**9. Interpretations**

#### **9.1 General**

FLOATGEN 2.3 m Hs berthing limit calculations compare precisely with feedback from offshore site test with full scale prototype [11]:

"Transfer up to 2.3 m significant wave height with no motion compensation". From **Table 1**, it may be noted that berthing limits are influenced by:


It is possible to propose an approximative analytical of the berthing limit due to the floater masking effect. Indeed, the calculated ratio T/N is (see Eq. (8)):

$$\mathbf{f}(\mathbf{T}\_{\pm}) = \frac{\mathbf{A}\mathbf{T}\_{\pm}^{2} + 2\mathbf{B}\mathbf{T}\_{\pm} - \mathbf{A}}{(\mathbf{C} - \mathcal{G})\mathbf{T}\_{\pm}^{2} + 2\mathbf{D}\mathbf{T}\_{\pm} - (\mathbf{C} + \mathcal{G})} \tag{16}$$


#### **Table 1.**

*CAT CTV berthing limits for monopile various floater designs.*

**Figure 17.** *Curves T/N and fgrip versus λ/B for 2 m Hs.*

#### where:

$$\begin{aligned} &\mathbf{T}\_{\pm} \stackrel{\text{def}}{=} \tan\left(\text{ot}\_{\pm}/2\right), \\ &\mathbf{A} \stackrel{\text{def}}{=} \mathbf{Z}\_{\text{m}} \cos\left(-\text{ot}\_{\text{T}} + \mathbf{q}\_{\text{x}}\right), \\ &\mathbf{B} \stackrel{\text{def}}{=} \mathbf{Z}\_{\text{m}} \sin\left(-\text{ot}\_{\text{T}} + \mathbf{q}\_{\text{x}}\right), \\ &\mathbf{C} \stackrel{\text{def}}{=} \mathbf{X}\_{\text{m}} \cos\left(-\text{ot}\_{\text{N}} + \mathbf{q}\_{\text{x}}\right) + \mathbf{Z}\_{\text{G}} \mathbf{\Theta}\_{\text{m}} \cos\left(-\text{ot}\_{\text{N}} + \mathbf{q}\_{\text{O}}\right), \\ &\mathbf{D} \stackrel{\text{def}}{=} \mathbf{X}\_{\text{m}} \sin\left(-\text{ot}\_{\text{N}} + \mathbf{q}\_{\text{x}}\right) + \mathbf{Z}\_{\text{G}} \mathbf{\Theta}\_{\text{m}} \sin\left(-\text{ot}\_{\text{N}} + \mathbf{q}\_{\text{O}}\right). \\ &\mathbf{G} = -\min\left(\mathbf{L}, |\mathbf{P}\_{\text{max}}|/[\text{mo}^{2}]\right) \\ &\mathbf{P}\_{\text{max}} = \max\left\{ (\mathbf{m} + \mathbf{n}\_{\text{a}}) \mathbf{a} \sqrt{\mathbf{g}/\text{h}} \bullet \text{o}, \text{CTV maximum thrust} \text{ force} \right\} \end{aligned} \tag{17}$$

At the large wave periods we have the following behavior:

$$\lim\_{\alpha \to 0} |\mathbf{f}\_{\text{max}}(\mathbf{T})| = \max \left\{ \lim\_{\alpha \to 0} |\mathbf{f}(\mathbf{T}\_{-})|, \lim\_{\alpha \to 0} |\mathbf{f}(\mathbf{T}\_{+})| \right\} \tag{18}$$

*Optimizing Berthing of Crew Transfer Vessels against Floating Wind Turbines… DOI: http://dx.doi.org/10.5772/intechopen.102012*

Where (see Eq. (11)):

$$\mathbf{T}\_{\pm} = \left( \mathbf{A} \mathcal{G} \pm \sqrt{\left\{ \mathbf{A}^{2} + \mathbf{B}^{2} \right\} \mathcal{G}^{2} - \left( \mathbf{A}\mathbf{D} - \mathbf{B}\mathbf{C} \right)^{2}} \right) / \left( \mathbf{A}\mathbf{D} - \mathbf{B}\mathbf{C} + \mathbf{B}\mathcal{G} \right) \tag{19}$$

