*2.1. Static Displacements and Equilibrium under the Steady Current and without the Wave Effect* The static displacements of the five elements are:

*x*<sup>0</sup> = 0, *y*<sup>0</sup> = 0 *x*1*<sup>s</sup>* = *Hbed* − *LC* = *LA* sin *θAs*, *y*1*<sup>s</sup>* = *LA* cos *θAs*; *x*2*<sup>s</sup>* = *Hbed* − *LD* = *x*1*<sup>s</sup>* − *LB* sin *θBs*, *y*2*<sup>s</sup>* = *y*1*<sup>s</sup>* + *LB* cos *θBs x*3*<sup>s</sup>* = *x*1*<sup>s</sup>* + *LC* = *Hbed*, *y*3*<sup>s</sup>* = *y*1*<sup>s</sup>*

$$\mathbf{x\_{4s}} = \mathbf{x\_{3s}} = \mathbf{x\_{2s}} + L\_D = H\_{\text{bed}} \cdot y\_{4s} = y\_{2s}$$

Due to *x*1*<sup>s</sup>* >> *x*1*d*, the global inclined angle *qA* can be expressed as:

$$
\sin \theta\_A = \frac{\mathbf{x}\_1}{L\_\mathbf{A}} = \frac{\mathbf{x}\_{1s} + \mathbf{x}\_{1d}}{L\_\mathbf{A}} \approx \frac{\mathbf{x}\_{1s}}{L\_\mathbf{A}} = \sin \theta\_{As} \tag{4}
$$

Due to *xis* >> *xid*, the global inclined angle *qB* can be expressed as:

$$
\sin \theta\_B = \frac{\mathbf{x}\_1 - \mathbf{x}\_2}{L\_\mathbf{B}} = \frac{(\mathbf{x}\_{1s} + \mathbf{x}\_{1d}) - (\mathbf{x}\_{2s} + \mathbf{x}\_{2d})}{L\_\mathbf{B}} \approx \frac{\mathbf{x}\_{1s} - \mathbf{x}\_{2s}}{L\_\mathbf{B}} = \sin \theta\_{Bs} \tag{5}
$$

Under the steady current and without the wave effect, the static horizontal and vertical equilibriums of the floating platform are written, respectively, as shown in Figure 1.

$$T\_{\rm Bs} \cos \theta\_{\rm Bs} + F\_{\rm DBs} = T\_{\rm As} \cos \theta\_{\rm As} \tag{6}$$

$$F\_{\rm B1s} = T\_{\rm As} \sin \theta\_{\rm As} + T\_{\rm Bs} \sin \theta\_{\rm Bs} + W\_1 \tag{7}$$

where *T*A*s*, *T*B*s*, *FB*<sup>1</sup>*<sup>s</sup>* and *W*<sup>1</sup> are the static tensions of ropes *A* and *B*, the buoyancy of the floating platform and the weight of the floating platform, respectively. The steady drag of the floating platform under current *FDFs* = <sup>1</sup> <sup>2</sup>*CDFyρAFYV*2.

The static horizontal and vertical equilibriums of the turbine are expressed, respectively, as:

$$T\_{\rm Bs} \cos \theta\_{\rm Bs} = F\_{\rm DTs} \tag{8}$$

where the steady drag of the turbine *FDTs* = *CDTy* <sup>1</sup> <sup>2</sup> *<sup>ρ</sup>ATyV*2.

$$F\_{\rm B2s} = \mathcal{W}\_{\rm 2} - T\_{\rm Ds} - T\_{\rm Bs} \sin \theta\_{\rm Bs} \tag{9}$$

where *T*D*s*, *FB*<sup>2</sup>*<sup>s</sup>* and *W*<sup>2</sup> are the static tensions of rope D, the static buoyancy and the weight of the turbine, respectively. The static vertical equilibrium of pontoon 3 is expressed as:

$$F\_{\text{B3s}} = \mathcal{W}\_{\text{\text{\textdegree}}} + T\_{\text{\textdegree}} \tag{10}$$

where *FB*<sup>3</sup>*<sup>s</sup>* and *W*<sup>3</sup> are the static buoyancy and the weight of pontoon 3, respectively. The static vertical equilibrium of the pontoon 4 is expressed as:

$$F\_{\rm B4s} = \mathcal{W}\_4 + T\_{\rm Ds} \tag{11}$$

where *FB*<sup>4</sup>*<sup>s</sup>* and *W*<sup>4</sup> are the static buoyancy and the weight of pontoon 4, respectively.

