Evaluation of Tuned Mass Damper for Offshore Wind Turbine Using Coupled Fatigue Analysis Method

Abstract

This study proposes an integrated fatigue life assessment methodology to accurately evaluate the time-domain evolution in tubular joint fatigue damage in offshore wind turbine (OWT) jacket structures under long-term combined wind and wave actions. A customized post-processing module was developed via secondary development on the MLife platform, employing a conditional probability distribution model to perform joint probabilistic modeling of measured marine environmental data, thereby establishing a long-term joint wind–wave distribution database. The reconstruction of hotspot stress time histories at the tubular joints was achieved through a hybrid analytical–numerical approach, integrating analytical formulations of nominal stress with a multi-axial stress concentration factor (SCF) matrix. Long-term fatigue damage assessment was implemented using the Palmgren–Miner linear cumulative damage hypothesis, where a weighted summation methodology based on joint wind–wave probability distributions rigorously accounted for the statistical contributions of individual design load cases. An ultimate bearing capacity analysis was also conducted based on S-N fatigue endurance characteristic curves. This research specifically investigates the influence mechanisms of tuned mass dampers (TMDs) on the time-domain-coupled fatigue performance of tubular joints subjected to long-term combined wind and wave loads. Numerical simulations demonstrate that parametrically optimized TMD systems significantly enhance the fatigue life metrics of critical joints in jacket structures.

1. Introduction

Offshore wind energy stands as a pivotal technology in the global energy transition, with large-scale deployment offering transformative potential to address escalating demands for decarbonized energy across industrial, commercial, and residential sectors. Projections indicate a 33% reduction in the global levelized cost of energy (LCOE) for offshore wind by 2035, including an immediate 10% decline by 2025 [1]. The Global Wind Energy Council (GWEC) forecasts the offshore wind capacity to surge to 380 GW by 2030 and 2000 GW by 2050 [2]. In China, data from the IRENA Renewable Energy Cost Database (RECD) reveals that operational offshore wind farms exhibit a weighted average water depth of 31 m, coastal proximity of 12 km, and mean project capacity of 245 MW [3], reflecting a predominant focus on nearshore shallow-water developments (20–40 m depth). Among foundation solutions for intermediate water depths (30–80 m), jacket structures emerge as the preferred choice due to their structural efficiency (lightweight design), hydrodynamic stability, technological maturity, and robustness in harsh marine environments. This study therefore prioritizes jacket foundations as its primary research focus.

Fatigue of offshore wind turbine foundations subjected to long-term exposure to stochastic wind and wave loading leads to gradually accumulated fatigue damage, which may eventually result in severe structural failure and substantial economic losses. Consequently, fatigue integrity assessment remains a critical research frontier in offshore wind turbine engineering. Chen et al. [4] using spectral fatigue analysis evaluated the fatigue damage of monopile-designed OWTs under individual and combined wind–wave loading. They argued that time-domain fatigue analysis is time-consuming and unsuitable for simulating various load conditions. However, due to its ability to capture structural nonlinearities and transient load effects under long-term cyclic loading, time-domain analysis is indispensable for final verification [5]. Dong et al. [6] conducted a decoupled time-domain long-term fatigue analysis of an OWT jacket support structure under wind and wave loads, identifying typical environmental conditions corresponding to the highest contributions to fatigue damage. In contrast to decoupled or simplified models, this study employs a fully coupled model that accounts for the aerodynamics of the blades [7], providing a more realistic fatigue assessment.

Previous time-domain fatigue analysis methods often focused on the study of load reduction and short-term dynamic response. Meng et al. [8] performed a time-domain fatigue evaluation of a floating OWT under various transient scenarios using damage equivalent loads (DELs). Li et al. [9] efficiently compared short-term and long-term fatigue damage of a monopile foundation OWT using different surrogate models to simulate less-load cases. Zhang et al. [10] proposed a modified nonlinear model to calculate the fatigue damage of a monopile foundation under long-term wind loading. The results indicated that the fatigue damage values considering the long-term probability distribution of wind loads were greater than those obtained by the linear superposition method. Therefore, the influence of long-term probability distribution should be considered in long-term fatigue damage analysis. As an extension of prior work, this study proposes a time-domain-coupled fatigue assessment framework that incorporates the site-specific joint probability model of wind and waves to account for long-term wind–wave interaction mechanisms for predicting the fatigue life of offshore wind turbines.

Vibration suppression strategies to extend the fatigue life of wind turbines have been extensively investigated. Based on fully coupled time-domain analysis using OpenFAST, Wang et al. [11] proposed a hybrid wind–wave energy system concept and optimized damping force for the optimal power take-off (PTO) under real sea conditions. Han et al. [12] conducted a comparative evaluation of the vibration suppression performance of TMDs and Tuned Liquid Column Dampers (TLCDs) on an FOWT. The results revealed that the TMD with a large stroke performed most prominently, effectively mitigating tower vibrations. Chan [13] compared the dynamic response of a monopile OWT equipped with a single TMD and multiple TMDs through a series of wind–wave load time histories. The results showed that the proposed multiplied inverse exchange element method (MulIEEM) outperformed all other methods in minimizing the structural top displacement of the OWT. Zhang et al. [14] investigated the vibration suppression mechanism of a bidirectional TMD installed in the nacelle on the dynamic response of a monopile OWT. The results indicated that the TMD exhibited excellent vibration reduction effects. Lu et al. [15] designed and manufactured a TMD model based on a 1/75 scale fully coupled test model of an OWT and evaluated the vibration suppression effect of the TMD on the structural response of a jacket OWT under typical wind–wave conditions. The results demonstrated that the TMD exhibited good vibration suppression effects on the structural response of the jacket OWT under normal operating conditions when the wind speed was not less than the rated wind speed.

In summary, although numerous studies have shown that TMDs have good vibration suppression performance on the dynamic response of OWTs under normal operating conditions, there is still insufficient research directly quantifying the impact of TMDs on OWT fatigue damage under combined wind and wave loading. Unlike previous studies, this framework directly quantifies the long-term cumulative fatigue damage at hot spots of different tubular joint types and the subsequent life extension effect brought by the TMD.