In the case of box barge of same displacement, length and draft, we have:

#### **9.2 Surge amplitude at large wave periods**

xm a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ ηω<sup>3</sup> � Imð Þ ϑ ω<sup>2</sup> þ Re ð Þκ ω <sup>2</sup> <sup>þ</sup> ð Þ ζω<sup>4</sup> <sup>þ</sup> Re ð Þ <sup>ϑ</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ<sup>κ</sup> <sup>ω</sup> <sup>þ</sup> Re ð Þ <sup>μ</sup> 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> ð Þ<sup>2</sup> <sup>þ</sup> ð Þ βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω 2 q Re ð Þ<sup>κ</sup> <sup>≝</sup> K55b3 <sup>þ</sup> ΛωH3 *=*3 � �k3 � �ð Þþ <sup>F</sup>xm*=*<sup>a</sup> ΛωH2 *=*2 � �k3 � � Mym*=*a � � Imð Þ<sup>κ</sup> <sup>≝</sup> <sup>þ</sup> ΛωH2 *<sup>=</sup>*<sup>2</sup> � �<sup>P</sup> � �ð Þ <sup>F</sup>zm*=*<sup>a</sup> Re ð Þ <sup>μ</sup> <sup>≝</sup> <sup>þ</sup> P2 <sup>þ</sup> K55k3 � �ð Þ <sup>F</sup>xm*=*<sup>a</sup> <sup>ε</sup><sup>≝</sup> � ð Þ <sup>m</sup> <sup>þ</sup> ma P2 � b3K55ð Þ� <sup>Λ</sup>H<sup>ω</sup> k3 ð Þ <sup>m</sup> <sup>þ</sup> ma K55 <sup>þ</sup> ΛωH<sup>2</sup> � �<sup>2</sup> *<sup>=</sup>*<sup>12</sup> n o h i <sup>ϵ</sup><sup>≝</sup> � ð Þ <sup>Λ</sup>H<sup>ω</sup> <sup>P</sup><sup>2</sup> <sup>þ</sup> k3K55 � � *<sup>K</sup>*<sup>55</sup> <sup>≝</sup> � ð Þþ <sup>Δ</sup>*<sup>g</sup>* <sup>∙</sup> *GC* <sup>Δ</sup>*g B*<sup>2</sup> *<sup>=</sup>*ð Þ <sup>12</sup>*<sup>H</sup>* � � <sup>þ</sup> *KZ*<sup>þ</sup> *A* <sup>2</sup> � � *k*<sup>3</sup> ≝ð Þ Δ*=H g* lim k ! 0 xm <sup>a</sup> <sup>¼</sup> lim ω ! 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ Re ð Þκ ω <sup>2</sup> <sup>þ</sup> ð Þ Imð Þ<sup>κ</sup> <sup>ω</sup> <sup>þ</sup> Re ð Þ <sup>μ</sup> 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εω<sup>2</sup> ð Þ<sup>2</sup> <sup>þ</sup> ð Þ ϵω 2 q lim <sup>ω</sup> ! <sup>0</sup> <sup>F</sup>xm*=*<sup>a</sup> <sup>¼</sup> <sup>O</sup> <sup>ω</sup><sup>3</sup> � � lim <sup>ω</sup> ! <sup>0</sup>Mym*=*<sup>a</sup> <sup>¼</sup> lim <sup>k</sup> ! <sup>0</sup> ð Þ <sup>m</sup> <sup>þ</sup> ma H <sup>2</sup> � <sup>Δ</sup> <sup>B</sup><sup>2</sup> 12H � � ffiffiffi g h r ω þ ð Þ m þ ma ffiffiffi g h r ωZ<sup>þ</sup> <sup>a</sup> <sup>þ</sup> <sup>O</sup> <sup>ω</sup><sup>3</sup> � � � � lim ω ! 0 ImFzm*=*a ¼ �ð Þ Δg*=*H f g sinh k z ½ � ð Þ <sup>0</sup> þ h *=* sinh kh ð Þ þ Oð Þ ω (20)