#### *2.2. Simulation of Irregular Wave*

The irregular wave is represented by the Jonswap wave spectrum. The Jonswap wave spectrum is given as a modification of the Pierson–Moskowitz spectrum in accordance with DNV [28,31].

The wave energy spectrum is:

$$S\_{\rm I}(f) = B\_{\rm I} H\_s^2 f\_p^4 f^{-5} \exp\left[\frac{-5}{4} \left(\frac{f}{f\_p}\right)^{-4}\right] \gamma^b \tag{12}$$

where *f* is the wave frequency, *fp* is the peak frequency and *Hs* is the significant wave height.

$$B\_{I} = \frac{0.06238 \times (1.094 - 0.01915 \text{lyr})}{0.230 + 0.0336 \gamma - \frac{0.0185}{1.9 + \gamma}}, \ b = \exp\left[ -0.5 \left( \frac{f - f\_{p}}{\sigma f\_{p}} \right)^{2} \right], \tag{13}$$

$$\sigma = \begin{cases} \ 0.07, \text{ for } f \le f\_{p} \\\ 0.09, \text{ for } f > f\_{p} \end{cases}, \gamma = 3.3$$

Referring to the information from the Central Meteorological Bureau Library of Taiwan about the typhoons that have invaded Taiwan from 1897 to 2019 [26,32] and selecting 150 typhoons that greatly affected Taiwan's Green Island, the significant wave height *Hs* during the 50-year regression period *Hs* =15.4 m, and the peak period *Pw* = 16.5 s.

Substituting the significant wave height *Hs* and the peak period *Pw* into Equations (12) and (13), the Jonswap wave spectrum is determined, as shown in Figure 3.

**Figure 3.** Jonswap wave spectrum.

The frequency domain of the wave spectrum is divided into *N* subdomains of (*ω*0, *ω*1), ... , (*ωN*−1, *ωN*). The sea surface elevation of an irregular wave can be generated by the superposition of the regular wave components:

$$\chi\_{\mathcal{W}} = \sum\_{i=1}^{N} a\_i \sin \left( \Omega\_i t - \stackrel{\rightarrow}{K}\_i \stackrel{\rightarrow}{\cdot \bar{R}} + \varphi\_i \right) \tag{14}$$

where *ai*, <sup>Ω</sup>*i*, *<sup>ϕ</sup><sup>i</sup>* and <sup>→</sup> *Ki* are the amplitude, angular frequency, phase angle and wave vector of the *i*-th regular wave, respectively. The angular frequency is Ω*<sup>i</sup>* ∈ (*ωi*−1, *ωi*). The amplitude can be determined by:

$$\frac{1}{2}a\_i^2(\Omega\_i) = \int\_{\omega\_{i-1}}^{\omega\_i} S(\omega)d\omega \tag{15}$$

The linear dispersion relation is considered [33]:

$$
\Omega\_i^2 = \lg \tilde{k}\_i \tanh \tilde{k}\_i H\_{bcd} \tag{16}
$$

where *g* is gravity. The wave number 1*ki* = → *Ki* . Based on Equation (16), the wave number is obtained. Further, the wave length *λ<sup>i</sup>* can be determined via the relation between the wave number 1*ki* and the wave length 1*ki* = 2*π*/*λi*. Based on Equations (14)–(16) and Figure 4, and letting *n* = 6, the irregular wave is simulated by regular waves and listed in Table 1. It is assumed that the total wave energy flow rate of the regular waves is equal to that of the Jonswap wave spectrum. In cases 1~4, the numbers of regular waves range from 3 to 6. Every energy flow rate of every regular wave is assumed to be the same. According to Equation (15), the amplitude of each regular wave is the same, and there is no distinction

between the dominated wave and the secondary wave. Moreover, the frequency of the regular wave and the given peak frequency are significantly different. In case 5, when six regular waves are used to simulate an irregular wave, every energy flow rate of every regular wave is different. It is obtained that the dominated wave frequency is consistent with the given peak frequency, and the amplitude of the dominated wave is significantly larger than that of other waves. The simulated results of case 6 are used later.


