## 2 FOWT Coupled Dynamics Theory
### Platform  
For the entire nonlinear system, all forces acting on the entire structure should be balanced. For floating platforms, the received external forces mainly include platform gravity, hydrodynamics, and mooring line tension of the mooring system. This part shows the forces modeled in in-house code when using a potential flow theory approach. The forces and added mass are applied at the platform reference point in the theory. The potential flow theory is used to calculate hydrodynamics and is expressed as follows [33]:  
\[\overset{\rightarrow}{F}_{WRP}(x)=\overset{\rightarrow}{F}_{W}(x)+\overset{ \rightarrow}{F}_{HS}(x)+\overset{\rightarrow}{F}_{RD}(x)+\overset{ \rightarrow}{F}_{AM}(x) \tag{3}\]  
in which,  
\[\overset{\rightarrow}{F}_{AM}(x)=-AM_{R}\overset{\rightarrow}{a}_{p} \tag{4}\]  
\[\overset{\rightarrow}{F}_{W}(x)=\frac{1}{N}\sum_{k=-\frac{N}{2}-1}^{\frac{N} {2}}W[k]\sqrt{\frac{2\pi}{\Delta t}S_{\xi}^{2-sided}(\omega_{0})}X(\omega_{0},\beta)|_{\omega_{0}=k\Delta\omega}e^{j\frac{2\pi\beta n}{N}} \tag{5}\]  
\[\overset{\rightarrow}{F}_{HS}(x)=\rho gV_{0}\delta_{3}-C^{Hydrostatic}x \tag{6}\]  
\[\overset{\rightarrow}{F}_{RD}(x)=-\int_{0}^{t}K(t-\tau)\dot{x}_{j}(\tau)d\tau \tag{7}\]  
where, \(\overset{\rightarrow}{F}_{WRP}(x)\) is the total load at the platform reference point; \(\overset{\rightarrow}{F}_{W}(x)\) is the incident-wave excitation force at the platform reference point; \(\overset{\rightarrow}{F}_{HS}(x)\) is the hydrostatic force at the reference node; \(\overset{\rightarrow}{F}_{RD}(x)\) is the radiation memory-effect force at the reference point; and \(\overset{\rightarrow}{F}_{AM}(x)\) is the total added mass force from all contributions.  
The terms of the general equation are discussed separately. \(AM_{R}\) is the added-mass due to radiation at the reference point; \(\overset{\rightarrow}{a}_{p}\) is the linear acceleration of the structure; \(W[k]\) is the Fourier transform of a white noise time series with zero mean and unit variance; \(S_{\xi}^{2-sided}(\omega_{0})\) is the desired power spectral density of the wave elevation per unit time, this paper uses the P-M spectrum; \(X(\omega_{0},\beta)\) is wave-excitation force array normalized per unit wave amplitude; \(\omega_{0}\) is the frequency; \(\beta\) is the incident wave direction angle; \(j\) is the number of degrees of freedom (DOF); \(n\) is the discrete-time-step counter; \(k\) is the discrete-frequency-step counter; \(N\) is the number of discrete steps; \(\rho gV_{0}\delta_{3}\) is the buoyancy and equals the platform weight; \(V_{0}\) is the volume of the immersed part of the platform; \(\delta_{3}\) is the component of the Kronecker-Delta function; \(C^{Hydrostatic}\) is the hydrostatic-restoring matrix; \(x\) is the platform motion; \(t\) is time; \(K\) is the radiation kernel from potential flow theory; and \(\dot{x}(\tau)\) is the platform velocity. The details of this method can be referred in the study by [PERSON] [33].  
Based on the theories introduced above, a program for coupling computation of the FOWT was developed in this work. The coupled dynamic response of FOWT consists of three parts, namely the aerodynamics of the wind turbine, the hydrodynamics of the floating platform, and the response of the mooring system. Since this paper mainly studies the dynamic response of floating platforms with different structural forms under the condition of the number of offset columns, only the calculation theory of the latter two parts with more important influences is considered. Figure 1 shows the structure of the in-house developed code, consisting of the calculations of the wind inflow, aerodynamics, waves, sea current, hydrodynamics, and dynamics of the floater.  
We regard the tower as a rigid body and only the low-frequency response of the platform with six degrees of freedom is studied. The high-frequency responses of the system, such as the high-frequency vibrations of the blade and the tower, are not considered. Therefore, the turbine-tower interaction is not taken into consideration in the present research.