Note: since the vertical incident wave loads are only the ones passing below the floater keel then, if its draft is |z0|, zm/a = zm(z = z0) = sinh[k(z0 + h)]/sinh(kh) [12] (refer to equations written in **Figures 6, 9, 12**, and **15**).

Therefore:

$$\begin{aligned} \varlimsup\_{\mathbf{u}\to\mathbf{0}} \mathbf{Re}\left(\kappa\right) &= \varlimsup\_{\mathbf{u}\to\mathbf{0}} \mathbf{O}\left(\mathbf{u}^{3}\right) + \left[\frac{\mathbf{A}\mathbf{H}^{2}}{2}\mathbf{k}\_{3}\right] \left\{ \left(\mathbf{m} + \mathbf{m}\_{\mathbf{a}}\right) \left[\mathbf{Z}\_{\mathbf{i}}^{+} + \frac{\mathbf{H}}{2}\right] - \Delta \frac{\mathbf{B}^{2}}{12\mathbf{H}} \right\} \sqrt{\frac{\mathbf{g}}{\mathbf{h}}} \mathbf{u}^{2} \\ \varlimsup\_{\mathbf{u}\to\mathbf{0}} \mathbf{Im}(\kappa) &= \varlimsup\_{\mathbf{u}\to\mathbf{0}} \mathbf{O}\left(\mathbf{u}^{2}\right) \end{aligned}$$

$$\begin{aligned} \lim\_{\substack{\mathbf{u}\to\mathbf{0}}\ }\mathrm{Re}\left(\mu\right) &= \lim\_{\mathbf{u}\to\mathbf{0}}\left[\mathrm{O}\left(\mathbf{u}^{2}\right) + \mathbf{K}\_{\mathfrak{S}5}\mathbf{k}\_{\mathfrak{J}}\right]\mathrm{O}\left(\mathbf{u}^{3}\right) \\ \lim\_{\substack{\mathbf{u}\to\mathbf{0}}\ }\mathrm{Re}\frac{1}{\mathbf{u}\to\mathbf{0}}\mathrm{e} &= \lim\_{\mathbf{u}\to\mathbf{0}} -\mathrm{k}\_{\mathfrak{J}}(\mathbf{m} + \mathbf{m}\_{\mathfrak{a}})\mathbf{K}\_{\mathfrak{S}5} + \mathrm{O}(\mathfrak{a}) \\ \lim\_{\mathbf{u}\to\mathbf{0}}\mathrm{e} &= \lim\_{\mathbf{u}\to\mathbf{0}} -\left(\Lambda\mathrm{H}\mathrm{o}\right)\left(\mathrm{P}^{2} + \mathrm{k}\_{\mathfrak{J}}\mathrm{K}\_{\mathfrak{S}5}\right) \\ \lim\_{\mathbf{k}\to\mathbf{0}}\ \frac{\mathbf{x}\_{\mathbf{m}}}{\mathbf{k}\to\mathbf{0}} &= \lim\_{\mathbf{u}\to\mathbf{0}} \frac{\sqrt{[\mathsf{O}(\mathbf{u}^{3})]^{2} + [\mathsf{O}(\mathbf{u}^{3}) + (\mathsf{O}(\mathbf{u}^{2}) + \mathsf{K}\_{\mathfrak{S}5}\mathrm{k}\_{\mathfrak{J}})\mathrm{O}(\mathbf{u}^{3})]^{2}} \\ \lim\_{\mathbf{k}\to\mathbf{0}}\frac{\lim\_{\mathbf{u}\to\mathbf{0}}\mathbf{x}\_{\mathbf{m}}/\mathbf{a}}{\mathbf{k}\to\mathbf{0}}\mathbf{x}\_{\mathbf{m}}/\mathbf{a} = \mathcal{O}(\mathsf{a}) \end{aligned} \tag{21}$$

