The spatial Fourier transform was performed on the results at the frequencies from Table . This gave the breakdown of the waves in each frequency. The following figures show the spatial wavenumber comparisons at each frequency for the different models. Note that the xaxis reports the spatial wavenumber (ks = 1/λ) as opposed to angular wavenumber. The solid model for Case 3b was animated at the peak frequencies to provide an example of the shape of the plate vibration. This model used the incompatible mode formulation brick elements, which according to the Abaqus user guide, [11], provides a solution close to the theoretical solution. Note that the line of nodes used for the Fourier analysis is highlighted on the model.
The models are compared for the first peak response in Figure . The first peak occurs at 37 Hz for all of the models. Additionally, all of the models except case 3a contain the same wavenumber content, and thus the same plate vibration shape, at peak 1. It is apparent from the matching results of Case 1 and Case 2 that using beam or shell elements as stiffeners on a plate modeled with shell elements produces consistent plate response.
Figure Peak 1 Wavenumber Comparison
The wavenumber plot for the second peak amplitude is shown in Figure . As with the first resonant response at 37 Hz, all models predict the same resonant frequency at 96 Hz. Once again Cases 1 and 2 are the same, and all cases contain the same wavenumber content and thus shared a similar plate response shape. However, Cases 3a and 3b predict different amplitudes, with case 3a predicting lower response than the plate and beam models and case 3b predicting higher response. It was expected that the full3D solid element models would begin to diverge from the plate and beam models as the excitation frequency of the plate increases. However, it is interesting to note that the two different solid element formulations have a large amplitude discrepancy with each other, and that they bound the shell and beam solutions.
Figure Peak 2 Wavenumber Comparison
Figure shows the wavenumber comparison for the third resonant peak. This is the first comparison where the resonant frequencies between the modeled cases do not match. For all cases, the wavenumber content is very similar, suggesting similar plate motion. Case 1 and Case 2 still line up exactly, and predict the response at 227 Hz. Case 3a has good agreement with the Case 1 and Case 2 model predictions. However, the resonance is predicted at 223 Hz. Case 3B predicts the resonance at 228 Hz, and also has a higher amplitude for the wavenumber content.
Figure Peak 3 Wavenumber Comparison
Figure shows the wavenumber comparison of the fourth resonance peak. The wavenumber content is similar for all cases, but the amplitudes are different. Cases 1 and 2 continue to predict matching wavenumber content and resonance frequencies. Cases 3a and 3b show more divergence. Cases 1 and 2 predict the lowest resonant frequency of 255 Hz. Case 3a predicts 259 Hz, while Case 3b predicts 262 Hz.
Figure Peak 4 Wavenumber Comparison
The fifth resonant peak wavenumber comparison is shown in Figure . Case 1 and Case 2 match in both the wavenumber content and the predicted resonant frequency. Cases3a and 3b are very similar in the wavenumber content, but Case 3a predicts the resonant frequency at 288 Hz whereas Case 3b predicts it at 280 Hz. Cases 1 and 2 predict 279 Hz which is close to the Case 3b prediction. All cases share the same wavenumber content trend.
Figure Peak 5 Wavenumber Comparison
Figure shows the wavenumber comparison for the sixth resonant peak. The wavenumber content differs between all the cases except Case 1 and Case 2, suggesting differences in the plate motion. Case 1 and Case 2 predict the same wavenumber content at low amplitudes for the same resonant frequency of 900 Hz. However, the resonant frequency for Cases 1 and 2 is approximate since it was difficult to determine from the drive response in Figure . Although there was an increased amplitude at 900 Hz, the amplitude was small compared to that seen in the solid element models. Checking the wavenumber content at surrounding frequencies did not reveal any other possible resonances. A higher frequency resolution, such as 0.25 Hz spacing, may have revealed a stronger resonance in the Case 1 and Case 2 models. However, it is likely that the beam and plate elements are not capable of capturing the higherorder mode shapes present at the frequency. The amplitude for the wavenumber content of Case 3a is also low, but with different wavenumber content than Case 1 and Case 2. For this case, the resonant frequency was predicted at 884 Hz. Case 3b has the highest amplitude and predicts the resonant frequency at 881 Hz.
Figure Peak 6 Wavenumber Comparison
The wavenumber comparison for the seventh peak is shown in Figure . The wavenumber content and the resonant frequencies differ for all the cases except between Case 1 and Case 2. Case 1 and Case 2 predict a resonant frequency of 966 Hz and share the smallest amplitude in wavenumber content. This is probably due to the low amplitude of the predicted resonance at 966 Hz. This is similar to what was observed in Figure . The resonant response is predicted at 945 Hz and 952 Hz for the Case 3a and Case 3b models, respectively. The wavenumber content for Case 3a has a large contribution from the k_{s} = 0.05 wavenumber, while the contribution is distributed across more wavenumbers for Case 3b. This means that multiple waves of different wavelengths are being predicted by the incompatible mode solid element mesh of Case 3b, while the reduced integration solid element mesh of Case 3a predicts one dominant wavelength.
Figure Peak 7 Wavenumber Comparison
The last two peaks show more of the wavenumber content being dominated by the higher wavenumbers. This corresponds to smaller wavelengths in the plate which is expected for the higher frequencies.