This study conducts a comprehensive fatigue integrity assessment of welded tubular joints in an OWT with a jacket foundation rated at a 50 m water depth designed for the South China Sea, focusing on quantifying fatigue damage accumulation induced by long-term synergistic wind–wave actions to ensure structural safety and lifecycle reliability. A probabilistic framework is employed to establish a joint wind–wave occurrence distribution through systematic load case probability computation and fatigue-critical scenario selection, leveraging measured metocean data for environmental characterization. Multi-physics fatigue loads derived from a fully coupled aero-hydro-servo-elastic model are processed via HSS time-history reconstruction, integrating nominal-stress analytical solutions with multi-axial SCF matrices to resolve stress gradients under combined loading. The methodology advances conventional time-domain analysis by discretizing environmental states into weighted load cases, enabling high-fidelity simulation of long-term coupled fatigue damage through the Palmgren–Miner linear cumulative damage theory and S-N-curve-based ultimate capacity evaluation. Further, the coupled fatigue analysis method is used for the quantitative analysis of TMD-induced vibration suppression effects, facilitating risk-informed inspection scheduling and lifecycle-oriented maintenance protocols.

2. The Framework for the Coupled Fatigue Analysis of an Offshore Wind Turbine

2.1. Brief Conceptual Explanation of the Proposed Coupled Fatigue Framework

The long-term time-domain-coupled fatigue assessment framework employed in this study, as illustrated in Figure 1, evaluates the enhancement of fatigue resistance in the tubular joints of OWT jacket foundations through TMD implementation. To address conventional conservatism from computational overhead and aleatory uncertainties in fatigue load case selection, the framework integrates probabilistic extrapolation scaling factors for short-term damage accumulation with fatigue-critical environmental state identification. It rigorously resolves multi-physics coupling mechanisms between environmental loads and the joint probability distribution of wind–wave actions on tubular joint fatigue degradation, enabling precise fatigue life quantification.

As shown in Figure 1, the coupled fatigue framework consists of four steps. Step 1: To address the inadequacy of traditional analyses that rely on a limited set of predefined design cases, which risk missing critical long-term environmental states, a joint probability distribution of metocean parameters (wind speed, wave height, spectral period) is constructed. This model utilizes statistical features of site-specific wind–wave data to preserve parameter interdependencies, thereby enabling the automated and rational identification of fatigue-critical load cases. Step 2: Overcoming the major limitation of frequency-domain analysis or decoupled simulations, which cannot capture nonlinear hydrodynamic–aerodynamic interactions; this step involves the execution of fully coupled aero-hydro-servo-elastic time-domain simulations (FAST). These simulations generate comprehensive component internal force databases, ensuring the complex, real-world dynamic responses of the integrated system are accurately captured. Step 3: To tackle the challenge of obtaining accurate local stress calculations for complex tubular joints, a hybrid analytical–numerical module is employed. This module combines nominal-stress formulations with multi-axial SCFs via coordinate-invariant transformations to efficiently produce precise HSS time histories at fatigue-prone locations. Step 4: This step solves the significant computational challenge of efficiently extrapolating numerous short-term results to the entire design life. A MATLAB R2021a -based framework discretizes the joint environmental probability distribution (from Step 1) using adaptive binning algorithms, assigning probabilistic extrapolation scaling factors (${\mathrm{f}}_{\mathrm{j}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}$) to each load case. Subsequently, rainflow counting, S-N curves, and the Palmgren–Miner rule are applied to compute short-term damage, which is then extrapolated to a 25-year cumulative fatigue damage (D_lt) through rigorous automated environmental weighting.

2.2. Detailed Workflow of the Coupled Fatigue Analysis Framework

The framework consists of four modules, each with a distinct function, as illustrated in Figure 2. The details of these modules are described in the following subsections.

2.2.1. Determination of Fatigue Load Cases

The fatigue assessment of tubular joints in OWT jacket structures must be grounded in site-specific marine environmental parameters in order to accurately simulate the interactive effects of wind–wave loads on fatigue design life. To probabilistically represent these multivariate environmental correlations, this study comprehensively evaluated advanced statistical frameworks including conditional joint distributions [16,17,18], Nataf transformation [19,20,21], and Copula models [22,23,24,25], ultimately adopting the conditional joint distribution method for its superior computational efficiency.

Based on the methodology reported by Johannessen et al. [26], this study parameterizes the multivariate joint probability distribution (wind speed, Uw; significant wave height, Hs; spectral peak period, Tp) using site-measured data (Equations (1)–(4)). The probabilistic screening of dominant wind–wave load cases was conducted using the cumulative occurrence probability thresholding method. The selection of the threshold value was determined with reference to the IEC [27] for the design requirements of OWTs concerning combined wind–wave assessment, alongside the DNV [28], in that fatigue limit state analysis should cover events down to a probability level of 10⁻⁴. In accordance with these standards, a cumulative occurrence probability threshold of no less than 96% was adopted for this study. This ensures that the identified load cases cover at least 96% of all possible environmental states. This approach enables the automated identification of critical fatigue load cases while maintaining computational efficiency by excluding extremely rare events. All the identified parameters were subsequently integrated directly into the coupled simulation framework.

$${\mathrm{f}}_{{\mathrm{U}}_{\mathrm{w}},{\mathrm{H}}_{\mathrm{s}},{\mathrm{T}}_{\mathrm{p}}}\left(\mathrm{u},\mathrm{h},\mathrm{t}\right)={\mathrm{f}}_{{\mathrm{U}}_{\mathrm{w}}}\left(\mathrm{u}\right)\cdot {\mathrm{f}}_{{\mathrm{H}}_{\mathrm{s}}|{\mathrm{U}}_{\mathrm{w}}}\left(\mathrm{h}|\mathrm{u}\right)\cdot {\mathrm{f}}_{{\mathrm{T}}_{\mathrm{p}}|{\mathrm{U}}_{\mathrm{w}},{\mathrm{H}}_{\mathrm{s}}}\left(\mathrm{t}|\mathrm{u},\mathrm{h}\right)$$

(1)

$${\mathrm{f}}_{{\mathrm{U}}_{\mathrm{w}}}\left(\mathrm{u}\right)={\displaystyle \frac{{\mathsf{\alpha }}_{\mathrm{U}}}{{\mathsf{\beta }}_{\mathrm{U}}}}{\left({\displaystyle \frac{\mathrm{u}}{{\mathsf{\beta }}_{\mathrm{U}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{U}}-1}\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{\mathrm{u}}{{\mathsf{\beta }}_{\mathrm{U}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{U}}}\right]$$

