# Experimental study of hydrodynamic loads induced by the motion of a vertical circular cylinder with heave plates in proximity of the free surface

Seung-Yoon Han, Benjamin Bouscasse, Vincent Leroy, Sylvain Delacroix, Jean-Christophe Gilloteaux, Félicien Bonnefoy, David Le Touzé

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## To cite this version:

Seung-Yoon Han, Benjamin Bouscasse, Vincent Leroy, Sylvain Delacroix, Jean-Christophe Gilloteaux, et al.. Experimental study of hydrodynamic loads induced by the motion of a vertical circular cylinder with heave plates in proximity of the free surface. Ocean Engineering, 2024, 310, pp.118712. 10.1016/j.oceaneng.2024.118712. hal-04782797

HAL Id: hal-04782797

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Submitted on 26 May 2025

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Research paper

# Experimental study of hydrodynamic loads induced by the motion of a vertical circular cylinder with heave plates in proximity of the free surface

Seung-Yoon Han<sup>a,*</sup>, Benjamin Bouscasse<sup>a</sup>, Vincent Leroy<sup>a</sup>, Sylvain Delacroix<sup>a</sup>, Jean-Christophe Gilloteaux<sup>b</sup>, Félicien Bonnefoy<sup>a</sup>, David Le Touzé<sup>a</sup>

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## ARTICLE INFO

**Keywords:**

Hydrodynamic loads

Added mass

Damping

Forced oscillation tests

Heave plates

Floating wind turbines

## ABSTRACT

This study presents an experimental investigation of hydrodynamic loads for surge, heave, and pitch oscillations of a vertical circular cylinder with heave plates. The study considers flow parameters (motion amplitude and frequency), the plate geometry (diameter and porosity), and the proximity of the free surface. A captive model technique is employed to obtain the hydrodynamic loads from the load cell. The load time series are analyzed to identify its harmonic components using Fourier analysis. The results show that the first harmonic is the dominant contributor to the loads in general, while the higher harmonics become more noticeable as motion amplitude increases and the submergence decreases. The added mass and damping coefficients associated with the first harmonic indicate that both the amplitude and frequency strongly influence the heave hydrodynamic coefficients as the plate approaches the free surface. The pitch hydrodynamic coefficients align with the general trend of heave motions. In the surge motion, the added mass depends less on amplitude and more on frequency, while damping is noticeably affected by the amplitude. This highlights the importance of motion frequency and higher harmonic contents, especially in shallow draft configurations where the flow induced by the heave plate interacts with the free surface.

## 1. Introduction

Reducing the motion responses of floating platforms in harsh environments is a major concern in ocean engineering as such responses may induce extreme or fatigue loads on the structure, mooring lines, and dynamic cables. Damping plates, acting as appendages to floating structures, can be employed to mitigate excessive motion responses. Their shape depends on the characteristics of the floater: bilge keels on ships, heave plates on floating offshore platforms, and skirts on buoys.

For ships, the roll motion is a key element in the design and operation of ships in waves for the safety of crew and cargo. The bilge-keel is designed to increase viscous damping and reduce roll motion. The viscosity carries non-linearity and makes the roll damping estimation difficult. The damping can be predicted by experimental or numerical analysis and also by the empirical formula called the ITH method (Ikeda, Tanaka, and Himeno). Himeno (1981) summarized the damping predictions for conventional ships based on damping components, including hull friction, vortices, waves, lift, and bilge keel. The method can predict dampings as a function of the amplitude and frequency of the roll motion and of the principal dimensions of the vessel with reasonable accuracy. ITTC (International Towing Tank

Community, ITTC (2021)) selected this method as a recommendation. In engineering design tools, the method is applied to estimate the ship roll damping for evaluating ship performance, such as seakeeping and maneuvering.

With offshore platforms, the heave response induced by non-linear wave loads is one of the quantities of interest. This is especially important with Tension Leg Platforms (TLP) and classic Spar platforms, which have very small damping in heave. Haslum and Faltinsen (1999) presented the vulnerability of the classic Spar platform to heave resonance in waves and discussed alternative Spar shapes to increase damping and added mass. Thiagarajan and Troesch (1998) and Thiagarajan et al. (2002) noted the effects of adding a plate on the heave motion of the TLP and Spar: it not only increased the damping of the system but also shifted the resonant period from the dominant wave periods due to the added mass.

