# Experimental investigation on wave-induced bending moments of a 6,750-TEU containership in oblique waves

Shinwoong Kim, Benjamin Bouscasse, Guillaume Ducrozet, Sylvain Delacroix, Guillaume de Hautesclocque, Pierre Ferrant

► To cite this version:

Shinwoong Kim, Benjamin Bouscasse, Guillaume Ducrozet, Sylvain Delacroix, Guillaume de Hautesclocque, et al.. Experimental investigation on wave-induced bending moments of a 6,750-TEU containership in oblique waves. Ocean Engineering, 2023, 284, pp.115161. 10.1016/j.oceaneng.2023.115161. hal-04491017

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Submitted on 27 May 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Experimental investigation on wave-induced bending
moments of a 6,750-TEU containership in oblique waves

Shinwoong KIMª, Benjamin BOUSCASSEª,*, Guillaume DUCROZETª, Sylvain DELACROIXª, Guillaume DE HAUTECLOCQUEᵇ, Pierre FERRANTª

ªNantes Université, École Centrale Nantes, CNRS, LHEEA, UMR 6598, F-44000 Nantes, France

ᵇBureau Veritas, Paris, France

**Abstract**

This study aims at the experimental investigation of wave-induced motions and loads of a containership model without forward speed in -120 degree oblique regular waves to study the influence of the wave steepness and provide reference data for future benchmark studies. A mooring system with 4 horizontally arranged spring lines was used to maintain the heading angle of a 1/65 scaled 9-segmented model designed to be as rigid as possible.

Focuses were on studying the nonlinear effects due to the wave steepness on the
vertical bending moment (VBM) and horizontal bending moment (HBM) near amid-
ships, and 6-DOF motions at the center of gravity (COG) of the model. Several wave
series that are distinguished by wave steepness were considered in the experiment
accordingly. Each series consists of waves with various periods that were intended
to cover the peak of the wave bending moment transfer functions. Through this,
the nonlinear wave effects on the responses of the rigid body at various periods were
determined. It was confirmed that the steeper wave contributes to the increase in
the higher-order harmonic components including slamming events in the bending

<footnote>*Corresponding author</footnote>
moments measured. The change in the characteristics of the wave bending moments
according to the change in wave steepness was found to be very different from the
linear response.

A detailed discussion was made on the influence of the mooring system on the
asymmetric horizontal bending moment (HBM) and the change in the average yaw
motion. Experiments with and without a mooring system under the same wave
condition were conducted, and the effect of the mooring system was identified. It
was qualitatively confirmed that the mooring system’s restoring moment correlates
with the asymmetric HBM and average yaw movement change.

# 1. Introduction

Wave bending moments amidships are key parameters to be considered in the
structural design of a ship. However, it is sometimes challenging to quantitatively
and qualitatively estimate the loads and motions of a ship facing highly nonlinear
waves, especially in oblique sea conditions inducing additional bending and torsional
moments. In the last decade, the effect of hull flexibility on wave-induced loads has
received much attention and has been one of the concerns in ship design. Therefore,
many experimental studies have been conducted highlighting the significant contri-
bution of hydroelasticity to the structural response (Hong and Kim (2014); Marón
and Kapsenberg (2014); Lin et al. (2020); Chen et al. (2017)).

Numerical investigations of the wave-induced motions and loads of flexible bod-
ies have also been conducted for several decades and have provided good estimates.
However, converged numerical results based on several numerical theories for various
environmental and operating conditions have been an issue to address. A bench-
mark study of the vessel’s vertical bending moment (VBM) response found that the
VBM responses estimated through various nonlinear numerical tools showed large
discrepancies in severe head sea conditions (Kim and Kim (2016)). In a previous
experimental study with a flexible model by Zhu et al. (2011a), numerical models
based on 2D strip theory and 3D potential theory predicted well the horizontal bend-
ing moment (HBM) and torsional moment (TM) in oblique seas with respect to the
experiments. Whereas, the numerical results showed significant differences in the
vertical shear force (VSF) and VBM compared to the experimental results.

To overcome the limitations of the numerical solvers mentioned above, wave-
induced bending moments have been studied using Computational Fluid Dynamics
(CFD), especially for extreme wave conditions. The hydroelastic effects of a flexible
ship have been studied with an approach based on a two-way coupling between the
CFD and a structural analysis solver (El Moctar et al. (2017)). However, high-fidelity
CFD methods are basically computationally expensive.

Given that there are only a few detailed studies on the response characteristics of
wave bending moments due to the wave nonlinearity, the present work uses a model
designed to be rigid in order to clearly understand the relevant physical phenom-
ena caused by high waves, i.e., additional nonlinear response characteristics such as
slamming and whipping that may occur in a flexible model were excluded.

For the VBM of the rigid model, the maximum is typically observed in head sea
conditions (180 degrees) and tends to decrease towards the oblique seas, but the
order of magnitude is usually comparable with each other (Zhu et al. (2011b); Si
et al. (2022)). For the HBM, on the other hand, the increase as the heading changes
from head sea to oblique sea tends to be visibly large, with a maximum amplitude
that could reach the one of the VBM.

In previous research by Zhu et al. (2011b), the wave heading effect on the wave
bending moments was experimentally analyzed, and it was found that the HBM
significantly increased at the 120-degree wave heading compared to the one at 150-
degree, with the maximum magnitude corresponding to the VBM (Zhu et al. (2010); Jiao et al. (2021); Si et al. (2022)). The same trend was observed in the preliminary calculation with Hydrostar in this present study. To summarize, it is clear that the wave bending moments under a 120-degree oblique wave condition can be of major concern in the estimation of the cross-sectional stress distribution of a ship, implying that the contribution of the HBM, as well as the VBM induced by given wave conditions, needs to be figured out in detail (Jiao et al. (2021)).

Taking the above into account, this present study sought to identify the response characteristics of a zero-speed rigid containership model to wave nonlinearity under -120 degree oblique regular wave conditions. Estimation of the response of interest under a given regular wave with a known specific wave height and period is considered a deterministic approach which is a commonly accepted method for the calculation of design loads as implemented in the IACS CSR (IACS (2014)). The focus was mainly on the effect of wave nonlinearity of a given wave on the wave bending moments, therefore, 7 wave steepness $H/λ$ series from 0.01 (1 %) to 0.105 (10.5 %) consisting of various $λ/L_{pp}$ cases each was applied.

This may allow a better understanding of the response characteristics of a zero-speed rigid containership model to wave nonlinearity. Ultimately, the results observed in oblique wave conditions would enable the intuitive validation of numerical codes currently under development and furthermore provide solid reference data for future benchmark studies.

