1. Introduction
With the rapid expansion of offshore wind power, wind turbines are increasing in tower height and blade length to harness more wind energy resources [
1,
2,
3]. However, this trend poses a significant challenge: taller, slenderer towers and larger, heavier rotors lead to increased flexibility and reduced damping in large wind turbines. Moreover, the turbine blades are vulnerable to damage, making the entire structure highly sensitive to wind and wave forces. Throughout their operational life cycle, these turbines can experience excessive vibration, which threatens structural integrity [
4,
5,
6,
7,
8].
Effective structural vibration control methods are imperative to ensure the stable operation and longevity of wind turbines. Recent years have seen significant advancements in structural vibration control techniques being successfully applied across various structures. These methods broadly fall into four categories: passive control, active control, hybrid control, and semi-active control.
Passive control systems (Chen et al. 2019) [
9] generate control forces directly from the primary structural motion, eliminating the need for external power and complex control algorithms. This characteristic makes passive control techniques ideal for vibration suppression in wind turbines, offering simplicity compared to other control methods. Consequently, passive control strategies for wind turbines have become a focal point of research.
Currently, passive control methods, such as conventional tuned mass dampers (TMDs) (Murtagh et al. 2015) [
10], dual TMD systems (Stewart and Lackner, 2017) [
11], three-dimensional pendulum TMDs (Sun and Jahangiri, 2018) [
12], tuned liquid column dampers (TLCDs) (Sun et al. 2016) [
13], and tuned liquid dampers (TLDs) (Zhang et al. 2016; 2017) [
14], have been proposed for mitigating vibration in wind turbines.
Effective vibration control of wind turbines using tuned mass dampers (TMDs) necessitates a thoughtful selection of TMD parameters, including mass and stiffness (Biglari and Fakhari, 2020) [
15]. During optimization, accurately predicting the wind turbine’s dynamic response and adhering to TMD travel limitations within confined spaces are crucial considerations (Chen et al., 2018) [
16]. Donagh McNamara et al. [
17] achieved good results by using multiple TMDs for damping. Zhenbo Lei et al. [
18] proposed a novel in-platform modified tuned mass damper (IP-TMD) to control the excessive vibration of steel wind turbine towers (WTTs) resulting in a reduction of more than 45% in the dynamic response of steel WTTs compared to the uncontrolled WTT case. Wang Jiao et al. (2019) [
19] enhanced the vibration stress reduction in blades by introducing a viscous damping block at the blade root. Cong Cong et al. (2021) [
20] explored integrating an active FM mass damper 50 m from the blade root to mitigate effects on the electrical system. Hu et al. [
21] developed a linear dynamics model for a barge-type offshore wind turbine, encompassing tower top displacement, platform pitch, and TMD degrees of freedom. Yang Yang et al. (2022) [
22] integrated a TMD into a 10 MW offshore wind turbine, achieving effective vibration damping under seismic excitation through mass and stiffness adjustments. There are also methods of adding vibration isolation for damping. Changning Liu et al. [
23] provide a detailed review of inertial machines from the perspective of network simulation and synthesis, mechanical domain, and power flow transfer, and summarize device improvements and control methods.
However, existing literature predominantly examines TMDs’ vibration damping effects on wind turbine towers or platforms, often overlooking TMD integration into turbine blades, the impacts of wind and wave loads on turbine vibration, and how these factors are influenced by load frequency characteristics. Therefore, understanding the frequency characteristics of wind and wave loads is crucial for comprehending wind turbine dynamic responses (with and without TMDs) and TMD damping mechanisms and performance.
This study investigates the frequency characteristics of various wind and wave loads, aiming to deepen insights into wind turbine dynamic responses and TMD damping effectiveness under these conditions.
4. Optimized Design of TMDs
In this section, the tuned mass damper (TMD) is initially optimized to enhance its damping performance on the wind turbine. Subsequently, the response characteristics of the TMD to the wind turbine are examined under five moderate operating conditions.
