A closer look at the tip oscillations of blade 2 (see Fig. e) reveals a vibration ramp up between -30 and -43° yaw misalignment. This ramp up can be visually split into two segments: from -30 to -33°, where amplitudes show a minor increase, and from -33 to -43°, where the growth rate of the tip displacement is larger, reaching a maximum tip displacement amplitude of 3.5 m at the end of the segment.

When looking into the total energy exchange between fluid and solid for this period, the edgewise excitation becomes more evident. In this context, the aerodynamic energy Enode is utilized, which is calculated via the force projection of each blade surface face onto the beam nodes as follows: 7Enode=∫t0n⋅TFaero⋅udt. Here, t0 denotes the time at which the first vibration cycle begins, while T signifies the duration of each deformation period. The number of cycles is represented by n, and u denotes the velocity of the node. The instantaneous aerodynamic force Faero acting on each beam node is computed using a weighted projection method, which employs the internal shape function of each beam element to project the forces from the surface faces onto the nodes of the element.

In Fig.  the summed edgewise (panel a) and flapwise (panel b) energy for each monitored spanwise position is shown. A positive energy value corresponds to energy being injected into the blade structure, and a negative energy value corresponds to energy being absorbed by the fluid (i.e., aerodynamic damping). For the edgewise component in Fig. a, a relatively weak damping for the inner half of the blade is visible, where the minimal negative value corresponds to -19 kJ. The majority of the exchanged edgewise energy is found for the outer half of the blade, with a maximum value for a relative spanwise position of 0.9 at a magnitude of 80 kJ. The total net energy for the ramp-up phase results in 240.86 kJ, which means that energy from the fluid is transferred into the blade and leads towards increasing edgewise deformations. A contrary picture is shown for the flapwise component in Fig. b. The majority of the blade is found to be exposed to negative energy exchange, which means that the flapwise vibration direction of the blade is being damped out. The total sum of flapwise energy is found to be -200.14 kJ.

For yaw misalignments between -43 and -53°, the edgewise vibration amplitudes of blade 2 decrease linearly until almost the whole vibration is damped out. This can also be seen in Fig. e. The paths of the displacement peaks almost mirror the blade behavior of the ramp-up phase.

A closer look at the edgewise energy over relative span shows two large damping regions on the blade (see Fig. a). From 0.05≤r/R≤0.45 the summed edgewise energy is negative, i.e., contributes to the damping of the motion in this direction. The highest damping energy magnitude is found around 0.19r/R, with a value of -17 kJ. At 0.61r/R the summed section energy contributes to excitation, with a peak value of ≈ 42 kJ. From 0.65≤r/R≤1.0, the blade sectional energy contributes to the blade vibration damping, with its highest damping magnitude being -27 kJ at r/R=0.7. Although the mid part of the blade is extracting energy from the fluid, the total edgewise energy exchange summed over the whole blade span within this period is found to be -87 kJ and therefore leads to damping overall, which is in accordance with the just-discussed decaying amplitude.

The flapwise accumulated energy, as shown in Fig. b, gives a similar pattern as for the ramp-up phase. From r/R=0.2 outwards, all sectional flapwise energy components show negative values reaching their peak between 0.9≤r/R≤0.95 and a value of -40 kJ. Over the whole blade, the flapwise aerodynamic energy sums up to -215 kJ. This means the flapwise component is being damped down during the whole investigated yawing period.

An analogous analysis of the other two blades indicates a much smaller fluid–solid energy transfer, consistently with the edgewise displacements of blade 1 (see Fig. d) and blade 3 (see Fig. f), where no vibration build up is found.

An even slower yaw rate would further amplify blade vibrations due to longer exposure to critical yaw misalignments, whereas a faster yaw rate would mitigate them. An extreme scenario would be for a blade at a fixed yaw angle within this region, which is analyzed in Sect. , where a more detailed analysis of the vibration modes is performed.

