**2. Method description**

## **2.1 General**

The calculation is based on a simplified linear diffraction-radiation model applied in the frequential domain [1].

The studied ship is a CATamaran CTV (CAT CTV) [2, 3]: 27 m long, 8.2 m wide, twin hulls 3.2 m wide. It is modelled with Wigley hulls (**Figure 1**) [1].

The software used are GMSH for meshing [4] and NEMOH for hydrodynamics [5].

### **2.2 Loads of a unidirectional wave on CTV (seakeeping)**

The Wigley hull, due to the wave excitation, moves in a vertical plane as follows (**Figure 2**) [1]:


**Figure 1.** *CTV berthing against monopile (3D view).*

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

**Figure 2.** *CTV sea keeping without berthing (elevation).*

The equations of dynamics are:

$$\begin{aligned} \mathbf{(I+I\_a)\ddot{X} + B\dot{X} + KX &= F\_{\text{excit}}} \\ \Rightarrow I\ddot{X} = F\_{\text{excit}} - I\_a\ddot{X} - B\dot{X} - KX \Rightarrow I\ddot{X} = \sum F\_{\text{ext}} \end{aligned} \tag{1}$$


#### **2.3 Loads of a unidirectional wave on CTV (berthing)**

The friction coefficient without sliding is, about the principle of action and reaction (**Figure 3**) [1]:

$$\mathbf{f} = \frac{\text{Tangential force at vertical wall}}{\text{Normal force at vertical wall}} = \frac{\mathbf{T}}{\mathbf{N}} \Longrightarrow \mathbf{f} = \left(-\sum \mathbf{F\_{ext \, x}}\right) / \left(-\sum \mathbf{F\_{ext \, x}}\right) \tag{2}$$

Assumptions:

1.A thrust P is added to N, in order never to reach N < 0:

**Figure 3.** *Coulomb's friction law.*

$$\overrightarrow{\mathbf{P}} = -\left\|\overrightarrow{\mathbf{P}}\right\|\overrightarrow{\mathbf{x}} = \text{mo}^2 \mathcal{G} \overrightarrow{\mathbf{x}} (\text{ave } \mathbf{P} < \mathbf{0} \text{ et } \mathcal{G} < \mathbf{0}) \tag{3}$$

2.For a low friction berthing:

$$
\Sigma \,\mathbf{F}\_{\text{ext }\mathbf{z}} = \mathbf{I}\_{33} \ddot{\mathbf{Z}} \tag{4}
$$

The equations at the berthing point A = A- become therefore:

$$\sum \mathbf{F\_{ext z}} = \mathbf{I\_{33}} \ddot{\mathbf{Z}} \text{ and } \sum \mathbf{F\_{ext x} + P} = \mathbf{I\_{11}} \ddot{\mathbf{X}} + \mathbf{I\_{15}} \ddot{\boldsymbol{\theta}} \tag{5}$$

$$\Rightarrow \mathbf{f} = \frac{-\mathbf{I}\_{33}\ddot{\mathbf{Z}}}{-\mathbf{I}\_{11}\ddot{\mathbf{X}} - \mathbf{I}\_{15}\ddot{\boldsymbol{\Theta}} + \mathbf{P}} \Longrightarrow \mathbf{f} = \frac{-\ddot{\mathbf{Z}}}{-\ddot{\mathbf{X}} - \mathbf{Z}\_{G}\ddot{\boldsymbol{\Theta}} + \mathbf{\mathcal{G}}\boldsymbol{\alpha}^{2}} \\ \text{where } \boldsymbol{\mathcal{G}} = \mathbf{P}/\left[\mathbf{m}\left(\mathbf{1} + \mathbf{C}\_{\mathbf{M1}}'\right)\boldsymbol{\alpha}^{2}\right] \tag{6}$$

We define tT and tN respectively as the time phase corrections required to get the calculated loads T(t) and N(t) in phase with HSVA test results THSVA (t) and NHSVA (t) [2].