## 2. Segmented hull: physical and mathematical model

### 2.1. *Description*

The ship benchmarked in this study is a 6750-TEU containership with a scale ratio of 1/65 (Bouscasse et al. (2022)). The main dimensions of the model are the
<table><caption>Table 1: Principal dimensions of the 6750-TEU containership</caption><thead><tr><th></th><th>Prototype</th><th>Model</th></tr></thead><tbody><tr><td>Scale</td><td>1/1</td><td>1/65</td></tr><tr><td>L<sub>pp</sub> (m)</td><td>286.6</td><td>4.409</td></tr><tr><td>B (m)</td><td>40</td><td>0.615</td></tr><tr><td>H (m)</td><td>24.2</td><td>0.372</td></tr><tr><td>Design d (m)</td><td>11.98</td><td>0.188</td></tr><tr><td>Displacement</td><td>85663.8 (ton)</td><td>311.93 (kg)</td></tr><tr><td>KG (m)</td><td>16.562</td><td>0.257</td></tr><tr><td>LCG from AP (m)</td><td>139.56</td><td>2.147</td></tr><tr><td>kxx (m)</td><td>14.4</td><td>0.222</td></tr><tr><td>kyy (m)</td><td>71.5</td><td>1.109</td></tr><tr><td>kzz (m)</td><td>71.4</td><td>1.106</td></tr></tbody></table>

same as those of the one used in the ITTC-ISSC benchmark study carried out by Kim and Kim (2016) (see Table 2.1). As presented in Figure 1, the model consists of 9 segments from the stern (Seg1) to the bow (Seg9), each of which is equipped with a 3-DOF load sensor ($C_i$), connecting the beam part ($b_i$) to the hull part ($h_i$). The 3-DOF load sensor measures $F_z$, $M_x$, and $M_y$, and the maximum measurement range of the sensor is 30 kg to 100 kg. Segments are connected to each other with bolts through their element $b_i$ and set as a rigid model. The measurement data of each 3-DOF load sensor enable the reconstruction of internal loads (VSF and VBM) at each intersegment (between segments), providing the longitudinal distribution of VSF and VBM of the model. A 6-DOF load/torque sensor, denoted as ATI is installed between beams $b_4$ and $b_5$ corresponding to intersegment No.4 to directly measure the 6-DOF internal loads at the location (see Figure 1). The ATI sensor is capable of measuring a large range of loads up to 1400 Nm for moments.

The 6-DOF motions at the center of gravity (COG) of the model are measured through an optical motion tracking system, Qualisys. The reference coordinate system of Qualisys is defined through its calibration at a location coinciding with
<figure><img src="image_2.png" /><figcaption>Figure 1: 9-segmented 6750-TEU containership model and load sensor location</figcaption></figure>

the ATI location (x=18.2 m from a wavemaker) and in the waterline. A total of 4 cameras are installed to track the motion of several reflective markers attached to the model to measure the motion of the COG with respect to the initially defined reference frame. Three accelerometers are installed at (3.66, 0, 0.385) on $b_1$, at (2.258, 0, 0.363) on $b_5$ near amidships, and at (0.163, 0, 0.340) on $b_9$ based on the aft perpendicular (AP) and the baseline of the model. The measured acceleration data are mainly used to calculate the inertial force component of each segment in the internal load reconstruction process with 3-DOF load sensor measurements.

Tension measurement of the four mooring lines is performed using a tensile/ compressive force transducer, measuring a nominal force of up to 500 N. The two mooring lines at the bow and the other two mooring lines at the aft are installed at (4.520, 0, 0.257) and (-0.091,0, 0.257) based on AP and baseline respectively, and each tensiometer (total of 4) is attached to the fairlead, connecting each mooring line to the model.

Figure 2 shows the process of data measurement and acquisition through sensors installed in the model and relevant devices. At the outset of the process, when the
<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic diagram of sensor position, data measurement, and acquisition process</figcaption></figure>

wavemaker is run through a wavemaker control PC, a PC for the 6-DOF motions and the other PC for acceleration and wave loads are triggered, and data measurement starts with the relevant sensors presented in Figure 2.

Thus, motion, acceleration, and wave load data measurements are automatically synchronized with wave elevation measurements. The data measured from the sensors are conditioned through a data acquisition system called QuantumX device and a wave gauge conditioner, and the data are finally transmitted to the corresponding PC. All the sensors measure signals with a 100 Hz sampling rate. More details about the model and sensor instrumentation can be found in previous research with the same model, where tests were performed in the head wave condition (Bouscasse et al. (2022); Horel et al. (2019)).
## 2.2. Rigid-body equations of motion and internal loads

For reference, the mathematical notations that allow for identifying the physical quantities such as position, velocity, and acceleration at a position of interest of the model are defined following Fossen (2011). The coordinate system {$n$} = ($x_n, y_n, z_n$) with origin $o_n$ is fixed in the wave tank and can be considered as an inertial reference frame. The moving body-fixed frame is referred to as {$b$} = ($x_b, y_b, z_b$) with origin $o_b$ amidships, in the longitudinal symmetry plane, and in the waterline plane (see Figure 1). Considering the Newton-Euler equations of motion for a rigid-body, shear forces and bending moments at a position of interest, $\mathbf{A} = (x_A, y_A, z_A)$, expressed in the body-fixed frame can be obtained as follows (1):

$$ 
\begin{aligned}
\mathbf{Q} &= -\mathbf{F}_{ext}^{x<x_A} + m^{x<x_A} \left( \frac{d\mathbf{V}_{b/n}^b}{dt} + \mathbf{\Omega}_{b/n}^b \times \mathbf{V}_{b/n}^b + \frac{d\mathbf{\Omega}_{b/n}^b}{dt} \times \mathbf{r}_A^{G^{x<x_A}} + \mathbf{\Omega}_{b/n}^b \times (\mathbf{\Omega}_{b/n}^b \times \mathbf{r}_A^{G^{x<x_A}}) \right) \\
\mathbf{M}^\mathbf{A} &= -\mathbf{M}_{ext}^\mathbf{A} + \mathbf{I}^{x<x_A} \frac{d\mathbf{\Omega}_{b/n}^b}{dt} + \mathbf{\Omega}_{b/n}^b \times (\mathbf{I}^{x<x_A} \mathbf{\Omega}_{b/n}^b) + (m\mathbf{r}_A^G)^{x<x_A} \times \left( \frac{d\mathbf{V}_{b/n}^b}{dt} + \mathbf{\Omega}_{b/n}^b \times \mathbf{V}_{b/n}^b \right)
\end{aligned}
\quad (1) $$

where $m^{x<x_A}$ and $\mathbf{I}^{x<x_A}$ are the mass and the moment of inertia of a $0 < x < x_A$ portion of the model. $\mathbf{V}_{b/n}^b = [u, v, w]^\top$ and $\mathbf{\Omega}_{b/n}^b = [p, q, r]^\top$ correspond to a linear velocity vector and an angular velocity vector of {$b$} relative to {$n$} expressed in the body-fixed frame {$b$} both. $\mathbf{r}_A^G = [(x_G - x_A), (y_G - y_A), (z_G - z_A)]^\top$ stands for the position vector of position A relative to G, the center of gravity of $x < x_A$ section of the model, expressed in {$b$}. $\top$ denotes the transpose, and this is applied to have the same expression sequence as the 6-DOF equations of motion below. The complete definition of the wave bending moments and the shear stresses with $y_G - y_A = z_G - z_A = 0$ for simplicity is as follows:
$$
\begin{equation}
\left( \begin{matrix} Q_x \\ Q_y \\ Q_z \\ M_x^A \\ M_y^A \\ M_z^A \end{matrix} \right)_{x=x_A} = - \left( \begin{matrix} F_{extx} \\ F_{exty} \\ F_{extz} \\ M_{extx}^A \\ M_{exty}^A \\ M_{extz}^A \end{matrix} \right)_{x<x_A} + \left( \begin{matrix} m[\dot{u} + qw - rv - (x_G - x_A)(q^2 + r^2)] \\ m[\dot{v} + ru - pw + (x_G - x_A)(qp + \dot{r})] \\ m[\dot{w} + pv - qu + (x_G - x_A)(rp - \dot{q})] \\ I_x \dot{p} - I_{xy} (\dot{q} - rp) - I_{zx} (\dot{r} + pq) + (I_z - I_y) qr + I_{yz} (r^2 - q^2) \\ I_y \dot{q} - I_{yz} (\dot{r} - pq) - I_{xy} (\dot{p} + qr) + (I_x - I_z) rp + I_{zx} (p^2 - r^2) - m[(x_G - x_A)(\dot{w} + pv - qu)] \\ I_z \dot{r} - I_{zx} (\dot{p} - qr) - I_{yz} (\dot{q} + rp) + (I_y - I_x) pq + I_{xy} (q^2 - p^2) + m[(x_G - x_A)(\dot{v} + ru - pw)] \end{matrix} \right)_{x<x_A} \tag{2}
\end{equation}
$$