To contrast situations of high- and low-blade-vibration amplitudes, single blade setups of yaw misalignments of -37 and -60° at an azimuth angle of 330° are being focused on. The first case reflects a scenario with negative aerodynamic damping, while the second case represents a clear damping period, where all vibrations have ceased (see Fig. e). Both cases are simulated in a rigid and flexible manner. For convenience, the four described cases are abbreviated as shown in Table .

An instantaneous flow field of the two yaw misalignments in rigid and flexible setups is depicted in Fig. , highlighting the in-plane velocity component. Figure a and b show the flexible and rigid simulations for a yaw misalignment of -37°, while, in Fig. c and d, the simulations with -60° are shown. The difference in flow structures between the two different yaw angles is apparent. Both cases with -37° show smaller structures of in-plane velocity, while the flow field with -60° inflow is described by larger structures expanding over almost the whole blade span.

At a fixed yaw angle, the edgewise, flapwise, and torsional components of the blade tip vibration of case FLEX37 are shown in Fig.  in the time and frequency domain. It is visible that the edgewise amplitude (Fig. a) reaches much higher values compared to the dynamic yaw maneuver shown in Fig. e as the blade stays in the critical region until it reaches a limit cycle oscillation. Here, the limit cycle is reached at 180 s, with an amplitude of 9.5 m, where the injected aerodynamic power and structural damping are balancing each other out. This is roughly 3 times the amplitude of the yawing case. From a frequency perspective, this motion is mainly dominated by the edgewise eigenfrequencies. The flapwise and torsional natural frequencies do not appear to impact the vibration behavior. This is shown in Fig. b, where the vertical red lines mark the blade's natural frequencies, and the second lowest corresponds to the first edgewise eigenfrequency. At this point, a slight shift in the blade's inherent oscillation frequencies towards smaller values can be noted relative to the theoretical eigenfrequencies that are based on a preceding modal analysis. This behavior is attributed to the non-linear characteristics of the blade structure, which become evident as a result of the significant deformation amplitudes. A similar behavior was also described in . Similar findings can be observed in both the flapwise and torsional components. The flapwise component reaches an oscillation amplitude of approximately 1.7 m at a mean deflection of -5.8 m. At the limit cycle, the torsional component oscillates between -6.1 and 7.7°. Both components appear to be largely influenced by the edgewise natural frequency (see Fig. d and f). Consequently, the investigations will mainly focus on the edgewise component of the blade vibrations.

The corresponding edgewise and flapwise energy contributions are highlighted in Fig. . The shape of the edgewise component in Fig. a deviates from the corresponding direction of the yaw maneuver in Fig. a. In the FLEX37 case, all analyzed sections in the range of 0.05≤r/R≤0.95 contribute to an energy transition into the blade structure and therefore contribute to blade vibrations. Only the last monitored section at the blade tip does not inject power into the blade. Similar findings are described in , which show that the aerodynamic energy at the blade tip changes with large vibration amplitudes relative to lower amplitudes. In the flapwise component in Fig. b, the deviations towards the full rotor case are even more dominant as compared to Fig. b. Here, positive and negative contributions of flapwise energy are being calculated for different spanwise positions, where the outer-tip part also contributes to energy injection into the blade structure. Specifically, the positive energy is concluded to act only on the amplification of the flapwise component within the first edgewise mode because a frequency analysis of the flapwise vibration component reveals contributions only from the first edgewise natural frequency and not from the first flapwise one.

For comparison purposes, the edgewise blade deformation for case FLEX60 is shown in Fig.  in the time and frequency domain. This case can be seen as stable since the vibrations are being damped down within the investigated time period. The initial deformation peak is the result of the fluid–structure coupling being turned on at 5 s, which leads to the blade being initially deformed by aerodynamic forces and gravitational loads. From a frequency perspective, in the stable case, the first two edgewise eigenfrequencies dominate the motion. Nevertheless, other frequencies are visible in the spectral analysis, in contrast to case FLEX37.

