*2.4. Solution Method*

#### 2.4.1. Free Vibration

Without the excitation of wave and under steady ocean current, the coupled motion of the system is in free vibration. According to Equations (26), (32), (34), (49), (52) and (56), the coupled equations of free vibration can be expressed as:

$$\mathbf{M}\ddot{\mathbf{Z}}\_d + \mathbf{C}\dot{\mathbf{Z}}\_d + \mathbf{K}\mathbf{Z}\_d = 0 \tag{53}$$

where

**Z***<sup>d</sup>* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *x*1*<sup>d</sup> x*2*<sup>d</sup> x*3*<sup>d</sup> x*4*<sup>d</sup> y*1*<sup>d</sup> y*2*<sup>d</sup>* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , **M** = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ - *M*<sup>1</sup> + *meff* ,*<sup>x</sup>* 000 0 0 0 *M*<sup>2</sup> 00 0 0 0 0 *M*<sup>3</sup> 000 0 00 *M*<sup>4</sup> 0 0 0 000 - *M*<sup>1</sup> + *meff* ,*<sup>y</sup>* 0 0 000 0 *M*<sup>2</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ **C** = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *C*<sup>1</sup> 00000 0 *C*<sup>2</sup> 0000 0 0 *C*<sup>3</sup> 000 000 *C*<sup>4</sup> 0 0 0000 *C*<sup>5</sup> 0 00000 *C*<sup>6</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , **K** = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *K*<sup>11</sup> *K*<sup>12</sup> *K*<sup>13</sup> 0 *K*<sup>15</sup> *K*<sup>16</sup> *K*<sup>21</sup> *K*<sup>22</sup> 0 *K*<sup>24</sup> *K*<sup>25</sup> *K*<sup>26</sup> *K*<sup>31</sup> 0 *K*<sup>33</sup> 000 0 *K*<sup>42</sup> 0 *K*<sup>44</sup> 0 0 *K*<sup>51</sup> *K*<sup>52</sup> 0 0 *K*<sup>55</sup> *K*<sup>56</sup> *K*<sup>61</sup> *K*<sup>62</sup> 0 0 *K*<sup>65</sup> *K*<sup>66</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (54)

*C*<sup>1</sup> = *C*<sup>2</sup> = *C*<sup>3</sup> = *C*4= 0, *C*<sup>5</sup> = *CDFyρAFyV*, *C*<sup>6</sup> = *CDTyρATyV K*<sup>11</sup> = - *KAd <sup>x</sup>*1*<sup>s</sup> LA* sin *<sup>θ</sup><sup>A</sup>* <sup>+</sup> *KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* sin *<sup>θ</sup><sup>B</sup>* <sup>+</sup> *KCd K*<sup>12</sup> = − - *KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* sin *<sup>θ</sup><sup>B</sup>* , *K*<sup>13</sup> = −*KCd K*<sup>15</sup> = - *KAd <sup>y</sup>*1*<sup>s</sup> LA* sin *<sup>θ</sup><sup>A</sup>* <sup>+</sup> *KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* sin *<sup>θ</sup><sup>B</sup>* , *K*<sup>16</sup> = − - *KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* sin *<sup>θ</sup><sup>B</sup> <sup>K</sup>*<sup>21</sup> <sup>=</sup> <sup>−</sup>*KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* sin *<sup>θ</sup>B*, *<sup>K</sup>*<sup>22</sup> = - *KDd* + *KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* sin *<sup>θ</sup><sup>B</sup>* , *<sup>K</sup>*<sup>24</sup> <sup>=</sup> <sup>−</sup>*KDd*, *<sup>K</sup>*<sup>25</sup> <sup>=</sup> <sup>−</sup>*KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* sin *<sup>θ</sup>B*, *<sup>K</sup>*<sup>26</sup> <sup>=</sup> *KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* sin *<sup>θ</sup><sup>B</sup> K*<sup>31</sup> = −*KCd K*<sup>42</sup> = −*KDd K*<sup>33</sup> = (*KCd* + *ABxρg*) *K*<sup>44</sup> = (*KDd* + *ABTρg*) *K*<sup>51</sup> = - *KAd <sup>x</sup>*1*<sup>s</sup> LA* cos *<sup>θ</sup><sup>A</sup>* <sup>−</sup> *KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* cos *<sup>θ</sup><sup>B</sup>* , *<sup>K</sup>*<sup>52</sup> = *KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* cos *<sup>θ</sup><sup>B</sup> K*<sup>55</sup> = - *KAd <sup>y</sup>*1*<sup>s</sup> LA* cos *<sup>θ</sup><sup>A</sup>* <sup>−</sup> *KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* cos *<sup>θ</sup><sup>B</sup>* , *<sup>K</sup>*<sup>56</sup> <sup>=</sup> *KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* cos *<sup>θ</sup><sup>B</sup> <sup>K</sup>*<sup>61</sup> = *KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* cos *<sup>θ</sup>B*, *<sup>K</sup>*<sup>62</sup> <sup>=</sup> <sup>−</sup>*KBd <sup>x</sup>*1*s*−*x*2*<sup>s</sup> LB* cos *<sup>θ</sup>B*, *<sup>K</sup>*<sup>65</sup> <sup>=</sup> *KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* cos *<sup>θ</sup>B*, *<sup>K</sup>*<sup>66</sup> <sup>=</sup> <sup>−</sup>*KBd <sup>y</sup>*1*s*−*y*2*<sup>s</sup> LB* cos *<sup>θ</sup>B*; (55)