(2)

$${\mathrm{f}}_{{\mathrm{H}}_{\mathrm{s}}|{\mathrm{U}}_{\mathrm{w}}}\left(\mathrm{h}|\mathrm{u}\right)={\displaystyle \frac{{\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}}{{\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}}}{\left({\displaystyle \frac{\mathrm{h}}{{\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}-1}\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{\mathrm{h}}{{\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}}\right]$$

(3)

$${\mathrm{f}}_{{\mathrm{T}}_{\mathrm{p}}|{\mathrm{U}}_{\mathrm{w}},{\mathrm{H}}_{\mathrm{s}}}\left(\mathrm{t}|\mathrm{u},\mathrm{h}\right)={\displaystyle \frac{1}{\sqrt{2\mathsf{\pi }}{\mathsf{\sigma }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}\mathrm{t}}}\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{l}\mathrm{n}\left(\mathrm{t}\right)-{\mathsf{\mu }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}}{{\mathsf{\sigma }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}}}\right)}^2\right]$$

(4)

where $\mathrm{f}\left(\right)$ is the probability density function (PDF); $\mathrm{u}$ is the wind speed; ${\mathsf{\alpha }}_{\mathrm{U}}$ and ${\mathsf{\beta }}_{\mathrm{U}}$ are the shape and scale parameters of the Weibull distribution, respectively; $\mathrm{h}$ is the wave height; ${\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}$ and ${\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}$ are the shape and scale parameters, respectively, as shown in Equation (5), using a power function to express the relationship with wind speed; $\mathrm{t}$ is the wave period; ${\mathsf{\mu }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}$ and ${\mathsf{\sigma }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}$ are the expectation and standard deviation of $\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)$, which can be expressed in Equation (6); and ${\mathsf{\mu }}_{{\mathrm{T}}_{\mathfrak{p}}}$ and ${\mathsf{\sigma }}_{{\mathrm{T}}_{\mathfrak{p}}}$ are the mean and standard deviation of Tp, ${\mathsf{\upsilon }}_{{\mathrm{T}}_{\mathfrak{p}}}={\mathsf{\sigma }}_{{\mathrm{T}}_{\mathfrak{p}}}/{\mathsf{\mu }}_{{\mathrm{T}}_{\mathfrak{p}}}$, where each combination of Hs and Uw can be calculated according to Equations (7)–(9):

$${\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}={\mathrm{a}}_1+{\mathrm{a}}_2\cdot {\mathrm{u}}^{{\mathrm{a}}_3}{,\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}={\mathrm{b}}_1+{\mathrm{b}}_2\cdot {\mathrm{u}}^{{\mathrm{b}}_3}$$

(5)

$${\mathsf{\mu }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathfrak{p}}\right)}=\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\mathsf{\mu }}_{{\mathrm{T}}_{\mathfrak{p}}}}{\sqrt{1+{\mathsf{\upsilon }}_{{\mathrm{T}}_{\mathfrak{p}}}^2}}}\right),{\mathsf{\sigma }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathfrak{p}}\right)}^2=\mathrm{l}\mathrm{n}\left({\mathsf{\upsilon }}_{{\mathrm{T}}_{\mathfrak{p}}}^2+1\right)$$

(6)

$${\mathsf{\mu }}_{{\mathrm{T}}_{\mathrm{p}}}=\stackrel{\mathrm{‾}}{\mathrm{t}}\left(\mathrm{u},\mathrm{h}\right)=\stackrel{\mathrm{‾}}{\mathrm{t}}\left(\mathrm{h}\right)\cdot \left[1+\mathsf{\theta }{\left({\displaystyle \frac{\mathrm{u}-\stackrel{\mathrm{‾}}{\mathrm{u}}\left(\mathrm{h}\right)}{\stackrel{\mathrm{‾}}{\mathrm{u}}\left(\mathrm{h}\right)}}\right)}^{\mathsf{\gamma }}\right]$$

(7)

$$\stackrel{\mathrm{‾}}{\mathrm{t}}\left(\mathrm{h}\right)={\mathrm{e}}_1+{\mathrm{e}}_2\cdot {\mathrm{h}}^{{\mathrm{e}}_3},\stackrel{\mathrm{‾}}{\mathrm{u}}\left(\mathrm{h}\right)={\mathrm{f}}_1+{\mathrm{f}}_2\cdot {\mathrm{h}}^{{\mathrm{f}}_3}$$

(8)

$${\mathrm{v}}_{{\mathrm{T}}_{\mathfrak{p}}}\left(\mathrm{h}\right)={\mathrm{k}}_1+{\mathrm{k}}_2\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{h}{\mathrm{k}}_3\right)$$

(9)

2.2.2. Fully Coupled Time-Domain Simulations of the OWT

The derived fatigue load cases drive high-fidelity aero-hydro-servo-elastic (AHSE) time-domain simulations in FAST (Figure 3), where a fully coupled model integrating the rotor–nacelle assembly, tower, and jacket foundation are established based on multibody dynamics and finite element modeling theory. Resolving global structural displacements and load distributions through ElastoDyn and SubDyn and capturing nonlinear hydro-aero-elastic interactions, a multi-axis force/moment time-history database for critical components is generated to support the subsequent HSS analysis.

Concurrently, parametrically optimized TMDs are integrated in ServoDyn at the nacelle location to suppress first-mode vibrational responses. Coupled equations of motion governing OWT-TMD dynamics are formulated by the member-local coordinate system in FAST, with TMD kinematic responses derived from OWT structural motion via Newmark-β time integration. The resultant inertial, damping, and stiffness forces generate counteracting control forces applied to the nacelle interface nodes, achieving vibration energy dissipation.

2.2.3. Hotspot Stress Calculation

Considering the multi-axial stress effects at the tubular joints, nominal stresses under multi-axial loads (axial/bending/shear) are calculated in MLife using FAST member-local coordinate systems based on the established member-level internal force database. HSS time histories for X-/Y-/K-type joints are reconstructed by integrating analytical solutions derived from the Euler–Bernoulli beam theory, with empirical SCFs specified in DNVGL-RP-C203 [28]. Eight critical positions (saddle/crown, Figure 4) are defined for each joint in order to conduct fatigue stress-state evaluations, directly interfacing with the damage quantification phase.

2.2.4. The Coupled Fatigue Damage

The rainflow counting algorithm [29] is applied to analyze HSS time histories, precisely capturing mean stress shifts induced by wave loads and high-frequency stress fluctuations from wind loads in OWTs. The short-term fatigue damage (D_st) for each load case is calculated using the S-N curve (Equation (10)) and the Palmgren–Miner linear cumulative damage hypothesis.