#### **9.3 Surge phase angle at large wave periods**

cos <sup>φ</sup><sup>x</sup> ¼ þ ηω<sup>3</sup> � Imð Þ <sup>ϑ</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Re ð Þ<sup>κ</sup> <sup>ω</sup> αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> � � � <sup>þ</sup> *ζω*<sup>4</sup> <sup>þ</sup> *Re*ð Þ *<sup>ϑ</sup> <sup>ω</sup>* ½ � <sup>2</sup> <sup>þ</sup> *Im*ð Þ*<sup>κ</sup> <sup>ω</sup>* <sup>þ</sup> *Re*ð Þ *<sup>μ</sup> βω*<sup>5</sup> <sup>þ</sup> *δω* ð Þ <sup>3</sup> <sup>þ</sup> *ϵω* � *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> þ γω<sup>4</sup> þ εω<sup>2</sup> ð Þι <sup>2</sup> <sup>þ</sup> ð Þ βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω 2 q *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � ηω<sup>3</sup> � Imð Þ ϑ ω<sup>2</sup> þ Re ð Þκ ω <sup>2</sup> <sup>þ</sup> ½ � ζω<sup>4</sup> <sup>þ</sup> Re ð Þ <sup>ϑ</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ<sup>κ</sup> <sup>ω</sup> <sup>þ</sup> Re ð Þ <sup>μ</sup> 2 q sin <sup>φ</sup><sup>x</sup> ¼ þ ηω<sup>3</sup> � Imð Þ <sup>ϑ</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Re ð Þ<sup>κ</sup> <sup>ω</sup> βω<sup>5</sup> <sup>þ</sup> δω ð Þ <sup>3</sup> <sup>þ</sup> ϵω � � ζω<sup>4</sup> <sup>þ</sup> Re ð Þ <sup>ϑ</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Imð Þ<sup>κ</sup> <sup>ω</sup> <sup>þ</sup> Re ð Þ <sup>μ</sup> αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> <sup>þ</sup> <sup>ι</sup> � �� *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> þ γω<sup>4</sup> þ εω<sup>2</sup> ð Þι <sup>2</sup> <sup>þ</sup> ð Þ βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω 2 q *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � ηω<sup>3</sup> � Imð Þ ϑ ω<sup>2</sup> þ Re ð Þκ ω <sup>2</sup> <sup>þ</sup> ½ � ζω<sup>4</sup> <sup>þ</sup> Re ð Þ <sup>ϑ</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ<sup>κ</sup> <sup>ω</sup> <sup>þ</sup> Re ð Þ <sup>μ</sup> 2 q *c*<sup>Λ</sup> ≝Λ*H=*ð Þ *m* þ *ma* lim *<sup>T</sup>*!þ<sup>∞</sup> cos *<sup>φ</sup><sup>x</sup>* ¼ �*c*Λ*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>c</sup>*Λ<sup>2</sup> <sup>p</sup> lim *<sup>T</sup>*!þ<sup>∞</sup> sin *<sup>φ</sup><sup>x</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>c</sup>*Λ<sup>2</sup> <sup>p</sup> (22)