The solution of Equation (53) is assumed to be:

$$\mathbf{Z}\_d = \begin{bmatrix} \ \ \overline{x}\_{1d} & \ \overline{x}\_{2d} & \ \overline{x}\_{3d} & \ \overline{x}\_{4d} & \ \overline{y}\_{1d} & \ \overline{y}\_{2d} \end{bmatrix}^T = \begin{pmatrix} \overline{x}\_{\mathbf{d}c}\cos\Omega t + \overline{z}\_{\mathbf{d}s}\sin\Omega t \end{pmatrix} \tag{56}$$

$$\text{where}\qquad \mathbf{\overline{z}}\_{\mathbf{dc}} = \begin{bmatrix} \ \mathbf{\overline{x}}\_{1d,c} & \ \mathbf{\overline{z}}\_{2d,c} & \ \mathbf{\overline{x}}\_{3d,c} & \ \mathbf{\overline{z}}\_{4d,c} & \ \mathbf{\overline{y}}\_{1d,c} & \ \mathbf{\overline{y}}\_{2d,c} \end{bmatrix}^{\mathrm{T}}$$

**zd***<sup>s</sup>* <sup>=</sup> <sup>3</sup> *<sup>x</sup>*1*d*,*<sup>s</sup> <sup>x</sup>*2*d*,*<sup>s</sup> <sup>x</sup>*3*d*,*<sup>s</sup> <sup>x</sup>*4*d*,*<sup>s</sup> <sup>y</sup>*1*d*,*<sup>s</sup> <sup>y</sup>*2*d*,*<sup>s</sup>* 4T . Substituting Equation (56) into Equation (53), one obtains:

$$\begin{cases} \left( \left( \mathbf{M}^{-1} \mathbf{K} - \boldsymbol{\Omega}^{2} \mathbf{I} \right) \overline{\mathbf{z}}\_{\mathbf{dc}} + \boldsymbol{\Omega} \mathbf{M}^{-1} \mathbf{C} \overline{\mathbf{z}}\_{\mathbf{ds}} \right) \cos \boldsymbol{\Omega} t \\ \quad + \left( \left( \mathbf{M}^{-1} \mathbf{K} - \boldsymbol{\Omega}^{2} \mathbf{I} \right) \overline{\mathbf{z}}\_{\mathbf{ds}} - \boldsymbol{\Omega} \mathbf{M}^{-1} \mathbf{C} \overline{\mathbf{z}}\_{\mathbf{dc}} \right) \sin \boldsymbol{\Omega} t = 0 \end{cases} \tag{57}$$

Due to the orthogonality of sin Ω*t* and cos Ω*t*, Equation (57) becomes:

$$\left(\mathbf{M}^{-1}\mathbf{K} - \Omega^2 \mathbf{I}\right)\overline{\mathbf{z}}\_{\mathbf{dc}} + \Omega \mathbf{M}^{-1} \mathbf{C} \overline{\mathbf{z}}\_{\mathbf{ds}} = 0 \tag{58}$$

,

$$-\Omega \mathbf{M}^{-1} \mathbf{C} \mathbf{z}\_{\mathbf{d}c} + \left(\mathbf{M}^{-1} \mathbf{K} - \Omega^2 \mathbf{I}\right) \mathbf{z}\_{\mathbf{d}s} = 0 \tag{59}$$

Further, Equation (58) can be expressed as:

$$\mathbf{Z\_{dc}} = -\left(\mathbf{M}^{-1}\mathbf{K} - \Omega^2 \mathbf{I}\right)^{-1} \boldsymbol{\Omega} \mathbf{M}^{-1} \mathbf{C} \mathbf{z\_{ds}} \tag{60}$$

Substituting Equation (60) into Equation (59), one obtains:

$$\mathbf{Q}\overline{\mathbf{z}}\_{\mathrm{ds}} = 0 \tag{61}$$

$$\text{where } \mathbf{Q} = \Omega^2 \mathbf{M}^{-1} \mathbf{C} \left( \mathbf{M}^{-1} \mathbf{K} - \Omega^2 \mathbf{I} \right)^{-1} \mathbf{M}^{-1} \mathbf{C} + \left( \mathbf{M}^{-1} \mathbf{K} - \Omega^2 \mathbf{I} \right). \text{ The frequency equation is:}$$

$$|\mathbf{Q}| = 0 \tag{62}$$

The natural frequencies of the system can be determined via Equation (62).

