Optimization Study of a Tuned Mass Damper for a Large Monopile Wind Turbine

Abstract

Passively tuned mass dampers (TMDs) are known to effectively mitigate the vibration of wind turbines. However, existing literature predominantly examines their application in damping vibrations of the tower or platform, overlooking the potential benefits of installing TMDs on the turbine blades themselves. This study investigates the impact of wind and wave loads on TMD damping effectiveness and proposes a comprehensive damping strategy involving TMDs installed in both the nacelle and the blades. The design optimized the mass and stiffness of these TMDs to enhance their performance. Results indicate that, as wind speeds increased from 12 m/s to 24 m/s, the power spectral density at the tower’s natural frequency (0.22 Hz) more than doubled. Notably, TMDs exhibited robust vibration damping capabilities under high wind speeds. Specifically, at wind speeds of 24 m/s, TMDs reduced anterior–posterior and lateral displacement at the tower top by 61.2% and 166.8%, respectively, when two TMDs were combined. Conversely, the study found that TMDs did not significantly improve vibration damping at lower to moderate wind speeds. This research underscores the importance of optimizing TMDs for high wind conditions to ensure wind turbine stability and mitigate potential vibration-related risks effectively under varying environmental loads, including wind and waves. It offers valuable insights for the refined design and deployment of TMDs in wind energy applications.

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.

2. Wind Turbine Parameters and TMD Parameter Design

Staff from the National Renewable Energy Laboratory (NREL) and the Technical University of Denmark (DTU) collaborated closely under the International Energy Agency’s (IEA) Wind Task on Wind Systems Engineering to design the 15 MW wind turbine.

2.1. Wind Turbine Parameters

In this study, a typical IEA 15 MW monopile wind turbine featuring a hub height of 150 m, rotor radius of 120 m, and a total nacelle mass of 1017 t, was selected for analysis. The key parameters are outlined in

Table 1. IEA 15 MW wind turbine parameters.

Parameter	Value	Parameter	Value
Rated power/MW	15	Gross nacelle mass/t	1017
Cut-in wind speed/(m/s)	3	Blade quality/t	65
Rated wind speed/(m/s)	10.59	Tower quality/t	860
Cut out air speed/(m/s)	25	Tower base diameter/m	10
Radius of the wind turbine/m	120	Monopile depth/m	45
cabin altitude/m	150	Wind turbine quality/t	1318

(Jonkman et al. 2021–2022) [24,25].

2.2. TMD Design Theory and Workflows

To minimize structural dynamic responses under wind and wave loads, a structural passive control module was developed using the tuned mass damper (TMD) method. The TMD operates on the principle that when a wind turbine experiences continuous external excitation, the vibration energy is transferred to the TMD. Due to the presence of the TMD, this vibration energy is dissipated as thermal energy. This mechanism effectively moderates the vibration of the wind turbine. Figure 1 illustrates a schematic diagram depicting TMD placement at the nacelle and within the turbine blades. A schematic of the TMD optimization process is provided in the flow chart shown in Figure 2. The process includes the determination of the mass ratio and frequency of the turbine using the turbine and the determination of the optimal parameters for each TMD using a set of measured ocean data.

In this study, the main structure is modeled as a single degree of freedom system. The equations of motion control for the structure–TMD system can be expressed as:

$$\left[M\right]\left\{\ddot{x}\right\}+\left[C\right]\left\{\dot{x}\right\}+\left[K\right]\left\{x\right\} =\left\{\mathrm{F}\right\}$$

(1)

where [M], [C] and [K] denote the mass, damping, and stiffness matrices of the structure–TMD system. The mass matrix is given by:

$$\left[M\right]=\mathrm{diag} [m_s,m_1,m_2,m_3,\cdots ,m_n]$$

(2)

where n is the total number of TMDs in the structure–TMD system. For the structure–TMD system, n = 1. The stiffness matrix can be formed as:

$$\left[K\right]=\left[\begin{array}{ccccc}{k}_s+\sum _{i=1}^n{k}_i& {-k}_1& {-k}_2& \dots & {-k}_n\\ {-k}_1& k_1& 0& \dots & 0\\ {-k}_2& 0& k_2& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {-k}_n& 0& 0& \dots & k_n\end{array}\right]$$

