# Eigenanalysis of an isotropic beam using BeamDyn

Dear all,

I am currently planning to use BeamDyn for the analysis of a composite rotors eigenfrequencies/ eigenmodes.

For this I first wanted to validate the tool and get an idea of deviations resulting from a computation with BeamDyn. To do so I started with the computation of a simple non-rotating aluminum beam (isotropic) and compared it to the analytical solution.
My procedure was as followed: I first created the mass and stiffness matrizes for BeamDyn using BECAS. Then I ran BeamDyn for one iteration/a single timestep with different settings (trapezoidal/gaussian, different order of interpolation basis function, number of members/key points). After that I used a python script that performs the eigenanalysis based on the stiffness and mass matrizes written to the .sum file. I did the analysis with the boundary conditions fixed-free and free-free as described by you in viewtopic.php?t=2175#top .

During the evaluation of the results I had some interesting insights, but also a few questions I want to share (some representative results can be found at the end of the forum post):

1.) Changing the order of the interpolation basis function seems to have a big influence on the results. When increasing the order the results steadily improve up to order_elem=6. For higher orders the results decline (especially for the first two eigenfrequencies, no matter which quadrature, no. of members and keypoints are chosen). Does this observation match your current experiences and how could this be explained?
From my understanding and interpretation of the presentation OpenFASTBeamDynLinearization_EnvisionMeeting181030_Jonkman_Presentation attached to the aforementioned and linked forum topic higher orders than order_elem=6 should rather improve the results.

2.) In order to get a satisfying resolution of the eigenmodes I started testing the gaussian quadrature with more members. This worked out fine for up to two members. For more members the script fails to detect the first natural frequency. Is there a way to get a fine resolution of the eigenmodes as well as a small deviation for the natural frequency? Or is there a way to get sound results for low frequent natural frequencies using the gaussian quadrature with multiple members

3.) When looking at the eigenfrequencies I could observe that the relative deviation from the analytical solution rises with increasing frequency. (For a order of interpolation function of 6 and fixed-free boundary conditions) the deviations of the first two frequencies were between 3 and 5 %. The deviations of the third, fourth and fifth frequencies were ~10-11, 17.5 and 24 per cent respectively (free-free: f1: 0-3 %, f2: 8-9 %, f3: 11-15 %). The high frequencies don’t matter too much to me. However, I wanted to share these findings and ask if the deviations seem to be reasonable.

4.) A variation of the number of key-points affects the results of the computation. I tested computations with 13, 20, 38 and 41 key-points. The results for 20 key-points were by far the closest to the analytical solution. Why could this be possible? Shouldn’t the isotropic beam be independent of the number of key-points?

I was always running a unmodified version of BeamDyn. The eigenmodes for all kinds of settings had the same shape as with a analytical solution. If you have any further suggestions on how I could improve the accuracy by using BeamDyn, I would be really thankful. The tool already helped me a lot so far and I hope that my results can help you a bit as well.

Best regards,
Michael

## Eigenfrequency: f1 [Hz] |f2 [Hz] |f3[Hz] Fixed-Free: Analytical: 82.9 |519.3 |1453 Trapez. Order 6: 87.4 (5.5%) |494.4 (4.8%) |1309 (9.9%) Gauss 1 Member: 86.7 (4.6%) |494.0 (4.9%) |1295 (10.9%) Gauss 2Mem O6: 79.9 (3.6%) |505.9 (2.6%) |1295 (10.9%) Gauss 2Mem O7: 101.8(22.9%) |505.9 (2.6%) |1295 (10.9%) Gauss 3Mem O6: 198.7 (140%) |523.7 (0.9%) |1303 (10.3%)

Free-Free:
Analytical: 527.2 |1453 |2849
Trapez. Order 6: 512.9 (2.7%) |1329 (8.5%) |2519 (11.6%)
Gauss 1 Member: 515.2 (2.3%) |1326 (8.7%) |2479 (13.0%)
Gauss 2Mem O6: 527.6 (0.1%) |1327 (8.7%) |2415 (15.2%)
Gauss 2Mem O7: 516.9 (2.0%) |1324 (8.9%) |2414 (15.3%)
Gauss 3Mem O6: 543.6 (3.1%) |1337 (8.0%) |2417 (15.1%)

Dear Michael,