where dot (') notation denotes the time derivative. The external force vector acting
on 0 < x < x_A of the ship model, **F**<sub>ext</sub><sup>x<x_A</sup>, and the external moment vector at x_A,
**M**<sub>ext</sub><sup>A</sup>, can be derived as follows:

$$
\begin{pmatrix} F_{extx} \\ F_{exty} \\ F_{extz} \\ M_{extx}^A \\ M_{exty}^A \\ M_{extz}^A \end{pmatrix}_{x<x_A} = \begin{pmatrix} \mathbf{F}_{hydro/h} + \mathbf{P} + \mathbf{F}_{moor} \\ \mathbf{M}_{hydro/h}^A + \mathbf{r}_A^G \times \mathbf{F}_{hydro/h} + \mathbf{r}_A^G \times \mathbf{P} + \mathbf{M}_{moor}^A \end{pmatrix}_{x<x_A} \quad (3)
$$

where **F**<sub>hydro/h</sub>, **M**<sub>hydro/h</sub>, and **P** = *mg* are the fluid force, moment and the weight
of the *x* < *x*<sub>A</sub> hull portion. **F**<sub>moor</sub> and **M**<sub>moor</sub><sup>A</sup> correspond to restoring forces and
moments by the mooring lines installed at $x < x_A$. Assuming that the x and y components horizontal to the mooring line are dominant in the restoring force by the mooring system, its restoring force for $x < x_A$ and moment at $x_A$ can be simplified as follows:

$$ \mathbf{F}_{\text{moor}}^{x<x_A} = \begin{cases} F_{x\text{moor}} \\ F_{y\text{moor}} \\ 0 \end{cases}^{x<x_A} \qquad \mathbf{M}_{\text{moor}}^{A} = \begin{cases} 0 \\ 0 \\ M_{z \text{ moor}}^{A} \end{cases} \qquad (4) $$

As mentioned, considering that the 3-DOF load sensor at each segment measures the hydrodynamic force in z and moment in x and y, equations (2) and (3) allow the reconstruction of the vertical shear force $Q_z$ and vertical bending moment $M_y$ at each intersegment. This enables cross-validation with the directly measured VSF and VBM through the 6-DOF ATI sensor at intersegment No.4 ($Q_z^4$ and $M_y^4 = VBM^4$ respectively). Detailed descriptions of the reconstruction procedure can be found in Bouscasse et al. (2022).

## 2.3. Sign conventions

Sign conventions of the physical quantities mentioned above are based on the Bureau Veritas (BV) Rule Note NR 625 (Bureau Veritas (2021)). In the case of 6-DOF motions, the first three motions correspond to translational motions namely, surge, sway, and heave along the $x_b$, $y_b$, and $z_b$ axes of the body-fixed reference frame, where the positive direction of the motions follows the positive $x_b$, $y_b$, and $z_b$ direction of the body-fixed frame. The last three called roll, pitch, and yaw refer to rotations about the respective x, y, and z axes, and their positive rotations follow the right-hand screw convention. In the case of the wave bending moments, as presented in Figure 3, the positive VBM (hogging) and HBM are when the compressive stress is induced in the bottom and the port side of a ship respectively. The wave heading
angle denoted as $\beta$ corresponds to the relative angle of a given wave direction based
on the positive $x_b$.

<figure><img src="image_4.png" /><figcaption>Figure 3: Sign conventions for VBM and HBM</figcaption></figure>

**3. Experimental setup**

**3.1. Test configuration**

The model test in the -120 degree (= $\beta$) oblique wave condition was conducted in
the hydrodynamic and ocean engineering tank of Ecole Centrale Nantes, consisting of
a flap-type wavemaker on one side and an absorbing beach on the opposite side of the
wave basin. The main dimensions are 30 m in width, 50 m in length, and 5 m in depth.
For the comparative analysis, a numerical calculation through a frequency domain
potential flow-based numerical solver, Hydrostar (Bureau Veritas (2016)) with and
without mooring system was performed first and results are presented together with
the model test results. As presented in Figure 4, four horizontally arranged soft
mooring lines in the air with stiffness of 58 N/m installed with $\theta_i$ of 45, -45, -135,
and 135 degrees (from line No.1 to 4) based on the $x_b$ of the body-fixed reference
frame are applied to maintain the heading of the model while minimizing its influence
on the global performance of the model. The spring applied in each mooring line
is longer and heavier than that used for the previous experiment in heading wave
conditions in Bouscasse et al. (2022). Thus, its contribution to an increase in overall
mass may be expected. The Hydrostar estimates the surge natural period of the
<figure><img src="image_5.png" /><figcaption>Figure 4: Experimental configuration for a 120deg oblique wave condition</figcaption></figure>

model as 10.4 s in model scale with consideration of the additional mooring stiffness
matrix calculated by reference to Al-Solihat and Nahon (2016), and it was observed
to be 10.1 s in the free decay test.

3.2. *Horizontal restoring moment by a mooring system*

The experiment is performed at zero speed with no other appendage controlling
the model, thus, the only possibility to perform the test is by using the mooring lines.
The mooring lines are not expected to affect the VBM in head sea conditions, but,
in oblique seas, the mooring system may contribute to the HBM and the motions.
In this study, the contribution of the mooring to the measured HBM and its effect
on the motions are thoroughly discussed

Based on Eq. (2), the horizontal bending moment ($M_z^A = \text{HBM}^A$) at position A
can be calculated by solving the simplified standard Euler law in the x-y plane of the body-fixed reference frame as expressed below:

$$ (M_z^A)^{x=x_A} = (-M_{extz}^A + I_z \dot{r} + m(x_G - x_A)(\dot{v} + ru))^x_{<x_A} \quad (5) $$

with

$$ M_{extz}^A = \sum_{i=1}^{j-1} (M_{hydroz/h_i} + \mathbf{r}_A^{G_i} \times F_{hydroy/h_i})^{x<x_A} + M_{z \text{ moor}}^A \quad (6) $$

where j-1 corresponds to the number of segments in $x < x_A$ range. $F_{hydroy/h_i}$ and $M_{hydroz/h_i}$ denote fluid force and moment acting on the $i^{th}$ segment, and $M_{z \text{ moor}}^A$ is the restoring moment at position A by the mooring line tension.

