# Vibration-Based Fingerprint Algorithm for Structural Health Monitoring of Wind Turbine Blades
## 3 Algorithm
### Spectrum Averaging  
In the following analysis, only measurements during stationary operational conditions were included, i.e., measurements during steering processes or non-stationary rotation speeds were not considered. Since measurements were conducted for a short period of 10 s in regular measurement campaigns, the Fourier transform was sufficient to monitor operational and environmental variability and no additional temporal resolution (e.g., by using the Short-time Fourier transform (STFT)) was needed. For each measurement campaign, the Discrete Fourier Transform (DFT) \(X(k)\) of the signal \(x[n]\) was computed, and single-sided power spectra \(S(k)\) were derived as  
\[X(k)=\frac{1}{\hat{N}}\sum_{n=0}^{N-1}x[n]e^{\frac{-2m\text{th}}{N}}, \tag{1}\]  
\[S(k)= \begin{cases}|X(k)^{2}|&\text{if}\ \ k=0\\ |2X(k)^{2}|&\text{if}\ \ 0<k\leq\frac{\hat{N}}{2},\end{cases}  \tag{2}\]  
with the discrete frequency bins \(k=0,1,...,\hat{N}-1\) corresponding to measured frequencies \(f=0,...,\frac{f_{s}}{2}\) with \(f_{k}=k\cdot\frac{f_{s}}{N}\) and \(f_{s}\) being the sampling rate of the accelerometer. The size of the Fourier transform was selected as \(\hat{N}=2^{16}\) to increase precision of frequency detection. The signal \(x[n]\) was derived from the vector sum \(av=\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}\) of triaxial acceleration measurements \(a_{i}\) with \(i=x,y,z\). Thereby, vibration information from three measurement axes could be combined to one fingerprint per sensor location. This procedure also enabled comparability between different sensors in case of a non-ideal alignment of sensor axes to the dimensions of the rotor blade due to mounting tolerances. Most signal energy was contained in the direct current (DC) component of the acceleration signal, which corresponded to centripetal acceleration. Since we focused on analysing vibrations with smaller amplitudes, the DC component was removed. In addition, a Hann window \(w\) was applied to account for discontinuities at beginning and end of each measurement campaign. Hence, the signal  
\[x[n]=w\cdot\left(av[n]-\frac{1}{N}\sum_{n=0}^{N-1}av[n]\right) \tag{3}\]  
was used to compute single-sided frequency spectra, in the following denoted as _spectra_. Figure 5 displays an exemplary triaxial acceleration measurement and the computed vector sum \(av\) (left) as well as the corresponding single-sided frequency spectrum (right). Figure 4: Overview of the proposed fingerprint algorithm in six analysis steps. Processing blocks are visualised for each step and presented in the corresponding sections. Next, environmental and operational conditions were taken into account to analyse spectrograms. To monitor operational and environmental variability, a separation of variables is needed and sufficient measurements have to be collected for each variable. In our case, measurements were collected during a large range of rotation frequencies, which additionally were expected to have the strongest impact on vibrations. Therefore, the demonstration of the algorithm was based on the rotation frequency as reference variable \(r\). However, any other reference can be applied to the algorithm if a sufficient amount of data is available. Since measurement campaigns were not scheduled with the reference variable, the number of available measurements was unevenly distributed across the reference. To facilitate visualisation and detection of vibrations, spectra were first sorted by the reference \(r\), i.e., original spectra \(S_{1},S_{2},...,S_{L_{M}}\) corresponded to reference measurements \(r_{1},r_{2},...,r_{L_{M}}\) with \(r_{j}\leq r_{j+1}\). The resulting set of spectra is denoted as _non-linear-scale spectrogram_ in the following.