# Dynamic Aeroelastic Response of Stall-Controlled Wind Turbine Rotors in Turbulent Wind Conditions
## 5 Energy-Transfer Characterization of the Pulse (Unstable Oscillatory Regime)  
In the unstable regime, \(\lambda_{1}\) becomes negative (that is, the attenuation of the aerodynamic damping becomes an amplification of the oscillatory amplitude), producing an initial exponential expansion that will be studied in detail in Section 5.1. The initial expansion continues until the axial displacement reaches a certain level. From then on, a second damping mechanism emerges that competes with the existing \(\lambda_{1}\) amplification associated with the first mechanism. This evolution then starts to depart from the exponential behavior until the amount of energy extracted from the flow by the first mechanism is equivalent to that dissipated by the second. At that point, the oscillation levels out at a fixed amplitude. This effect shall be discussed in Section 5.2. As in the stable regime, the axial displacement of the blade \(U_{hx}\) around the equilibrium position (depicted in the upper panel in Figure 15 for a wind-speed regime of 18 m/s) reflects the main component of the instantaneous amount of elastic energy stored in the rotor as an oscillatory system. The time evolution of the blade deflection shows the same qualitative behavior for all of the wind speeds in this regime (17.5 to 25 m/s), but with different values of the initial exponential expansion. As in the stable regime, \(\lambda_{1}\) depends exclusively on the mean wind speed. The effect of the kinetic energy content of different pulses manifests as a delay in time, and all of the curves for a certain wind speed  
Figure 15: (**a**) Time evolution of the axial displacement of the blade at a wind-speed regime of 18 m/s for pulses combining different values of amplitude and time span. (**b**) The same curves, but delayed in time for the exponential expansion to build up the same amount of energy contained in their respective pulses. Figure 16: (**a**) Time evolution of the axial displacement of the blade at a wind-speed regime of 18 m/s for pulses combining different values of amplitude and time span. (**b**) The same curves, but delayed in time for the exponential expansion to build up the same amount of energy contained in their respective pulses. The red and blue lines indicate respectively the enveloping curves for the exponential component and maximum amplitudes of the different signals. Then, the value of \(t_{d}\) for each pulse can be directly computed by using the parameters obtained by fitting each pulse's exponential curve. Equations (1) through (5) show the evaluation of \(t_{d}\) by equating the exponential expression for each individual pulse with the exponential expression for the reference pulse (that is, collapsing the curves for all of the pulses). \[U_{h}=A_{0}e^{-\lambda_{1}(t_{mf}-t_{d})} \tag{1}\]  
\[U_{h_{ref}}=A_{0_{ref}}e^{-\lambda_{1}t_{mf}} \tag{2}\]  
\[A_{0_{ref}}e^{-\lambda_{1}t_{mf}}=A_{0}e^{-\lambda_{1}(t_{mf}-t_{d})} \tag{3}\]  
\[\frac{A_{0_{ref}}}{A_{0}}=\frac{e^{(-\lambda_{1}t_{mf}+\lambda_{1}t_{d})}}{e^{ -\lambda_{1}t_{ref}}} \tag{4}\]  
\[t_{d}=\frac{1}{\lambda_{1}}Ln\left(\frac{A_{0_{ref}}}{A_{0}}\right) \tag{5}\]  
Validation for the Unstable Regime of the Hypothesis Connecting the Time Delay and the Kinetic Energy Content of the Pulse  
In order to validate the hypothesis that the time delay, \(t_{d}\), is indeed associated with the time required for the exponential expansion to build up the same amount of energy contained in the respective pulse, we analyzed the relationship between two parameters: pulse kinetic energy and time delay. In this derivation, we also validate the hypothesis that the energy-transfer mechanism is exactly the same as in the stable regime, but with a negative value of \(\lambda_{1}\), i.e., an amplification instead of attenuation of the oscillatory blade deflection.