Note that, as the 3-DOF load sensors in each segment only measure $F_z$, $M_x$, and $M_y$, the $M_{hydroz/h_i}$ and $F_{hydroy/h_i}$ cannot be obtained. Therefore, the reconstruction of $M_z^A$ at any intersegment is limited, and $M_z^A$ can be obtained only at the intersegment No.4 through the direct measurement of the ATI sensor (see Figure 1). Considering that the model is moored with four spring lines, the $M_z^4$ measured by the ATI corresponds to the wave-induced moment containing the additional restoring moment by the mooring system as one component of $M_{extz}^A$. Quantification of the mooring system's contribution to the horizontal bending moment at $x_A$ can be estimated with mooring line tension data.

The mooring tension-induced bending moment in the z-axis at the position $x_A$, $M_{z \text{ moor}}^A$, can be obtained by the cross product of the distance vector, $\mathbf{r}$ and the tension vector, $\mathbf{T}_i$ of the $i^{th}$ mooring line as follows (see Figure 5):
<figure><img src="image_6.png" /><figcaption>Figure 5: Schematic view of mooring lines in the x-y plane and decomposition of line tension components</figcaption></figure>

$$ 
\begin{aligned}
M_{z_i} &= (-r_{y_i} T_{x_i} + r_{x_i} T_{y_i}) z_b \\
&= (r_{x_i} T_{y_i}) z_b \quad (\because r_{y_i} = 0) \\
&= (r_{x_i} \| \mathbf{T}_i \| \sin \theta_i) z_b
\end{aligned} 
\quad (7) $$

With this in mind, the bending moment due to the mooring line tension at intersegment No.4 can be estimated with the two lines located at the bow or stern respectively. With the two lines located at $x < x_4$, for example, $M_{z \text{moor}}^4$ can be calculated as:

$$ M_{z \text{moor}}^4 = \sum_{i=1}^{2} (r_A^R \| \mathbf{T}_i \| \sin \theta_i)^{x<x_4} \quad (8) $$

This makes it possible to determine the contribution of the mooring system to $M_z^4$ at intersegment No.4 with the direct measurement from the ATI. It is important to note that the model geometry is y-axis asymmetric, and the ATI position is not the center of the moment of the two mooring line sets at the bow and stern (i.e. COG of the model), thus each mooring set may induce different moments in magnitude. In addition, as the wave heading is in a -120 degree oblique condition, generating yaw motion, the phase of each signal is expected to be different.
### 3.3. Test cases

To check the effect of wave nonlinearity in detail, a test matrix consisting of several cases in $H/\lambda$ from 1 % to 10.5 % was considered following Bouscasse et al. (2022). As presented in Table 2, in each wave steepness case, a number of cases corresponding to $\lambda/L_{pp}$ varying from a minimum of 0.25 to a maximum of 2.0 were considered, which enables a comparison of the RAO curves of the responses of interest with respect to the wave steepness. That is, with the fixed $\lambda/L_{pp}$ ratio and steepness series, the corresponding wave height $H$ and nonlinear wave period $T$ were calculated based on the stream function theory and reproduced in the ocean wave tank.

The effect of the mooring system on the wave-induced responses was performed under a less steep regular wave ($H/\lambda=3.1 \%$) with $H=0.062$ m, $T=1.116$ s, corresponding to the peak period of the HBM RAO. Intuitive comparisons of the resulting responses with and without the mooring system were performed, and the results are discussed in detail in Section 4.5.

To check and correct the wave height with respect to the corresponding target values of a given wave, experimental wave calibration was first performed at the target location ($x = 18.2$ m) before the model installation.

<table><caption>Table 2: Description of regular wave cases (model scale)</caption><thead><tr><th>Objective</th><th>Case in H/λ(%)</th><th>λ/L<sub>pp</sub></th><th>T (s)</th><th>H (m)</th><th>Mooring</th></tr></thead><tbody><tr><td rowspan="7">Wave nonlinearity</td><td>1.0 %</td><td>0.313 ~ 2.0</td><td>1.03 ~ 2.67</td><td>0.017 ~ 0.115</td><td rowspan="7">with</td></tr><tr><td>2.1 %</td><td>0.25 ~ 2.0</td><td>0.84 ~ 2.37</td><td>0.023 ~ 0.188</td></tr><tr><td>3.8 %</td><td>0.25 ~ 1.5</td><td>0.83 ~ 2.04</td><td>0.042 ~ 0.252</td></tr><tr><td>5.2 %</td><td>0.25 ~ 1.5</td><td>0.83 ~ 2.03</td><td>0.058 ~ 0.346</td></tr><tr><td>7.0 %</td><td>0.25 ~ 1.125</td><td>0.82 ~ 1.74</td><td>0.077 ~ 0.346</td></tr><tr><td>8.7 %</td><td>0.25 ~ 1.125</td><td>0.81 ~ 1.72</td><td>0.096 ~ 0.433</td></tr><tr><td>10.4 %</td><td>0.25 ~ 0.875</td><td>0.80 ~ 1.49</td><td>0.115 ~ 0.404</td></tr><tr><td>Mooring effect</td><td>3.1 %</td><td>0.446</td><td>1.116</td><td>0.062</td><td>with/without</td></tr></tbody></table>
# 4. Experiment

## 4.1. Data post-processing

Each test in the wave tank starts from a calm water condition. The wavemaker motion is ramped and it introduces transient responses of the model. In the measurement, all types of signals are synchronized and measured from the start of the wavemaker. Thus, the response time signals contain successive intervals of i) calm water conditions, ii) transients, iii) periodic oscillations around the dynamic equilibrium position reached by the model, and iv) deceleration. Note that the data analysis considers only the periodic oscillations around the dynamic equilibrium position reached by the model.

The response of the model at rest in calm water corresponds to the effect of the hydrostatic pressure. Thus, to analyze only the hydrodynamic responses induced by a given wave condition, 'zeroing' is first performed by subtracting the signals at rest from the total measurement. The hydrostatic results are used not only to check the experimental setup at the initial stage but also the total force estimation. In addition, based on the quality of the wave generated and depending on whether the responses reached a stable period, a time window for data analysis was manually selected case by case.

Basically, the time signal of the selected time window is used without filtering in the Fourier Transform analysis, while the zero-crossing analysis is performed with a filtered signal to remove the influence of high-frequency components. A filtering frequency of 7 Hz, which was selected based on a hammer test result in calm water, is applied.

## 4.2. Hydrostatic VSF and VBM distribution in calm water

The reconstruction of internal loads at every intersegment was first performed to ensure that the experimental setup was properly set. Figure 6 shows the hydro-
<figure><img src="image_7.png" /><figcaption>Figure 6: Comparison of reconstructed hydrostatic VSF and VBM distribution with Hydrostar result and direct measurement (model scale)</figcaption></figure>

static distribution of the reconstructed VSF and VBM in the calm water calculated. Hydrostar results and the direct measurement with the ATI are presented together for comparison. Overall, it can be seen that the VSF distribution results show good agreement between the different methodologies, but small discrepancies are observed in the Hydrostar results at some locations and the difference is more visible in the moment distribution. Considering that the experimental results in the heading wave condition, which used the lighter spring with the same stiffness, agreed very well with the Hydrostar results (Bouscasse et al. (2022)), the use of longer and heavier springs on the mooring line to cover the large drift expected in the oblique wave condition appears to have contributed to the change in the overall weight of the model. That is, the increase in the system's displacement involves the change of buoyancy as a result of the draft change, which eventually affects the hydrostatic load distribution of the model. As mentioned above, the mooring system is considered in Hydrostar only through a stiffness matrix.

