# Vibration-Based Fingerprint Algorithm for Structural Health Monitoring of Wind Turbine Blades
## 3 Algorithm
### Export and Visualisation  
Finally, detected Type 1 and Type 2 vibrations were combined to a characteristic vibration-based fingerprint. Each fingerprint consisted of the following properties:  
* Type 1 vibrations \(V_{1,j_{1}}\) with \(j_{1}=1,...,J_{1}\) including detected peaks \(\hat{p}_{i,n}\in P_{v1,j_{1}}\), first-degree fitting models \(f_{v1,j_{1}}\) and corresponding data ranges \(\epsilon_{2,j_{1}}\).
* Type 2 vibrations \(V_{2,j_{2}}\) with \(j_{2}=1,...,J_{2}\) including detected peaks \(\hat{p}_{i,n}\in P_{v2,j_{2}}\), second-degree fitting models \(f_{v2,j_{2}}\) and corresponding data ranges \(\epsilon_{2,j_{2}}\).  
Those fingerprints were stored for the subsequent analysis of results. Algorithm results were visualised by displaying averaged spectrograms, fingerprints and sensor positions (see, for example, Figure 11). Thereby, characteristic vibrations of Type 1 (\(v=v_{1},j=j_{1}\)) and Type 2 (\(v=v_{2},j=j_{2}\)) were visualised by displaying fitting models \(f_{v,j}\) as solid lines. The prominence of assigned peaks and the range of vibrations were considered by: (i) normalising the prominence \(p_{i,n}\) of assigned peaks \(\hat{p}_{i,n}\) to [0,1] (relative width); and (ii) scaling the range \(\epsilon_{2,j}\) of each vibration (absolute width) with the normalised prominence. In doing so, the maximum width of each visualised vibration corresponds to the deviation of peak locations \(\hat{l}_{i,n}\in P_{v_{2},j_{1}}\) of the respective vibration. In addition, the variation of the width over the reference variable \(\hat{r}\) corresponds to the prominence of assigned peaks. Thereby, the following properties can be visualised:  
* A large absolute width of vibrations indicates that the vibration is difficult to localise, e.g., due to an additional dependency on further environmental or operational parameters.
* A small relative width of vibrations indicates that the vibration is difficult to detect, e.g., either not excited or hidden in noise for the related reference \(\hat{r}\).
* A large relative width of vibrations indicates that the vibration can clearly be identified in the linear-scale spectrogram.  
In addition, fingerprints were compared for several sensors. Two vibrations \(V_{m_{1}}\) and \(V_{m_{2}}\) were detected as _joint_ or _overlapping_, i.e., being caused by the same physical behaviour of the blade with high probability if any of the following conditions applied:  
* At least 30% of assigned peaks overlapped: A number of joint peaks \(N_{\beta,joint}\geq 0.3\cdot\min(N_{\beta,m_{1}},N_{\beta,m_{2}})\) was found with \(N_{\beta,m1}\) being the number of peaks \(\hat{p}_{i,n}\in V_{m_{1}}\) and \(N_{\beta,m2}\) being the number of peaks \(\hat{p}_{i,n}\in V_{m_{2}}\).
* The difference at any point along the reference grid \(\hat{r}_{i}\) was smaller than 1.5%, i.e., \(|V_{m_{1}}(\hat{r}_{i})-V_{m_{2}}(\hat{r}_{i})|<0.015\hat{r}\).  
In the case of joint vibrations being detected, vibrations \(V_{m_{1}}\) and \(V_{m_{2}}\) were averaged to increase clarity of visualisation. Averaged vibrations are displayed by alternating coloured dots and error bars represent the width of single vibrations. An example for the visualisation of joint vibrations can be seen in Figure 12.  
Figure 10: Refinement of detected Type 2 vibrations by using 2 nd degree models, examples shown for vibration \(V_{2,3}\) (**left**) and \(V_{2,5}\) (**right**) of an exemplary sensor. Model range \(\epsilon_{2}\) was identified by using vibration scanning and vibration filtering, as described in Figure 8. Robust fitting with spectrogram amplitudes was used as weights, with coloured areas within range \(\epsilon_{2}\) corresponding to non-zero weights.