### 2.4.2. Forced Vibration

Considering the excitation of the wave, the coupled Equations (26), (32), (34), (49), (52) and (56) can be rewritten in the matrix format as follows:

$$\mathbf{M}\ddot{\mathbf{Z}}\_d + \mathbf{C}\dot{\mathbf{Z}}\_d + \mathbf{K}\mathbf{Z}\_d = \mathbf{F}\_d\tag{63}$$

,

where

$$\begin{aligned} \mathbf{F}\_d &= \begin{bmatrix} 0 & 0 & \sum\_{i=1}^N [f\_{\text{B};j} \sin \Omega\_i t + f\_{\text{B};j} \cos \Omega\_i t] & \sum\_{i=1}^N [f\_{\text{T};j} \sin \Omega\_i t + f\_{\text{T};j} \cos \Omega\_i t] & 0 & 0 \end{bmatrix}^T, \\\ f\_{\text{B};j} &= A\_{\text{B};t} \rho g\_i i \cos \cdot\_{\text{B}\_i} \rho\_{\text{I};j} = A\_{\text{B};t} \rho\_{\text{S}} g\_i \sin \varphi\_i \\\ f\_{\text{T};j} &= A\_{\text{B};t} \rho\_{\text{S}} g\_i \cos(\rho\_i + \phi\_i), \ f\_{\text{T};j} = A\_{\text{B};t} \rho g\_i \sin(\rho\_i + \phi\_i) \end{aligned} \tag{64}$$

The solution of Equation (63) is assumed:

$$\mathbf{Z}\_d = \begin{bmatrix} \ \mathbf{x}\_{1d} & \mathbf{x}\_{2d} & \mathbf{x}\_{3d} & \mathbf{x}\_{4d} & y\_{1d} & y\_{2d} \end{bmatrix}^\mathrm{T} = \sum\_{i=1}^N (\mathbf{z}\_{\mathbf{d}c,i}\cos\Omega\_i t + \mathbf{z}\_{\mathbf{d}s,i}\sin\Omega\_i t)\_\prime \tag{65}$$

where **zd***c*,*<sup>i</sup>* = <sup>3</sup> *<sup>x</sup>*1*d*,*<sup>c</sup> <sup>x</sup>*2*d*,*<sup>c</sup> <sup>x</sup>*3*d*,*<sup>c</sup> <sup>x</sup>*4*d*,*<sup>c</sup> <sup>y</sup>*1*d*,*<sup>c</sup> <sup>y</sup>*2*d*,*<sup>c</sup>* 4T

**zd***s*,*<sup>i</sup>* = <sup>3</sup> *<sup>x</sup>*1*d*,*<sup>s</sup> <sup>x</sup>*2*d*,*<sup>s</sup> <sup>x</sup>*3*d*,*<sup>s</sup> <sup>x</sup>*4*d*,*<sup>s</sup> <sup>y</sup>*1*d*,*<sup>s</sup> <sup>y</sup>*2*d*,*<sup>s</sup>* 4T . Substituting Equation (65) into Equation (63), one obtains:

$$\begin{aligned} &-\sum\_{i=1}^{N} \Omega\_{i}^{2} (\mathbf{z}\_{\mathbf{d}c,i} \cos \Omega\_{i} t + \mathbf{z}\_{\mathbf{d}s,i} \sin \Omega\_{i} t) + \mathbf{M}^{-1} \mathbf{C} \sum\_{i=1}^{N} \left(-\Omega\_{i} \mathbf{z}\_{\mathbf{d}c,i} \sin \Omega\_{i} t + \Omega\_{i} \mathbf{z}\_{\mathbf{d}s,i} \cos \Omega\_{i} t\right) \\ &+ \mathbf{M}^{-1} \mathbf{K} \sum\_{i=1}^{N} \left(\mathbf{z}\_{\mathbf{d}c,i} \cos \Omega\_{i} t + \mathbf{z}\_{\mathbf{d}s,i} \sin \Omega\_{i} t\right) = \sum\_{i=1}^{N} \left(\mathbf{F}\_{s,i} \sin \Omega\_{i} t + \mathbf{F}\_{c,i} \cos \Omega\_{i} t\right) \end{aligned} \tag{66}$$

Multiplying Equation (66) by cos Ω*mt* and integrating it from 0 to the period *Tm*, 2π/Ω*m*, Equation (66) becomes:

$$\sum\_{i=1}^{N} \mathbf{a}\_{im} \mathbf{z}\_{\mathbf{d}c,i} + \sum\_{i=1}^{N} \mathbf{b}\_{im} \mathbf{z}\_{\mathbf{d}s,i} = \chi\_{cm'} \ m = 1,2,\dots,N\tag{67}$$

$$\begin{aligned} \text{where} \\ \mathbf{a}\_{im} &= \left[ a\_{im} \left( \mathbf{M}^{-1} \mathbf{K} - \Omega\_i^2 \mathbf{I} \right) - \beta\_{im} \Omega\_i \mathbf{M}^{-1} \mathbf{C} \right], \mathbf{b}\_{im} = \left[ \beta\_{im} \left( \mathbf{M}^{-1} \mathbf{K} - \Omega\_i^2 \mathbf{I} \right) - a\_{im} \Omega\_i \mathbf{M}^{-1} \mathbf{C} \right] \\ \mathbf{x}\_{cm} &= \sum\_{i=1}^N \left( \mathbf{F}\_{iS} \beta\_{im} + \mathbf{F}\_{c,i} a\_{im} \right) \\ a\_{im} &= \begin{cases} \frac{\mathbf{T}\_m}{\sum}, & i = m \\ \frac{\Omega\_i \sin(\Omega\_i T\_m)}{(\Omega\_i + \Omega\_m)(\Omega\_i - \Omega\_m)}, & i \neq m \end{cases}, \beta\_{im} = \begin{cases} 0, & i = m \\\\ \frac{\Omega\_i (1 - \cos(\Omega\_i T\_m))}{(\Omega\_i + \Omega\_m)(\Omega\_i - \Omega\_m)}, & i \neq m \end{cases} \end{aligned} \tag{68}$$