**Table 1.** Irregular wave simulated by regular waves [*Hs* = 15.4 m, *Pw* = 16.5 s, *n* = 6, *Hbed* = 1300 m].

*2.3. Dynamic Equilibrium with the Effects of the Steady Current and Irregular Wave* The dynamic equilibrium in the vertical direction for pontoon 3 is:

$$M\_3 \ddot{\chi}\_{3d} - F\_{B3} + W\_3 + T\_\mathbb{C} = 0 \tag{17}$$

where *M*<sup>3</sup> is the mass of pontoon 3. *TC* is the tension of rope C. Substituting Equations (2) and (10) into Equation (17), one obtains:

$$M\_{\rm 3}\ddot{\mathbf{x}}\_{\rm 3d} + T\_{\rm Cd} - F\_{\rm B3d} = 0 \tag{18}$$

where the dynamic tension of the rope C is:

$$T\_{\mathbb{C}d} = K\_{\mathbb{C}d} (\mathbb{x}\_{\mathbb{S}d} - \mathbb{x}\_{1d}) \tag{19}$$

in which *KCd* is the effective spring constant. *x*3*<sup>d</sup>* − *x*1*<sup>d</sup>* is the dynamic elongation between floating platform 1 and pontoon 3. Considering the safety of the rope, some buffer springs are used to serially connect the rope between elements 1 and 3. The effective spring constant of the rope–buffer spring connection is obtained:

$$K\_{\mathbb{C}d} = \frac{K\_{\text{rope }\mathbb{C}}}{1 + K\_{\text{rope }\mathbb{C}} / K\_{\mathbb{C},spring}}\tag{20}$$

where *KC*,*spring* is the constant of the spring connecting with rope C. The effective spring constant of the rope C, *K*rope C = *ECAC*/*LC*, where *EC*, *AC*, and *LC* are the Young's modulus, cross-sectional area and length of rope C, respectively.

**Figure 4.** Relation among phase *fi*, wave length *λ<sup>i</sup>* and relative direction *α* of wave and current and distance *LE* between the two pontoons.

Assuming the coordinates at pontoon 3 are as shown in Figure 3:

$$
\overrightarrow{R}\_{\text{pontocon 3}} = 0,\tag{21}
$$

The sea surface elevation at pontoon 3 is:

$$\chi\_{w\_i \text{pontoson } \mathfrak{Z}} = \sum\_{i=1}^{N=6} a\_i \sin \left( \Omega\_i t + q\_i \right) \tag{22}$$

The coordinates at pontoon 4 are as shown in Figure 4:

$$
\stackrel{\rightarrow}{R}\_{\text{pontcon 4}} = L\_E \stackrel{\rightarrow}{\cdot} \tag{23}
$$

The sea surface elevation at pontoon 4 is:

$$x\_{w, \text{pontoon } 4} = \sum\_{i=1}^{N=6} a\_i \sin(\Omega\_i t + \varphi\_i + \phi\_i) \tag{24}$$

where the phase angle *<sup>φ</sup><sup>i</sup>* = <sup>2</sup>*πLE <sup>λ</sup><sup>i</sup>* cos *<sup>α</sup>* and *LE* = + *L*2 *<sup>B</sup>* − (*LC* − *LD*) 2 . The values of the relative angle α and the wave length *λ<sup>i</sup>* are naturally determined. Nevertheless, the length *LE* can be changed to obtain the desired phase angle *φi*.

The wave force on the pontoons should include horizontal force and vertical force. Because the length of the ropes connecting the pontoon to the turbine and the carrier is more than 60 m, and the rope can only transmit the axial force and cannot transmit the lateral force, the effect of horizontal force on the dynamic stability of system can be ignored. The volume of the surfaced pontoon should be reduced as much as possible to reduce the wave force, which can increase the dynamic stability and safety of the system and can also be analyzed by the one-way-coupled FSI method. Because the volume of the pontoon is considered small, the horizontal wave force to the pontoon is small. In addition, the length of the ropes connecting the pontoon to the turbine and the carrier is more than 60 m, and the rope can only transmit the axial force and cannot transmit the lateral force; the effect