The edgewise and flapwise aerodynamic energies over the blade span of case FLEX60 are shown in Fig. . For the edgewise component (see Fig. a) the outer 55 % of the blade contributes to aerodynamic damping since all values are negative. As the impact of the aerodynamic energy increases linearly with span, the outer part is known to be more important with respect to blade vibration analysis.

In conclusion, depending on the yaw misalignment configuration, i.e., -37 or -60°, the flow can either amplify or dampen the blade vibrations, underlining the importance of understanding these dynamics from both the fluid dynamics and structural perspectives. Therefore, Sect. – provide a detailed interpretation of these differences.

Understanding how the aerodynamic power needs to overcome the inherent structural damping for blade excitation is essential for optimizing rotor performance and ensuring structural integrity under extreme wind conditions. Following the method of , the structural damping of the blade can be estimated from the force balance of an FSI simulation reaching a pure-mode limit cycle oscillation, where PAERO is equal to PSTRUC. As this power is proportional to the square of the vibration amplitude, the power dissipated PSTRUC by structural damping can be described as follows: 8PSTRUC=CSTRUC⋅A2, with A being the amplitude of the blade tip limit cycle oscillations and CSTRUC being a proportionality constant linking the dissipated power to the vibration amplitude. Assuming that the vibration of FLEX37 almost reflects the pure first edgewise mode, CSTRUC can be estimated from the last vibration cycles as follows: 9CSTRUC=PAEROA2, which is, for the last 18 cycles, calculated to be 555.44 Wm-2. If, now, the difference in power between aerodynamic contribution and structural damping, 10ΔP=PAERO-PSTRUC, is being plotted for blade 2 of the yaw maneuver of the full rotor (see Fig. ), it can be seen that the blade vibrations are increasing for positive values of ΔP and are being damped for negative values for most parts of the yaw range. Based on the results shown in , one would expect the ΔP curve to start and end at 0 kW. However, in the present case, a different behavior is observed, indicating a non-zero net energy exchange at the beginning and end of the yaw maneuver. It can be noted that, for smaller vibrations in the range of -30 to -35°, increasing amplitudes are also visible even for negative ΔP values. This can be explained by the fact that, in this range, the inner part of the blade is exposed to negative aerodynamic power, while the outer part is exposed to positive aerodynamic power. In total, this gives a negative value for the whole blade, although the outer part is being excited by the flow, leading to smaller but increasing blade tip vibrations.

The largest ΔP that is calculated as 7.5 kW is reached at -40° in the regime of growing blade vibration amplitude. This is mirrored during the damping phase of the vibration at -47° yaw misalignment and ΔP≈-7.5 kW. The largest tip displacement is reached at -43°, where PAERO and PSTRUC seem to be balanced out at ΔP≈0 kW.

The analysis of the integrated aerodynamic energy on the single blade in the two examined scenarios revealed that FLEX37 exhibits unstable blade vibrations, whereas FLEX60 demonstrates stability. The vibrations are primarily influenced by the first edgewise eigenfrequency, resulting in the most significant displacements at the blade tip. To gain deeper insights into the mechanisms that cause the blade to transition from standstill to either unstable or stable motion, the subsequent focus will be on examining the blade tip displacements in both cases.

In Fig. , the displacements of the two blade tips are illustrated. Figure a and b depict the motion within the horizontal x–y plane across various phases of displacement. In contrast to that, Fig. c and d present the edgewise component over time. The motion time series is divided into four distinct phases, each highlighted with a different color-coding. The first phase, designated as the settling phase, ranges from 6 to 20 s. It is visible that both simulation cases describe a similar motion pattern, which is mainly dominated by the coupling of fluid and structure, where the motion from the undeformed blade towards the final flapwise displacement prevails. The second phase, occurring between 20 and 40 s, is referred to as the decision phase, during which the motion may either transition to being predominantly edgewise or gradually diminish. The third phase, highlighted in dark blue, represents a typical duration of one edgewise motion cycle that takes 1.47 s. Lastly, the fourth phase extends from 41.47 to 90 s and captures the excitation of blade vibrations in the context of the unstable case, as well as the near-stabilization observed in the stable case.

