## 4.Conclusions

The results of the wavenumber analysis show that there is little difference in wavenumber content for all modeled cases for excitation frequencies below 300 Hz. By 800 Hz, the wavenumber content varies greatly between the modeled cases. Different wavenumber content means that the actual shape of vibration is different between models. Since the models compared have identical

geometry and mesh resolution, the changes in wavenumber content are due to the element formulations.

The Case 1 model (shell element plate with beam element stiffening) and the Case 2 model (shell element plate with shell element stiffening) provided the same response throughout the entire frequency range. This suggests that refining the stiffeners by switching from beams to plates did not affect the modal response of the plate. However, at frequencies above 800 Hz, there appeared to be little to no resonant character present in the Case 1 and Case 2 models, whereas the Case 3a (solid element, reduced integration) and Case 3b (solid element, incompatible mode formulation) continued to predict resonant character.

It should be noted that the small amplitude differences seen at low frequencies between model wavenumber responses that otherwise followed the same trend is due to the frequency resolution of the analysis. An attempt was made to plot the wavenumber response of each model at its resonant frequency. However, the 1 Hz frequency resolution was not sufficient to fully align all resonant frequencies. For a future study, it is recommended that an Eigen study be performed throughout the frequency range to establish the exact natural frequencies of vibration for each model. Performing the Fourier analysis at the exact resonant frequency of each model should result in better amplitude correlation.

In addition to isolating the exact natural frequencies, future work should include evaluation of different plate geometries and stiffening arrangements, such as an isogrid stiffening arrangement, curved plates, and thick plate structures (plate thickness less than 1/10 the length). Higher order elements, such as the 20-node brick element, should also be evaluated. Finally, a two-dimensional wavenumber analysis technique could be used to characterize the entire surface of the plate.

## 5.References

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