of horizontal force on the dynamic stability of system can be ignored. The corresponding dynamic vertical buoyance of pontoon 3 can be expressed as:

$$F\_{B3d}(t) = \sum\_{i=1}^{N} \left[ f\_{B\varepsilon,i} \sin \Omega\_i t + f\_{B\varepsilon,i} \cos \Omega\_i t \right] - A\_{Bx} \rho g \chi\_{3d} \tag{25}$$

where *fBs*,*<sup>i</sup>* = *ABxρgai* cos *ϕi*, *fBc*,*<sup>i</sup>* = *ABxρgai* sin *ϕi*. Substituting Equations (19) and (22) into Equation (18), one obtains:

$$(M\_3\ddot{\mathbf{x}}\_{3d} - K\_{\mathbb{C}d}\mathbf{x}\_{1d} + (K\_{\mathbb{C}d} + A\_{Bx}\rho\mathbf{g})\mathbf{x}\_{3d} = \sum\_{i=1}^{N} [f\_{Bc,i}\sin\Omega\_i t + f\_{Bc,i}\cos\Omega\_i t] \tag{26}$$

The pontoon is composed of two parts: (1) the floating section on the water surface and (2) the underwater container. The floating section on the water surface is cylindrical with equal diameters, and the dynamic buoyancy of the pontoon is related to its dynamic displacement. The mass of the pontoon can be controlled by the water in the underwater container. Therefore, the mass and cross-sectional area of the pontoon can be considered, and their independent, individual effects can be studied.

The dynamic equilibrium in the vertical direction for pontoon 4 is:

$$M\_4\ddot{\mathbf{x}}\_{4d} - F\_{\text{B4}} + W\_4 + T\_D = 0 \tag{27}$$

where *M*<sup>4</sup> is the mass of the pontoon 4, and *TD* is the tension of the rope D. Substituting Equations (2) and (11) into Equation (29), one obtains:

$$M\_4 \ddot{\mathbf{x}}\_{4d} - F\_{B4d} + T\_{Dd} = 0 \tag{28}$$

where the dynamic tension of rope D is:

$$T\_{Dd} = K\_{Dd}(\mathbf{x}\_{4d} - \mathbf{x}\_{2d}) \tag{29}$$

in which *KDd* is the effective spring constant. *x*4*<sup>d</sup>* − *x*2*<sup>d</sup>* is the dynamic elongation between floating platform 2 and pontoon 4. Considering the safety of the rope, some buffer springs are used to serially connect the rope between elements 2 and 4. The effective spring constant of the rope–buffer spring connection is obtained:

$$K\_{Dd} = \frac{K\_{\text{rope D}}}{1 + K\_{\text{rope D}}/K\_{D,\text{spring}}} \tag{30}$$

where *KD*,*spring* is the constant of the spring connecting with rope D. The effective spring constant of rope D, *K*rope D = *ED AD*/*LD*, where *ED*, *AD*, and *LD* are the Young's modulus, cross-sectional area and length of the rope D, respectively.

According to Equation (24), the dynamic buoyance of pontoon 4 is:

$$F\_{B4d}(t) = \sum\_{i=1}^{N} \left[ f\_{Ts,i} \sin \Omega\_i t + f\_{Tc,i} \cos \Omega\_i t \right] - A\_{BT} \rho\_{\mathcal{S}} \chi\_{4d} \tag{31}$$

where *fTs*,*<sup>i</sup>* = *ABTρgai* cos(*ϕ<sup>i</sup>* + *φi*), *fTc*,*<sup>i</sup>* = *ABTρgai* sin(*ϕ<sup>i</sup>* + *φi*). Substituting Equations (29) and (31) into Equation (28), one obtains:

$$(M\_4\ddot{\mathbf{x}}\_{4d} - K\_{Dd}\mathbf{x}\_{2d} + (K\_{Dd} + A\_{BT}\rho g)\mathbf{x}\_{4d} = \sum\_{i=1}^{N} [f\_{\text{Ts},i}\sin\Omega\_i t + f\_{\text{Tc},i}\cos\Omega\_i t] \tag{32}$$

