Can you please explain if you have an idea why I got the gra

Can you please explain if you have an idea why I got the graph below looks different?

I wrote a matlab program to find the mode shape and natural frequency of a fixed-free beam. The graph in red is the result of the finite element method and the blue is the analytical solution. The first mode matches 16.7103 for the natural frequency but the second mode 104.7219 for FEM and 104.7223 for the analytical.

I don\'t understand why the graph looks so different but form another perceptive you can see that they are basically the inverse if you flip them.

Cantilever beam Cantilever Beam 0 oOOOOoo O O O O FEM - Mode 1- alphaL 1.8751 - Freq (Hz) 16.7103 0.1 O O o O -0.2 o -10 -0.3 -15 0.4 -20 -0.5 -0.6 30 -0.7 35 -0.8 0.9 -40 Analytical-Mode 1-Freq (Hz) = 16.7103 -45 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.25 Beam span 0.05 0.1 0.15 0.2 0.3 0.35 0.4 0.45 0.5 Beam span

Solution

First mode is same because there is no discretization in calculation. As element increases result will vary.

If you Discretized it very fine you will get very acurate result.

Secondly, flipping is due to problem in choosing the elements in mode calculation.

It also depends upon what kind of method you are using is it numerical functions such as Runga-kutta method, ode?

If you want to get exact solution you have to reduce the step in MATLAB.

For e.g.: the gaps seen in the MATLAB plot for modes show the accuracy according to step.

if you had given step of 0.0001 you get the answer in four decimal places.

Note: I have research in the same field as per experience we have taken frequency two decimal places because these result will go on upto as many decimal places.