To precisely quantify the long-term cumulative fatigue damage under coupled wind–wave loading, a robust extrapolation methodology was implemented within the MLife framework. This process begins by employing the bin method to discretize the continuous joint probability distribution of environmental parameters into manageable segments. For each unique environmental bin (a specific combination of wind and wave conditions), its occurrence probability (p_j) is calculated (Equation (11)), representing its statistical likelihood over the service life. This probability is then used to derive a crucial extrapolation scaling factor (${\mathrm{f}}_{\mathrm{j}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}$, Equation (12)), which acts as a multiplier to project the short-term results from each bin to the entire 25-year design lifespan, effectively weighting each load case’s contribution based on how frequently it occurs.

The analysis then seamlessly links the short-term simulations to long-term damage prediction. D_st obtained from a fully coupled simulation under a specific bin’s conditions is automatically matched with its corresponding ${\mathrm{f}}_{\mathrm{j}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}$. This pairing allows for the calculation of the equivalent number of stress cycles for that bin over 25 years (Equation (13)). Finally, the long-term cumulative fatigue damage (D_lt) is computed with high fidelity (Equation (14)) by summing the damage contributions from all bins, thus providing a comprehensive assessment that rigorously accounts for the full spectrum of environmental conditions and their probability of occurrence.

$$\mathrm{l}\mathrm{g}\mathrm{N}=\mathrm{l}\mathrm{g}\overline{\mathrm{a}}-\mathrm{m}\mathrm{l}\mathrm{g}\left[\Delta \mathsf{\sigma }{\left({\displaystyle \frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}\right)}^{\mathrm{k}}\right]$$

(10)

$${\mathrm{p}}_{\mathrm{j}}={\mathrm{p}}_{\mathrm{l}}^{\mathrm{u}}{\mathrm{p}}_{\mathrm{m}}^{\mathrm{h}}{\mathrm{p}}_{\mathrm{n}}^{\mathrm{t}}$$

(11)

$${\mathrm{f}}_{\mathrm{j}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}={\displaystyle \frac{{\mathrm{T}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}\mathrm{A}{\mathrm{p}}_{\mathrm{j}}}{{\mathrm{T}}_{\mathrm{j}}}}$$

(12)

$${\mathrm{n}}_{\mathrm{j}\mathrm{i}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}={\mathrm{f}}_{\mathrm{j}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}{\mathrm{n}}_{\mathrm{j}\mathrm{i}}$$

(13)

$${\mathrm{D}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}=\sum _{\mathrm{j}}\sum _{\mathrm{i}}{\displaystyle \frac{{\mathrm{n}}_{\mathrm{j}\mathrm{i}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}}{{\mathrm{N}}_{\mathrm{j}\mathrm{i}}}}$$

(14)

where ${\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the reference thickness, with welded tubular joints taking a value of 32 mm; $\mathrm{t}$ is the thickness most likely to crack, taking $\mathrm{t}={\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ if the thickness is less than the reference thickness; k is the thickness index; ${\mathrm{p}}_{\mathrm{l}}^{\mathrm{u}},{\mathrm{p}}_{\mathrm{m}}^{\mathrm{h}},\mathrm{and}{\mathrm{p}}_{\mathrm{n}}^{\mathrm{t}}$ can be calculated according to Equations (15)–(17); ${\mathrm{T}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}$ is the design life period, which is 25 years; ${\mathrm{T}}_{\mathrm{j}}$ is the total elapsed time for the j-th load case; A is the availability factor of the turbine, which is generally taken as 1; ${\mathrm{n}}_{\mathrm{j}\mathrm{i}}$ is the i-th cycle count for the j-th time series, which corresponds to stress range ${∆\mathsf{\sigma }}_{\mathrm{i}}$; and ${\mathrm{D}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}$ is the total cumulative fatigue damage, with fatigue failure occurring when ${\mathrm{D}}^{\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{e}}=1$.

$${\mathrm{p}}_{\mathrm{l}}^{\mathrm{u}}={\mathrm{e}}^{-{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{l}}-\frac{{\Delta }_{\mathrm{l}}^{\mathrm{V}}}{2}}{{\mathsf{\beta }}_{\mathrm{U}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{U}}}}-{\mathrm{e}}^{-{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{l}}+\frac{{\Delta }_{\mathrm{l}}^{\mathrm{V}}}{2}}{{\mathsf{\beta }}_{\mathrm{U}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{U}}}}$$

(15)

$${\mathrm{p}}_{\mathrm{m}}^{\mathrm{h}}={\mathrm{e}}^{-{\left({\displaystyle \frac{{\mathrm{H}}_{\mathrm{m}}-\frac{{\Delta }_{\mathrm{m}}^{\mathrm{H}}}{2}}{{\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}}}-{\mathrm{e}}^{-{\left({\displaystyle \frac{{\mathrm{H}}_{\mathrm{m}}+\frac{{\Delta }_{\mathrm{m}}^{\mathrm{H}}}{2}}{{\mathsf{\beta }}_{\mathrm{H}\mathrm{C}}}}\right)}^{{\mathsf{\alpha }}_{\mathrm{H}\mathrm{C}}}}$$

(16)

$${\mathrm{p}}_{\mathrm{n}}^{\mathrm{t}}=\mathrm{e}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\left(\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{n}}+\frac{{\Delta }_{\mathrm{n}}^{\mathrm{T}}}{2}\right)-{\mathsf{\mu }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}\right)}{\sqrt{2}{\mathsf{\sigma }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}}}\right]-\mathrm{e}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\left(\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{n}}-\frac{{\Delta }_{\mathrm{n}}^{\mathrm{T}}}{2}\right)-{\mathsf{\mu }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}\right)}{\sqrt{2}{\mathsf{\sigma }}_{\mathrm{l}\mathrm{n}\left({\mathrm{T}}_{\mathrm{p}}\right)}}}\right]$$

(17)

where ${\mathrm{V}}_{\mathrm{l}},{\mathrm{H}}_{\mathrm{m}}$, and ${\mathrm{T}}_{\mathrm{n}}$ are the wind speed Uw, significant wave height Hs, and spectral peak period Tp at the midpoint of bins l, m, and n, and ${\Delta }_{\mathrm{l}}^{\mathrm{V}},{\Delta }_{\mathrm{m}}^{\mathrm{H}}$, and ${\Delta }_{\mathrm{n}}^{\mathrm{T}}$ are the widths of bin l, m, and n.

