# Weird FEM issue (nat freqs)

• jeffziggy
He divides the natural frequencies by the number of elements in the model to get smaller values. He tries different values for the number of elements, but the frequencies always come out smaller. He finds out that the frequencies are the literature values divided by the number of elements.

#### 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

Code:
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)

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 $\sqrt{k/m} = 1/l^2$.

AlephZero said:
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

## What is a natural frequency in FEM analysis?

A natural frequency in FEM analysis refers to the frequency at which a system or structure will naturally vibrate when disturbed. It is determined by the mass, stiffness, and damping properties of the system.

## Why is it important to analyze natural frequencies in FEM?

Analyzing natural frequencies in FEM allows us to understand how a structure will respond to external forces and identify potential resonance or instability issues. It also helps in designing and optimizing structures for specific applications.

## What can cause weird natural frequencies in FEM analysis?

Weird natural frequencies in FEM analysis can be caused by a variety of factors, such as inaccurate modeling, boundary conditions, material properties, or numerical errors. It is important to carefully review and validate all inputs and assumptions in the analysis to avoid these issues.

## How can weird natural frequencies be resolved in FEM analysis?

To resolve weird natural frequencies in FEM analysis, it is necessary to identify and address the root cause of the issue. This may involve revising the model, adjusting boundary conditions, or refining the mesh. It is also important to verify the results using different methods and sensitivity analyses.

## Can weird natural frequencies affect the accuracy of FEM analysis?

Yes, weird natural frequencies can significantly affect the accuracy of FEM analysis. They can lead to incorrect predictions of structural behavior and compromise the reliability of the analysis results. Therefore, it is crucial to identify and resolve these issues to ensure accurate and reliable FEM analysis.