The dynamic equilibrium in the vertical direction for the floating platform is:

$$\left(M\_1 + m\_{eff,x}\right)\ddot{\mathbf{x}}\_{1d} - F\_{\text{B1s}} + W\_1 - T\_\text{C} + T\_A \sin \theta\_A + T\_B \sin \theta\_B = 0\tag{33}$$

where *M*<sup>1</sup> is the mass of the platform. The dynamic, effective mass of rope 1 in the xdirection, *meff* ,*<sup>x</sup>* <sup>=</sup> <sup>4</sup> *fgLAs* sin *<sup>θ</sup>*<sup>1</sup> *<sup>π</sup>*<sup>2</sup> , was derived by Lin and Chen [15]. Substituting Equations (2) and (7) into Equation (33), one obtains:

$$\left(M\_1 + m\_{eff,x}\right)\ddot{x}\_{1d} + T\_{\mathbb{C}d} + T\_{Ad}\sin\theta\_A + T\_{Bd}\sin\theta\_B = 0\tag{34}$$

where *TA* is the dynamic tension of rope A.

$$T\_{Ad} = K\_{Ad} \delta\_{Ad} \tag{35}$$

The dynamic elongation is *δAd* = *LAd* − *LA* where *LA* and *LAd* are the static and dynamic lengths of rope A, respectively. The effective spring constant of the rope–buffer spring connection is:

$$K\_{Ad} = \frac{K\_{\text{rope A}}}{1 + K\_{\text{rope A}} / K\_{A, \text{spring}}} \tag{36}$$

where *KA*,*spring* is the constant of the spring connecting to rope A. The effective spring constant of rope A is *K*rope A = *EA AA*/*LA*, where *EA* and *AA* are the Young's modulus and the cross-sectional area of rope A, respectively. The static and dynamic lengths are:

$$L\_A = \sqrt{\mathbf{x}\_{1s}^2 + y\_{1s}^2},\ L\_{Ad} = \sqrt{(\mathbf{x}\_{1s} + \mathbf{x}\_{1d})^2 + (y\_{1s} + y\_{1d})^2} \tag{37}$$

The approximated dynamic elongation is proposed by using the Taylor formula:

$$
\delta\_{Ad} = \frac{\mathfrak{x}\_{1s}}{L\_A} \mathfrak{x}\_{1d} + \frac{\mathfrak{y}\_{1s}}{L\_A} \mathfrak{y}\_{1d} \tag{38}
$$

where the dynamic tension of rope B is:

$$T\_{Bd} = K\_{Bd} \delta\_{Bd} \tag{39}$$

and where the dynamic elongation *δBd* = *LBd* − *LBs*. *LBs* and *LBd* are the static and dynamic lengths of rope B. The effective spring constant of the rope–buffer spring connection is:

$$K\_{Bd} = \frac{K\_{\text{rope B}}}{1 + K\_{\text{rope B}} / K\_{B,spring}} \tag{40}$$

where *KB*,*spring* is the constant of the spring connecting with rope B. The effective spring constant of rope B is *K*rope B = *EBAB*/*LB*, in which *EB* and *AB* are the Young's modulus and the cross-sectional area of rope B. The static and dynamic lengths are:

$$L\_B = \sqrt{\left(\mathbf{x}\_{1s} - \mathbf{x}\_{2s}\right)^2 + \left(y\_{1s} - y\_{2s}\right)^2},\ L\_{Bd} = \sqrt{\left(\mathbf{x}\_1 - \mathbf{x}\_2\right)^2 + \left(y\_1 - y\_2\right)^2} \tag{41}$$

Using the Tylor formula, one can obtain the approximated dynamic elongation:

$$\delta\_{Bd} = \frac{\mathbf{x}\_{1s} - \mathbf{x}\_{2s}}{L\_B} (\mathbf{x}\_{1d} - \mathbf{x}\_{2d}) + \frac{y\_{1s} - y\_{2s}}{L\_B} (y\_{1d} - y\_{2d}) \tag{42}$$

