Analytical approach to rectangular membrane vibration

[pitchtwit]

Homework Statement

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 https://dl.dropboxusercontent.com/u/11341635/%2A%20Theoretical%20investigation%20of%20the%20sound%20attenuation%20of%20membrane-type%20acoustic%20metamaterials%20-%20Y%20Zhang%2C%20J%20Wen%2C%20Y%20Xiao%2C%20X%20Wen%20and%20J%20Wang%20%282012%29.pdf.

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 realized 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 W_m)..."Hence, the equation of motion including the acoustic loading can therefore be written as,

ρ_s χ ∂²w/∂t² + ρ_massh(x,y,x₀,y₀,l_x,l_y) χ ∂²w/∂t² + 2ρ₁c₁ χ ∂w/∂t - T∇²w = 2Ae^jωt (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) = ∑^N_n=1W_n(x,y)q_n(t) (Eqn. 4)

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

W_n(x,y) = sin(r∏x/L_x χ sin(s∏y/L_y) (Eqn. 5)

where,
r = 1,2,...,N_x; s = 1,2,...,N_y; n = N_y(r - 1) + s

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

q_n(t) = q_n χ e^jωt (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 ≤ L_x, 0 ≤ y ≤ L_y), we obtain the following equation

-ω² M_m q_m - ω² ρ_mass ∑^N_n=1I_m,nq_n + jωC_mq_m + K_mq_m = 2AH_m

where,
m = 1,2,...,N
M_m = ρ_s∫₀^L_xW_m∑^N_n=1W_ndx dy"

It goes on to define I_m,n, C_m, K_m and H_m, but at no point does it define W_m.

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

I've tried running the code, just having W_m as the same as W_n, using,

W_m(x,y) = sin(a∏x/L_x χ sin(b∏y/L_y) (Eqn. 5)

where, a & b are still just numbers from 1 to N_x and 1 to N_y, 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 W_m also appears in https://dl.dropboxusercontent.com/u/11341635/%2A15%20FREE%20VIBRATIONS%20OF%20A%20RECTANGULAR%20PLATE%20%20CARRYING%20A%20DISTRIBUTED%20MASS%20-%20O.%20KOPMAZ%20AND%20S.%20TELLI%20%282002%29.pdf paper.

Homework Equations

If anyone can explain what W_m 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.

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

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

Many thanks.

Tom

 

[AlephZero]

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

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.

 

[pitchtwit]

The introduction of W_m and what it represents.

Thanks for taking the time AlephZero.

I did know about W_i,j and W_r,s, and how the first paper abbreviates those to W_m and W_n (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 W_m is introduced and what it represents really.

W_n represents the mode functions, but what does W_m represent?