Recently, heave plates have found widespread adoption in the design of the foundations of floating wind turbines (FWTs). For semi-submersible floaters (Roddier et al., 2010; Lopez-Pavon and Souto-Iglesias, 2015; Pegalajar-Jurado et al., 2018; Robertson et al., 2020),

<footnote><sup>a</sup> Nantes Université, École Centrale Nantes, CNRS, LHEEA, UMR 6598, F-44000 Nantes, France</footnote>

<footnote><sup>b</sup> INNOSEA, 11 rue Arthur 3, 44200 Nantes, France</footnote>

<footnote>* Corresponding author.
E-mail address: seung-yoon.han@ec-nantes.fr (S.-Y. Han).</footnote>
the heave plates have a larger diameter than the column, while barge-liked floaters (Choisnet et al., 2016) are equipped with small plates called skirts to reduce the heave, roll and pitch motion responses.

The effects of heave plates on hydrodynamic loads have been explored with two categories of governing parameters: flow parameters and geometry parameters. The flow is mainly oscillatory around the plate, so the Keulegan-Carpenter number ($KC$) and the frequency parameter ($\beta$) are typically selected as flow parameters to investigate their influence on the hydrodynamic loads, especially with the heave motion. They are defined as:

$$KC = \frac{UT}{L}, \quad \beta = \frac{(L)^2 f}{\nu} \qquad (1)$$

where $U$ is the amplitude of flow velocity, $T$ and $f$ are the period and frequency of oscillation. $L$ is the characteristic length of the object and $\nu$ is the kinematic viscosity of the fluid.

Graham (1980) studied the force induced by flow separation and vortex shedding on a sharp-edged cylinder in oscillatory flow based on the discrete vortex method. It was shown that the forces under the small oscillations of the fluid are proportional to $KC$ defined by body diameter. Details of the flow behavior of the oscillating plate were presented by Tao and Thiagarajan (2003) with the motion amplitude Keulegan-Carpenter number, $KC_d$ ($=2\pi\xi/D_d$) where $\xi$ is the motion amplitude of interest, and $D_d$ is the diameter of the plate. In the low $KC_d$ regime ($KC_d < 1$), three different vortex shedding regimes were observed with their numerical model, visually identified as independent, interactive and uni-directional vortex shedding, depending on the $KC_d$ number.

Low $KC_d$ numbers were considered in these previous studies of oscillating flat plates. This is reasonable for TLPs and Spar platforms, which have a moderate heave response in ocean waves owing to their mooring system or large drafts. However, when considering a floater with a large water plane area, such as a barge or a semi-submersible platform in extreme sea states, the $KC_d$ number should be increased. For $KC_d$ numbers in the range $0.15 \le KC_d \le 3.15$, an experimental study by Tian et al. (2017) provided hydrodynamic characteristics for various geometries. In addition, Thiagarajan and Moreno (2020) introduced a relative $KC$ number, $KC_{rel} = \frac{2\pi A_{rel}}{D_d}$, to describe the wave effects on the hydrodynamic coefficients, where $A_{rel}$ is a relative amplitude of motion with respect to a wave elevation.

The second flow parameter, the frequency parameter $\beta$, is shown to have a smaller influence on the hydrodynamic coefficients than the $KC_d$ number when the plate is deeply submerged (An and Faltinsen, 2013; Lopez-Pavon and Souto-Iglesias, 2015). In shallow draft conditions, Molin et al. (2008) carried out tank testing with an isolated plate with 5 cm immersion, corresponding to 0.08 of the submerged depth to the plate diameter ratio. The experimental measurements showed that the hydrodynamic coefficients depend on the frequency in that case. Especially in short motion periods, numerical models based on potential flow estimate the added mass and damping well. They observed that the added mass is negative over the lower periods from both the numerics and the measurements. An and Faltinsen (2013) also determined in their experimental and numerical study that the drag coefficients of solid and perforated plates are largely affected by the free surface.