Substituting Equations (19), (35), (38), (39) and (42) into Equation (34), one obtains:

$$\begin{cases} \left(M\_{1} + m\_{eff,x}\right)\bar{\mathbf{x}}\_{1d} + \left(K\_{Ad}\frac{\mathbf{x}\_{1i}}{L\_{A}}\sin\theta\_{A} + K\_{Ed}\frac{\mathbf{x}\_{1i} - \mathbf{x}\_{2i}}{L\_{B}}\sin\theta\_{B} - K\_{Cd}\right)\mathbf{x}\_{1d} - \left(K\_{Ed}\frac{\mathbf{x}\_{1i} - \mathbf{x}\_{2i}}{L\_{B}}\sin\theta\_{B}\right)\mathbf{x}\_{2d} \\ \quad + K\_{Cd}\mathbf{x}\_{3d} + \left(K\_{Ad}\frac{y\_{1i}}{L\_{A}}\sin\theta\_{A} + K\_{Ed}\frac{y\_{1i} - y\_{2i}}{L\_{B}}\sin\theta\_{B}\right)y\_{1d} - \left(K\_{Ed}\frac{y\_{1i} - y\_{2i}}{L\_{B}}\sin\theta\_{B}\right)y\_{2d} = 0 \end{cases} \tag{43}$$

The dynamic equilibrium in the vertical direction for the turbine is:

$$-M\_2\ddot{\mathbf{x}}\_{2d} - \mathcal{W}\_2 + F\_{\mathcal{B}2s} + T\_D + T\_B \sin \theta\_B = 0\tag{44}$$

Substituting Equations (2) and (9) into Equation (44), one obtains:

$$-M\_2\ddot{\mathbf{x}}\_{2d} + T\_{Dd} + T\_{Bd}\sin\theta\_B = 0\tag{45}$$

Substituting Equations (29), (39) and (42) into Equation (45), one obtains:

$$\begin{aligned} M\_{2} \ddot{\mathbf{x}}\_{2d} - K\_{Bd} \frac{\mathbf{x}\_{1i} - \mathbf{x}\_{2i}}{L\_{B}} \sin \theta\_{B} \mathbf{x}\_{1d} + \left( K\_{Dd} + K\_{Bd} \frac{\mathbf{x}\_{1i} - \mathbf{x}\_{2i}}{L\_{B}} \sin \theta\_{B} \right) \mathbf{x}\_{2d} - K\_{Dd} \mathbf{x}\_{4d} \\ -K\_{Bd} \frac{y\_{1i} - y\_{2i}}{L\_{B}} \sin \theta\_{B} y\_{1d} + K\_{Bd} \frac{y\_{1i} - y\_{2i}}{L\_{B}} \sin \theta\_{B} y\_{2d} = 0 \end{aligned} \tag{46}$$

The dynamic equilibrium in the horizontal direction for the floating platform is:

$$-\left(M\_1 + m\_{eff,y}\right)\ddot{y}\_{1d} + F\_{DFy} - T\_A \cos\theta\_A + T\_B \cos\theta\_B = 0\tag{47}$$

where *y*1*<sup>d</sup>* is the dynamic horizontal displacement of the floating platform. The dynamic effective mass of rope A in the y-direction is *meff* ,*<sup>y</sup>* <sup>=</sup> <sup>4</sup> *fgLA* cos *<sup>θ</sup><sup>A</sup> <sup>π</sup>*<sup>2</sup> [26]. The horizontal force on the platform due to the current velocity *<sup>V</sup>* and the horizontal velocity . *y*1*<sup>d</sup>* of the platform are expressed as [34]:

$$F\_{\rm DFT} = \frac{1}{2} \mathbb{C}\_{\rm DFy} \rho A\_{\rm FY} \left( V - \dot{y}\_{1d} \right)^2 = \frac{1}{2} \mathbb{C}\_{\rm DFy} \rho A\_{\rm FY} \left( V^2 - 2V\dot{y}\_{1d} + \dot{y}\_{1d}^2 \right) \approx F\_{\rm DFs} - \mathbb{C}\_{\rm DFy} \rho A\_{\rm FY} V \dot{y}\_{1d} \tag{48}$$

Because . *<sup>y</sup>*1*<sup>d</sup>* << *<sup>V</sup>*, the term . *y* 2 <sup>1</sup>*<sup>d</sup>* is negligible. The drag coefficient of the floating platform is considered close to that of a bullet, i.e., *CDFy* ≈ 0.3, [26].