4.1. Wind Turbine Response Characteristics in the Absence of TMD
In this section, the frequency characteristics of the dynamic response of the wind turbine without a tuned mass damper (TMD) are investigated.
Figure 7 and
Figure 8 illustrate the power spectral densities of the forward and backward displacement at the top of the tower, as well as the lateral displacement, for the five different operational conditions.
From
Figure 7, it is evident that at low wind speeds (
Figure 7a–c), the vibration energy of the wind turbine tower was primarily concentrated in the low frequencies rather than at its natural frequency. As wind speeds increased (
Figure 7d,e), the vibration energy of the tower increased significantly and shifted closer to its natural frequency, mirroring the behavior observed in wind loading.
Figure 8 shows that, across all cases, the tower exhibited dominant vibration energy at its natural frequency. This can be attributed to the wind predominantly acting perpendicular to the plane of blade rotation, resulting in minimal lateral wind velocity components and correspondingly low aerodynamic loads and damping effects. Therefore, lateral tower vibrations were predominantly driven by the 1 p component, whose dominant frequency aligned closely with the tower’s natural frequency rather than with the dominant frequency of wind loading.
In both
Figure 7 and
Figure 8, it is notable that the vibration amplitude of the tower is most pronounced at its natural frequency in case e (DLC5). Consequently, DLC5 was selected as the operational condition for optimizing the tuned mass damper (TMD).
4.2. Optimized Design of the TMD at the Nacelle under Optimal Operating Conditions
For the optimized design of the tuned mass damper (TMD) at the nacelle, the primary objective was to minimize the tower top displacement. Initially, after excluding transient effects within the first 30 s, the amplitude and standard deviation of the tower top front–back displacement, as well as the lateral displacement without the damper, measured 2.399 m, 0.2604 m, 0.75, and 0.37, respectively.
Substituting 20 parameter sets from
Table 3 into the servo system calculation in FAST, the integrated tower top displacement amplitude and standard deviation indicate that the most effective vibration damping occurred at a mass ratio of 1.8. Specifically, the amplitude and standard deviation of the tower top front–back displacement and lateral displacement were reduced to 1.819 m, −0.18 m, 0.38, and 0.11, respectively. This represents reductions of 24.1% and 169.1% in amplitude, and 49.3% and 70.2% in standard deviation, respectively.
Figure 9 and
Figure 10 depict the tower top front–back displacement and lateral displacement, along with the power spectral densities for these displacements. It is evident that the wind turbine’s vibration at its natural frequency was significantly dampened with a damper mass ratio of 1.8 compared to the operation without the damper.
The frequency of the wind turbine decreases when the overall mass of the nacelle increases, defining
γ as the ratio of the frequency of the target TMD to the first-order intrinsic frequency of the wind turbine tower:
where
is the damper frequency and
is the tower first order intrinsic frequency.
Taking 0.22 as the fan frequency and 10 groups of γ from 0.8 to 0.98, the design parameters are shown in
Table 7.
Figure 11 illustrates that the tuned mass damper (TMD) achieves optimal damping effectiveness with a frequency ratio of 0.86 and a mass ratio of 1.8%. After excluding transient effects in the first 30 s, the amplitude and standard deviation of both the tower top front-to-back and lateral displacements were measured at 1.524 m, −0.24 m, 0.23, and 0.10, respectively.
Compared to the configuration without the TMD, these values represent reductions of 16.2% and 33.3% in amplitude for the tower top front-to-back displacement and lateral displacement, respectively. Additionally, standard deviations were reduced by 39.4% and 9.1% for front-to-back and lateral displacements, respectively.
Furthermore, the tower top lateral displacement and its standard deviation exhibited reductions of 36.5% and 69.3%, respectively, compared to their counterparts without the TMD installed. These findings highlight the significant effectiveness of the TMD in mitigating vibrations in the wind turbine system.
4.3. Optimized Design of TMDs in Blades at Optimal Operating Conditions
For the optimal design of the tuned mass damper (TMD) in the blade, the primary objective was to minimize front-to-back displacement at both the blade tip and tower top. Based on the 20 parameter sets listed in
Table 5, results are illustrated in
Figure 12. After disregarding transient effects in the initial 30 s, the amplitudes and standard deviations of the tower top and tip displacement without the damper were measured at 2.399 m, 29.83 m, 0.75, and 2.77, respectively.