2.3. Design Load Cases for TMD Parameters

The fatigue mitigation efficacy of a TMD on tubular joints within the target OWT infrastructure is quantified through long-term coupled fatigue damage. The TMD comprises a passive damping assembly (mass block, linear spring, viscous damper), neglecting rotational coupling effects. Its vibration suppression performance is governed by three primary design variables, as defined in Equation (18): mass ratio (μ), frequency tuning ratio (f), and damping coefficient (ζ). To optimize resonant response attenuation, Den Hartog’s classical optimization criterion (Equation (19)) is employed to determine f and ζ, minimizing the fatigue stress range at critical hotspot locations [30].

$$\mathsf{\mu }={\displaystyle \frac{{\mathrm{m}}_{\mathrm{T}\mathrm{M}\mathrm{D}}}{{\mathrm{m}}_1}},\mathrm{f}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{T}\mathrm{M}\mathrm{D}}}{{\mathsf{\omega }}_1}},{\mathsf{\zeta }}_{\mathrm{T}\mathrm{M}\mathrm{D}}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{T}\mathrm{M}\mathrm{D}}}{2\sqrt{{\mathrm{M}}_{\mathrm{T}\mathrm{M}\mathrm{D}}{\mathrm{K}}_{\mathrm{T}\mathrm{M}\mathrm{D}}}}}$$

(18)

$${\mathrm{f}}_{\mathrm{o}\mathrm{p}\mathrm{t}}={\displaystyle \frac{1}{1+\mathsf{\mu }}},{\mathsf{\zeta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}=\sqrt{{\displaystyle \frac{3\mathsf{\mu }}{8{\left(1+\mathsf{\mu }\right)}^3}}}$$

(19)

where ${\mathrm{m}}_{\mathrm{T}\mathrm{M}\mathrm{D}}$ is the TMD mass and ${\mathrm{m}}_1$ is the total OWT mass; ${\mathsf{\omega }}_{\mathrm{T}\mathrm{M}\mathrm{D}}$ is the natural frequency for the TMD, ${\mathsf{\omega }}_{\mathrm{T}\mathrm{M}\mathrm{D}}=\sqrt{{\mathrm{K}}_{\mathrm{T}\mathrm{M}\mathrm{D}}/{\mathrm{M}}_{\mathrm{T}\mathrm{M}\mathrm{D}}}$ is the natural frequency for the TMD, and ${\mathsf{\omega }}_1$ is the structural natural frequency for OWT; and ${\mathrm{f}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ and ${\mathsf{\zeta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ are the most natural frequency ratio and damping ratio of the TMD, respectively.

3. Wind Turbines and Marine Environmental Parameters

3.1. Wind Turbine Parameters

High-power units can capture more wind energy and reduce the LCOE accordingly, so the DTU10MW wind turbine prototype was selected from the benchmark [31] released by the Danish University of Science and Technology (DUST), which has been widely used in domestic and international research on ultra-large OWTs; the main performance parameters of the DTU10MW turbine are shown in

Table 1. General system properties of DTU 10 MW.

Parameter	Units	Value
Turbine rating	MW	10.0
Cut-in, rated wind turbine speed	rpm	6.9, 9.6
Cut-in, rated cut-out velocity	m/s	4.0, 11.4, 25.0
Rotor diameter	m	178.3
Hub height	m	135.0
Tower base location	m	20.5

. The specific parameters of the jacket infrastructure in the overall coupling model are shown in Figure 5.

3.2. Marine Environmental Parameters

The environmental conditions for the long-term fatigue calculation of the jacket foundation structure were defined based on wind–wave data observed in the South China Sea. Fluctuating wind and irregular wave models were adopted to simulate external environmental loads, with characteristic environmental parameters summarized in

Table 2. Design wind and wave parameters.

Parameter	Units	Value
Annual average wind speed	m/s	10.0
Mean sea level	m	0.37
Maximum significant wave height (1 in 2 year)	m	8.42
Maximum peak period (1 in 2 year)	s	12.52

.

To realistically simulate long-term sea-state variations in marine environments, a conditional joint probability model was implemented. The statistical characteristics of wind–wave interactions were analyzed through scatter diagram analysis, and the joint wind–wave distribution was fitted, yielding distribution parameters as presented in

Table 3. Parameters for marginal distribution of Uw, ${\mathrm{f}}_{{\mathrm{U}}_{\mathrm{w}}}\left(\mathrm{u}\right)$, conditional distribution of Hs given Uw, ${\mathrm{f}}_{{\mathrm{H}}_{\mathrm{s}}|{\mathrm{U}}_{\mathrm{w}}}\left(\mathrm{h}|\mathrm{u}\right)$, and conditional distribution of Tp given both Uw and Hs, ${\mathrm{f}}_{{\mathrm{T}}_{\mathrm{p}}|{\mathrm{U}}_{\mathrm{w}},{\mathrm{H}}_{\mathrm{s}}}\left(\mathrm{t}|\mathrm{u},\mathrm{h}\right)$.

Distributions	Parameter	Value
Marginal Uw	${\mathsf{\alpha }}_{\mathrm{U}},{\mathsf{\beta }}_{\mathrm{U}}$	1.864, 7.992
Conditional Hs given Uw	${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$	1.179, 0.005, 1.516
	${\mathrm{b}}_1,{\mathrm{b}}_2,{\mathrm{b}}_3$	0.346, 0.002, 2.260
Conditional Tp given Uw and Hs	$\mathsf{\theta },\mathsf{\gamma }$	–0.003, 1.000
	${\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3$	3.449, 1.520, 0.796
	${\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3$	0.664, 9.337, 0.488
	${\mathrm{k}}_1,{\mathrm{k}}_2,{\mathrm{k}}_3$	0.001, 0.248, –0.543

. Critical load cases with high occurrence probabilities and significant contributions to fatigue damage were selected from the joint wind–wave distribution of the target site. The bin method was subsequently applied to determine probability weights for these load cases, enabling systematic long-term fatigue calculations.

