## 2 Research Methodology and TMD Fundamentals
### Theoretical Framework of TMD  
The dynamic behavior of offshore jacket structures under environmental loads such as waves, wind, etc., necessitates a rigorous analysis of structural vibrations. To characterize the time-dependent response of offshore platforms, the governing equations of motion under external excitation have the following expression. \[\text{M}\ddot{\delta}+\text{C}_{\text{T}}\dot{\delta}+\text{K}\delta=F \tag{1}\]  
where M represents the structural mass matrix. \(C_{T}\) indicates the damping matrix. K refers to the structural stiffness matrix. \(\delta\), \(\dot{\delta}\), and \(\ddot{\delta}\) define the deformation, velocity, and acceleration  
Figure 1: Overall research flow. (**a**) Main process; (**b**) Platform structure modeling process; (**c**) Environmental load simulation process. of the structure, respectively. \(F\) denotes the external forces, including wind load \(F_{wind}\), wave and current load \(F_{wc}\)  
\[\textbf{F}=\textbf{F}_{wind}+\textbf{F}_{wc} \tag{2}\]  
For the TMD control device, this study employs a node and two zero-length elements for modeling and simulation. The computational model is shown in Figure 2. The node is connected to the structural node via two zero-length elements to model \(K_{TMD}\) (stiffness) and C\({}_{TMD}\) (damping), respectively, with M\({}_{TMD}\) (mass) applied to this node. The stiffness coefficient and damping coefficient of the TMD, considering linear spring and linear damping characteristics, are determined using elastic and viscous materials, respectively. As shown in Equation (1), the motion equation of the TMD under external excitation on the offshore platform can be expressed as:  
\[\text{M}_{\text{TMD}}\ddot{\text{Z}}+\text{C}_{\text{TMD}}\left(\dot{\text{Z}} -\dot{\text{Z}_{\text{s}}}\right)+\text{K}_{\text{TMD}}(\text{Z}-\text{Z}_{ \text{s}})=0 \tag{3}\]  
where the equation, \(\ddot{Z}\), \(\dot{Z}\) and \(Z\) represent the horizontal acceleration, velocity, and displacement of the TMD, respectively, while \(\dot{\text{Z}_{\text{s}}}\) and \(\dot{\text{Z}_{\text{s}}}\) denote the horizontal velocity and displacement of the primary structure. This TMD modeling method modifies the global quality matrix (M), damping matrix (C), and stiffness matrix (K) in Equation (1) in the following equations:  
\[\text{M}= \begin{bmatrix}\text{M}&0\\ 0&\text{M}_{\text{TMD}}\end{bmatrix}  \tag{4}\]  
\[\text{C}= \begin{bmatrix}\text{C}+\text{C}_{\text{TMD}}&-\text{C}_{\text{TMD}} \\ -\text{C}_{\text{TMD}}&\text{C}_{\text{TMD}}\end{bmatrix}  \tag{5}\]  
\[\text{K}= \begin{bmatrix}\text{K}+\text{K}_{\text{TMD}}&-\text{K}_{\text{TMD}} \\ -\text{K}_{\text{TMD}}&\text{K}_{\text{TMD}}\end{bmatrix}  \tag{6}\]  
In this study, a mass ratio of 0.02 was adopted for the TMD. This ratio balances three key criteria: vibration suppression efficacy, [9] indicated that a mass ratio between 1% and 3% optimizes the attenuation of dominant low-frequency structural modes in the floating platform's main deck, ensuring optimal control of dynamic responses under wave excitation; economic viability, ratios exceeding 3% incur substantial costs due to increased deck weight and reinforcement needs [12], making the 2% selection a pragmatic compromise between performance and budget constraints; and physical feasibility, the chosen TMD mass adheres to spatial limitations during actual deck installation, ensuring structural compatibility and operational feasibility without compromising usable workspace. The frequency and optimal damping percentage of the TMD are calculated using Equations (7) and (8) proposed by [PERSON] and [PERSON] [23]:  
Figure 2: TMD computational model. \[f_{opt}=\frac{f_{1}}{1+\mu} \tag{7}\]  
\[\xi_{opt}=\sqrt{\frac{3\mu}{8(1+\mu)^{3}}} \tag{8}\]  
where \(f_{opt}\) and \(f_{1}\) are the frequencies of the Tuned Mass Damper (TMD) and the structure, respectively \(\mu\) is the mass ratio, and \(\xi_{opt}\) represents the optimal damping percentage of the TMD.