School of Mechanical and Manufacturing Engineering

MECH9325 Fundamentals of Acoustics and Noise

Assignment – Acoustic performance of a multilayered coating

Due date: Monday 31 March 5pm (Moodle online submission)

Aim

The aim of this assignment is to theoretically and numerically investigate the acoustic performance of multilayered coating.

Background

A silicone material called polydimethylsiloxane (PDMS) is often used as a noise control material in underwater applications due to its acoustic impedance being relatively close to that of water, thus allowing sound waves to transition into the material with minimal reflection. In this assignment, both the transfer matrix method and the finite element method will be used the characterise the acoustic performance of a multilayered coating using PDMS as the host material. Part 1 of this assignment will consider a multilayered homogeneous material comprising multiple layers of the same PDMS material.

Transfer Matrix Method

The transfer matrix method is a common method to model acoustic elements. Consider an acoustic element of length L. An incident harmonic plane wave propagating in the x-direction impinges on the left hand side of the element. An expression for the harmonic plane wave of unity amplitude is given by pinc = ei(wt-kx) where w is the angular frequency, k is the longitudinal wavenumber (also known as the acoustic wavenumber), t is the time variable, x is the spatial variable, and i = √一1.

The acoustic pressure and particle velocity on the incidence side of the element are denoted by pinc and uinc . Similarly, the acoustic pressure and particle velocity on the transmission side of the element are denoted by ptr and utr. Using the transfer matrix method, the acoustic pressure and particle velocity on the transmission side can be found in terms of the acoustic pressure and particle velocity on the incidence side as follows

(1)

where M is the transfer matrix given by

(2)

In Eq. (2), z = Pc is the characteristic impedance of the material where P is the density of the material and c is the speed of sound in the material, and k = w/c is the acoustic wavenumber. For multiple acoustic elements, the transfer matrix of the total design is obtained by multiplying together the transfer matrix of each element in sequential order of the elements as follows

M = M1 . M2 … Mn                                                                               (3)

Consider the homogeneous material of length L divided in N identical layers each of length Ln, that is, Ln = L/N. The transfer matrix method of each layer becomes

(4)

The total transfer matrix is obtained by multiplying together the transfer matrix  of each layer in sequential order of the layers, which yields

(5)

Using the elements of the transfer matrix, the transmission and reflection coefficients are obtained as follows

(6)

(7)

where zinc and ztr are the characteristic impedances ofthe fluid on incidence and transmission sides of the material. In this case, zinc  = ztr  = Pwcw where Pw , cw are the density and speed of sound in water. From the transmission coefficient, the transmission loss (in dB) is obtained as

TL = -10log10  |T|                                                      (8)

Single layer and Multilayered Designs

In Part 1 you are required to model a single layer PDMS material and a multilayered PDMS material as shown in Figs. 1(a) and 1(b), with water on the incidence and transmission sides of the material. Consider an  incident  harmonic  plane  wave.  Obtain  the  reflection  coefficient,  transmission  coefficient  and transmission loss of the single and multilayered homogeneous materials theoretically (using the transfer matrix method programmed in Matlab) and numerically (using COMSOL). A Matlab script. is appended to this document.

(a)

(b)

Figure  1 (a) Homogeneous material and (b) multilayered homogeneous material with water on the incidence and transmission sides.

Numerical Model

Using the finite element software COMSOL, develop a 3D numerical model to calculate the reflection and transmission coefficients, and the transmission loss. You can model either a square duct or a circular duct (dimensions are given below for a square duct). Use the following parameters.

Dimensions of the square duct cross-section: 0.05m by 0.05m

Length of single layer PDMS material: 0.2m

Length of duct before/after material: 0.5m

Length of Perfectly Matched Layer before/after duct: 0.4m

Maximum number of layers: 10

Density of water: 1000 kg/m3

Speed of sound in water: 1500 m/s

Density of PDMS: 1000 kg/m3

Speed of sound in PDMS: 1000 m/s

Frequency range: 1 Hz up to 5000 Hz in steps of 25 Hz

COMSOL Installation

Download and install the COMSOL software. To do this, the first step is to register for a COMSOL Access Account at

www.comsol.com/access/

Once you have registered, you can then download COMSOL 6.3 from the following link

www.comsol.com/product-download/

Use the ISO for offline installation (down the bottom of the product download page). If you attempt to download via the online installer, you may run into connectivity issues.