The results presented illustrate the structural behavior as a result of the surrounding fluid conditions. To fully understand the interaction between structure and fluid, it is essential to analyze both aspects. To investigate the fluid side in more detail, firstly, the subsequent discussion will focus on the local inflow conditions and the corresponding sectional loads, while, secondly, fluid characteristics in the near wake are discussed in Sect. .

To achieve this, all four cases presented in Table  are examined at r/R=0.9 since this spanwise position is expected to experience a major influence of aerodynamic energy (see Fig. ). In this context, cases RIG37 and RIG60 represent the moment just before the settling phase begins, while cases FLEX37 and FLEX60 capture a later state during the excitation phase. Each case is analyzed over a duration of three theoretical edgewise motion cycles, amounting to 4.41 s.

Figure  shows the angle of attack time series of the two rigid cases using the three-point method introduced by . On the one hand, for case RIG37 in Fig. a, the angle of attack fluctuates around -30.2° by amplitudes in the range of ±0.5°. The color-coding depicts the corresponding aerodynamic edgewise force, summed from the pressure and friction forces. Blue corresponds to a force component towards the leading edge, while red points towards the trailing edge. It is visible that, for RIG37, the edgewise force is negative throughout the whole investigated time period, ranging from 0 to -100 Nm-1. No connection between the small angle-of-attack fluctuations and the edgewise force is drawn. On the other hand, Fig. b shows the angle of attack over time for case RIG60. It is considered to be almost constant at a value of ≈-55.7°, where minor changes in the predicted edgewise force of 40 Nm-1 are also apparent.

To highlight the connection between lift and angle of attack, Fig.  shows the corresponding three theoretical edgewise motion cycle periods for the rigid cases RIG37 and RIG60. Here, the time is color-coded from dark blue to yellow. While the RIG37 describes a more circular pattern for the lift coefficient (Cl) values in the range of -0.93≤Cl≤-0.48, RIG60 shows a linear increasing Cl for decreasing angle of attack and increasing time. Here, Cl values increase from -0.8≤Cl≤-0.55.

Once the blade flexibility is considered and the vibration behavior reaches the excitation phase, a similar analysis is done. Figure  shows again the local angle of attack at r/R=0.9 for case FLEX37 and FLEX60. It is noted that the torsional deformation also contributes to a change in angle of attack (AoA). FLEX37 shows a different behavior compared to its corresponding rigid case as it describes three distinct motion cycles that are directly reflected in terms of the angle of attack. Also, the connection between edgewise force and the angle of attack is visible. With decreasing angle of attack, the edgewise force component is found to be negative, while it flips directions when the angle of attack increases again. This change in force direction shows the contribution of the force to the direction of motion that excites the blade to vibrate. However, case FLEX60 in Fig. b barely differs from its corresponding rigid setup. Again, angle of attack is considered to be constant at AoA = -44° and an edgewise force of 50 Nm-1.

Lift coefficients over time and angle of attack are shown in Fig. . For case FLEX37 in Fig. a, the circular hysteresis effect between Cl and AoA is even more pronounced. The lift describes three cycles in time that correspond to the three edgewise motion periods. This hysteresis pattern is well known and could indicate the existence of SIV and was, among others, shown in . For the FLEX60 case in Fig. b, the pattern just deviates from its corresponding rigid case but still draws its linear connection, confirming that no hysteresis effect is apparent at -60° yaw.