Several geometric configurations have been investigated for an isolated plate, and with the addition of a column. Tian et al. (2017) experimentally investigated the influence of the plate thickness ratio ($t_d/D_d$ with the plate thickness, $t_d$) on the added mass and damping. They found a critical ratio defining thin and thick plates based on the relationship between the thickness ratio and the hydrodynamic coefficients. The influence of the edge corner shape was also studied in Li et al. (2013) and Tian et al. (2017). It showed that plates with sharp edges could provide more added mass than plates with rounded and triangular corners, whereas damping is almost independent of the shape, at least for $KC_d < 1.0$. Lopez-Pavon and Souto-Iglesias (2015) studied the influence of the reinforced plate on the hydrodynamic coefficient by adding a vertical flap at the edge. It was found that

the hydrodynamic damping was significantly decreased due to the reduction of the vortex on the reinforced plates.

Perforated plates were proposed to enhance hydrodynamic performances by Downie et al. (2000). The perforation ratio of the plate, $\tau$, was investigated by forced heave oscillation model tests with isolated plates in Tao and Dray (2008). From the results, the plates with porosity were found to produce less added mass and more damping. Sandvik et al. (2006), Molin et al. (2008) and An and Faltinsen (2013) proposed a numerical model based on potential flow theory, considering a pressure drop condition and viscous drag forces to model porosity on the plate surface and flow separation at the plate edge, respectively. They showed that the hydrodynamic damping is primarily due to flow separation through the perforated openings when the $KC_d$ number is small, whereas the flow separating from the outer edge of the plate becomes the dominant factor of the damping when $KC_d$ is large. Tian et al. (2017) reported that damping of the perforated plate is larger than that of the solid plate at low $KC_d$ but found the opposite at large $KC_d$. On the other hand, the perforated plates produce lower added mass than the solid ones at all $KC_d$ numbers. Furthermore, it was found that the perforated hole size is not significant for the hydrodynamic coefficients.

The influences of free surface and seabed effects on the added mass and damping coefficients were studied experimentally (Wadhwa and Thiagarajan, 2009; Wadhwa et al., 2010). In the experiments, an isolated plate in forced heave oscillation was considered with different geometric ratios ($h/D_d$ and $d/D_d$, where $h$ is the elevation from the seabed and $d$ is the draft of the plate). The added mass and damping coefficients were found to increase with the proximity to the boundary (seabed and free surface). Furthermore, a critical $KC_d$ value was found where the linear trend of the coefficients with $KC_d$ is broken even at a low $KC_d$ number. Numerical simulations by Garrido-Mendoza et al. (2014, 2015) followed, and the numerical results presented a similar behavior to the experiments. It was found that the effect of the free surface on the damping is more dominant than that of the seabed at the same distance ratio ($h/D_d = d/D_d$). In the case of shallow drafts, the drag coefficients of the solid and perforated plate are greatly affected by the free surface (An and Faltinsen, 2013).

In actual engineering applications, the heave plate is necessarily connected to the platform and often to a column, and this influences the flow. The effect of the diameter ratio $D_d/D_c$ (heave plate diameter over column diameter, $D_c$) on the added mass and damping was presented by Thiagarajan et al. (2002) for $D_d/D_c = 1.0, 1.1, 1.2$ and $1.3$. From the experimental and numerical results, the added mass and damping normalized with, respectively, the displaced mass of the model and the critical damping, increase with the disk diameter ratio as $(D_d/D_c)^3$ for the added mass and $(D_d/D_c)^2$ for the damping. Recently, Zhang et al. (2023) carried out forced oscillation tests in a calm water tank with different diameter ratios ranging from 1.5 to 2.5. The study confirmed that the added mass and damping are dependent on the diameter ratio. According to the theoretical added mass formulation proposed by Tao et al. (2007), the non-dimensional added mass and damping decrease with the diameter ratio at a given heave amplitude.

Extensive experimental studies helped accumulate some physical understanding of the hydrodynamic loads on heave plates in various engineering configurations, from classical offshore platforms to floating wind turbines (see Table 1). The characteristics of the motion response and geometries were considered depending on the types of floaters. Most of the studies employed the forced oscillation technique in calm water to measure the hydrodynamic loads and provided the added mass and damping (or drag) coefficients with a focus on the heave motion.

