## 2 Materials and Methods
### Laser Scanning Vibrometry  
The axial blade-surface velocity distribution was captured using a PSV-500 Scanning Vibrometer coupled with a PSV-A-400 derotator (Polytec GmbH, Waldbronn, Germany). The method is also known as tracking laser (scanning) vibrometry, and has been applied to rotating machinery over the recent years in helicopter or automotive research; see for example, [22; 23]. The method facilitates the comparatively fast capturing of a rotating object's surface velocity at many probe locations. In comparison to accelerometers, this non-intrusive method also has the advantage of not imposing additional weight or aerodynamic obstacles in the form of sensors and wiring onto the measuring object. The fan blades were sprayed with a reflective coating, such that their local surface velocity could be computed from the Doppler frequency shift of the reflection of a continuous-wave 633 nm HeNe laser beam. Successively, equally distributed probe locations (\(N_{\mathrm{ScPts}}=400\)) with a typical spacing of 10 mm between them, as shown in Figure 2a, were measured by the co-rotating beam at a sampling rate of 12.5 kHz. On each scanning point, 30 samples with 640 ms velocity signals (around 27 full fan rotations) were successively captured and transferred to the frequency domain. Hanning windows with 70% overlap were used to average the 30 samples, resulting in frequency domain data sets with a bandwidth of \(B=5\) kHz resolved to discrete frequencies \(k\cdot\Delta f\), where \(\Delta f=1.5625\) Hz and integer values \(k\) are defined as in Equation (6). With the respective fan rotation rate \(f_{0}\), discrete dimensionless frequency coefficients  
\[n_{k}=k\cdot\Delta f/f_{0} \tag{5}\]  
result for \(n=f/f_{0}\). Plain kinematic effects were observed at rotation rate \(f_{0}\), caused by the uncertainty in the parallel alignment of the laser beam and fan axis. To exclude such rigid-body rotation effects from the blade vibration analysis, the frequency domain was filtered to a minimum integer \(k_{0}\) above the second rotation-rate harmonic, such that  
\[k\;\epsilon\;\left\{k_{0},\,k_{0}+1,\,\ldots,\,B/\Delta f\right\},\;\text{where }k_{0}=\left\lceil\frac{2.2\,f_{0}}{\Delta f}\right\rceil\;\text{and }B/\Delta f=3200 \tag{6}\]  
The scanning laser was accompanied by a reference beam signal \(v_{z,\text{ref}}(n_{k})\), which remained fixed on one probe location for phase reference to the individual measured velocities \(v_{z,i}(n_{k})\) (for \(i=1\ldots N_{\mathrm{ScPts}}\) scan points). Using the recorded cross-spectral density between the two signals \(G_{i,\text{ref}}(n_{k})\), a phase offset of point \(i\) at \(n_{k}\),  
\[\theta_{i,\text{ref}}(n_{k})=\tan^{-1}\left(\frac{\text{Im}\left\{G_{i,\text {ref}}(n_{k})\right\}}{\text{Re}\left\{G_{i,\text{ref}}(n_{k})\right\}}\right) \tag{7}\]  
could be computed. This allowed for the reconstruction of a relative motion pattern:  
\[\tilde{v}_{z,i}(n_{k})=v_{z,i}\cdot\cos\left(\theta_{i,\text{ref}}(n_{k})\right) \tag{8}\] \begin{table}
\begin{tabular}{l l c c c c} \hline \hline  & & \(\mathbf{U_{0}}\) **in m/s** & \(\mathbf{f_{0}}\) **in Hz** & \(\mathbf{\hat{V}}\) **in m3/s** & \(\mathbf{\Delta p_{\text{tt}}}\) **in Pa** \\ \hline \multirow{2}{*}{OP1} & OP10 & 0.0 & \(43.27\pm 0.04\) & \(0.412\pm 0.008\) & \(141.1\pm 2.2\) \\  & OP11 & \(10.0\pm 0.29\) & \(41.13\pm 0.04\) & \(0.408\pm 0.008\) & \(95.8\pm 2.0\) \\ \hline \multirow{2}{*}{OP2} & OP20 & 0.0 & \(44.18\pm 0.04\) & \(0.706\pm 0.013\) & \(90.8\pm 2.0\) \\  & OP21 & \(10.0\pm 0.29\) & \(43.42\pm 0.04\) & \(0.717\pm 0.013\) & \(14.8\pm 1.6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Examined operating points with expanded combined uncertainties (dimensional).