3.3. Validation of Wind–Wave Joint Distribution

This study systematically validated the wind–wave joint distribution model’s accuracy through statistical hypothesis testing and empirical distribution alignment, ensuring its reliability in short-term extrapolation for fatigue damage calculation. For statistical verification, the Chi-square test, a robust method for multivariate joint distributions, was employed to evaluate the consistency between the modeled predictions and observed wind–wave scatter data. Environmental parameters (Uw, Hs, and Tp) were discretized into predefined bins, with their observed frequency (Oi) and predicted frequency (Ei) statistically compared. The calculated Chi-square statistic (Equation (20)) yielded χ² = 1.21, which is significantly lower than the critical value χ²_critical = 132.89, confirming the model’s statistical validity at a 95% confidence level.

$${\mathsf{\chi }}^2=\sum {\displaystyle \frac{{\left({\mathrm{O}}_{\mathrm{i}}-{\mathrm{E}}_{\mathrm{i}}\right)}^2}{{\mathrm{E}}_{\mathrm{i}}}}$$

(20)

Furthermore, a quantile–quantile (Q-Q) analysis was conducted to visually assess the distribution consistency between predictions and field measurements. Key parameters (Uw, Hs, and Tp) were analyzed by comparing their predicted and empirical quantiles (

Table 4. Theoretical quantiles and observed quantiles for Uw, Hs, and Tp.

Empirical Quantiles	Theoretical Quantiles			Observed Quantiles
	Uw	Hs	Tp	Uw	Hs	Tp
1%	5.000	0.250	2.500	4.135	0.246	2.378
10%	7.000	0.250	3.500	6.360	0.340	3.482
50%	15.000	1.000	5.500	14.895	1.096	5.363
90%	21.000	2.750	6.500	20.571	2.800	6.902
99%	23.000	4.450	8.500	23.698	4.556	8.615

), as illustrated in Figure 6. The scatter points for all parameters exhibit close alignment with the diagonal line, demonstrating that the model effectively captures the joint distribution characteristics of real-world wind–wave conditions. This alignment provides high-confidence inputs for subsequent fatigue life predictions, ensuring engineering applicability.

4. The Validation of the Coupled Fatigue Analysis Method

4.1. The Coupled Fatigue Load Cases

The coupled fatigue analysis method emphasizes the processing of long-term environmental statistical data, employing a combined environmental contour method based on joint long-term wind–wave distributions to rationally partition load cases and streamline analysis. In accordance with the fatigue design load case DLC 1.2 for normal turbine operation specified in IEC 61400-3 [29], this study analyzed wind–wave parameter relationships and identified 110 high-probability wind–wave combinations (Figure 7) based on field monitoring data. These combinations collectively account for 97.53% of the total fatigue damage contribution.

4.2. The Comparison with the Direct Time-Domain Fatigue Damage

To validate the accuracy of the long-term coupled time-domain fatigue methodology, this study selected the highest-probability combined operational condition (11 m/s-0.75 m-4.5 s, 3.2%; Figure 7) and performed direct coupled time-domain simulations and fatigue damage calculations for all 167 cases within the corresponding environmental parameter cell (10–12 m/s, 0.5–1.0 m, 4–5 s; as illustrated in Figure 8). The resulting fatigue damage value at the tower base location was determined as 1.63 × 10⁻¹ A comparative analysis of the data in Figure 9 reveals consistent long-term fatigue damage trends across hotspot locations at the tower base, with the maximum fatigue damage occurring at N5 and a prediction error of approximately 6.34%. These results confirm that the proposed methodology enhances computational efficiency while maintaining accuracy. Notably, the long-term coupled time-domain fatigue damage values slightly exceed those from the full-load case simulations, a discrepancy hypothesized to originate from the long-term fatigue extrapolation factor derived from discretized environmental distributions, which marginally overestimates the actual probabilistic weights under operational conditions.

Furthermore, the calculations for all coupled fatigue load cases are carried out using the linear superposition method and the integral method considering the long-term wind speed probability distribution model, respectively. The linear superposition method is obtained based on the time of a year by direct multiplication with D_st, which is 3.99 × 10⁻¹, whereas the coupled method is calculated by combining D_st and f_Tp|Uw,Hs (t|u,h) of the jacket foundation. Therefore, the long-term damage can be calculated as 5.14 × 10⁻¹ by superimposing the fatigue damage values under different conditions.

From the above analysis, the long-term fatigue damage value obtained by the coupled method is larger than the result of the linear superposition method for this OWT foundation due to the influence of the wind–wave joint distribution. It can be seen from

Table 5. Long-term fatigue damage and fatigue life of tower base.

Computing Method	Long-Term Fatigue Damage	Fatigue Life (Year)
Linear superposition method	3.99 × 10−1	62.7
Coupled fatigue method	5.14 × 10−1	48.5
Spectral fatigue method	7.65 × 10−1	32.7

that the effect of the long-term wind–wave joint probability distribution should be considered when performing long-term fatigue damage analysis.

4.3. Comparison with the Spectral Fatigue Method

To validate the reliability of the computational results from the time-domain-coupled long-term fatigue methodology, the traditional spectral fatigue method was employed to calculate the long-term fatigue damage at the tower base under combined wind–wave loading. A finite element model of the jacket foundation offshore wind turbine structure was developed using the SACS V16 software. The fatigue damage effects induced by wave loads were calculated based on wave spectrum analysis (Equation (21)), while those caused by wind turbine equivalent fatigue loads were computed via deterministic methods (Equation (22)). The overall fatigue damage level and service life of the wind turbine structure were subsequently obtained through linear superposition (Equation (23)).

$${\mathrm{D}}_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}=\sum _{\mathrm{i}}{\displaystyle \frac{{\mathrm{n}}_{\mathrm{i}}}{{\mathrm{N}}_{\mathrm{i}}\left({\mathrm{S}}_{\mathsf{\sigma },\mathrm{i}}\right)}}=\sum _{\mathrm{i}}{\displaystyle \frac{{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{S}}_{\mathsf{\sigma }}\left(\mathrm{f}\right)}{{\mathrm{S}}_{\mathsf{\eta }}\left(\mathrm{f}\right)}\cdot {\mathrm{S}}_{\mathsf{\eta }}\left(\mathrm{f}\right)\mathrm{d}\mathrm{f}}{\cdot {\mathrm{S}}_{\mathsf{\sigma },\mathrm{i}}^{-\mathrm{m}}}}$$

(21)

$${\mathrm{D}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}=\sum _{\mathrm{j}}{\displaystyle \frac{{\mathrm{n}}_{\mathrm{j}}}{{\mathrm{N}}_{\mathrm{j}}\left({\mathrm{S}}_{\mathrm{e}\mathrm{q},\mathrm{j}}\right)}}=\sum _{\mathrm{j}}{\displaystyle \frac{{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{e}}\cdot {\mathrm{f}}_{\mathrm{j}}\cdot {\mathrm{S}}_{\mathrm{e}\mathrm{q},\mathrm{j}}^{\mathrm{m}}}{\mathrm{A}}}$$

(22)

$${\mathrm{D}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\mathrm{D}}_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}+{\mathrm{D}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}$$

(23)

where Sσ(f) = |H_σ(f)|²⋅S_η(f) is the stress power spectral density, H_σ(f) is the stress transfer function, S_η(f) is the wave spectrum, A and m are S-N curve parameters, S_eq,j is the equivalent stress amplitude, f_j is the load cycle frequency, and T_life is the design lifetime.