Multiplying Equation (66) by sin Ω*mt* and integrating it from 0 to the period *Tm*, 2π/Ω*m*, Equation (66) becomes:

$$\sum\_{i=1}^{N} \mathbf{c}\_{im} \mathbf{z}\_{\mathbf{d}c,i} + \sum\_{i=1}^{N} \mathbf{d}\_{im} \mathbf{z}\_{\mathbf{d}s,i} = \chi\_{sm}, m = 1, 2, \dots, N \tag{69}$$

where

where

$$\begin{aligned} \mathbf{c}\_{\text{im}} &= \left[ \beta\_{\text{mi}} \left( \mathbf{M}^{-1} \mathbf{K} - \Omega\_i^2 \mathbf{I} \right) - \gamma\_{\text{im}} \Omega\_i \mathbf{M}^{-1} \mathbf{C} \right], \mathbf{d}\_{\text{im}} = \left[ \gamma\_{\text{im}} \left( \mathbf{M}^{-1} \mathbf{K} - \Omega\_i^2 \mathbf{I} \right) - \beta\_{\text{mi}} \Omega\_i \mathbf{M}^{-1} \mathbf{C} \right] \\ \chi\_{\text{sur}} &= \sum\_{i=1}^N \left( \mathbf{F}\_{\text{s},i} \gamma\_{\text{im}} + \mathbf{F}\_{\text{c},i} \beta\_{\text{mi}} \right) \\ &\gamma\_{\text{im}} = \begin{cases} \frac{\mathbf{T}\_m}{\mathbf{T}\_s}, & i = m \\\ \frac{\Omega\_n \sin(\Omega\_i \mathbf{T}\_m)}{(\Omega\_i + \Omega\_m)(\Omega\_i - \Omega\_m)}, & i \neq m \end{cases} \end{aligned} \tag{70}$$

Equations (67) and (69) can be written as:

$$\mathbf{BZ} = \mathbf{F} \tag{71}$$

**B** = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣ **a**<sup>11</sup> **a**<sup>21</sup> ··· **a**N1 **a**<sup>12</sup> **a**<sup>22</sup> ··· **a**N2 . . . . . . ··· . . . **a**1N **a**2*<sup>N</sup>* ··· **a**NN ⎤ ⎥ ⎥ ⎥ ⎦ 6N×6N ⎡ ⎢ ⎢ ⎢ ⎣ **b**<sup>11</sup> **b**<sup>21</sup> ··· **b***N*<sup>1</sup> **b**<sup>12</sup> **b**<sup>22</sup> ··· **b***N*<sup>2</sup> . . . . . . ··· . . . **b**1*<sup>N</sup>* **b**2*<sup>N</sup>* ··· **b***NN* ⎤ ⎥ ⎥ ⎥ ⎦ <sup>⎡</sup> 6N×6N ⎢ ⎢ ⎢ ⎣ **c**<sup>11</sup> **c**<sup>21</sup> ··· **c***N*<sup>1</sup> **c**<sup>12</sup> **c**<sup>22</sup> ··· **c***N*<sup>2</sup> . . . . . . ··· . . . **c**1*<sup>N</sup>* **c**2*<sup>N</sup>* ··· **c**N*<sup>N</sup>* ⎤ ⎥ ⎥ ⎥ ⎦ 6N×6N ⎡ ⎢ ⎢ ⎢ ⎣ **d**<sup>11</sup> **d**<sup>21</sup> ··· **d***N*<sup>2</sup> **d**<sup>12</sup> **d**<sup>22</sup> ··· **d***N*<sup>2</sup> . . . . . . ··· . . . **d**1*<sup>N</sup>* **d**2*<sup>N</sup>* ··· **d***NN* ⎤ ⎥ ⎥ ⎥ ⎦ 6N×6N ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 12N×12N , **Z=** ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣ **zd***c*,1 **zd***c*,2 . . . **zd***c*,*<sup>N</sup>* ⎤ ⎥ ⎥ ⎥ ⎦ <sup>⎡</sup> 6N×<sup>1</sup> ⎢ ⎢ ⎢ ⎣ **zd***s*,1 **zd***s*,2 . . . **zd***s*,*<sup>N</sup>* ⎤ ⎥ ⎥ ⎥ ⎦ 6N×1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 12N×1 , **F** = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣ χ*c*<sup>1</sup> χ*c*<sup>2</sup> . . . χ*c*<sup>N</sup> ⎤ ⎥ ⎥ ⎥ ⎦ <sup>⎡</sup> 6N×<sup>1</sup> ⎢ ⎢ ⎢ ⎣ χs1 χs2 . . . χsN ⎤ ⎥ ⎥ ⎥ ⎦ 6N×1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 12N×1 (72)

