## 3 Results and Discussion
### Reference Results and Computed Mode Shapes
#### 3.1.2 Spectral Decomposition of the Surface-Averaged Velocity Amplitude  
Reference results show the response of the fan vibration without the presence of a uniform ambient cross-flow field at the fan inlet (\(U_{0}=0\,\mathrm{m/s}\)). Dominant amplitude peaks could be determined, and a comparison of the two operating points OP10 and OP20 could be made. Selected frequencies \(n\) were examined more closely, and the respective reconstructed motion pattern could be compared to associated natural modes determined from a FEM simulation. Figure 3 shows the spectral decomposition of the surface-averaged fan blade velocity amplitude \(v_{z}\) from all \(N_{\mathrm{ScPts}}\) scan points, as defined in Equation (9), for the frequency range \(2.2<n\leq 50\). As annotated above, the first two rotation-rate harmonics \(n=1\) and \(2\) are cut off because of distortion by kinematic effects. In Figure 3, black dotted vertical lines indicate the harmonics of the blade-pass frequency \(n=5\). The data sets' natural frequencies \(n_{\mathrm{e},i}\) are added in dashed vertical lines for \(i=1\) to \(4\). Because of the different rotation rates \(f_{0}\) of the operating points' natural frequencies, these dashed lines do not coincide for the data sets. The base-level noise was detected at \(20\cdot\log_{10}(v_{z,\mathrm{noise}}\cdot 1\,\mathrm{s/mm})\approx-3.75\) from the data. It becomes clear from Figure 3 that the measured mean surface vibration was dominated by the rotation-rate harmonics, that is, where \(n\) had integer values, with \(v_{z}\) amplitudes close to zero in between. Although operating at distinctly different flow rates \(\Phi\), the reference spectra of OP10 and OP20 coincided intensely; they have almost identical integral vibration measures of \(v_{\mathrm{rms}}=362.52\,\mathrm{mm/s}\) and \(363.40\,\mathrm{mm/s}\), respectively. The dominant peak for both operating points is \(n=3\), with similar velocity amplitudes, but the blade-pass frequency harmonics \(n=5\) and \(15\) are also prominent. Noteworthy non-integer values \(n\) are located in the proximity of the first three natural frequencies, especially at \(n_{\mathrm{e},1}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(j\)} & \multirow{2}{*}{\(f_{\mathrm{e},j}\) in Hz} & \multicolumn{6}{c}{**Distinct Associated Frequency in Hz**} \\ \cline{2-7}  & & **Blade 1** & **Blade 2** & **Blade 3** & **Blade 4** & **Blade 5** \\ \hline
1 & 184.7 & 184.96 & 184.70 & 184.72 & 184.73 & 184.77 \\
2 & 432.8 & 432.71 & 432.80 & 432.84 & 432.88 & 432.94 \\
3 & 942.8 & 942.02 & 942.10 & 943.03 & 943.17 & 943.57 \\
4 & 1123 & 1122.4 & 1122.5 & 1122.6 & 1123.6 & 1123.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: First four blade-specific distinct natural frequencies. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\(j\)} & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \(f_{\mathrm{e},j}\) in Hz & 184.7 & 432.8 & 942.8 & 1123 & 1657 & 1948 & 2034 & 2190 \\ \(n_{\mathrm{e},j}\) (FEM) & 4.181 & 9.796 & 21.34 & 25.42 & 37.51 & 44.09 & 46.04 & 49.57 \\ \hline \(n_{\mathrm{e},j}\) (OP10) & 4.269 & 10.00 & 21.79 & 25.95 & 38.29 & 45.02 & 47.01 & 50.61 \\ \(n_{\mathrm{e},j}\) (OP11) & 4.491 & 10.52 & 22.92 & 27.30 & 40.29 & 47.36 & 49.45 & 53.25 \\ \(n_{\mathrm{e},j}\) (OP20) & 4.181 & 9.796 & 21.34 & 25.42 & 37.51 & 44.09 & 46.04 & 49.57 \\ \(n_{\mathrm{e},j}\) (OP21) & 4.254 & 9.968 & 21.71 & 25.86 & 38.16 & 44.86 & 46.85 & 50.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: First eight mean natural frequencies computed under centripetal load from rotation at \(f_{0,\mathrm{e}}=44.18\,\mathrm{Hz}\), including natural frequency numbers respective to the four operating points. FEM: Finite Element Method. In the experimental realization of the commercial fan, natural frequencies of the distinct blades were expected to disperse more than in the numerical model, as a result of manufacturing and installation uncertainties.