Thus, it is concluded that, in the case of -37° yaw misalignment, unsteady aerodynamic effects are the driving mechanism behind the vibration build-up since a hysteresis is already present in the rigid configuration. This hysteresis becomes more pronounced when blade flexibility is introduced (FLEX37), leading to lock-in with the edgewise vibration mode. This mutual influence is not evident in case FLEX60, which therefore remains stable. Because the angle of attack is extracted from velocity probes upstream and downstream of the blade, vortex shedding is inferred to make a possible contribution to the apparent vibrations, a mechanism analyzed in more detail in Sect. .

To this point, blade structural deformations, as well as local aerodynamic quantities, i.e., angle of attack, lift coefficient, and edgewise force, have been studied. To understand the relation between fluid and structure, flow quantities in the near wake of the blade also have to be analyzed. For this purpose, the Hilbert–Huang transformation is used to highlight the impact of normal to inflow velocity frequencies during the different phases of the blade motion analysis. The reason for choosing this method is that, in contrast to a simple fast Fourier transformation (FFT), a clearer picture of the underlying frequency signals has been observed for various processed signals. To form a link between the local aerodynamic quantities and the near wake, this method is firstly applied on the sectional loads in contrast to the power spectrum of an FFT before investigating the near wake.

Figure  presents the monitored edgewise force signal for 70 s of case RIG37 at a spanwise position of r/R=0.9, along with its corresponding frequency contributions obtained through a fast Fourier transform. The challenge of establishing a clear connection between the dominant frequencies and the force signal is evident (see Fig. b). The power spectrum of frequencies within the range of 0.0 Hz ≤ f ≤ 2.2 Hz appears to be uniformly distributed. No frequency stands out in the region of the first two natural frequencies. To address this issue, the Hilbert–Huang Transform is applied.

In accordance with the procedure described in Sect. , the first eight IMFs of the edgewise sectional force can be obtained from the original signal as shown in Figs.  and . A closer look on the power of the frequencies of each IMF is shown in Figs.  and . The solid lines show the power spectrum of each IMF. The dotted and dashed-dotted lines represent the first two eigenfrequencies, i.e., first flapwise and first edgewise, of the blade, respectively. According to Sect. , vibrations are driven by the first edgewise natural frequency. It can be seen that this frequency is already inherent in the edgewise force of case RIG37, as shown via IMF-6 (see Fig. a), although the blade is analyzed rigidly. In the flexible simulation (Fig. b), once the blade reached the excitation phase, a lock-in phenomenon of that particular frequency is visible. The whole edgewise force signal is dominated by IMF-6 in the first edgewise natural frequency, highlighting that both force and deformation are aligned in phase. Here, again, a slight shift towards the left (lower frequencies) of the natural frequency is noted. This aligns with the findings of the blade tip displacements shown in Fig. .

This lock-in phenomenon does not occur in the case of -60° yaw misalignment, as shown in Fig. . Although IMF-7 has its peak in the vicinity of the first edgewise frequency in the rigid setup (Fig. a), the signal's power distribution does not lock into the first edgewise natural frequency during the flexible simulation phase (Fig. b). It even moves further away from the frequency of scope. The slight differences between RIG60 and FLEX60 can be explained by the minor change in AoA, when the blade adjusts to the loads. This confirms the instability of FLEX37 and the stability of FLEX60 as the forces either align with the frequency of motion and contribute the vibrations or play no significant role in the ramp-up of blade vibrations being too far away from the edgewise eigenfrequency.

To see a similar frequency impact on the flow in the wake of the blades, the HHT is applied to the v velocity component that is pointing horizontally, normal to the inflow direction. For that purpose, a velocity probe is analyzed, located three chord lengths downstream of the r/R=0.9 position. A probing rate of 10 Hz is used to save computational costs and focus on frequencies within 0.0 Hz ≤ f ≤ 2.2 Hz. Figure  shows the impact of this frequency range on the probed velocity signal for the two rigid setups, RIG37 and RIG60. Even more pronounced than in the investigated sectional force signal, case RIG37 consists of a dominant frequency component in the near wake at the edgewise eigenfrequency, as shown by IMF-3 in Fig. a. Additionally, frequencies around 1.5 Hz show larger relative power peaks compared to the sectional loads. On the other hand, Fig. b shows the corresponding velocity frequency spectrum for the RIG60 case. It is visible that the edgewise frequency component makes a minor contribution to the frequency–power distribution of the velocity signal in contrast to the wide spread of IMF-1, which dominates the velocity component.