The solution of Equation (65) is:

$$\mathbf{Z} = \mathbf{B}^{-1} \mathbf{F} \tag{73}$$

Further, one can derive the dynamic tensions of ropes under irregular wave as follows: The dynamic tension of rope A is:

$$T\_{Ad} = \sum\_{i=1}^{N} T\_{Adc,i} \cos \Omega\_i t + T\_{Ads,i} \sin \Omega\_i t \tag{74}$$

where *TAdc*,*<sup>i</sup>* <sup>=</sup> *KAd*- *x*1*<sup>s</sup> LA <sup>x</sup>*1*dc*,*<sup>i</sup>* <sup>+</sup> *<sup>y</sup>*1*<sup>s</sup> LA y*1*dc*,*<sup>i</sup>* , *TAds*,*<sup>i</sup>* <sup>=</sup> *KAd*- *x*1*<sup>s</sup> LA <sup>x</sup>*1*ds*,*<sup>i</sup>* <sup>+</sup> *<sup>y</sup>*1*<sup>s</sup> LA y*1*ds*,*<sup>i</sup>* . The dynamic tension of rope B is:

$$T\_{Bd} = \sum\_{i=1}^{N} T\_{Bdc,i} \cos \Omega\_i t + T\_{Bds,i} \sin \Omega\_i t \tag{75}$$

where

$$\begin{cases} T\_{Bdc,i} = K\_{Bd} \left[ \frac{\mathbf{x}\_{2i} - \mathbf{x}\_{1i}}{L\_B} (\mathbf{x}\_{2dc,i} - \mathbf{x}\_{1dc,i}) + \frac{y\_{2i} - y\_{1s}}{L\_B} (y\_{2dc,i} - y\_{1dc,i}) \right], \\ T\_{Bds,i} = K\_{Bd} \left[ \frac{\mathbf{x}\_{2i} - \mathbf{x}\_{1i}}{L\_B} (\mathbf{x}\_{2ds,i} - \mathbf{x}\_{1ds,i}) + \frac{y\_{2i} - y\_{1s}}{L\_B} (y\_{2ds,i} - y\_{1ds,i}) \right]. \end{cases}$$

The dynamic tension of rope C is:

$$T\_{\mathbb{C}d} = \sum\_{i=1}^{N} T\_{\mathbb{C}dc,i} \cos \Omega\_i t + T\_{\mathbb{C}ds,i} \sin \Omega\_i t \tag{76}$$

where *TCdc*,*<sup>i</sup>* = *KCd*(*x*3**d***c*,*<sup>i</sup>* − *x*1*dc*,*i*), *TCds*,*<sup>i</sup>* = *KCd*(*x*3**d***s*,*<sup>i</sup>* − *x*1*ds*,*i*).

The dynamic tension of rope D is:

$$T\_{Dd} = \sum\_{i=1}^{N} T\_{Ddc,i} \cos \Omega\_i t + T\_{Dds,i} \sin \Omega\_i t \tag{77}$$

where *TDdc*,*<sup>i</sup>* = *KDd*(*x*4**d***c*,*<sup>i</sup>* − *x*2*dc*,*i*), *TDds*,*<sup>i</sup>* = *KDd*(*x*4**d***s*,*<sup>i</sup>* − *x*2*ds*,*i*).

#### **3. Numerical Results and Discussion**

This study investigates the dynamic response of two kinds of mooring system under the typhoon irregular wave: (1) the diving depth of the turbine *LD* = 60 m, the diving depth of the floating platform *LC* ≥ 60 m and (2) the diving depth of the floating platform *LC* = 60 m, the diving depth of the turbine *LD* ≥ 60 m. Meanwhile, the effects of several parameters on the dynamic response are investigated.

Firstly, the first kind of mooring system is investigated. Consider the conditions in Figure 5a,b: (1) the depth of seabed *Hbed* = 1300 m, (2) the cross-sectional area of pontoon 3 connecting to floating platform *ABX* = 2.12 m2, (3) the cross-sectional area of pontoon 4 connecting to turbine *ABT* = 2.12 m2, (4) no buffer spring, (5) the ropes A, B, C and D are made of some commercial, high-strength PE dyneema; Young's modulus *EPE* = 100 GPa, weight per unit length *fg,PE* = 16.22 kg/m, diameter *DPE* = 154 mm, cross-sectional area *APE* = 0.0186 m2, fracture strength *Tfracture* = 759 tons, (6) the static diving depth of the turbine *LD* = 60 m, (7) the horizontal distance between the turbine and floating platform *LE* = 100 m, (8) the inclined angle of the rope A, *θ*<sup>A</sup> = 30◦ , (9) the current velocity *V* = 1 m/s, (10) the irregular wave is simulated by six regular waves which are listed in Table 1, (11) the wave phase angles *ϕi*, *i* = 1, 2, . . . , 6 are assumed as 30◦ , 60◦ , 90◦ , 120◦ , 170◦ , 270◦ , (12) the masses of turbine, floating platform and pontoons *M*<sup>1</sup> = 300 *tons*, *M*<sup>2</sup> = 838 *tons*, *M*<sup>3</sup> = *M*<sup>4</sup> = 250 *tons*, (13) the cross-sectional area of the floating platform and turbine *AFY* = 23 m<sup>2</sup> and *ATY* = 500 m2, (14) the effective damping coefficients *CDFy* = 0.3 and *CDTy* = 8/9, (15) the static axial force to turbine *FDTs* = 180 tons and (16) the relative orientation between current and wave α = 60◦.

**Figure 5.** Dynamic tensions of ropes and Dynamic displacements of elements under the typhoon irregular wave. (**a**) Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length for *LE* = 100 m; (**b**) Dynamic displacements of the four elements at resonance for *LC* = 66 m.