This study focuses on the hydrodynamic loads induced by an oscillating circular cylinder with a heave plate in pure surge, heave and pitch motions. The following section provides information on a dedicated experimental setup, which encompasses a captive system, experimental model design, and acquisition systems. The test matrix
<table><caption>**Table 1**
Previous experimental studies on floating structures with heave plates.</caption><thead><tr><th>Author(s)</th><th>Floater</th><th>Test type</th><th>Motions</th><th>Configurations</th></tr></thead><tbody><tr><td>Thiagarajan and Troesch (1998)</td><td>Offshore platform</td><td>Forced oscillation</td><td>Heave</td><td>Column with plate</td></tr><tr><td>Cozijn et al. (2005)</td><td>CALM buoy</td><td>Forced oscillation</td><td>Surge, heave, pitch</td><td>Column with plate</td></tr><tr><td>Tao and Dray (2008)</td><td>Offshore platform</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Molin et al. (2008)</td><td>Subsea equipment</td><td>Forced Oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Wadhwa and Thiagarajan (2009)</td><td>Subsea equipment</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Sudhakar and Nallayarasu (2011)</td><td>Offshore platform</td><td>Free decay</td><td>Heave, pitch</td><td>Column with plate</td></tr><tr><td>An and Faltinsen (2013)</td><td>Subsea equipment</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Li et al. (2013)</td><td>Offshore platform</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Lopez-Pavon and Souto-Iglesias (2015)</td><td>FWT</td><td>Forced oscillation</td><td>Heave</td><td>Column with plate</td></tr><tr><td>Tian et al. (2017)</td><td>Offshore platform</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Zhu and Lim (2017)</td><td>FWT</td><td>Forced oscillation, free decay</td><td>Heave</td><td>Column with plate</td></tr><tr><td>Zhang and Ishihara (2018)</td><td>FWT</td><td>Forced oscillation</td><td>Heave</td><td>Column with multiple plates</td></tr><tr><td>Mentzoni and Kristiansen (2019)</td><td>Subsea equipment</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Thiagarajan and Moreno (2020)</td><td>FWT</td><td>Forced oscillation</td><td>Heave</td><td>Column with plate</td></tr><tr><td>Robertson et al. (2020)</td><td>FWT</td><td>Forced oscillation, free decay</td><td>Surge, heave, pitch</td><td>Full substructure</td></tr><tr><td>Abazari et al. (2021)</td><td>Offshore platform</td><td>Forced oscillation</td><td>Heave</td><td>Isolated plate</td></tr><tr><td>Zhang et al. (2023)</td><td>FWT</td><td>Forced oscillation</td><td>Heave</td><td>Column with plate</td></tr><tr><td>Present study</td><td>FWT</td><td>Forced oscillation</td><td>Surge, heave, pitch</td><td>Column with plate</td></tr></tbody></table>

is provided for the pure surge, heave, and pitch motions with the proximity of the free surface. Section 3 introduces the method that was used to obtain the hydrodynamic loads from the measurements. A procedure is introduced to obtain mass and moment of inertia experimentally, and a mathematical description is given for the fluid loads by defining the dynamic and static contributions.

The experimental results are presented in Sections 4 to 6 for forced heave, surge, and pitch motions, respectively. The hydrodynamic loads are examined at various submerged drafts first, and observation of the harmonic content of the loads is discussed. In particular, the first harmonics are interpreted as added mass and damping coefficients. The coefficients are investigated by studying the influence of flow parameters, plate geometry and submerged plate depth. Lastly, conclusions are drawn in Section 7.

## 2. Experimental setup

Captive model tests were conducted to measure the hydrodynamic loads induced by body motions in the hydrodynamic and ocean engineering tank of Ecole Centrale Nantes illustrated in Fig. 1. The dimensions of the tank are 46.4 m in length, 30.0 m in breadth, and a water depth of 5.0 m. The tank is equipped with a flap-type wave maker and a wave-breaking beach on the front and back sides. The wave maker was not used during the experimental campaign.