In the final step, use the following passcode

17211400000004001-PAUS-250706-5083305-70EA692E560A

You will also be able to access a wide range of helpful and innovative resources.

Useful Resources

Please use the following resources useful to generate your numerical models:

1.    Geometry tutorial:https://www.youtube.com/watch?v=I6te1NFnS44

2.    Meshing tutorial:https://www.youtube.com/watch?v=UK8O-LV1uoU

3.    Frequency   analysis: Tutorial Models for Various Analyses of    a   Bracket Download bracket_basic.mph and models.sme.bracket_frequency.pdf, following the instructions in the PDF to build the full model. The final model is also provided on this webpage as bracket_frequency.mph.

Results

Theoretically and numerically calculate the absolute value for the reflection and transmission coefficients, and the transmission loss, for the single layer PDMS and the multilayered PDMS. Present your results as 2D plots as a function of frequency. You can superimpose your results for the single and multiple layers, as well as superimpose your results obtained theoretically and numerically. Discuss the values of the coefficients.

Poster

Provide your results in a poster. A couple of poster templates have been provided. You can also design your own poster layout. Your poster should include an introduction, description of the numerical model, results and discussion. A poster should not be too detailed or busy. For example, the lecture slides for this course are visually much easier to read than the lecture notes. The presentation of a poster should aim to be similar to the lecture slides by using large font size, bullet points where possible, no large chunks of text and large clear figures with easy to read axis labels. Your poster can also include references in smaller font size.

Matlab Script

Matlab script. of a single layer

clc

clear

close all

z_water = rho_water*c_water;     %characteristic impedance of water

z_PDMS = rho_PDMS*c_PDMS;   %characteristic impedance of PDMS

L = 0.2;                                          %total thickness of PDMS block

j=1;

for f=10:10:5000                          %frequency sweep

k = 2*pi*f/c_PDMS;                      %acoustic wavenumber of PDMS

%Transfer matrix

M = [cos(k*L), 1i.*z_PDMS.*sin(k*L); 1i.*sin(k*L)./z_PDMS, cos(k*L)];     %1i is the imaginary unit

%Elements of the transfer matrix

M11 = M(1,1);

M12 = M(1,2);

M21 = M(2,1);

M22 = M(2,2);

%Transmission coefficient

T(1,j) = 2./(M11 + M12/z_water + z_water*M21 + M22);

T(1,j) = sqrt(real(T(1,j)).^2 + imag(T(1,j)).^2);

%Reflection coefficient

R(1,j) = (M11 + M12/z_water - z_water*M21 - M22)/(M11 + M12/z_water + z_water*M21 + M22); R(1,j) = sqrt(real(R(1,j)).^2 + imag(R(1,j)).^2);

%Transmission loss

TL(1,j)= -10*log10(T(1,j));

freq(1,j)=f;

j=j+1;

end

% Figures

figure(1)

plot(freq,TL,'k-')

xlabel('Frequency (Hz)')

ylabel('Transmission loss (dB)')

figure(2)

plot(freq,T,'k-')

xlabel('Frequency (Hz)')

ylabel('Transmission coefficient')

figure(3)

plot(freq,R,'k-')

xlabel('Frequency (Hz)')

ylabel('Reflection coefficient')

Modification to Matlab script. for multiple layers

%Transfer matrix of one layer

Mn = [cos(k*Ln), 1i.*z_PDMS.*sin(k*Ln); 1i.*sin(k*Ln)./z_PDMS, cos(k*Ln)];

%Total transfer matrix for N number of layers M=Mn*Mn*Mn*Mn;