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I recently used Matlab to find the natural frequencies of a clamped-clamped beam. It was fairly simple, as I construct the global mass and stiffness matrices. Then it's just a matter of using the eig() function. (For sake of simplicity I put beam properties all to 1.)

When I choose how many elements (lets call it N) to use, it seems to give me natural frequencies which, seem to be smaller than usual. I somehow stumbled upon the fact that these frequencies are the literature values I have divided by N^2.

Anyone got any ideas as why this is so? Here is my code. Thanks!! Edit: For example when N = 10, my first natural frequency is 0.2237 when the exact value is 22.37.

Jeff

Code (Text):syms y;

syms z;

area = 1;

L = 1;

p = 1;

E = 1;

Ig = 1;

n = 10; %number of elements

if n == 0

stop

end

size = 4+((n-1)*2);

%define global matrices

Mg = zeros(size);

Kg = zeros(size);

%beam function

A = [1 0 0 0 ; 0 1 0 0 ; 1 L L^2 L^3 ; 0 1 2*L 3*L^2];

Ainv = inv(A);

Aitrans = transpose(Ainv);

Yh = [1 ; y ; y^2 ; y^3];

Ya = [1 ; y];

Ydotdot = diff(diff(Yh));

%Solving for Mass matrix (kinetic energy)

prod1 = Yh*transpose(Yh);

M = p*area*Aitrans*int(prod1, y, 0, L)*Ainv;

%solving for stiffness matrix (potential energy)

K = (E*Ig)*Aitrans*int(Ydotdot*transpose(Ydotdot), y, 0, L)*Ainv;

%creating the global mass matrix

Mg(1:4, 1:4) = M(1:4, 1:4);

Kg(1:4, 1:4) = K(1:4, 1:4);

if n > 1

i = 1;

j = 1;

for i=1:n-1

Mg(j+2:j+5, j+2:j+5) = Mg(j+2:j+5, j+2:j+5) + M(1:4, 1:4);

Kg(j+2:j+5, j+2:j+5) = Kg(j+2:j+5, j+2:j+5) + K(1:4, 1:4);

j = j+2;

end

end

%delete rows/columns based on clamped ends (boundary conditions)

Mg(size,:) = [];

Mg(size-1,:) = [];

Mg(:,size) = [];

Mg(:,size-1) = [];

Mg(:,1) = [];

Mg(:,1) = [];

Mg(1,:) = [];

Mg(1,:) = [];

Kg(size,:) = [];

Kg(size-1,:) = [];

Kg(:,size) = [];

Kg(:,size-1) = [];

Kg(:,1) = [];

Kg(:,1) = [];

Kg(1,:) = [];

Kg(1,:) = [];

eigs = eig(Kg,Mg);

sqrt(eigs)

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# Weird FEM issue (nat freqs)

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