# Dynamic Aeroelastic Response of Stall-Controlled Wind Turbine Rotors in Turbulent Wind Conditions
## 4 Energy-Transfer Characterization of the Pulse (Stable Oscillatory Regime)  
The axial displacement of the blade \(U_{lx}\) (depicted in Figure 5a for a wind-speed regime of 8 m/s) reflects the main component of the instantaneous amount of elastic energy stored in the rotor as an oscillatory system. We used it as an observable to construct a reduced-order characterization of the complex nonlinear oscillatory behaviors in terms of only one degree of freedom. We conducted this experiment for a wide variation of pulses in the short-duration zone, as in the classification chart shown in Figure 4, by combining different values of pulse amplitude and pulse time span. Figure 6 shows the code of the colors and markers used to identify the results for the pulses that combined different values of amplitude and time span. Figure 5: (**a**) Time evolution of the axial displacement of the blade at a wind-speed regime of 8 m/s for pulses that combine different values of amplitude and time span. (**b**) The same curves normalized by the kinetic energy content of their respective pulses. When normalized by the kinetic energy content of the pulses, as shown in Figure 5b, all of the displacement curves collapse into a unique signal, which scales as the time evolution of the rotor's elastic vibrational energy. This has the units of an equivalent flexibility, N\({}^{-1}\), or the inverse of a stiffness \(1/K_{el}\). We repeated the same experiment for several wind speed conditions that covered the entire stable regime and found the same systematic behavior. For example, Figure 7 shows the results of the kinetic energy normalization for a wind speed of 12 m/s, and Figure 8 shows the results for a wind speed of 16 m/s. The value of the equivalent stiffness mentioned above, \(K_{el}\), which serves as a scaling factor for the rotor's elastic vibrational energy, depends exclusively on the mean wind speed. This is consistent with the average deflection of the blades around which the oscillations occur. The higher the mean wind speed is, the higher the value of this \"pre-stiffening\" will be. Figure 6: Code of colors and markers used to identify the results for pulses that combine different values of amplitude and time span. Figure 7: _Cont._  
Figure 8: _Cont._  
Figure 7: (**a**) Time evolution of the axial displacement of the blade at a wind-speed regime of 12 m/s for pulses that combine different values of amplitude and time span. (**b**) The same curves normalized by the kinetic energy content of their respective pulses. Frequency Content in the Stable Oscillatory Regime  
In analyzing the frequency content in the stable regime, we found the appearance of five frequencies (only three of them were dominant), which showed different proportional contributions in four different subranges along the stable wind-speed range. Figure 9 shows the spectra for the first subrange, which extends from 5 to 8 m/s, where only the first two frequencies (0.87 and 1.2 Hz) are dominant. The upper panel shows the frequency content along the time evolution (i.e., until oscillations are totally dissipated), where the 0.87 Hz frequency dominates, with a relatively small contribution of the 1.2 Hz frequency. The lower panel shows the spectrum for the long-term part of the same time signals. The 0.87 Hz frequency rapidly decays, while the 1.2 Hz component--even though it is smaller in the beginning--persists longer. In the long term, there is also a minimal contribution of the non-dominant 4.2 Hz frequency. Figure 10 shows the spectra for the second subrange, which extends from 9 to 10 m/s. In this subrange, the 0.87 Hz frequency is dominant in the early stages, but the contribution of the 1.2 Hz frequency becomes more prominent. The long-term spectra for the same signals only show the 1.2 Hz component, which persists even longer than in the first subrange. Again, there is also a minimal contribution of the non-dominant 4.2 Hz frequency. Figure 11 shows the spectra for the third subrange, which extends from 12.5 to 12.5 m/s.