(3)

The modeling of the TMD needed to be modified in OpenFast (version 3.5), as the force generated by the motion of the TMD was added to the generalized force; the TMD force $F_{TMD}$ in each direction is derived as shown in Equation (4):

$$F_{TMD}=-k_T·x_{TMD}-c_T·{\dot{x}}_{TMD}$$

(4)

where $x_{TMD}$ and ${\dot{x}}_{TMD}$ are the displacement and velocity of the TMD, respectively. $k_T$ and $c_T$ are the stiffness and damping values of the TMD.

The motion of the TMD is affected by the mechanics associated with centrifugal, Eulerian, and Coriolis forces. The TMD acceleration ${\ddot{x}}_{TMD}$can be expressed as in Equation (5):

$${\ddot{x}}_{TMD}=-{\ddot{x}}_N-{\omega }_N\times \left({\omega }_N\times x_{TMD}\right)-{\alpha }_N\times x_{TMD}-2{\omega }_N\times {\dot{x}}_{TMD}-F_{TMD}/m_T$$

(5)

where ${\omega }_N$ and ${\alpha }_N$ are denoted as the translational acceleration and rotational angular velocity of the nacelle. ${\omega }_N\times x_{TMD}$, ${\alpha }_N\times x_{TMD}$ and ${\omega }_N\times {\dot{x}}_{TMD}$ indicate the contribution of centrifugal, Eulerian, and Coriolis forces, respectively.

2.3. Design of TMD Parameters at the Nacelle

Given that wind turbine tower vibration is primarily influenced by first-order modes, this paper employs TMDs to specifically attenuate first-order vibrations.

Table 2. Frequency results of OpenFast and Bladed calculations.

Vibration Pattern	OpenFast Calculation Results	Bladed Calculation Results
First-order mode (fore–aft)	0.2236	0.2275
First-order mode (side–side)	0.2236	0.2275
Second-order mode (fore–aft)	1.0418	1.0583
Second-order mode (side–side)	1.0418	1.0583

presents a comparison of frequencies calculated by OpenFast linearization and Bladed, revealing that the first-order frequency differed by 1.7% and the second-order frequency by 1.6%. Results from OpenFast linearization fell within acceptable error margins.

For the design of TMDs at the nacelle, a frequency of 0.22 was adopted. The design optimization involved employing an exhaustive method to refine 20 TMD configurations, varying the mass ratios from 0.1% to 2%. Detailed design parameters for the TMDs are outlined in

Table 3. Design of TMDs at nacelle for different mass ratios.

Quality Ratio/%	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Mass/kg	1017	2034	3051	4068	5085	6102	7119	8136	9153	10,170
Stiffness/(N/m)	1943	3886	5830	7773	9716	11,659	13,603	15,546	17,489	19,432
Quality ratio/%	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2.0
Mass/kg	11,187	12,204	13,221	14,238	15,255	16,272	17,289	18,306	19,323	20,340
Stiffness/(N/m)	21,376	23,319	25,262	27,205	29,149	31,092	33,035	34,978	36,922	38,865

.

2.4. Design of TMD Parameters in Blades

The wind turbine model incorporated rotating pre-twisted blades, which were represented as continuous beams with variable mass and stiffness in OpenFast. Vibration in the cantilever beam is primarily driven by first-order modes. Due to the impracticality of placing them at the blade tips, TMDs were strategically placed at distances of 0 m, 58.5 m, 74 m, 82 m, 90 m, and 93 m from the root of the blades. The analysis focuses on fore-and-aft displacement at the blade tip under rated conditions, with results summarized in

Table 4. Displacement of leaf tip at different positions.