Figure 5a demonstrates the effect of the diving depth of the floating platform *LC* on the maximum dynamic tensions of the four ropes, *TA,max*, *TB,max*, *TC,max* and *TD,max* under the typhoon irregular wave when the diving depth of the turbine *LD* = 60 m. The irregular wave is simulated by six regular waves which are listed in Table 1. When the depth *LC* increases from 60 m, the dynamic tensions of the ropes increase significantly. If *LC,res* = 66 m , the resonance happens, and the maximum dynamic tensions *TA,max* = 2422 tons, *TC,max* = 7329 tons and *TD,max* = 18,835 tons, which is over that of the fracture strength of rope, *Tfracture* = 759 tons. Figure 5b demonstrates the vibration mode at the resonance. It is found that the displacements x2d and x4d of turbine 2 and pontoon 4 are largest. Therefore, the maximum dynamic tension is that of rope D, *TD,max*.

When *LC* increases further, the dynamic tension decreases sharply. If *LC* > 80 m, all the dynamic tensions are significantly less than the fracture strength of rope, *Tfracture* = 759 tons. If *LC* = 80 m, *TA,max* = 442 tons, *TB,max* = 27 tons, *TC,max* = 187 tons, then*TD,max* = 478 tons. The maximum one among the four dynamic tensions is *Tmax* = *TD,max* = 478 tons. If *LC* = 150 m, the maximum dynamic tension *Tmax* = *TA,max* = 367 tons. This is because the natural frequency changes with the length *LC*. The excitation frequencies of the irregular wave are different to the natural frequency of the mooring system. Therefore, the resonance does not exist. It is found that the greater the diving depth of the floating platform *LC*, the smaller the maximum dynamic tension. In other words, the mooring system of the diving depth of the floating platform *LC* = 150 m is better than that of *LC* = 80 m. Because the diving depth of the floating platform is different to that of turbine, the water flowing through the floating platform does not interfere with the flow field of the turbine. Moreover, for *LC* > 80 m, the dynamic tension *TA,max* of rope A decreases with the diving depth *LC*. This is because the angle *θ<sup>A</sup>* of rope A decreases with the diving depth *LC*. The towing force is horizontal due to the ocean velocity. Meanwhile, the dynamic tension *TB,max* of rope B increases with the diving depth *LC*. It is because the angle *θ<sup>B</sup>* of rope B increases with the diving depth *LC*.

Figure 6 presents the relation between the diving depth *LC* of the floating platform and the dynamic tensions of ropes under the typhoon irregular wave for the distance *LE* = 200 m. Aside from the distance *LE* =200 m, all other parameters are the same as those of Figure 5. It can be observed in Figure 6 that, when *LE* = 200 m, the maximum resonant position *LC,res* = 80 m is different to *LC,res* = 66 m for *LE* = 100 m in Figure 5. The effect of the horizontal distance between the turbine and floating platform *LE* on the dynamic tension with *LC* = 150 m is negligible.

**Figure 6.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length *LC* and the horizontal distance between the turbine and floating platform *LE*.

Figure 7 shows the effects of the diving depth of the floating platform *LC* and the mass of pontoons *M*<sup>3</sup> and *M*<sup>4</sup> on the dynamic tensions of the four ropes, *TA,max*, *TB,max*, *TC,max* and *TD,max* under the typhoon irregular wave. In this case, the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> = 150 tons; other parameters are the same as those of Figure 6. It is found that, if the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> = 150 tons, the resonance occurs at several diving depths of the floating platform *LC*, and the maximum dynamic tensions are over that of the fracture

strength of rope, *Tfracture* = 759 tons. In other words, if the weight of the pontoon is too low, the dynamic displacement of the system is too intense, resulting in the excessive dynamic tension of the rope.

**Figure 7.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length *LC* and the mass of pontoons *M*<sup>3</sup> and *M*4.

Further, the second kind of mooring system is investigated. Figure 8 demonstrates the effect of the diving depth of the floating platform *LD* and the mass of pontoons *M*<sup>3</sup> and *M*<sup>4</sup> on the maximum dynamic tensions of the four ropes, *TA,max*, *TB,max*, *TC,max* and *TD,max* under the typhoon irregular wave when the diving depth of the turbine *LC* = 60 m and the horizontal distance between the turbine and floating platform *LE* = 100 m. All the other parameters are the same as those of Figure 5. It is found that there is no resonance. The dynamic tension increases with the diving depth of the floating platform *LD*, especially in the case where *M*<sup>3</sup> = *M*<sup>4</sup> = 150 tons. The maximum tension is that of rope A, *TA,max*, which is close or over that of the fracture strength of rope, *Tfracture* = 759 tons. It is concluded that this mooring system should not be proposed.

Figure 9 demonstrates the effect of the diving depth of the turbine *LD* and the mass of pontoons *M*<sup>3</sup> and *M*<sup>4</sup> on the maximum dynamic tensions of the four ropes, *TA,max*, *TB,max*, *TC,max* and *TD,max* under the typhoon irregular wave when the diving depth of the floating platform *LC* = 60 m and the horizontal distance between the turbine and floating platform *LE* = 200 m. All the other parameters are the same as those in Figure 8. It is found that the maximum tension of the four ropes is the dynamic tension of rope A, *TA,max*. If the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> =150 tons, the maximum tension *TA,max* decreases with the diving depth of the turbine *LD*. However, it is the reverse for the case of the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> = 250 tons. Moreover, the dynamic tension *TA,max*, with the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> = 150 tons, is obviously less than that of the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> = 250 tons.