$$Z = Z\_m \text{COS}[o(t - t\_T) + \rho\_x]$$

$$X = X\_m \text{COS}[o(t - t\_N) + \rho\_x]$$

$$\theta = \theta\_m \text{COS}[o(t - t\_N) + \rho\_\theta] \tag{7}$$

We also define the following notations:

$$\begin{aligned} &T \stackrel{\text{def}}{=} \tan\left(\alpha t/2\right) \\ &A \stackrel{\text{def}}{=} Z\_m \cos\left(-\alpha t\_T + \varphi\_x\right), \\ &B \stackrel{\text{def}}{=} Z\_m \sin\left(-\alpha t\_T + \varphi\_x\right), \\ &C \stackrel{\text{def}}{=} X\_m \cos\left(-\alpha t\_N + \varphi\_x\right) + Z\_G \theta\_m \cos\left(-\alpha t\_N + \varphi\_\theta\right), \\ &D \stackrel{\text{def}}{=} X\_m \sin\left(-\alpha t\_N + \varphi\_x\right) + Z\_G \theta\_m \sin\left(-\alpha t\_N + \varphi\_\theta\right) \\ &\Rightarrow f = \left[-AT^2 - 2BT + A\right] / \left[-\left(C - \mathcal{G}\right)T^2 - 2DT + \left(C + \mathcal{G}\right)\right] \end{aligned}$$

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

In order for the function f(t) to get relative extremes, the numerator of the quotient in Eq. (8) must have a positive discriminant δ<sup>0</sup> :

$$\mathbf{S}' > \mathbf{0} \Leftrightarrow \left| \mathbf{P} \right| > \mathbf{m} \alpha^2 \sqrt{\left(\mathbf{A}\mathbf{D} - \mathbf{B}\mathbf{C}\right)^2} \Big/ \sqrt{\mathbf{A}^2 + \mathbf{B}^2} \tag{9}$$

Moreover, the denominator in Eq. (8) must never be null. Physically that means that the CTV propeller thrust P should be great enough in order never to get N < 0. Mathematically that implies both that its discriminant Δ<sup>0</sup> must be negative and thatð Þ G � C <0:

$$\Delta'<0 \text{ and } (\mathcal{G}-\mathbb{C})<\mathbf{0} \Leftrightarrow |\mathbb{P}| > \mathbf{m}\alpha^2\sqrt{\mathbf{D}^2+\mathbb{C}^2} \tag{10}$$

If the conditions (3) and (4) are met, then, over a wave period, the friction coefficient will reach its extremes at the instants t+ and t�, which correspond to the following values T+ and T�:

$$\mathbf{T}\_{\pm} = \left[ \mathbf{A} \mathcal{G} \pm \sqrt{\left( \mathbf{A}^2 + \mathbf{B}^2 \right) \mathcal{G}^2 - \left( \mathbf{A}\mathbf{D} - \mathbf{B}\mathbf{C} \right)^2} \right] / \left[ \mathbf{A}\mathbf{D} - \mathbf{B}\mathbf{C} + \mathbf{B}\mathcal{G} \right] \tag{11}$$

Then the maximum friction coefficient over a wave period is:

$$|\mathbf{f}\_{\max}(\mathbf{T})| = \max\left[|\mathbf{f}(\mathbf{T}\_{+})|, |\mathbf{f}(\mathbf{T}\_{-})|\right] \tag{12}$$

We must therefore choose. Physically, that is the CTV surge over which the CTV captain has the time to adjust the propeller thrust P, in order for the fender never to lose contact with the boat landing. Nevertheless, the CTV is limited by her maximum thrust Pmax. We infer that:

$$|\mathbf{P}| = \min\left(\mathbf{m}|\mathcal{G}|\mathbf{o}^2, |\mathbf{P}\_{\text{max}}|\right) \tag{13}$$

Or, in other words:

$$\mathcal{G} = -\min\left(\mathbf{L}, \left| \mathbf{P}\_{\max} \right| / \left[ \mathbf{m} \mathbf{o}^{2} \right] \right) \tag{14}$$

The selected criterion for CTV boarding at the berthing point is that the friction coefficient must never exceed the grip factor:

$$|\mathbf{f}\_{\text{max}}(\mathbf{T})| < \mathbf{f}\_{\text{grip}} \text{ with } \mathbf{f}\_{\text{grip}} = \mathbf{0}, \mathbf{8} \text{ (number -- very wet soil [6])}\tag{15}$$

## **3. CTV against monopile**

For benchmarking purpose, the first calculation models the "bump and jump" against a monopile (**Figures 1** and **4**). The water depth is 29 m.

The studied monopile has a 5 m diameter [2].

It may be noted that the CAT CTV is wider than the monopile: therefore the former is not masked from the waves by the latter.

**Figure 5** compares for 2 m significant wave height (Hs) the calculated ratio of wave vertical force over wave horizontal force (T/N) with the grip coefficient of rubber against very wet soil (fgrip) [6].

For comparison reference [7] estimates the berthing limit to be for a ratio wavelength over boat length of 1.85 (λ/B): both results meet with 5% accuracy.

**Figure 4.**

*CTV berthing against monopile (plane view).*

**Figure 5.**

*Curves T/N and fgrip versus wavelength over boat length for 2 m Hs.*