Substituting Equations (2), (7), (35), (38), (39), (42) and (48) into Equation (47), one obtains:

$$\begin{cases} \begin{aligned} & \left(M\_{1} + m\_{\varepsilon f f, y}\right) \ddot{y}\_{1d} + \mathbb{C}\_{D \mathbf{F} \mathbf{y}} \rho A\_{\mathbf{F} \mathbf{y}} V \dot{y}\_{1d} \\ & + \left(K\_{Ad} \frac{x\_{1s}}{L\_{A}} \cos \theta\_{A} - K\_{Ed} \frac{x\_{1s} - x\_{2s}}{L\_{B}} \cos \theta\_{B} \right) x\_{1d} + K\_{Ed} \frac{x\_{1s} - x\_{2s}}{L\_{B}} \cos \theta\_{B} x\_{2d} \\ & + \left(K\_{Ad} \frac{y\_{1s}}{L\_{A}} \cos \theta\_{A} - K\_{Ed} \frac{y\_{1s} - y\_{2s}}{L\_{B}} \cos \theta\_{B} \right) y\_{1d} + K\_{Ed} \frac{y\_{1s} - y\_{2s}}{L\_{B}} \cos \theta\_{B} y\_{2d} = 0 \end{aligned} \end{cases} \tag{49}$$

It is discovered from Equation (49) that the second term is the damping effect for vibration of the system. The damping effect depends on the parameters: (1) the damping coefficient *CDyF*, (2) the damping area *ABY* and (3) the current velocity *V*.

The dynamic equilibrium in the horizontal direction for the turbine is:

$$-M\_2\ddot{y}\_{2d} + F\_{DT\bar{y}} - T\_B\cos\theta\_B = 0\tag{50}$$

where *y*2*<sup>d</sup>* is the dynamic, horizontal displacement of the turbine. The horizontal force on the platform caused by the current velocity *<sup>V</sup>* and the horizontal velocity . *y*2*<sup>d</sup>* of the turbine is expressed as [34]:

$$F\_{\rm DYy} = \mathbb{C}\_{\rm DYy} \frac{1}{2} \rho A\_{\rm Ty} \left( V - \dot{y}\_{2d} \right)^2 = \mathbb{C}\_{\rm DYy} \frac{1}{2} \rho A\_{\rm Ty} \left( V^2 - 2V \dot{y}\_{2d} + \dot{y}\_{2d}^2 \right) \approx F\_{\rm DTs} - \mathbb{C}\_{\rm DyT} \rho A\_{\rm Ty} V \dot{y}\_{2d} \tag{51}$$

where *ATy* is the effective operating area of the turbine. The theoretical effective drag coefficient of optimum efficiency is *CDTy* = 8/9, [27]. Considering . *<sup>y</sup>*2*<sup>d</sup>* << *<sup>V</sup>*, the term . *y* 2 <sup>2</sup>*<sup>d</sup>* is negligible. Substituting Equations (2), (8), (39), (42) and (51) into Equation (50), one obtains:

$$\begin{cases} M\_2 \ddot{y}\_{2d} + C\_{D7y} \rho A\_{Ty} V \dot{y}\_{2d} + K\_{Bd} \frac{x\_{1i} - x\_{2i}}{L\_B} \cos \theta\_B x\_{1d} \\\ -K\_{Bd} \frac{x\_{1i} - x\_{2i}}{L\_B} \cos \theta\_B x\_{2d} + K\_{Bd} \frac{y\_{1i} - y\_{2i}}{L\_B} \cos \theta\_B y\_{1d} - K\_{Bd} \frac{y\_{1i} - y\_{2i}}{L\_B} \cos \theta\_B y\_{2d} = 0 \end{cases} \tag{52}$$

It is discovered from Equation (52) that the second term is the damping effect for vibration of the system. The damping effect depends on the parameters: (1) the damping coefficient *CDTy*, (2) the damping area *ATY* and (3) the current velocity *V*.

Finally, the coupled equations of motion in terms of the dynamic displacements *x*1*d*, *x*2*d*, *x*3*d*, *x*4*d*, *y*1*d*, and *y*2*<sup>d</sup>* are discovered as Equations (26), (32), (34), (49), (52) and (56).