**Figure 8.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length *LD* and the mass of pontoons *M*<sup>3</sup> and *M*<sup>4</sup> for *LE* = 100 m.

**Figure 9.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length *LD* and the mass of pontoons *M*<sup>3</sup> and *M*<sup>4</sup> for *LE* = 200 m.

On the eastern coast of Taiwan, the average velocity of the Kuroshio at a depth of 150 m is 0.65 m/s, and that at a depth of 30 m is 1.1 m/s [35]. It is well known that the potential energy of ocean current can be estimated by using the formula *η* <sup>1</sup> <sup>2</sup> *<sup>ρ</sup>AV*3, where *<sup>h</sup>* is the efficiency, *r* is the density, *A* is the operating area and *V* is the flow velocity. Based on the formula, the ratio of the potential power generation of the diving depth of the turbine *LD* = 30 m to that of *LD* = 150 m is about 4.85. In other words, the deeper the diving depth of the turbine *LD*, the smaller the power generation.

Figure 10a demonstrates the dynamic displacements of the turbine, floating platform and pontoons.

**Figure 10.** (**a**) Dynamic displacements of the four elements and (**b**) dynamic tensions of ropes under typhoon irregular wave for *LC* = 60 m, *LD* = 70 m.

The diving depth of the floating platform *LC* = 60 m, the diving depth of the turbine *LD* = 70 m and the mass of pontoons *M*<sup>3</sup> = *M*<sup>4</sup> = 150 tons. The other parameters are the same as those in Figure 9. Dynamic displacements are multi-frequency coupled. The horizontal displacements of the turbine and the floating platform *y*1*<sup>d</sup>* and *y*2*<sup>d</sup>* are very small, the amplitude is about 0.30 m, the vertical displacements *x*1*<sup>d</sup>* and *x*3*<sup>d</sup>* are large and the amplitude is about 15.5 m, which is close to the significant wave *HS* = 15.4 m. The amplitudes of vertical displacements *x*2*<sup>d</sup>* and *x*4*<sup>d</sup>* are about 15.5 m. The amplitudes of vertical displacements *x*1*<sup>d</sup>* and *x*3*<sup>d</sup>* are about 9.69 m. The vertical displacements of pontoon 3 and the floating platform directly connected by using rope C are synchronized and similar. The vertical displacements of pontoon 4 and the turbine directly connected by using rope D are synchronized and similar.

Figure 10b demonstrates the dynamic tension of the rope. The maximum dynamic tension *TA,max* of rope A connecting the floating platform and the mooring foundation is about 589 tons. The maximum dynamic tension *TB,max* of rope B connecting the turbine and the floating platform is about 38 tons. The maximum dynamic tension *TC,max* of rope C connecting pontoon 3 and the floating platform is about 322 tons. The maximum dynamic tension *TD,max* of rope D connecting pontoon 4 and the turbine is about 75 tons.

Figure 11a demonstrates the dynamic displacements of the turbine, floating platform and pontoons. The diving depth of the floating platform *LC* = 150 m, the diving depth of the turbine *LD* = 60 m. The other parameters are the same as those in Figure 6. Dynamic displacements are multi-frequency coupled. The horizontal displacements of the turbine and the floating platform *y*1*<sup>d</sup>* and *y*2*<sup>d</sup>* are very small, the amplitude is about 0.14 m, the amplitudes of vertical displacements *x*1*<sup>d</sup>* and *x*3*<sup>d</sup>* are about 8.6 m and the amplitudes of vertical displacements *x*2*<sup>d</sup>* and *x*4*<sup>d</sup>* are about 9.6 m, which are significantly lower than the significant wave *HS* = 15.4 m. The vertical displacements of pontoon 3 and the floating platform directly connected by using rope C are synchronized and similar. The vertical displacements of pontoon 4 and the turbine directly connected by using rope D are synchronized and similar.

**Figure 11.** (**a**) Dynamic displacements of the four elements and (**b**) dynamic tensions of ropes under typhoon irregular wave for *LC* = 150 m, *LD* = 60 m.

Figure 11b demonstrates the dynamic tension of rope. The maximum dynamic tension *TA,max* of rope A connecting the floating platform and the mooring foundation is about 375 tons. The maximum dynamic tension *TB,max* of rope B connecting the turbine and the floating platform is about 58 tons. The maximum dynamic tension *TC,max* of rope C connecting pontoon 3 and the floating platform is about 143 tons. The maximum dynamic tension *TD,max* of rope D connecting pontoon 4 and the turbine is about 131 tons.

Figure 12 demonstrates the effects of the diving depth of the floating platform *LC* and the buffer spring connected in series with ropes C and D on the dynamic tension of the rope. The diving depth of the turbine *LD* = 60 m. The effective spring constants of the two buffer springs are *KC,spring* = *KD,spring* = *K*rope A. The other parameters are the same as those in Figure 5. Compared with Figure 5, it is found that the dynamic tensions *TA,max*, *TB,max* and *TC,max* of the ropes A, B and C are significantly reduced at the resonance point, but the effect on *TD,max* is not obvious and is still over the fracture strength *Tfracture*. If the diving depth of the floating platform *LC* > 72 m, the effect of the buffer springs on the dynamic tensions is negligible. It is concluded that the effect of the buffer springs on the dynamic tensions of this mooring system is slight.

