4446. 67 1. 99812E- 14 2. 78515E- 16 1. 07244E- 14

10 5000 1. 58489E- 13 3. 19803E- 13 1. 29858E- 13

Table 3.8 Total Energy of Silicon Rubber

For observing the trend of total energy, two plot of total energy of two materials would be made, as shown below in Fig 3.60 and Fig 3.61:

Figure 3.60 Total Energy of PU foam

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As shown in Fig 3.60, the trend of three positions picked is very similar with the one shown in Fig 3.52 above, it increases and decreases rapidly at the beginning then becoming steady, but increases again at about 4000Hz. After further observation, it could be noticed that the amount of total energy at 25mm is smaller than the one at 50mm under most circumstances, and both the data of 25mm and 50mm are smaller than the one at 50mm under most circumstances.

Figure 3.61 Total Energy of Silicon Rubber

As shown in Fig 3.61, the trend of three positions picked is very similar with the one shown in Fig 3.53 above, it is steady at the beginning but increases and decreases rapidly among two continuous range, which is 2200Hz to 3400Hz and 3400Hz to 4500Hz. After further observation, it could be noticed that the amount of total energy at 25mm is smaller than the one at 50mm at the range of 0Hz to 4500Hz, but the data of 25mm becomes the biggest of all at 5000Hz.

Combining both Fig 3.60 and 3.61, it could be noticed that the amount of total energy of silicon rubber is smaller than PU foam’s. The trend of silicon rubber is steadier than PU foam at lower frequency (about 0Hz to 2000Hz), PU foam performs steadier than the silicon rubber at about 2500Hz to 4000Hz, but total energy data of both the two materials increase after 4500Hz.

III.3 Total Energy Absorption Rate

After discussed the total energy shown in 3.2 above, the rate data of total energy absorbed by the material has been calculated. It is the ratio of energy absorbed and the

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amount of energy coming from the source. Table 3.9 and 3.10 below shows the detail data:

PU Total 0- 25mm PU Total 25- 50mm PU Total 0- 50mmFrequency [Hz] Energy absobt i on % Energy absobt i on % Energy absobt i on %

20 93. 66% 90. 57% 99. 40%573. 333 33. 53% - 63. 60% - 8. 74%1126. 67 75. 90% 7. 92% 77. 81%

1680 78. 76% - 355. 00% 3. 35%2233. 33 91. 03% - 386. 04% 56. 41%2786. 67 - 1279. 54% 26. 34% - 916. 13%

3340 50. 98% - 81. 09% 11. 24%3893. 33 83. 99% - 581. 42% - 9. 10%4446. 67 44. 35% - 52. 18% 15. 31%

5000 80. 10% - 246. 92% 30. 98%

Table 3.9 Rate of PU foam

Rubber Total 0- 25mm Rubber Total 25- 50mm Rubber Total 0- 50mmFrequency [Hz] Energy absobt i on % Energy absobt i on % Energy absobt i on %

20 84. 32% - 1. 51% 84. 08%573. 333 78. 98% - 24. 18% 73. 90%1126. 67 57. 59% - 62. 54% 31. 06%

1680 - 8. 88% - 107. 75% - 126. 19%2233. 33 - 165. 52% - 218. 51% - 745. 70%2786. 67 88. 38% 96. 69% 99. 62%

3340 55. 30% - 447. 37% - 144. 66%3893. 33 91. 05% - 703. 61% 28. 08%4446. 67 98. 61% - 3750. 58% 46. 33%

5000 - 101. 78% 59. 39% 18. 07%

Table 3.10 Rate of Silicon Rubber

Combining both Table 3.9 and 3.10, it could be noticed that both PU foam and silicon rubber performs worse in energy absorption with the increase of distance and frequency respectively, especially among the range of 25mm to 50mm, but energy has been absorbed at the total range (0mm-50mm) under most frequencies used in this simulation.

Comparing the data at the range of 0mm to 25mm, it could be noticed that the energy has been aborted under almost all frequencies used in this simulation, but it seems like the PU foam could provide a steadier absorption rate than silicon rubber.

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IV. Conclusion

Based on the result shown above, it could be concluded that silicone rubber would let less energy pass through the material than the PU foam among the frequency range of 20Hz to 5000Hz. On the other hand, PU foam would absorb more energy than silicon rubber and could provide steadier performance in cancelling noise with frequency range from 20Hz to 5000Hz by absorption than silicon rubber. The performance for cancelling noise of both PU foam and silicon rubber would become worse when the frequency or distance increasing, and the distance range of 0mm to 25mm would be a better choice for both PU foam and silicon rubber to canceling noise.

This would help engineers have a clear idea when designing a passive noise canceling headphones, it would also do some help when designing other passive noise canceling equipment such as sound insulation earplugs, which is widely used in protecting workers’ ears in industries.

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V. References

1. ABAQUS Documentation 6.14, http://abaqus.software.polimi.it/v6.14/index.html2. B C Duncan, A S Maxwell, L E Crocker and R Hunt, “Failure Criteria and their

Application to Visco-Elastic/Visco-Plastic Materials, PAJ1 Report No 17”, NPL Report CMMT(A)226, October 1999

3. Nigel J. Mills, “Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide”, Butterworth-Heinemann, 2007

4. Qibai Huang, “Foundation of Engineering Acoustics”, Huazhong University of Science and Technology

5. MatWeb: Material Overall Property Data. Retrieved from http://www.matweb.com/

6. MATLAB Audio File Processing, http://m.blog.csdn.net/article/details?id=6655408

7. Michael R. Gosz, “Finite Element Method Applications in Solids, Structures, and Heat Transfer”. CRC Press Taylor & Francis Group, 2011.

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