The best damping effect was achieved with a mass ratio of 1.8. The amplitudes and standard deviations of both the tower top and tip displacement before and after installing the TMD were 1.64 m, 27.41 m, 0.23, and 2.13, respectively. This represents reductions of 31.6% and 8.1% in amplitude, and 69.3% and 23.1% in standard deviation for the tower top and tip displacement, respectively.
Figure 12 displays the front-to-back displacement of the tower top and blade tip for visual comparison.
Using 0.55 as the fan frequency, 10 groups of
were taken from 0.8 to 0.98, and the design parameters are shown in
Table 8.
Figure 13 demonstrates that the tuned mass damper (TMD) achieved optimal damping at a frequency ratio of 0.86 and a mass ratio of 1.8%. After eliminating transient effects within the first 30 s, the amplitudes and standard deviations of the tower top displacement and blade tip displacement were measured at 1.573 m, 26.87 m, 0.196, and 1.77, respectively.
These values represent reductions of 0.4%, 2.0%, 14.8%, and 73.9% in amplitude and standard deviation compared to configurations without a TMD at a 1.8% mass ratio. Compared to configurations without a TMD, at a 2.0% mass ratio, these reductions are 34.4%, 10.0%, 73.9%, and 16.9% for tower top displacement, and 36.1% for blade tip displacement.
Figure 13 also illustrates the anterior–posterior displacement and standard deviations of the tower top and blade tip with and without a TMD for clear comparison.
4.4. Vibration Damping Effect of the Two TMDs under Optimal Operating Conditions
To investigate the effectiveness of dampers in the model, two optimized tuned mass dampers (TMDs) were integrated, and their impact on tower top trajectories and time courses are presented in
Figure 14. The results showed that, after eliminating transient effects within the first 30 s, the tower top displacement trajectory exhibited significantly reduced variability compared to the pre-optimization scenario.
With the addition of the two dampers, the front-to-back and side-to-side displacement of the tower top were measured at 1.633 m and −0.174 m, with standard deviations of 0.21 and 0.12, respectively. This represents a reduction in amplitude of 61.2% and 166.8%, and a decrease in standard deviation of 72% and 67.6%, respectively.
The impact of two TMDs on tower top displacement reveals improvements in forward and backward movements compared to a single TMD. However, the reduction in lateral displacement of the tower top was noted as 25.4%. This outcome may stem from the TMDs on the blades focusing solely on the first-order blade bending frequency, neglecting the first-order pendulum oscillation frequency. Consequently, their effectiveness in mitigating lateral displacement was less pronounced compared to the TMDs installed at the nacelles, where only one nacelle was present.
4.5. The Vibration Damping Effect of TMDs under Other Operating Conditions
Based on the findings in DLC5, it was observed that TMDs with a mass ratio of 1.8 and a frequency ratio of 0.86 exhibited the most effective damping. However, for cases 1 to 4, the improvement in damping effectiveness was minimal due to the concentration of vibration energy at low frequencies at the top of the tower. Conversely, in case 5, where the dominant vibration frequency aligned closely with the natural frequency of the tower, TMDs showed a significant enhancement in damping effectiveness.
This observation is supported by
Figure 15 and
Figure 16, which depict the power spectral densities of tower top displacement in both front-to-back and side-to-side directions across cases 1 to 4.
Figure 15 illustrates that, despite optimizing TMD mass and stiffness for each case, damping performance remained largely unchanged in cases 1 to 4 but notably improved in case 5. In
Figure 16, a minimal improvement in lateral displacement reduction was observed across all cases, indicating that dampers had limited impact due to wind wave energies predominantly manifesting at lower frequencies rather than coinciding with the tower’s natural frequencies.
These results underscore the necessity for further TMD optimization tailored to different operational conditions, accounting comprehensively for wind and wave load frequency characteristics.