All good questions. I will answer them when I am back at my computer. Unfortunately, eigenvalues and eigencectors go hand in hand with matrix equations so a little bit of that is unavoidable.

**Moderator:** bernd

All good questions. I will answer them when I am back at my computer. Unfortunately, eigenvalues and eigencectors go hand in hand with matrix equations so a little bit of that is unavoidable.

I looked into this and I need to make a few corrections to my earlier posts on this topic (which I will do through EDITS). Here is how I got myself confused:

1) The eigenvalue reported in Calculix (second column in dat file) is the square of the natural frequency and not the natural frequency itself.

2) So if this squared value is negative it is either due to numerical rounding of a near-zero value or something very special is going on

3) The Calculix manual gives an example of such a special case for a spinning disk on a slender shaft. The shaft becomes unstable above a certain rotational speed and the motion changes from oscillatory to exponential (unstable/explosive). This exponential behavior corresponds to a negative eigenvalue (and therefore imaginary natural frequency)

4) The only case I can imagine where we could hit a similar issue with FreeCAD is when we load a part or structure with forces that exceed the elastic buckling load (but I would need to check the math of such a case to be sure)

5) The references that I made to the bridge calculation are irrelevant. There all eigenvalues were positive, but the eigenmodes were complex due to damping. This required me to square and add the real and imaginary response amplitudes of the resulting motion.

A clear reference for understanding free vibration and eigenvalue analysis is the book Finite Element Procedures by Klaus Jurgen Bathe.

I can confirm this is the case. Analysis of a rigid column on a rotational spring shows that the frequency eigenvalue depends as follows on the vertical load F:

(Omega_F) ^ 2= (1 - F/FE) * (Omega_0) ^ 2

where FE is the Euler buckling load of the column. So the natural frequency reduces to zero as the load approaches the Euler buckling load and then becomes imaginary (sqrt of a negative number). This means that the behaviour changes from oscillatory to exponential (uncontrolled collapse).

So I stick to the advice above (in bold) but would not display zero, but rather Omega = Sqrt(Abs(eigenvalue)) and display the warning when eigenvalue < 0.0.

For who is interested: http://web.mit.edu/kjb/www/Books/FEP_2n ... inting.pdf

HarryvL wrote: ↑Tue Jul 09, 2019 6:54 amI can confirm this is the case. Analysis of a rigid column on a rotational spring shows that the frequency eigenvalue depends as follows on the vertical load F:

(Omega_F) ^ 2= (1 - F/FE) * (Omega_0) ^ 2

where FE is the Euler buckling load of the column. So the natural frequency reduces to zero as the load approaches the Euler buckling load and then becomes imaginary (sqrt of a negative number). This means that the behaviour changes from oscillatory to exponential (uncontrolled collapse).