On the other hand, it is found that the reconstructed results from the different segment load cells are very comparable to the ATI measurement at intersegment No.4 (red dot). This means that the variables considered in the simplified equations
<figure><img src="image_8.png" /><figcaption>Figure 7: Time signal of hydrodynamic VSF and VBM (model scale)</figcaption></figure>

of the reconstruction process represent sufficiently well the motion and load response
characteristics of the model. The hydrodynamic time series of the reconstructed and
directly measured VSF and VBM are presented in Figure 7 as an example, showing
that the reconstructed result can even capture local oscillations with very similar
periodic responses in terms of shape and magnitude.

4.3. Response time series of the cases near the HBM peak period

The nonlinear response characteristics of 7 cases in H/λ corresponding to the
configuration close to the peak period of the HBM response (λ/Lpp = 0.4375) are
first discussed. For clarity, Figures 8 and 9 present only the results for H/λ = 1
%, 5.2 %, and 10.5 %. Note that the time series responses presented correspond to
7 Hz filtered (blue line) and unfiltered (red line) measurements. This allows us to
estimate the potential influence of the high-frequency vibration due to wave impacts
that may occur in the most severe conditions on the wave bending moments. The
time range presented corresponds to the time window selected for the data analysis.

Looking at the surge motion, a low-frequency response corresponding to the nat-
ural period measured in the free decay test (10.1 s) is observed when H/λ is more
than 5 %. In the sway, the mean drift motion steadily increases, reaching up to about
<figure><img src="image_9.png" /><figcaption>Figure 8: 6-DOF motion time signals of the cases with $H/\lambda = 1\%$, $5.2\%$, and $10.5\%$ having the same $\lambda/L_{pp}$ of 0.4375 (model scale)</figcaption></figure>

0.9 m when $H/\lambda = 10.5\%$. A typical periodic response is observed in the heave and rotational motions, while the roll motion exhibits a relatively unstable response with local oscillations. The average value of the yaw motion gradually moves away from 0 as the $H/\lambda$ becomes steeper, indicating that it is related to the increase in the asymmetry of the HBM response in Figure 9.

In the case of VBM⁴ and HBM⁴ measured by the ATI sensor at intersegment No.4, the most sinusoidal response in which the first harmonic component seems to be dominant is observed in both signals when the $H/\lambda = 1\%$. As the steepness increases, both responses are found to have an asymmetric shape based on $y = 0$, and the high-frequency components are clearly observed in time signals for the $H/\lambda = 10.5\%$.
<figure><img src="image_10.png" /><figcaption>Figure 9: Moment time signals of the cases with $H/λ = 1 \%, 5.2 \%, \text{and } 10.5 \%$ having the same $λ/L_{pp}$ of 0.4375 (model scale)</figcaption></figure>

4.3.1. *Slamming effect on rigid body assumption*

Two snapshots of each test case of when the yaw motion is minimum and maxi-
mum are presented for clear understanding. In short, in the $H/λ = 10.5 \%$ case, the
slamming response occurs very strongly at the bow part of the model, and the effect
is measured as a high-frequency impact in the VBM$^4$ and HBM$^4$ (see Figure 9). The
VBM even reaches a magnitude similar to the head wave case (see Bouscasse et al. (2022)). The magnitude of the filtered response applied in the RAO calculation has a significant difference of about up to 35 % in VBM and 45 % in HBM, respectively, compared with their unfiltered response in Figure 9.

Figure 10 shows all test cases applied to this present experimental study and the case where the slamming was observed. It can be seen that the slamming mainly occurs around $\lambda/L_{pp} = 0.4375$ corresponding to the HBM peak period when the wave steepness is considerably steeper than $H/\lambda = 8 \%$. Taking the FFT result of the HBM time series for the most severe wave condition with $H/\lambda = 10.5 \%$ and $\lambda/L_{pp} = 0.4375$ as an example (see Figure 11), it can be seen that the influence of the high-frequency components even above the filtering frequency of 7 Hz account for a significant proportion of the overall response. This leads to a significant difference in magnitude between the filtered and unfiltered signals. The same trend was observed in the FFT of the other four test cases where the slamming was observed. Although it is limited only to very steep wave cases at near the HBM peak period, this leads to questioning the validity of the rigid model assumption in the oblique case and explains the need for further investigation.

### 4.3.2. Mooring system's effect on ship behavior

The change in the yaw motion can be related to the moment $M_{z \text{moor}}^4$ induced by the two mooring line sets at the bow and stern. Specifically, looking at the moment time series induced by each mooring line set at the bow and stern in Figure 9, it can be seen that the steeper the wave, the clearer the results of opposite phases are shown. This demonstrates that the mooring lines play a role in balancing out the dynamics (heading angle of the model). However, the moment induced by the lines of the bow is, on average, greater than the moment due to the stern lines. At the
<figure><img src="image_11.png" /><figcaption>Figure 10: All wave conditions applied in the experiment, and test cases where slamming events were observed (marked in red points)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 11: FFT result of a wave case with $H/λ$= 10.5 % and $λ/L_{pp}$=0.4375</figcaption></figure>

same time, it shows a negative sign-biased response, showing that it is related to the asymmetric $HBM^4$ and the gradual shift of the average model's heading to the negative yaw direction.

The contribution of the mooring lines to the $HBM^4$ can be computed but as the mooring lines influence the motion, it is not easy to directly subtract the mooring
system from the experiment to estimate what the HBM would be without mooring lines. On the other hand, the order of magnitude of the mooring line's contribution can still be a good indicator and it is observed that $M_{z\text{moor}}^4$ does not exceed 10 % in the most severe cases in Figure 9. Further analysis and detailed discussion of this issue are covered in Section 4.5.

### 4.4. Response amplitude operator (RAO) estimation

The calculation of motion and bending moment RAOs was performed following Bouscasse et al. (2022). All the RAO results are expressed in dimensionless quantities, and the variables applied to the denominator are theoretical values including half the wave height, $H/2$. In the case of the numerator, the physical quantity of a numerator takes the 1st and 0th harmonic components from the Fourier decomposition in Eq. (9) or the average of maximum/minimum values measured in the selected time window. For the former case,

$$X(t) = \mathbb{R} \left[ \sum_{n=-\infty}^{\infty} c_n e^{i\omega_n t} \right] \quad (9)$$

where $c_n$ is the complex amplitude of the n-th frequency component. The n-th harmonic RAO of the arbitrary response $X(t)$ is thus expressed as:

$$|\text{RAO}^{(n)}| = \frac{\epsilon_n |c_n|}{\text{Denominator}} \quad (10)$$

$$\epsilon_n = \begin{cases} 1, & n = 0 \\ 2, & n > 0 \end{cases} \quad (11)$$

For the latter case, by applying direct measurement data that include all harmonic components, the average of positive or negative peak events in a selected time window ($T_w$) can be applied as a numerator of the RAO.
$$ \overline{X^{(+)}} = \frac{1}{N} \sum_{i=1}^{N} (X_{i}^{(+)}) \quad \text{for N positive peaks in } T_{w} \quad (12) $$