#### **9.4 Heave amplitude at large wave periods**

zm a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � �Imð Þ Γ ω<sup>4</sup> � Imð Þ Π ω<sup>2</sup> þ Re ð Þ Σ ω <sup>2</sup> <sup>þ</sup> ½ � Imð Þ <sup>Ξ</sup> <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>Π</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ <sup>Σ</sup> <sup>ω</sup> 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> ½ �<sup>2</sup> <sup>þ</sup> ½ � βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω 2 q Re ð Þ <sup>Π</sup> <sup>≝</sup> ½ � IGA15P ð Þ� <sup>F</sup>xm*=*<sup>a</sup> ½ � ð Þ <sup>m</sup> <sup>þ</sup> ma <sup>P</sup> <sup>M</sup>ym*=*<sup>a</sup> � � Imð Þ <sup>Π</sup> <sup>≝</sup> ð Þ <sup>m</sup> <sup>þ</sup> ma K55 <sup>þ</sup> B11B55 � <sup>B</sup><sup>2</sup> <sup>15</sup> � �Imð Þ <sup>F</sup>zm*=*<sup>a</sup> Re ð Þ <sup>Σ</sup> <sup>≝</sup> � ½ � B15P ð Þþ <sup>F</sup>xm*=*<sup>a</sup> ½ � B11P <sup>M</sup>ym*=*<sup>a</sup> � � Imð Þ Σ ≝ � ½ � KB55 þ K55B11 Imð Þ Fzm*=*a

*Optimizing Berthing of Crew Transfer Vessels against Floating Wind Turbines… DOI: http://dx.doi.org/10.5772/intechopen.102012*

$$\begin{aligned} \text{Ag}\_{1\text{A}5} & \stackrel{\text{def}}{=} \text{m}\_{\text{A}1} \text{H}/2) \\\\ \text{B}\_{11} & \stackrel{\text{def}}{=} \lambda \text{H} \\\\ \text{B}\_{15} & \stackrel{\text{def}}{=} -\left(\lambda \text{H}^{2}/2\right) \\\\ \text{B}\_{55} & \stackrel{\text{def}}{=} \lambda \text{o} \text{H}^{3}/3 \end{aligned} \tag{23}$$
 
$$\frac{\lim}{k \to 0} z\_{m}/a = \sinh\left[k(z\_{0} + h)\right]/\sinh\left(kh\right)$$

#### **9.5 Heave phase angle at large wave periods**

cos <sup>φ</sup><sup>z</sup> ¼ ��Imð Þ <sup>Γ</sup> <sup>ω</sup><sup>4</sup> � Imð Þ <sup>Π</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Re ð Þ <sup>Σ</sup> <sup>ω</sup> αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> � � � � Imð Þ <sup>Ξ</sup> <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>Π</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Imð Þ <sup>Σ</sup> <sup>ω</sup> βω<sup>5</sup> <sup>þ</sup> δω ð Þ <sup>3</sup> <sup>þ</sup> ϵω � *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � �Imð Þ <sup>Γ</sup> <sup>ω</sup><sup>4</sup> � Imð Þ <sup>Π</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Re ð Þ <sup>Σ</sup> <sup>ω</sup> <sup>2</sup> <sup>þ</sup> ½ � Imð Þ <sup>Ξ</sup> <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>Π</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ <sup>Σ</sup> <sup>ω</sup> <sup>2</sup> q *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> ½ �<sup>2</sup> <sup>þ</sup> ½ � βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω <sup>2</sup> q sin <sup>φ</sup><sup>z</sup> ¼ þ�Imð Þ <sup>Γ</sup> <sup>ω</sup><sup>4</sup> � Imð Þ <sup>Π</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Re ð Þ <sup>Σ</sup> <sup>ω</sup> βω<sup>5</sup> <sup>þ</sup> δω ð Þ <sup>3</sup> <sup>þ</sup> ϵω � <sup>þ</sup> Imð Þ <sup>Ξ</sup> <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>Π</sup> <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Imð Þ <sup>Σ</sup> <sup>ω</sup> αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> � �� *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � �Imð Þ <sup>Γ</sup> <sup>ω</sup><sup>4</sup> � Imð Þ <sup>Π</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Re ð Þ <sup>Σ</sup> <sup>ω</sup> <sup>2</sup> <sup>þ</sup> ½ � Imð Þ <sup>Ξ</sup> <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>Π</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ <sup>Σ</sup> <sup>ω</sup> <sup>2</sup> q *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> ½ �<sup>2</sup> <sup>þ</sup> ½ � βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω <sup>2</sup> q limT!þ<sup>∞</sup> *φ<sup>z</sup>* ¼ 0° (24)

