Why are the natural frequencies of a clamped-clamped beam smaller than usual?

[jeffziggy]

Hello,

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

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)

 

[AlephZero]

I haven't checked every line of the code, but I think you have each element of length 1, so the length of your beam depends on the number of elements in the model.

If the length of the beam is l, the global mass is proportional to l and the global stiffness proportional to 1/l^3, so the frequences would be proportional to \sqrt{k/m} = 1/l^2.

 

[jeffziggy]

I haven't checked every line of the code, but I think you have each element of length 1, so the length of your beam depends on the number of elements in the model.

If the length of the beam is l, the global mass is proportional to l and the global stiffness proportional to 1/l^3, so the frequences would be proportional to \sqrt{k/m} = 1/l^2.

That was exactly it! Thank you kind sir!

Jeff