TMD Location/m	Minimum Leaf Tip Displacement/m	Median Leaf Tip Displacement/m	Maximum Leaf Tip Displacement/m
0	9.559	17.07	21.81
58.5	9.563	17.07	21.8
74	9.558	17.06	21.76
82	9.543	17.02	21.71
90	9.508	16.96	21.62
93	9.487	16.92	21.58

. It is observed that displacement decreased as the TMD was positioned farther from the root, although practical considerations favor placement at 82 m, balancing damping effectiveness and TMD displacement.

The first-order flapping frequency of the blade in the model was 0.555 Hz [26]. Here, a frequency of 0.55 was adopted for TMD design in the blade, and an exhaustive method optimized TMD configurations across 20 mass ratios ranging from 0.1% to 2%. Detailed design parameters are provided in

Table 5. Design of TMD in leaves at different mass ratios.

Quality Ratio/%	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Mass/kg	65	130	195	260	325	390	455	520	585	650
Stiffness/(N/m)	790	1581	2371	3162	3952	4743	5533	6323	7114	7904
Quality Ratio/%	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2.0
Mass/kg	715	780	845	910	975	1040	1105	1170	1235	1300
Stiffness/(N/m)	8695	9485	10,275	11,066	11,856	12,647	13,437	14,228	15,018	15,808

.

3. Load Design of Wind Turbines

This section explores various scenarios of wind and wave loads chosen as external excitations for the wind turbine following the guidelines outlined in IEC 61400-3 [25]. The characteristics of these wind and wave loads across different scenarios are analyzed to elucidate their impact on the dynamic response of the wind turbine, both with and without TMDs.

3.1. Selection of Working Conditions

This section adheres to the design load cases (DLC) outlined in IEC 61400-3, which are depicted in Figure 3. Here, the average wind speed at the hub satisfies $v_{in}$ < $v_{hub}$ < $v_{out}$, where $v_{in}$,$v_{hub}$, and $v_{out}$, representing the cut-in, mean, and cut-out wind speeds at the hub, respectively. Under these conditions, the wind turbine operates under normal circumstances.

Table 6. Environmental parameters under different working conditions.

	Wind Velocity/(m/s)	Significant Wave Height/m	Peak Spectral Period/s	Sea Speed/(m/s)
DLC1	12	1.836	7.441	0.83
DLC2	14	2.188	7.461	1
DLC3	18	3.061	8.047	1.2
DLC4	20	3.617	8.521	1.4
DLC5	24	4.516	9.452	1.6

illustrates the five designated operating conditions.

3.2. Wind Load Characteristics

Firstly, the 600-second wind speed time series generated by Turbsim for the five cases is presented in Figure 4. To mitigate transient effects during the wind turbine’s startup phase, the generated 600-s wind model was extracted from 30 s to 630 s. It is evident from the figure that maximum wind speeds exceeded average wind speeds and that they escalated with increasing average wind speeds ranging from 12 m/s to 24 m/s. As the mean wind speed became larger, the amplitude and peak-to-peak values of turbulent winds increased relatively.

In the investigation of wind load effects on wind turbines, two key parameters were computed: the standard deviation of wind speed and the power spectral density. Firstly, as depicted in Figure 5a, the standard deviation of wind speed tended to increase with higher wind speeds. This indicated that the variability or uncertainty in wind speed became more pronounced as wind speeds rose. Figure 5b illustrates a crucial characteristic of wind load, namely, its low-frequency components. The power spectral density distribution revealed that wind load energy predominantly resided in the low-frequency range. This suggests that the structural impact of wind load on the wind turbine was primarily influenced by low-frequency wind movements.

Moreover, as wind speeds escalated from 12 m/s to 24 m/s, the power spectral density at the tower’s natural frequency (0.22 Hz) increased by more than 100%. This observation highlights that, when wind frequencies align closely with or match the tower’s natural frequencies, the effect of wind load on the tower intensified significantly at high wind speeds. Hence, the tower exhibited heightened sensitivity to low-frequency wind motions during high wind conditions.