#### **9.6 Pitch amplitude at large wave periods**

θ<sup>m</sup> a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f g Re ð Þ ω<sup>3</sup> þ Re ð Þ ω<sup>2</sup> þ Re ð Þ ω <sup>2</sup> <sup>þ</sup> f g Imð Þ <sup>ω</sup><sup>4</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> f g<sup>2</sup> <sup>þ</sup> f g βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω 2 q Re ð Þ ≝ � �f g ½ � ð Þ m þ ma P Imð Þ Fzm*=*a ¼ �ð Þ m þ ma PΔg*=*H Imð Þ ≝ IGA15 � ð Þ m þ ma Z<sup>þ</sup> a � �k3ð Þ <sup>P</sup>*=*<sup>a</sup> Re ð Þ <sup>≝</sup> � ½ � k3B15 ð Þ� <sup>F</sup>xm*=*<sup>a</sup> ½ � k3B11 <sup>M</sup>ym*=*<sup>a</sup> � � � � Imð Þ ≝ � �f g ½ � B11P Imð Þ Fzm*=*a lim k ! 0 θ<sup>m</sup> <sup>a</sup> <sup>¼</sup> lim ω ! 0 ð Þ m þ ma ffiffiffiffiffiffiffiffi g*=*h p ω K55 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H <sup>2</sup> <sup>þ</sup> <sup>Z</sup><sup>þ</sup> a � � � ½ � Δ*=*ð Þ m þ ma B2 12H � �<sup>2</sup> þ a2 s *lim <sup>k</sup>* ! <sup>0</sup> *<sup>θ</sup>m=<sup>a</sup>* <sup>¼</sup> <sup>0</sup>

#### **9.7 Pitch phase angle at large wave periods**

cos φθ ¼ þ Re ð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup> αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> � � � <sup>þ</sup> Imð Þ <sup>ω</sup><sup>4</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> βω<sup>5</sup> <sup>þ</sup> δω ð Þ <sup>3</sup> <sup>þ</sup> ϵω � *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f g Re ð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup><sup>2</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup> <sup>2</sup> <sup>þ</sup> f g Imð Þ <sup>ω</sup><sup>4</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> <sup>2</sup> q *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> f g<sup>2</sup> <sup>þ</sup> f g βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω <sup>2</sup> q sin φθ ¼ þ Re ð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup> βω<sup>5</sup> <sup>þ</sup> δω ð Þ <sup>3</sup> <sup>þ</sup> ϵω � � Imð Þ <sup>ω</sup><sup>4</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> ½ � <sup>2</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> � �� *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f g Re ð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup><sup>2</sup> <sup>þ</sup> Re ð Þ <sup>ω</sup> <sup>2</sup> <sup>þ</sup> f g Imð Þ <sup>ω</sup><sup>4</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>3</sup> <sup>þ</sup> Imð Þ <sup>ω</sup><sup>2</sup> <sup>þ</sup> Imð Þ <sup>ω</sup> <sup>2</sup> q *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αω<sup>6</sup> <sup>þ</sup> γω<sup>4</sup> <sup>þ</sup> εω<sup>2</sup> f g<sup>2</sup> <sup>þ</sup> f g βω<sup>5</sup> <sup>þ</sup> δω<sup>3</sup> <sup>þ</sup> ϵω <sup>2</sup> q lim <sup>T</sup>!þ<sup>∞</sup> cos φθ <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ � Z<sup>þ</sup> <sup>a</sup> *<sup>=</sup>*<sup>a</sup> � � � ð Þ� <sup>1</sup>*=*<sup>2</sup> <sup>B</sup><sup>2</sup> *=* 12H<sup>2</sup> \_ ð Þ <sup>1</sup> <sup>þ</sup> Cm1 h i n o h i ð Þ <sup>H</sup>*=*<sup>a</sup> n o<sup>2</sup> r lim <sup>T</sup>!þ<sup>∞</sup> sin φθ <sup>¼</sup> � Z<sup>þ</sup> <sup>a</sup> *<sup>=</sup>*<sup>a</sup> � � � ð Þ� <sup>1</sup>*=*<sup>2</sup> B2 *=* 12H<sup>2</sup> \_ ð Þ <sup>1</sup> <sup>þ</sup> Cm1 h i n o h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ � Z<sup>þ</sup> <sup>a</sup> *<sup>=</sup>*<sup>a</sup> � � � ð Þ� <sup>1</sup>*=*<sup>2</sup> B2 *=* 12H<sup>2</sup> \_ ð Þ <sup>1</sup> <sup>þ</sup> Cm1 h i n o h i ð Þ <sup>H</sup>*=*<sup>a</sup> n o<sup>2</sup> r 8 >>>>>>>>>< >>>>>>>>>: *φ*000 *<sup>x</sup>*<sup>1</sup> ≝ *Z*<sup>þ</sup> *<sup>a</sup> <sup>=</sup><sup>a</sup>* � � <sup>þ</sup> ð Þ� <sup>1</sup>*=*<sup>2</sup> *<sup>B</sup>*<sup>2</sup> *=* 12*H*<sup>2</sup> ð Þ 1 þ *Cm*<sup>1</sup> � � � � � � ð Þ *<sup>H</sup>=<sup>a</sup>* � � lim*ω*!<sup>0</sup> cos *φθ* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *φ*000<sup>2</sup> *x*1 p lim*ω*!0 sin *φθ* ¼ �*φ*<sup>000</sup> *<sup>x</sup>*1*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*000<sup>2</sup> *x*1 p (26)

