## 2 Materials and Methods
### Wind Tunnel Fan Test Rig  
The influence of an inlet cross-flow on the fan blade vibration was measured using LSV inside a customized fan test rig inside a Gottinger-type wind tunnel with a 2.80 m long open test section and an exit nozzle of \(1.87\cdot 1.40\) m\({}^{2}\). A commercial low-pressure axial fan (\(D_{\mathrm{fan}}=300\) mm diameter) with five forward-skewed blades and a hub-to-tip ratio of \(\
u_{\mathrm{hub}}=0.342\), operating with its axis perpendicular to a uniform ambient velocity \(U_{0}\), was examined. Figure 1 shows the experimental setup, while the fan design is illustrated in Figure 2.  
Figure 1: Experimental setup in wind tunnel test section. Figure 2: Experimental subject axial fan with forward-skewed blades. The test fan was examined, operating in its standard shroud without a guard grille and held in place by four thin struts with its outlet mounted to a square duct section, \(1.5D_{\text{fan}}\) above the wind tunnel test section's floor. Behind a short flow-straightening device, the stagnation pressure \(p_{\text{m},i}\) was captured relative to the wind tunnel ambient pressure with 81 Kiel-type stagnation pressure probes, and was surface averaged to the total-to-total fan pressure \(\Delta p_{\text{tt}}=\sum_{i=1}^{S_{\text{f}}}p_{\text{m},i}/81\). The equally spaced \(9\times 9\) grid of pressure probes inside the square duct (\(D_{\text{duct}}\times D_{\text{duct}}=300\times 300\,\text{mm}^{2}\)) is described in [19]. The fan flow rate \(\dot{V}\) was computed from static pressure loss over five standard nozzles inside a settling chamber at the end of the outlet duct, in accordance to the fan test rig standard ISO 5801 (not displayed in Figure 1). The experimental setup was previously used to investigate the integral inlet flow field influence on the characteristic fan curve, as shown in [20]. To measure the fan rotation rate \(f_{0}\) and to synchronize the test fan with the co-rotating LSV's derotator, a rotary encoder was mounted directly to the fan axis at its center. With the wind tunnel either turned off or operating at a \(U_{0}=10\,\text{m/s}\) wind velocity, the characteristic cross-flow coefficient  
\[\mu=\frac{U_{0}}{\pi D_{\text{fan}}f_{0}} \tag{1}\]  
differed slightly around 0.25, depending on the rotation rate \(f_{0}\) of the commercial fan. At one low flow rate (\(\Phi\approx 0.17\)) called operating point 1 (OP1), and at a higher flow rate, OP2 (\(\Phi\approx 0.28\)), with similar flow-rate coefficients  
\[\Phi=\frac{4\,\,\dot{V}}{\pi D_{\text{fan}}^{2}\cdot(1-\
u_{\text{hub}}^{2}) \cdot\pi D_{\text{fan}}f_{0}} \tag{2}\]  
a total of four operating points were measured with \(\mu=0\) (OP10 and OP20) and \(\mu\approx 0.25\) (OP11 and OP21), as listed in Table 1 along with the respective total fan pressure coefficients  
\[\Psi_{\text{tt}}=\frac{2\,\Delta p_{\text{tt}}}{\rho\,(\pi D_{\text{fan}}f_{0} )^{2}} \tag{3}\]  
Tables 1 and 2 list the expanded combined uncertainties \(U_{\text{c}}(x)\) of the quantities \(x\) for a 95% level of confidence, as defined by  
\[U_{\text{c}}(x)=\sqrt{\sum_{x_{i}}\left(\frac{\partial x}{\partial x_{i}}\cdot U (x_{i})\right)^{2}} \tag{4}\]  
in [21] with the measured quantities \(x_{i}\) and their expanded uncertainties \(U(x_{i})\). The measured quantities \(x_{i}\) were the pressure measurements of the total fan pressure \(\Delta p_{\text{tt}}\), the wind tunnel velocity stagnation pressure used for \(U_{0}\), and the multi-nozzle pressure loss for flow rate \(\dot{V}\); the density \(\rho\), the rotation rate \(f_{0}\), the fan dimensions \(D_{\text{fan}}\) and \(\
u_{\text{hub}}\), and the coefficient uncertainties for the computation of \(\dot{V}\) with multi-nozzles listed in ISO 5801. Because the fan dimensions were a dominant source of uncertainty in the computation of \(\mu\), \(\Phi\) and \(\Psi_{\text{tt}}\) but did not change between data sets, the combined uncertainties of Table 2 provide a better estimate of possible uncertainties caused by random error.