The corresponding RAO can be expressed as follows:

$$ |RAO^{(+)}| = \frac{\overline{X^{(+)}}}{\text{Denominator}} \quad (13) $$

The same goes for $|RAO^{(-)}|$ calculated with the negative peak events measured in the selected time window, $T_w$. Note that, in this study, the measurement data applied to the RAO calculation correspond to low-pass filtered data with a cutoff frequency of 7 Hz. Regarding the denominator of the RAO for translational and rotational motion, and wave bending moments, the following physical quantities are applied for each:

$$ \begin{aligned} \text{Translational motion} &= H/2 \\ \text{Rotational motion} &= kH/2 \\ \text{Bending moment} &= \rho g L_{pp}^2 B H/2 \end{aligned} \quad (14) $$

where $k$ is the wave number calculated through the linear dispersion relation corresponding to the depth of the wave tank (5 m). For reference, the latter applied filtered data, and as mentioned, measurements in calm water conditions were intentionally excluded so as to consider only the hydrodynamic response.

### 4.4.1. 6-DOF motions

The estimated 1st harmonic 6-DOF motion RAOs at the COG of the model for each $H/\lambda$ series are presented together with the HydroStar results with and without the mooring system for comparison in Figure 12.

Overall, it can be said that the experimental results are in very good agreement with the linear RAO results considering the mooring stiffness matrix in most motions, although some fluctuations appear to occur in sway for short-wave cases inducing
<figure><img src="image_13.png" /><figcaption>Figure 12: 1st harmonic 6-DOF motion RAOs at the COG of the model for each H/λ series</figcaption></figure>

large drift motion. The experimental roll RAO follows very well the linear result in relatively less steep cases and shows a large changing rate as the wave steepness of a series increases. A similar trend is found in heave, pitch, and yaw results, but with a relatively small magnitude in the changing rate of each curve. In the yaw motion, the Hydrostar results with and without the mooring system show relatively noticeable discrepancies for the long-wave cases with a wave period T = 1.5 s or more. Given that the horizontal mooring setup applied in this study generates additional resilience in surge, sway, and yaw motions, inconsistency in these motions can be naturally expected. Here, it can be said that the mooring system effect is particularly evident in the yaw motion for large periods.
4.4.2. VBM and HBM

1st and 0th harmonic VBM RAOs at intersegment No.4 estimated with the ATI data for each H/λ case are presented in Figure 13 as a dimensionless quantity. The wave's nonlinear effect can be clearly observed near the peak period range of the 1st harmonic result. The H/λ = 1 % case, which can be said to contain very small nonlinearity in a given wave, shows a result comparable with that of Hydrostar in most wave period ranges. Some discrepancy is observed, however, near the peak period, implying that the VBM induced by a H/λ = 1 % wave contains nonlinearity as observed in the previous experimental study by Bouscasse et al. (2022).

The magnitude of the peak of the curve was expected to increase in proportion to the wave steepness of the series, but the maximum value of the curve is observed in the $H/λ = 5$ % case, then the curve gradually decreases as it goes higher than $H/λ = 5$ %.

<figure><img src="image_14.png" /><figcaption>Figure 13: 1st and 0th harmonic VBM RAO for each H/λ case</figcaption></figure>

Meanwhile, a clear increasing trend is confirmed in the 0th harmonic VBM RAO. One thing to note here is that for very steep wave cases ($H/λ = 10.5$ %) near the peak period, the 0th harmonic contribution is almost 35 % of the 1st harmonic one.
The 0ᵗʰ and 1st harmonic VBM components are found to account for the major part of the total VBM RAO presented in Figure 14 where the RAO was calculated with filtered data with a low pass cutoff frequency of 7 Hz. Thus their contribution is likely to be lowered if the high-frequency responses such as the previously observed slamming are considered in the analysis.

In Figure 14, overall increasing and decreasing trends in sagging and hogging, corresponding to the average of maxima and minima respectively in the selected time window, are observed. Since the sagging RAO of each *H*/*λ* series is always higher than its hogging RAO shown in Figure 14, it can be said that the 0ᵗʰ harmonic component (mean VBM) is biased to the negative sign based on VBM = 0 Nm in the time signal. One thing to note is that the sagging result has a clear tendency to increase in proportion to the *H*/*λ*, whereas the hogging RAO near the peak period increases up to the case *H*/*λ* = 5 % with a maximum value, and the RAO starts to decrease when the case is *H*/*λ* > 5 %. Also, in both RAO results, the peak of each curve tends to shift gradually as the *H*/*λ* changes.

<figure><img src="image_15.png" /><figcaption>Figure 14: Hogging (average of maxima measured) and sagging (average of minima measured) RAO estimated with filtered (7 Hz) ATI measurements for each *H*/*λ* case</figcaption></figure>

In the case of the HBM results presented in Figure 15, the 1st harmonic HBM
<figure><img src="image_16.png" /><figcaption>Figure 15: 1st and 0th harmonic HBM RAO for each $H/λ$ case</figcaption></figure>

RAO for $H/λ = 1\%$ has the same tendency as the one observed in the VBM results, showing a magnitude larger than the linear prediction near the peak period. It seems that the influence of the disturbed waves such as waves that propagate from the bow to the stern by the diagonally arranged model and wave configuration sensitively affected the responses in mild conditions. The overall shape of the curve in the other period ranges is in good agreement with the linear result, and the RAO curve continuously decreases with the increase in $H/λ$. The 0th harmonic RAO shows the opposite trend to the 1st harmonic result, namely a distinct increase with respect to the $H/λ$ (see Figure 15) as that observed in the 0th harmonic VBM result.

On the other hand, the positive and negative HBM, which correspond to the average of maxima and minima containing all the frequency components, show different patterns of curve fluctuations near the peak period in Figure 16, indicating that not only the 1st and the 0th harmonics but also the higher harmonic components, i.e. above the second-order or high-frequency responses (e.g. slamming impact) need to be identified. Figure 17 shows the hogging (positive) and sagging (negative) VBM (HBM) RAO of $λ/L_{pp} = 0.4375$ corresponding to the near peak period of 1.116 s for
<figure><img src="image_17.png" /><figcaption>Figure 16: Positive HBM (average of maxima measured) and negative HBM (average of minima measured) RAO estimated with filtered (7 Hz) ATI measurements for each $H/λ$ series</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 17: Hogging and sagging RAO (top) and positive HBM and negative HBM RAO (bottom) for the $λ/L_{pp} = 0.4375$ case</figcaption></figure>

all $H/λ$ series. An interesting point is that except for the sagging RAO, the remaining three responses have the maximum or minimum RAO value when $H/λ = 5 \%$, and the RAO curve increases or decreases based on this.