#### **9.8 Friction coefficient at large wave periods**

Therefore:

$$\begin{aligned} \lim\_{a \to 0} A &= \lim\_{a \to 0} a \{ \sinh \left[ k(z\_0 + h) \right] / \sinh \left( k h \right) \} \star \cos 360^\circ = a(z\_0 + h) / h, \\ \lim\_{a \to 0} B &= \lim\_{a \to 0} a \{ \sinh \left[ k(z\_0 + h) \right] / \sinh \left( k h \right) \} \star (-at\_T) = O(w) \\ \lim\_{a \to 0} C &= O(w) \left( -c\_\Lambda / \sqrt{1 + c\_\Lambda^2} \right) + Z\_G O(w) \left( 1 / \sqrt{1 + q\_{\ge 1}^{\eta \pi 2}} \right) = O(w), \\ \lim\_{a \to 0} D &= O(w) \left( 1 / \sqrt{1 + c\_\Lambda^2} \right) + Z\_G O(w) \left( -q\_{\ge 1}^{\eta \pi} / \sqrt{1 + q\_{\ge 1}^{\eta \pi 2}} \right) = O(w) \\ \lim\_{a \to 0} \mathcal{G} &= (-L) \end{aligned} \tag{27}$$

$$\begin{split} \lim\_{\boldsymbol{\theta}\to\boldsymbol{0}}T\_{\pm} &= \lim\_{\boldsymbol{w}\to\boldsymbol{0}} \frac{A(-L) \mp \sqrt{\left\{A^{2} + O(\boldsymbol{w})^{2}\right\} \mathrm{L}^{2} - \left[AO(\boldsymbol{w}) - O(\boldsymbol{w})O(\boldsymbol{w})\right]^{2}}}{\left[AO(\boldsymbol{w}) - O(\boldsymbol{w})O(\boldsymbol{w}) + O(\boldsymbol{w})(-L) + O(\boldsymbol{w})(-L)\right]} \\ \lim\_{\boldsymbol{w}\to\boldsymbol{0}}T\_{\pm} &= \lim\_{\boldsymbol{w}\to\boldsymbol{0}} \left[-AL \mp \left\{AL + O(\boldsymbol{w}^{2})\right\}\right] / O(\boldsymbol{w}) \\ \lim\_{\boldsymbol{w}\to\boldsymbol{0}}T\_{-} &= \underset{\boldsymbol{w}\to\boldsymbol{0}}{\text{IM}}[-[\boldsymbol{a}(\boldsymbol{z}\_{0} + \boldsymbol{h})/\boldsymbol{h}]L - [\boldsymbol{a}(\boldsymbol{z}\_{0} + \boldsymbol{h})/\boldsymbol{h}]L + O(\boldsymbol{w})]/O(\boldsymbol{w}) = \boldsymbol{\infty} \end{split}$$