3.3. Characterization of Wave Loads

The waves were generated using the Hydrodyn module in OpenFAST, employing Janswap spectra with a step size consistent with the main fst file. In assessing the impact of wave loads on the wind turbine, a key parameter analyzed was the wave frequency. As depicted in Figure 6, it is evident that, as wave height increased, the amplitude of wave forces also increased at the same frequency, underscoring their low-frequency characteristics. The amplitude–frequency plot reveals that the energy of wave loads is primarily concentrated in the low-frequency range. This indicates that the structural influence of wave loading on the wind turbine is predominantly driven by low-frequency wave motion.

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:

$$\gamma =f_{TMD}/f$$

(6)

where $f_{TMD}$ is the damper frequency and $f$ 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. TMD parameters at the nacelle for different γ.

$\mathit{\gamma }$/%	0.8	0.82	0.84	0.86	0.88	0.9	0.92	0.94	0.96	0.98
Mass/kg	18,306	18,306	18,306	18,306	18,306	18,306	18,306	18,306	18,306	18,306
Stiffness/(N/m)	22,386	23,519	24,681	25,870	27,087	28,332	29,606	30,907	32,236	33,593

.

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 $\gamma$ were taken from 0.8 to 0.98, and the design parameters are shown in

Table 8. TMD parameters in leaves at different $\mathit{\gamma }$.

$\mathit{\gamma }$/%	0.8	0.82	0.84	0.86	0.88	0.9	0.92	0.94	0.96	0.98
Mass/kg	1170	1170	1170	1170	1170	1170	1170	1170	1170	1170
Stiffness/(N/m)	8942	9395	9856	10,334	10,820	11,318	11,826	12,346	12,877	13,419

.

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.

5. Conclusions

In this study, we addressed the reduction of vibration at the top of wind turbine towers by integrating an optimized tuned mass damper (TMD) in both the nacelle and the blades. Our investigation comprehensively examined the impact of these TMDs on the dynamic response of the wind turbine under varying wind and wave loads as specified in the IEC 61400-3 standards. Further optimization and design of the TMD were conducted under optimal operational conditions, leading to the following conclusions:

• Wind spectrum analysis revealed that energy is predominantly concentrated in the low-frequency range, with higher wind speeds corresponding to increased energy in higher frequencies. This characteristic resulted in tower vibration energy being concentrated primarily in the low-frequency range under wind loading.
• TMD placement on the blade is critical for effective vibration mitigation. Considering the blade as a cantilever beam, we found that the top displacement was most significant under first-order vibration. Thus, the TMD was strategically positioned at 82 m from the root of the blade tip, a decision reached through iterative optimization using the dichotomous method.
• We propose a combined strategy for vibration reduction by installing TMDs in both the nacelle and the blades. Our simulations showed that TMDs achieve optimal damping effects when the mass ratio was 1.8 and the frequency ratio was 0.86. This combined installation significantly reduced the amplitude and standard deviation of both anterior–posterior and lateral displacement at the tower top by 61.2%, 166.8%, 72%, and 67.6%, respectively. Notably, the TMD in the blades, targeting first-order swing frequencies, contributed less to lateral displacement reduction compared to the nacelle-based TMD, which addressed broader oscillation frequencies.
• Under low-to-moderate wind speeds typical of wind turbine operation, TMD performance in damping was moderate, reflecting the dominance of low-frequency vibrations. However, as wind speeds increased and higher frequencies in the wind spectrum became prominent, TMDs effectively dampened vibrations, thereby substantially improving tower stability.

These findings underscore the effectiveness of TMDs in mitigating wind-induced vibrations in wind turbine structures, with implications for optimizing TMD placement and configurations under varying operational conditions. This study provides some references for the design of dampers in wind turbines, which can be done by investigating a range of operating conditions in pre-installed TMDs, as well as wind turbine data, and designing TMDs with reference to the strategy in Figure 2.