Asked by Tom
on 19 Sep 2013

I struggle with this generally and have come up against a barrier again.

I'm trying to create vibrational mode functions for a rectangular membrane. The equation is simple enough: -

W_n = sin(x*pi*i/L_x)*sin(y*pi*i/L_y)

Here's where I'm up to: -

L_x = 27.4e-3; %membrane width (m) L_y = 27.4e-3; %membrane height (m) N_x = 10; %no. of eigenfreqs considered in x dimension N_y = 10; %no. of eigenfreqs considered in y dimension N = N_x*N_y; %total no. of eigenfreqs considered

x = (linspace(0.5*L_x/N_x,L_x - 0.5*L_x/N_x,N_x))'; %membrane x-dimension mapping points %each at the centre of an element y = (linspace(0.5*L_y/N_y,L_y - 0.5*L_y/N_y,N_y)); %membrane y-dimension mapping points %each at the centre of an element x_ref = 1:length(x); y_ref = 1:length(y); W_n = zeros(N_x,N_y,N); W_n(:,:,01) = (sin(x(x_ref,:,:)*1*pi/L_x)... *sin(y(:,y_ref,:)*1*pi/L_y)); W_n(:,:,02) = (sin(x(x_ref,:,:)*1*pi/L_x)... *sin(y(:,y_ref,:)*2*pi/L_y)); W_n(:,:,18) = (sin(x(x_ref,:,:)*2*pi/L_x)... *sin(y(:,y_ref,:)*8*pi/L_y)); W_n(:,:,46) = (sin(x(x_ref,:,:)*5*pi/L_x)... *sin(y(:,y_ref,:)*6*pi/L_y)); W_n(:,:,89) = (sin(x(x_ref,:,:)*9*pi/L_x)... *sin(y(:,y_ref,:)*9*pi/L_y)); W_n(:,:,100) = (sin(x(x_ref,:,:)*10*pi/L_x)... *sin(y(:,y_ref,:)*10*pi/L_y));

The numbers on the left hand side - 01, 02, 18, 46, 89, 100 - actually need to be 1,2,3,...,100, i.e.

n = 1:100; %mode number counting from 1 to 100

These refer to the mode numbers - although this is a bit odd - but still not too complicated. Basically if n = 01, then the mode is (1,1). If n = 59 the mode is (6,9), i.e. i = 6 & j = 9. I have solved this using the following: -

i = floor((n/10 - n/1000)) + 1; %mode number from 1 to 10 in x dimension j = n - 10*(floor((n/10 - n/1000))); %mode number from 1 to 10 in y dimension

Now I just need to put n, i and j into my W_n equation, but I can't figure out how!

Any help would be greatly appreciated :)

Answer by Simon
on 19 Sep 2013

Accepted answer

Hi!

I don't understand why the x and y spacing depends on the number of modes/frequencies.

Take a look at the following code:

L_x = 27.4e-3; %membrane width (m) L_y = 27.4e-3; %membrane height (m) N_x = 5; %no. of eigenfreqs considered in x dimension N_y = 10; %no. of eigenfreqs considered in y dimension N = N_x*N_y; %total no. of eigenfreqs considered

NumX = 50; % number of mapping points in x-direction NumY = 50; % number of mapping points in y-direction

% create X and Y array in 2d [X,Y] = meshgrid(... (linspace(0.5*L_y/NumY, L_y - 0.5*L_y/NumY, NumY)), ... (linspace(0.5*L_x/NumX, L_x - 0.5*L_x/NumX, NumX)));

% modify X and Y array for 3d XFull = repmat(X, [1 1 N]); YFull = repmat(Y, [1 1 N]);

% create mode array for X Mx = ones(NumX, NumY, N_y); MxFull = []; for n = 1:N_x MxFull = cat(3, MxFull, n*Mx); end

% create mode array for Y My = ones(NumX, NumY); MyFull = []; for n = 1:N_y MyFull = cat(3, MyFull, n*My); end MyFull = repmat(MyFull, [1 1 N_x]);

% modes for each direction Wx_n = arrayfun(@(x,mx) sin(x .* (mx *pi/L_x)), XFull, MxFull); Wy_n = arrayfun(@(y,my) sin(y .* (my *pi/L_x)), YFull, MyFull);

% superposition of modes W_n = Wx_n .* Wy_n;

% plot figure(1); cla surface(W_n(:,:,23));

Show 4 older comments

Tom
on 19 Sep 2013

That's perfect - thank you!!

Tom
on 20 Dec 2013

Hi again Simon, I'm just getting into this again and I've noticed that when I examine the 1st mode, it appears off-centre - whereas it should be completely central on the membrane.

This can be seen when the end of the surface command is

W_n(:,:,1);

Do you know why this is?

Many thanks,

Tom

Tom
on 20 Dec 2013

I think I may have solved this by using

(linspace(L_y/NumY, L_y, NumY))

instead of

(linspace(0.5*L_y/NumY,L_y - 0.5*L_y/NumY,NumY))

Do you think this solution is correct?

Thanks

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