*Optimizing Berthing of Crew Transfer Vessels against Floating Wind Turbines… DOI: http://dx.doi.org/10.5772/intechopen.102012*


#### **Table 2.**

*CAT CTV analytical berthing limits for monopile various floater designs.*

**Figure 18.** *Floater wave masking performances (cylinder versus FLOATGEN).*

$$\lim\_{k \to 0} |f(T\_{-})| = \left| \frac{-A - 2 \bullet 0 + \{A \bullet \mathbf{0}^{2}\}}{-(\mathbf{0} - [-L]) - 2 \bullet 0 + [(\mathbf{0} + [-L]) \bullet \mathbf{0}^{2}]} \right| = \frac{[a(z\_{0} + h)/h]}{L}$$

$$\lim\_{a \to 0} T\_{+} = \underset{a \to 0}{\text{LIM}} [-AL + AL + O(a^{2})] / O(a) = O(a) = \mathbf{0} \tag{28}$$

$$\lim\_{k \to 0} |f(T\_{+})| = \left| \frac{-A \bullet \mathbf{0}^{2} - 2 \bullet \mathbf{0} \bullet \mathbf{0} + A}{-(\mathbf{0} - [-L]) \bullet \mathbf{0}^{2} - 2 \bullet \mathbf{D} \bullet \mathbf{0} + (\mathbf{0} + [-L])} \right| = \frac{[a(z\_{0} + h)/h]}{L}$$

Eventually, we get the following formula for berthing to possible [1]:

$$\lim\_{k \to 0} |f(T\_{\pm})| = [a(\mathbf{z}\_0 + h)/h]/L < f\_{\text{grip}} \tag{29}$$
 
$$\text{if } f\_{\text{grip}} = \mathbf{0}, \mathbf{8} \text{ (number - very wet soil) (refer to Eq.(9))}$$

**Table 2** sums up the analytical Hs found for berthing to occur whatever the wavelength, by using Eq. (29).

As can be seen results from **Tables 1** and **2** meet with less than 12% discrepancy. The reason why is that the analytical calculation does not account for the diffraction forces.

Eventually, some practical considerations must be accounted for (see **Figure 18**):

The more the wave direction varies, the more a large floater width becomes necessary to allow berthing by masking the waves.

### **10. Conclusions and recommendations**

The present study results find that berthing a CTV against an offshore wind turbine is not yet optimized, at least from a marine maintenance point of view: most designs do not provide sheltered waters.

Some proposals focus on improving the CTV:

• ESNA promote the use of Surface Effect Ships (SES), to minimize their heave. However, the lack of commercial success of the solution seems related to the high fuel consumption of such boats [7].

Other proposals focus rather on the floater side:


Another possible axis of development would be to design an additional wall to existing boat landings, providing a sheltered water.

Eventually, since the present study only applies to a unidirectional wave, the next studies will focus on multidirectional waves, in order represent a more realistic sea state.

### **Acknowledgements**

The author gratefully acknowledges the support from ENSM and its director of research and industrial relations, Mr. Dominique FOLLUT, for throughout the present research work.

*Optimizing Berthing of Crew Transfer Vessels against Floating Wind Turbines… DOI: http://dx.doi.org/10.5772/intechopen.102012*
