extraction of negative eigenvalues

About the development of the FEM module/workbench.

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HarryvL
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Re: extraction of negative eigenvalues

Postby HarryvL » Mon Jul 08, 2019 11:18 am

bernd wrote:
Mon Jul 08, 2019 10:03 am
HarryvL wrote:
Mon Jul 08, 2019 9:50 am
This is surprising and different from what I remember. I will look into it and compare to what I did for the bridge.
great
HarryvL wrote:
Mon Jul 08, 2019 9:50 am
PS: yes the natural frequency in rad/s is the eigenvalue of the free vibration problem.
than what is the column eigenvalue for? Do you know some paper which explains all this without tons of integrals and matrix operations ... :oops: wikipedia does not a good job here IMHO.
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.
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HarryvL
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Re: extraction of negative eigenvalues

Postby HarryvL » Mon Jul 08, 2019 11:43 pm

bernd wrote:
Mon Jul 08, 2019 10:03 am
HarryvL wrote:
Mon Jul 08, 2019 9:50 am
This is surprising and different from what I remember. I will look into it and compare to what I did for the bridge.
great
HarryvL wrote:
Mon Jul 08, 2019 9:50 am
PS: yes the natural frequency in rad/s is the eigenvalue of the free vibration problem.
than what is the column eigenvalue for? Do you know some paper which explains all this without tons of integrals and matrix operations ... :oops: wikipedia does not a good job here IMHO.
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.

So what does it all mean for the FEM work bench? Simply use the square root of the eigenvalue (second column) and divide this number by 2*PI to get the natural frequency in Hz. If the eigenvalue is negative then report 0.0 and display a warning ("Imaginary natural frequency found. This could either be due to numerical rounding of rigid body modes or physical instability, e.g. buckling. Please check input").

A clear reference for understanding free vibration and eigenvalue analysis is the book Finite Element Procedures by Klaus Jurgen Bathe.
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HarryvL
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Re: extraction of negative eigenvalues

Postby HarryvL » Tue Jul 09, 2019 6:54 am

HarryvL wrote:
Mon Jul 08, 2019 11:43 pm
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)
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.
Jee-Bee
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Re: extraction of negative eigenvalues

Postby Jee-Bee » Tue Jul 09, 2019 8:46 am

HarryvL wrote:
Mon Jul 08, 2019 11:43 pm
A clear reference for understanding free vibration and eigenvalue analysis is the book Finite Element Procedures by Klaus Jurgen Bathe.
For who is interested: http://web.mit.edu/kjb/www/Books/FEP_2n ... inting.pdf
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HarryvL
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Re: extraction of negative eigenvalues

Postby HarryvL » Tue Jul 09, 2019 8:15 pm

HarryvL wrote:
Tue Jul 09, 2019 6:54 am
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).
natural frequency and instability.jpg
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