# Analytical approach to rectangular membrane vibration

1. Jul 8, 2014

### pitchtwit

1. The problem statement, all variables and given/known data

I'm following a procedure in a particular paper as part of my Masters project. The paper is called 'Theoretical investigation of the sound attenuation of membrane-type acoustic metamaterials - Y Zhang, J Wen, Y Xiao, X Wen and J Wang (2012)' and can be seen here.

It's quite hard to sum up the problem without looking at the paper - I know the rule above says I should write the problem here and not attach it - so I'll try to do that.

W_n is used to in the paper to refer to mode functions - actually it should be W_n(x,y), but it's abbreviated.

The membrane is square in this case and simply supported at all four sides. It is 27.4 mm along each side and made of flexible material.

In practice (in MATLAB) these are realised as a 3D array of numbers - say 28 by 28 by 5 if I use 28 values for both x & y (the x and y values represent points on the membrane from 0 to 27.4 mm) and choose to consider 5 modes of vibration (so it'd be 28x28x5).

The values in the 5 column actually represent distance in the z-direction - so giving the shape of the membrane in each of the 1st 5 modes. Not surprisingly the 1st one just has one peak at the centre.

From the paper: -
"The basic unit of the membrane-type acoustic metamaterials can be described as an elastic membrane carrying a small mass, the rim of the membrane was fixed. Consider a cell of membrane- type acoustic metamaterials in an x, y coordinate system. It is sub- ject to biaxial tension T per unit length, which will be assumed to remain constant when the membrane vibrates. The width, length, density per unit area of the membrane are Lx, Ly, ρs, and the width, length, density per unit area of the mass are lx, ly, ρmass. The point (x0, y0) represents the corner of the mass which is clos- est to the origin. In Cartesian coordinates, w(x, y,t) represents the transverse displacement in z direction of a point (x, y) at time t."...(equations 1 & 2 would have gone here, but they are not important to this particular question - and do not mention Wm)..."Hence, the equation of motion including the acoustic loading can therefore be written as,

ρs χ ∂2w/∂t2 + ρmassh(x,y,x0,y0,lx,ly) χ ∂2w/∂t2 + 2ρ1c1 χ ∂w/∂t - T∇2w = 2Aejωt (Eqn. 3)

Using the mode superposition theory [15,16], w(x, y,t) can be written as a superposition of the mode functions Wn(x, y) multi- plying the corresponding time-dependent, generalized co-ordinate qn(t), i.e.,

w(x,y,t) = ∑Nn=1Wn(x,y)qn(t) (Eqn. 4)

The mode function of the membrane with fixed rim can be written as

Wn(x,y) = sin(r∏x/Lx χ sin(s∏y/Ly) (Eqn. 5)

where,
r = 1,2,...,Nx; s = 1,2,...,Ny; n = Ny(r - 1) + s

The membrane performing harmonic vibration under the harmonic acoustic excited, the generalized co-ordinate qm(t) has the form

qn(t) = qn χ ejωt (Eqn. 6)

Substituting Eqs. (4), (6) into Eq. (3), multiplying each term by Wm(x, y) and integrating all terms over the domain (0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly), we obtain the following equation

2 Mm qm - ω2 ρmassNn=1Im,nqn + jωCmqm + Kmqm = 2AHm

where,
m = 1,2,...,N
Mm = ρs0LxWmNn=1Wndx dy"

It goes on to define Im,n, Cm, Km and Hm, but at no point does it define Wm.

This is what I'm stuck on - what is Wm?

I've tried running the code, just having Wm as the same as Wn, using,

Wm(x,y) = sin(a∏x/Lx χ sin(b∏y/Ly) (Eqn. 5)

where, a & b are still just numbers from 1 to Nx and 1 to Ny, and I've got the attached plot. But it's not the same as theirs, and when I change the number of x and y points I use, the plot shifts along the frequency (x) access left or right (and bit in the y-direction too).

This is the 2nd thing that I'm stuck on - but I'm hoping solving one with solve the other.

The appearance and use of Wm also appears in this paper.

2. Relevant equations

If anyone can explain what Wm is, that would really help, and if someone could explain why the plot shifts from left to right when I use different numbers of x & y points that would be great too.

3. The attempt at a solution

Here's the image showing that I have attempted (and gotten fairly close I suppose) to the solution. https://dl.dropboxusercontent.com/u/11341635/Screen%20Shot%202014-07-08%20at%2008.47.33.png [Broken]

If you look at the first paper you can see that it's quite close to there one (Fig. 1).

Many thanks.

Tom
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: May 6, 2017
2. Jul 8, 2014

### AlephZero

The second paper uses $W_{ij}$ which seems a more logical notation, since the mode shapes are simple functions of $i$ and $j$.. $W_{rs}$ are the same set of functions, replacing $i$ with $r$ and $j$ with $s$.

In the first paper they have chosen an arbitrary way to "label" the $W_{ij}$ with one subscript $n$ in equation 5. $W_m$ will be the same labelling system corresponding to $W_{rs}$ in the second paper.

For the small number of modes they used (N = 9?) the results will depend on exactly how many modes are in the solution. I would have guessed 9 was much to small for this sort of model ($N_x = N_y = 3$?) I would probably have used 100 or more (i.e. $N_x = N_y = 10$ or more).

If the results don't converge as you increase $N_x$ and $N_y$, there is probably something wrong with your code.

From a quick reading I don't see what is original about these papers. They look like applications of the well known standard method for fluid-solid coupling in modal coordinates in commercial software like Abaqus, Nastran, etc.

3. Jul 9, 2014

### pitchtwit

The introduction of W_m and what it represents.

Thanks for taking the time AlephZero.

I did know about Wi,j and Wr,s, and how the first paper abbreviates those to Wm and Wn (actually the 2nd paper does this too, but later on in the calculation from equation 25 to 26).

Also, although the number of modes used seems small, convergence is achieved at this low number. Presumably this is because of the prominence (radiation efficiency) of these early modes.

The original concept looked at by the 1st paper is that the transmission loss for this type of acoustic metamaterial significantly increases above that predicted by the mass law - for a narrow band at low frequencies, effectively overcoming the mass law for this type of material.

I am grateful for the response, but I still don't understand why Wm is introduced and what it represents really.

Wn represents the mode functions, but what does Wm represent?