As shown in Table 5, the time-domain-coupled long-term fatigue damage value at the tower base was 5.14 × 10⁻¹, representing a 32.81% reduction compared to the spectral fatigue analysis results; the corresponding fatigue life was nearly one-third longer than that obtained via spectral analysis. This demonstrates that the conventional assumption of independence between wind and wave actions, combined with the linearization inherent in frequency-domain analysis, leads to overly conservative predictions of long-term fatigue damage. Such conservatism may result in excessive overdesigning of tubular joints in offshore wind turbine foundation structures.

5. Evaluation of Tuned Mass Damper for Offshore Wind Turbine

5.1. TMD Parameters

Because the OWT in this study operates at a rated rotational speed of 9.6 rpm, to mitigate resonance risks, the natural frequency of the integrated structure must avoid the 1P (0.16 Hz) and 3P (0.48 Hz) excitation frequency bands. Free decay analysis of the fully coupled jacket model identified a first-order natural frequency of 0.264 Hz for the structure, which complies with the safety interval requirement (0.21–0.45 Hz).

The modal analysis (Figure 10) demonstrates that the tower displacement amplitude exhibits a linear growth pattern along the vertical axis, reaching a maximum value of 1.82 m at the nacelle position. This displacement pattern characterizes a dominant first-order lateral vibration mode. To suppress low-frequency vibration responses, a TMD with a tuned mass ratio of 1% (27,676.676 kg,

Table 6. TMD parameters.

Parameter	Units	Value
TMD Location	-	Nacelle
Mass Ratio	-	1%
Frequency	Hz	0.256
Mass	kg	27,676.676
Stiffness	N/m	61,423.677
Damping	N/ms−1	4123.109

) was optimally designed and integrated into the nacelle, specifically targeting the enhancement of first-order modal damping. Post-installation monitoring showed an 85.7% reduction in the tower-top displacement amplitude, decreasing from 1.82 m to 0.26 m, thereby validating its efficacy in suppressing coupled dynamic instabilities under operational conditions.

5.2. Hotspot Stress Analysis

Three tubular joint configurations (K-type: JD1; X-type: JD2; Y-type: JD3) were selected for this study. Their stress concentration factors (SCFs) were calculated using Efthymiou’s semi-empirical parametric formulations [32], with their geometric parameters (d: brace diameter; D: chord diameter; t: brace thickness; T: chord thickness; L: chord length; g: gap length) defined as per

Table 7. Basic dimensionless parameters used for stress concentration factor and influence function calculations.

Tubular Joints	JD1 (M1/M2)	JD2 (M3/M4)	JD3
$\mathsf{\beta }=\mathrm{d}/\mathrm{D}$	0.595	0.974	0.522
$\mathsf{\tau }=\mathrm{t}/\mathrm{T}$	0.5	0.714	0.692
$\mathsf{\gamma }=\mathrm{D}/2\mathrm{T}$	10.5	10.857	10.928
$\mathsf{\alpha }=2\mathrm{L}/\mathrm{D}$	25	19.737	15.217
$\mathsf{\zeta }=\mathrm{g}/\mathrm{D}$	0.089	-	-

. Based on the derived SCFs, 40 HSS locations were analyzed, encompassing critical fatigue-prone regions of the upper/lower braces (M1/M2 at JD1; M3/M4 at JD2), as illustrated in Figure 11.

The HSS ranges and standard deviations at critical tubular joint locations under all fatigue load cases are presented in Figure 12, Figure 13 and Figure 14. A comparative analysis reveals significantly larger HSS ranges and standard deviations for JD1 and JD2 compared to JD3, with both parameters exhibiting parallel trends. Following TMD implementation, all of the monitored positions demonstrated reduced HSS ranges and standard deviations, with the most pronounced reductions observed at critical hotspot locations. While TMD integration minimally altered the fundamental patterns of the HSS range and standard deviation distributions, it achieved substantial mitigation at critical hotspots. As detailed in

Table 8. Maximum hotspot stress range and standard deviation of tubular joints.

Key Joints	Range		Standard Deviation
	Without TMD	With TMD	Without TMD	With TMD
JD1M1N5 (K-joint)	34.3	26.5	3.48	3.23
JD1M2N1 (K-joint)	24.4	20.0	2.57	2.30
JD2M3N4 (X-joint)	54.2	42.5	5.34	4.79
JD2M4N5 (X-joint)	73.8	66.7	9.27	8.77
JD3N7 (Y-joint)	8.94	8.94	0.88	0.82

, the HSS range at JD1M1 decreased by 22.7%, while JD2M4 exhibited reductions of 9.62% in the HSS range (to 73.8 MPa) and 5.39% in the standard deviation (to 9.27 MPa). This systematic attenuation confirms TMDs’ efficacy in suppressing vibration-induced stress concentrations while preserving inherent fatigue-response characteristics.

5.3. Long-Term Coupled Fatigue Damage Analysis

The equivalent fatigue stress range of the tubular joints in the jacket foundation structure was statistically determined through rainflow counting of HSS time histories. The short-term equivalent fatigue stress ranges and corresponding fatigue damage values for each joint are summarized in

Table 9. Maximum equivalent fatigue stress ranges and fatigue damage.