Given that the model is rigid, the changes in the RAOs near the peak period with respect to the $H/λ$ can be related to the wave's and/or response's nonlinear effect, which can be defined as a ratio of measurement to its linear value, $HBM_{NL}/HBM_L - 1$. The same goes for the VBM responses. To check this, linear values were first
calculated by considering the linear RAO from Hydrostar. The linear indicator of both responses for a given wave frequency (f) is defined as follows:

$$VBM_L(f) = |RAO^{VBM}(f)| \times A(f) \qquad (15)$$

$$HBM_L(f) = |RAO^{HBM}(f)| \times A(f)$$

where A is the amplitude of a given wave and RAO is a linear RAO calculated with Hydrostar. The nonlinear effects of the wave on the magnitude of VBM⁴ and HBM⁴ including all harmonic components are then evaluated, and the results are presented in Figure 18.

<figure><img src="image_19.png" /><figcaption>Figure 18: Nonlinear factor for the 7 cases in H/λ with λ/Lpp = 0.4375</figcaption></figure>

It can be seen that the experimental results already contain nonlinearity even for H/λ = 1 %. In the VBM result, as seen in the RAO result in Figure 17, the ratio of the nonlinear sagging response to the linear value shows a continuous increase, whereas the hogging response has a maximum value in the H/λ = 5 %
case, and thereafter a trend of decreasing nonlinear factor results is observed until
the $H/\lambda = 10 \%$. A sagging response greater than 80 % compared to the linear
value is observed in the most severe condition, and hogging is 20 % less than the
linear value. Nonlinear HBM also shows a large difference compared to the linear
value in the negative response, showing an increase or decrease in nonlinear factor
based on $H/\lambda = 5 \%$. The negative HBM response tends to decrease by up to 40 %
compared to the linear value. The strong nonlinear responses seem to be prominent
in the case of $H/\lambda = 5 \%$ or more, and it can be expected that the proportion of
higher harmonic components would increase significantly.

As shown in Figure 19, to check the contribution of harmonic components in 7
wave steepness cases with $\lambda/L_{pp} = 0.4375$, component decomposition was performed
for both VBM and HBM that are directly measured through the ATI. Interestingly,
as for the relationship between the 1st harmonic and other harmonics of HBM$^4$, it
is confirmed that their relative contribution to the response tends to change overall
in the opposite way to each other with the increase in wave steepness. In the VBM
results, it is observed that the 1st harmonic VBM increases up to $H/\lambda = 5 \%$ then
decreases, while the remaining harmonic components are found to gradually increase
overall with the increase in wave steepness. In addition, for the case where $H/\lambda$ is
more than 7 %, the higher harmonic components more than third-order are found
to account for a significant part of the overall response. This is a result of the wave-
structure interaction, showing that the more severe the given wave condition, the
stronger the nonlinearity of the interaction.

Given the results about the harmonic components and the nonlinear factor for
the 7 wave cases near the peak period, it can be said that when a given wave's
steepness is greater than $H/\lambda = 5 \%$, a more intense wave-structure interaction
occurs, resulting in a response with strong nonlinearity. The nonlinearity of the
<figure><img src="image_20.png" /><figcaption>Figure 19: Harmonic components of VBM⁴ and HBM⁴ of all H/λ series with λ/Lpp = 0.4375</figcaption></figure>

response can be even stronger if the high-frequency slamming impact is considered
in the analysis as observed in the very steep wave case. In other words, the response
of the resulting wave bending moments in the peak period varies greatly depending
on the wave steepness.

4.5. Contribution of mooring system to HBM

As mentioned earlier, the HBM, $M_z^4 = \text{HBM}^4$, measured by the ATI includes
the additional effect induced by the mooring line, and the corresponding internal
moment $M_{z \text{moor}}^4$ at the ATI position can be estimated using the tension data of
the two lines of the bow or stern. To qualitatively evaluate the contribution of the
mooring system to the $\text{HBM}^4$, the z-axis moment by mooring lines was calculated.
Note that only the hydrodynamic tension by a given wave was considered; the mean
tension of the selected time window was excluded. The corresponding results are
presented in Figure 20.

Overall, the $M_{z \text{moor}}^4$ RAOs of the stern and bow lines are similar to each other,
but in the case where $H/\lambda$ is relatively steep, it is observed that the mooring lines
at the bow, which encounters the undisturbed wave first, appear to estimate a more
<figure><img src="image_21.png" /><figcaption>Figure 20: z-axis bending moment RAO at intersegment No.4, estimated with the mooring tension data of the stern (left) and bow (right)</figcaption></figure>

conservative RAO in particular for the cases with $T < 1.7$ s. Considering that each mooring line set at the bow and the stern is supposed to act as a counterbalance to each other and thus maintain the heading angle of the model, the result showing a larger magnitude of the moment by the mooring lines of the bow possibly contributes to making the HBM$^4$ signal asymmetric, and consequently, the average heading angle of the model is likely to be changed in those cases. The distribution of the mean yaw and the difference between the $M_{z \text{moor}}^4$ RAO of the bow and stern for all cases is presented in Figure 21.

In Figure 21, it is observed that the mean yaw of each case corresponding to ‘the measured mean yaw - the target yaw (-120 degree)’ stays around 0 degree in relatively mild conditions (up to $H/\lambda = 3.8 \%$). $M_{z \text{moor}}^4$ difference shows the same trends as well. When the $H/\lambda$ increases more than 5.2 %, both the mean yaw and $M_{z \text{moor}}^4$ difference starts to change. It is found to be -2 degree when $M_{z \text{moor}}^4$ difference is around 15 Nm in $H/\lambda = 10.5 \%$. This implies that the difference between the two $M_{z \text{moor}}^4$ of the two mooring line sets at stern and bow at the ATI position may relate to the changes in average yaw angle. Overall, the more the mean yaw motion of a
given wave case differs from the initial -120 degree, the greater the $M_{z \text{moor}}^4$ difference,
showing that there is a correlation.

<figure><img src="image_22.png" /><figcaption>Figure 21: Measured mean yaw - target yaw (-120 degree) for all RW cases (top) and difference of $M_{z \text{moor}}^4$ of the bow and stern mooring lines (bottom)</figcaption></figure>

### 4.5.1. Model test with and without mooring

The presence or absence of the mooring system contributes to the HBM, but it is
maybe not the only factor having an influence. As already mentioned, intersegment
No.4 is close to the COG of the model but does not coincide with it, and the model
geometry is y-axis asymmetric. The interaction between the mooring system and the
y-axis asymmetric model, the resulting different wave quality at bow and stern, and
the position of interest to be evaluated may play a part and thus has to be taken
into consideration.

A more intuitive way to compare the effects of a mooring system is to compare
test results with and without mooring lines. However, in reality, an accurate com-
parison between the motion of the unconstrained model, which has drift caused by
waves and changes in the bow angle, and the case of the moored model, is impossi-
ble. Nevertheless, if the motion change between the two configurations is within an
acceptable range, it will be likely that the comparative analysis of the HBM response
time series includes meaningful results.

To check that, one special test was performed. The wave period selected is 1.116
s, which is the same as the peak period of the HBM RAO. As for the wave slope, 3.1
% is selected considering that the drift effect is not excessive and the expected HBM
has a certain analyzable size. In the case where the steepness is more than 5 % with
the same period, large drift occurs from the beginning, and it affects the rest of the
motions, making it impossible to compare the results.