For the two flexible cases, FLEX37 and FLEX60, the investigation is again split into the three time periods, i.e., the settling phase, the decision phase, and the excitation phase. Firstly, Fig.  shows the frequency–power distribution for the settling phase. As this case is seen as the transition from rigid towards flexible setups, it can be seen in Fig. a that, especially for the FLEX37 case, most frequency components in the vicinity of the first two natural frequencies are being damped out. This can be explained by the drag-driven deformation trajectory described in Sect.  that does not allow for periodic repetitions of deformations.

Within the decision phase, flow characteristics seem to have adjusted towards the given blade deformation conditions and, in the case of FLEX37, feed back with aeroelastic frequencies at the blade by showing increasing impact of the first edgewise frequency (see Fig. a). However, this is not the case for FLEX60, where frequencies barely move closer to the edgewise eigenfrequency, as shown in Fig. b.

Within the excitation phase in Fig. , the picture becomes more clear. Case FLEX37 clearly locks-in on the edgewise eigenfrequency at 0.68 Hz, while FLEX60 only shows a minor peak at this particular frequency compared to the whole given frequency power distribution.

In summary, case FLEX37 exhibits large edgewise vibration amplitudes, in contrast to the stable FLEX60 case. This difference arises from the frequency contribution near the first edgewise eigenfrequency already present in the rigid setup, RIG37, whereas RIG60 is driven by components around 2 Hz. In Fig. c and d, it can be seen that the wakes of RIG60 and FLEX60 are dominated by strongly correlated structures. As shown by , such correlated shedding can be triggered by inclined inflow at specific inflow angles, which seems to apply here as well.

Using a characteristic vortex shedding Strouhal number for wind turbine blades at high angles of attack of Sr = 0.15 (see ), an inflow speed of 35 ms-1, and the observed shedding frequency of f=2 Hz in FLEX60, the characteristic blade length can be calculated as follows: 11L=Sr⋅Uf=2.6m. It corresponds to the chord lengths in the spanwise range of 0.8≤r/R≤0.9, the region that is primarily responsible for the injected aerodynamic power (see Figs. a and a). In , these correlated structures were created from inflow speeds that produced shedding near the edgewise eigenfrequency, leading to large blade vibrations. Here, however, opposing behavior can be observed. The dominant correlated frequencies are roughly 3 times higher than the first edgewise mode due to the much higher inflow speed of 35 ms-1. Consequently, the dominating wake correlation appears to stabilize the blade by washing out frequency contributions near the first edgewise frequency. A preceding study showed that a FLEX60 setup with a uniform inflow velocity of one-third of the original speed, i.e., 11.77 ms-1, could indeed trigger small vibrations (slightly larger amplitudes than in the original FLEX60) in the pure first edgewise eigenmode. However, the lack of growth to larger amplitudes was attributed to insufficient energy in the flow. Although the critical vortex shedding frequency was met, the fluid did not transfer enough energy to overcome structural damping.

It is therefore concluded that instability occurs when wake structures lock in with the first edgewise eigenfrequency (FLEX37), whereas higher-frequency correlated wake shedding that avoids this lock-in suppresses energy near the mode and stabilizes the blade (FLEX60). In the FLEX37 case, the observed coherent wake frequencies cannot be attributed to classical Strouhal shedding from a bluff body. Instead, the instability appears to originate from a three-dimensional separated shear layer that develops under yawed and inclined inflow, generating periodic wake structures whose characteristic frequency approaches the first edgewise eigenfrequency. This frequency alignment promotes lock-in and results in negative aerodynamic damping.