Figure 13 demonstrates the effects of the cross-sectional area of pontoon *ABX*, *APX* and the diving depth of the floating platform *LC* on the dynamic tensions of the four ropes. The cross-sectional area of the two pontoons is *ABX* = *APX* =4m2. The other parameters are the same as those in Figure 5. Compared with Figure 5, it is found that the dynamic tensions are significantly increased. At the resonance point, the dynamic tension is over the fracture strength *Tfracture*. If the diving depth of the floating platform *LC* > 85 m, the dynamic tension is close to the fracture strength *Tfracture*. It is concluded that the larger the cross-sectional area of the pontoon, the larger the dynamic tension.

**Figure 12.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length *LC* and the buffer springs *KC,spring*, *KD,spring*.

**Figure 13.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the cross-sectional area of the two pontoons *ABX*, *APX* and the diving depth of the floating platform *LC*.

Figure 14 demonstrates the effects of the significant wave height *Hs* and the peak period *Pw* on the dynamic tension *TA,max*. Based on Equations (14)–(16), the irregular wave is simulated by six regular waves, i.e., *n* = 6. The six regular waves share according to the ratio of energy {2,35,8,4,3,1}. The amplitude *a*i, frequency *f* i, the wave number *k*<sup>i</sup> and wave length *l*<sup>i</sup> can be determined. The diving depths *LC* = 60 m and *LD* =150 m. The horizontal distance between the turbine and floating platform *LE* = 200 m. Two buffer springs are connected in series with ropes C and D. The effective spring constants of the two buffer springs are *KC,spring* = *KD,spring* = *K*rope A. The other parameters are the same as those of Figure 5. It is found that the more the significant wave height *Hs*, the larger the dynamic tension *TA,max*. For the peak period *Tp* = 13.5 s, the dynamic tension *TA,max* increases dramatically with the significant wave height Hs. With the increase of the peak period *Tp*, the increase rate of the dynamic tension *TA,max* becomes low.

**Figure 14.** Dynamic tension *TA,max* under the typhoon irregular wave as a function of the significant wave height *Hs* and the peak period *Tp*.

Figure 15 demonstrates the effect of the relative angle a of the wave and ocean current on the dynamic tension. The significant wave height *Hs* = 15 m, and the peak period *Pw* = 16.5 s. The other parameters are the same as those of Figure 14. It is observed that the effect of the relative angle a of the wave and ocean current on the dynamic tension is significant. *TA,max*(*α* = 180◦) and *TC,max*(*α* = 180◦) are much larger than *TA,max*(*α* = 0◦) and *TC,max*(*α* = 0◦), respectively. Moreover, *TD,max*(*α* = 90◦) is significantly larger than *TA,max*(*α* = 0◦) and *TA,max*(*α* = 180◦).

**Figure 15.** Dynamic tension of the four ropes under the typhoon irregular wave as a function of the relative angle *α*.

Based on the frequency Equation (62), the effects of the masses *M*3, *M*<sup>4</sup> and *M*<sup>2</sup> and the distance *LE* and the areas on the natural frequencies are investigated and listed in Table 2. It is found that the larger the cross-sectional areas of pontoon *ABx* and *ABT*, the higher the natural frequencies of the system. The larger the masses of pontoon *M*<sup>3</sup> and *M*4, the lower the natural frequencies of the system. The larger the mass of turbine *M*2, the lower the first natural frequency of the system. However, the effect of the mass of turbine *M*<sup>2</sup> on the second natural frequency of the system is negligible. The larger the distance between the turbine and the floating platform *LE*, the higher the second natural frequency of the system. However, the effect of the distance *LE* on the first natural frequency of the system is negligible.

**Table 2.** The first two natural frequencies *f <sup>n</sup>*<sup>1</sup> and *f <sup>n</sup>*<sup>2</sup> as a function of the masses *M*3, *M*<sup>4</sup> and *M*2, the distance *LE* and the areas *ABx*, *ABT* for *M*<sup>1</sup> = 300 tons.


### **4. Conclusions**

This paper studies the safe design of a mooring system for an ocean current generator that is working under the impact of typhoon waves. Two mooring designs are investigated, and one safe and feasible mooring system is proposed. The proposed mooring design can stabilize the turbine and platform around a certain predetermined water depth, thereby, maintaining the stability and safety of the ocean current generator. The effects of several parameters on the dynamic response under irregular wave impact were discovered as follows:


The coupled translational and rotational motions will be studied in another manuscript. Moreover, the transient response of the system subjected to impact force will be investigated in the future.

**Author Contributions:** Conceptualization, S.-M.L. and C.-T.L.; methodology, S.-M.L.; software, S.-M.L. and D.-W.U.; validation, S.-M.L.; formal analysis, S.-M.L.; investigation, S.-M.L. and D.-W.U.; resources, S.-M.L. and C.-T.L.; data curation, D.-W.U.; writing—original draft preparation, S.-M.L.; writing—review and editing, C.-T.L.; visualization, S.-M.L.; supervision, S.-M.L.; funding acquisition, S.-M.L. and C.-T.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was financially supported by the Green Energy Technology Research Center from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan and the National Academy of Marine Research of Taiwan, R. O. C. (NAMR110051).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The figures and the tables in this manuscript have clearly described all the data of this study.

**Acknowledgments:** This work was financially supported by the Green Energy Technology Research Center from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan and the National Academy of Marine Research of Taiwan (NAMR110051).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