<figure><img src="image_23.png" /><figcaption>Figure 22: Test case with *H* = 0.062 m *T* = 1.116 s *H*/*λ* = 3.1 %. 4-DOF motion time signals measured with/without mooring system</figcaption></figure>

The comparative analysis results of the model motions with and without the
mooring system are presented in Figure 22. For reference, 6 DOF motion measure-
ments were performed via LED markers attached to the model, which sometimes
showed unphysical peak responses (see Figure 22) when the model was getting out-
of-sight of the cameras and/or was obscured by surrounding objects. However, note
that the motion signals were only used as an indicator to select the time range for
the wave bending moment analysis and only the ATI sensor measurement results
were taken into account.

To enable comparison of physically reasonable responses, a criterion was set in
yaw motion. The allowable range was set to 5 degrees based on the designated
heading angle (-120 degrees), and responses above the criterion were excluded from
the analysis.

As can be seen in Figure 22, from around 20 s, when the wave generated by the
wavemaker seems to have reached the model, to around 40 s, in the overall trend of
the response, the heave and pitch show similar results to the response with mooring.
On the other hand, in the case of roll and yaw, which are more affected by model
drift, it can be observed that they deviate from the response with the mooring system
earlier in time.

The top two graphs in Figure 23 show the result of the wave bending moments
(VBM and HBM) measured by the ATI sensor, and the last graph shows the horizon-
tal moment at the ATI position by the mooring lines' tension at the same moment.

In the case of the wave bending moments measured, the results without mooring
lines are observed to be generally low. Considering that the model is continuously
drifting without restoration by the mooring system, the zero-crossing period is con-
tinuously longer over time, and its size is gradually reduced compared to the result
of the case with mooring. However, the signal shows a response of a similar period
and size in the beginning as observed in the motion results. It can be seen that the
long-period response of the moment time series due to the mooring may affect the
HBM response (see the last graph in Figure 23).

For the mooring effect analysis, given that the heading of the ship changed no-
<figure><img src="image_24.png" /><figcaption>Figure 23: Time signals measured with/without mooring system: VBM$^4$, HBM$^4$, and $M_{z \text{moor}}^4$ at stern and bow</figcaption></figure>

ticeably as expected after a certain period after facing the wave, showing completely different roll and yaw responses from the results of the moored model (see Figure 22), it was decided to consider the wave bending moment in the range of 35 s to 40.5 s.

The time interval selected for analysis is shown in Figure 24, and it can be roughly estimated that the difference between the two cases in the HBM results is comparable to the magnitude of the moment caused by the mooring line. The phase difference between the two signals corresponds to about 0.5 s, and it seems that the difference between the two HBM (with and without the mooring lines) is of a similar order to the $M_{z \text{moor}}^4$ of each mooring line set.

To demonstrate the influence of the presence or absence of mooring lines, Figure 25 is presented. The black dots of each graph correspond to the difference between the peak values of the blue curve and the gray curve and the peak values of the
<figure><img src="image_25.png" /><figcaption>Figure 24: The moment signals of the selected time interval for the comparative analysis: VBM⁴ (top), HBM⁴ (middle), and $M_{z \text{moor}}^4$ at stern and bow (bottom)</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure 25: Correlation between HBM⁴ difference by the presence and absence of mooring and $M_{z \text{moor}}^4$ of two mooring sets.</figcaption></figure>

moment generated by each mooring line set (red and green). It can be observed that the HBM (+)peak difference is directly related to the sign of the difference between
the moment by the bow and stern mooring line sets. When the two moments are equal, the HBM (+)peak difference is also close to 0, and the change in sign is consistent with both results. For the HBM (-)peak results, the mean is probably not zero and the roll motion starts to deviate quite a lot after $t = 37.5$ s. Thus it is difficult to interpret the result in detail. However, it can be confirmed that the overall changing trend of the HBM (-)peak difference is similar compared to the moments by the bow and stern mooring line sets.

Given that the experimental conditions are not strictly identical, the possibility that other factors may affect the results cannot be excluded. The reason why the range of the maximum and minimum values of the additional moment by the mooring lines is relatively small is also likely to be influenced by several factors including the change in bow angle due to the drift.

## 5. Conclusions

Through regular wave experiments in a -120 degree oblique sea, nonlinear trends on motions and wave bending moments were observed with increasing wave steepness (7 wave steepnesses ranging from $H/λ = 1$ % and 11 %). These were observations whose results cannot be accurately predicted by linear theory.

The effect of the wave steepness caused an increase and (or) decrease in the estimated RAO curves of the 6-DOF motions, HBM and VBM, and as expected most responses showed the largest difference from the linear estimation when $H/λ = 10.5$ %. In particular, in the VBM and HBM responses of the cases corresponding to near the peak period of the HBM RAO, it was confirmed that strong nonlinear responses from the intense wave-structure interaction and in particular slamming were observed when $H/λ$ was 7 % or higher containing a large proportion of higher harmonic components. This emphasized the importance of considering higher harmonic
responses for steep waves.

In low steepness wave conditions, the positive HBM was larger than the negative one in absolute value. This trend was changed by the slamming event in the larger steepness cases and the negative HBM increased. For these steep wave conditions over $H/\lambda = 7\%$, the VBM reached similar values as the ones observed in head waves. The assumption of a rigid model was validated in head waves in that the filtered data were shown to be very close to what the rigid solution would be.

However, this should be carefully verified for the oblique wave conditions where slamming is stronger and structural vibrations are more visible. At the peak period of the HBM RAO, very intensive slamming events occurred when $H/\lambda$ is more than 8%, which additionally affected the wave bending moments of the rigid body model. Results showed that the unfiltered signals of VBM and HBM containing slamming impacts could increase by 35% and 45% of the respective filtered signals. This led to the conclusion that the assumption of a rigid model in the oblique wave conditions should be further investigated perhaps with CFD computations and consequences attributed to hydrodynamic nonlinearities should be further discussed.

A detailed discussion was also made on the influence of the mooring system on the asymmetric HBM and the change in the average yaw motion. The mooring effect was qualitatively identified through experiments with and without a mooring system under the same wave condition. The restoring moment of the soft mooring system was found to be correlated with the asymmetric HBM and the mean yaw motion, and it is expected to be used as useful data for either experimental studies of similar settings or numerical simulations.

The soft mooring system at the bow and stern provided different magnitudes of restoring moments from each other, which became more pronounced in steep wave conditions. In particular, under the wave condition of $H/\lambda = 10.5\%$ with $\lambda/L_{pp} =$
0.4375 corresponding to the vicinity of the HBM peak period, it was observed that the magnitude of $M_{z\text{moor}}^4$ by the bow lines had a difference of -15 Nm compared to $M_{z\text{moor}}^4$ by the stern lines. The $M_{z\text{moor}}^4$ difference by the two mooring line sets at the bow and stern was approximately less than 10 % of the HBM response measured by the ATI sensor.

The time series response of $M_{z\text{moor}}^4$ by the bow line tension was in phase with the negative-biased HBM signal measured by the ATI sensor, and the corresponding mean yaw was found to decrease as the difference in $M_{z\text{moor}}^4$ increased. Although it cannot be concluded that the mooring system is the ‘cause’ of the asymmetric HBM and the yaw change, it has been shown that there is a correlation. The y-axis asymmetric model geometry may be one of the main influences as well and can be partially responsible for the asymmetric HBM response and changes in mean yaw motion.

# 6. Acknowledgment

The work presented in this paper was supported by Centrale Nantes - Bureau Veritas research chair.

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