I found my error, I had used the "plane rotation" constraint and not the "fixed" constraint. Icons are bit hard to differentiate.

Interesting enough, the "Mechanical analysis solver window output" reports that the analysis was successful, but CalculiX finds that the matrix is singular due to the improperly chosen constraints and cannot calculate the problem. I have to check another FE solver, but I think that the eigenvalue problem should be able to be solved for this case. Nonetheless, that is a CalculiX issue. There seems to be a problem with the FreeCAD interface to CalculiX: we still get a successful response from the mechanical analysis window: 0.8: CalculiX done without error!

Here is the output in the Mechanical analysis solver window:

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```
0.0: Check dependencies...
0.1: Write completed.
0.0: CalculiX binary: /usr/bin/ccx
0.0: Run CalculiX...
0.0: Starting CalculiX...
0.0: CalculiX is running...
0.8: CalculiX stopped.
0.8: b"\n************************************************************\n\nCalculiX Version 2.11, Copyright(C) 1998-2015 Guido Dhondt\nCalculiX comes with ABSOLUTELY NO WARRANTY. This is free\nsoftware, and you are welcome to redistribute it under\ncertain conditions, see gpl.htm\n\n************************************************************\n\nYou are using an executable made on So 31. Jul 13:26:31 CEST 2016\n\n The numbers below are estimated upper bounds\n\n number of:\n\n nodes: 1743\n elements: 1332\n one-dimensional elements: 0\n two-dimensional elements: 0\n integration points per element: 4\n degrees of freedom per node: 3\n layers per element: 1\n\n distributed facial loads: 0\n distributed volumetric loads: 0\n concentrated loads: 0\n single point constraints: 162\n multiple point constraints: 163\n terms in all multiple point constraints: 2095\n tie constraints: 0\n dependent nodes tied by cyclic constraints: 0\n dependent nodes in pre-tension constraints: 0\n\n sets: 5\n terms in all sets: 5879\n\n materials: 1\n constants per material and temperature: 2\n temperature points per material: 1\n plastic data points per material: 0\n\n orientations: 0\n amplitudes: 0\n data points in all amplitudes: 0\n print requests: 0\n transformations: 0\n property cards: 0\n\n *INFO reading *MPC: nonlinear geometric\n effects are turned on\n\n\n STEP 1\n\n Frequency analysis was selected\n\n Nonlinear geometric effects are taken into account\n\n Decascading the MPC's\n\n Determining the structure of the matrix:\n number of equations\n 4585\n number of nonzero lower triangular matrix elements\n 167576\n\n Using up to 1 cpu(s) for the stress calculation.\n\n Using up to 1 cpu(s) for the symmetric stiffness/mass contributions.\n\n Factoring the system of equations using the symmetric spooles solver\n Using up to 1 cpu(s) for spooles.\n\n"
0.8: CalculiX done without error!
0.8: Loading result sets...
```

Here the Python console output:

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```
runCalculix
run CalculiX at: /usr/bin/ccx with: /_tmp/_TestEigFreeCAD-CalculiX/calc/FEMMeshGmsh
calculixStarted()
PySide2.QtCore.QProcess.ProcessState.Running
calculixFinished() 255
PySide2.QtCore.QProcess.ProcessState.NotRunning
PySide2.QtCore.QRegExp('\*ERROR.*\n\n', 1, 0)
CalculiX done without error!
We will load the ccx frd and dat result file.
Read ccx results from frd file: /_tmp/_TestEigFreeCAD-CalculiX/calc/FEMMeshGmsh.frd
FEM: No nodes found in Frd file.
Problem on frd file import. No nodes found in frd file.
FEM: No result object in active Analysis.
Read ccx results from dat file: /_tmp/_TestEigFreeCAD-CalculiX/calc/FEMMeshGmsh.dat
```

My details:

Ubuntu 19.04

FreeCAD 0.18.3