Key Joints	Damage Equivalent Stress		Fatigue Damage
	Without TMD	With TMD	Without TMD	With TMD
JD1M1N5 (K-joint)	5.62	4.90	1.68 × 10−6	1.11 × 10−6
JD1M2N1 (K-joint)	4.24	3.70	1.11 × 10−6	7.46 × 10−7
JD2M3N4 (X-joint)	8.91	7.80	2.96 × 10−6	1.98 × 10−6
JD2M4N5 (X-joint)	12.80	11.50	3.76 × 10−6	2.83 × 10−6
JD3N7 (Y-joint)	1.84	1.74	5.86 × 10−8	4.97 × 10−8

, indicating a direct correlation between higher short-term equivalent stress ranges and increased fatigue damage. For the K-joint (JD1), the fatigue damage at the M1-chord connection exceeds that at M2 by approximately 51.4%, while for the X-joint (JD2), the M4-chord connection exhibits 27.2% higher fatigue damage compared to M3. These results designate M1 (JD1) and M4 (JD2) as critical members for long-term fatigue analysis. The X-joint (JD2) demonstrates the highest equivalent fatigue stress range (12.8 MPa), surpassing values observed in the K- and Y-joints by 18–22%. Following TMD implementation, all joints exhibit reduced equivalent fatigue stress ranges while retaining consistent spatial distribution patterns. As illustrated in Figure 15, the comparative analysis confirms the alignment between the hotspot stress ranges and equivalent fatigue stress trends, with identical critical hotspot locations identified across both evaluations.

As demonstrated in Figure 16, Figure 17 and Figure 18, significant discrepancies exist in the time-domain-coupled long-term cumulative fatigue damage across tubular joint types. The long-term fatigue damage for JD1 (K-joint) and JD2 (X-joint), located in the splash zone of the jacket foundation near the mean sea level, substantially exceeds that of JD3 (Y-joint), with JD2 exhibiting approximately twice the cumulative damage of JD1. An analysis of the short-term-to-long-term fatigue damage accumulation ratio (Dst/Dlt) in Figure 16b and Figure 18b reveals a near-inverse trend compared to that shown in Figure 16a and Figure 18a. Under long-term combined wind–wave loading, the fatigue damage accumulation rates at the hotspot locations of the tubular joints vary significantly during operational periods, with critical hotspots demonstrating the highest rates. Notably, the short-term-to-long-term fatigue damage accumulation ratio at hotspot location N5 of JD2 (X-joint) reaches a minimum value of 2.05 × 10⁻⁵. After TMD implementation, Figure 16, Figure 17 and Figure 18 indicate that the spatial distribution patterns of long-term cumulative fatigue damage for JD1, JD2, and JD3 retain their original critical hotspot locations, albeit with reduced overall accumulation rates. As summarized in

Table 10. Characteristic long-term fatigue damage and fatigue life of tubular joints.

Tubular Joints	Long-Term Fatigue Damage		Fatigue Life (Year)
	Without TMD	With TMD	Without TMD	With TMD
JD1 (K-joint)	9.50 × 10−2	6.19 × 10−2	2.63 × 102	4.03 × 102
JD2 (X-joint)	1.83 × 10−1	1.27 × 10−1	1.37 × 102	1.97 × 102
JD3 (Y-joint)	5.55 × 10−3	4.68 × 10−3	4.50 × 103	5.33 × 103

, the X-joint (JD2) exhibits the highest fatigue damage reduction during the operational period, decreasing from 0.183 to 0.127, corresponding to a 44% fatigue life extension. The K-joint (JD1) achieves the most pronounced improvement, with a 53% fatigue life extension due to the effectiveness of the TMD.

5.4. Limitations of the Proposed Coupled Fatigue Analysis Method

Although the coupled fatigue analysis method proposed in this study has yielded successful conclusions regarding the performance evaluation of the TMD and fatigue life extension, the framework is subject to several limitations that should be addressed in future research.

Firstly, the validation of the proposed damage calculation methodology requires further strengthening. The current validation is primarily based on comparisons with other numerical methods. Future work urgently needs quantitative validation against experimental model tests or field monitoring data from offshore wind farms to unequivocally demonstrate its accuracy in practical engineering applications. Furthermore, this study did not systematically analyze the impact of the inherent scatter in S-N curve data on fatigue prediction uncertainty, nor did it investigate the quantitative influence of different design safety factors on the final fatigue life results.

Secondly, regarding local stress analysis, empirically derived SCF formulas recommended by standards were employed to balance computational efficiency and engineering accuracy. While suitable for a system-level parametric study, they might not fully capture the precise stress fields under complex joint geometries. The accuracy of local stress evaluation could be further enhanced in future studies by applying more advanced parametric SCF models or conducting detailed finite element (FE) analysis on the critical hotspot locations identified herein.

Finally, the TMD parameters in this study were optimized based on the classical Den Hartog formula and validated within the fully coupled model. However, certain modeling assumptions, such as neglecting rotational coupling effects in the TMD model and assuming a turbine availability factor of 1, could influence the absolute values of the predicted fatigue life. Although sensitivity analysis suggests that the impact of these assumptions on the comparative conclusions of this study is limited, their effects should be considered for specific design projects.

Notwithstanding these limitations, the proposed fatigue assessment framework based on a site-specific joint wind–wave probability model effectively overcomes the constraints of traditional design load cases. It provides a robust tool for evaluating the life-extension performance of TMDs, demonstrating significant potential for engineering applications. Future research could beneficially incorporate a sensitivity analysis of the fatigue life predictions to different probabilistic models to enhance the robustness of the framework. In addition, investigating the applicability of this framework for fatigue assessment of a next-generation 15–20 MW large-scale OWT will constitute a critical and promising research direction in the future.

6. Conclusions

This study presents a time-domain-coupled long-term fatigue analysis of an OWT with a jacket foundation rated at a 50 m water depth in China’s coastal waters, focusing on developing an engineering-oriented methodology for quantifying tubular joint fatigue damage under combined wind–wave stress. Using this framework, the fatigue life enhancement effect of TMD integration on the tubular joints of the jacket foundation was systematically investigated. Our key findings include the following:

• Significant discrepancies were observed in time-domain-coupled long-term cumulative fatigue damage across joint types. In the splash zone near the mean sea level, the JD1 (K-type) and JD2 (X-type) joints exhibited substantially higher cumulative damage than the submerged JD3 (Y-type) joints, with JD2 demonstrating the maximum cumulative damage (0.183) during the operational period.
• TMD implementation minimally altered critical hotspot locations or fundamental trends in stress amplitude distribution (hotspot stress, equivalent fatigue stress) and short-/long-term fatigue accumulation patterns across all joints.
• However, the TMD effectively reduced fatigue accumulation rates at hotspot locations under long-term wind–wave loading, achieving notable life extension for the K-, X-, and Y-type joints. The K-joint (JD1) exhibited the most significant improvement, with a 53% fatigue life extension during the operational period.