The test campaign involved forced oscillations in pure surge, heave, and pitch motions at various submerged depths. The motions are imposed based on the Galilean reference frame {$O$}, which is the earth-fixed coordinate system given by ($O_0, X_0, Y_0, Z_0$). The origin of the frame $O_0$ is located on the free surface at rest, 17.2 m from the wave maker in the x-direction and 15.0 m from the sidewall in the y-direction.

### 2.1. Captive system

A captive model technique was employed to impose body motion as a function of amplitude $\xi$ and frequency $\omega$. The captive system consists of a tripod and hexapod, as shown in Fig. 2(a). The system is installed in the center of the tank. The tripod is a rigid, heavy jacket structure measuring 12.6 m wide and 7.2 m high. In particular, the free surface piercing part of the tripod has a truss structure to minimize interference caused by wave diffraction. The whole structure is firmly fixed at the bottom of the tank.

The hexapod, a forced motion generator, is equipped with fixed and mobile platforms, as illustrated in Fig. 2(b). The fixed platform is installed on the tripod, and the hexapod is placed above the free surface. The mobile platform is moved by six electric actuators within the operational limits described in Table 2. The experimental model is mounted on the mobile platform, and the hexapod imposes motion with six degrees of freedom.

### 2.2. Experimental models

The experimental model is designed as a single vertical, free-surface piercing circular cylinder with a circular plate mounted at the bottom. This model has three main components, as illustrated in Fig. 2(b): an outer cylinder shell, a base plate, and a heave plate. The cylinder shell is designed to be divided into three parts in order to consider three submerged drafts: deep, middle, and shallow. The shell is connected to the base plate and completely sealed for waterproofing. Additionally, the heave plate is mounted on the base plate and can be replaced by another plate.

The entire model is bolted to the hexapod through an inner beam structure, and a force transducer is installed between the base plate and the beam structure. The detailed connection is shown in Fig. 4. The model has a high structural stiffness to obtain a natural frequency far from the frequency of interest. For example, the natural frequency of the cylinder measured from hammer tests is 16 Hz in dry mode and 15 Hz in wet mode.

Fig. 3 presents the model in the tank and five heave plate configurations mounted at the bottom of the cylinder. The diameter ratios of the plate $D_d/D_c$ range from 1.0 to 3.5, including a single circular cylinder ($D_d/D_c = 1.0$). A porous plate with 10 % of porous ratio $\tau$ is considered with $D_d/D_c = 2.9$ based on Tian et al. (2017), where they showed that the perforated plate decreases the added mass but increases the hydrodynamic damping more than the solid plate at low $KC_d$ numbers.

The main dimensions of the model configurations are summarized in Table 3. All the plates are made of 5 mm thickness ($t_d$) stainless steel to increase the plate rigidity. The effect of the aspect ratio $t_d/D_d$ is neglected in this study. Furthermore, the wet natural frequencies of the plates were identified through hammer tests. The lowest mode was found at 5.1 Hz for the largest plate ($D_d/D_c = 3.5$), which was used as a reference for signal filtering to avoid structural responses. The center of the cylinder at rest is located at the origin of the Galilean reference frame, which is 17.2 m and 15.0 m from the wave maker and the side wall, respectively. The distance between the model and the tank bottom is 4.225 m based on the maximum submerged draft $d = 0.775$ m. Therefore, the wall boundary effect on the bottom of the tank (Wadhwa et al., 2010) is negligible in this study.

### 2.3. Instrumentation

The measurements include motion, motion-induced external loads, free surface elevation, and accelerations. The motions are recorded by a Qualisys motion tracking system utilizing the cameras following the movement of reflectable balls on the hexapod (see Fig. 2(b)). It is used to verify the movement against the prescribed motions of the hexapod.
<figure><img src="image_5.png" /><figcaption>Fig. 1. The hydrodynamic and ocean engineering tank at Ecole Centrale Nantes. The Galilean reference system, ($O_0$, $X_0$, $Y_0$, $Z_0$) is defined as 17.2 m away from the wave maker and 15.0 m from the side wall. The positive x-axis is towards the wave-breaking beach, and the z-axis is pointing upward.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. The captive system consists of the tripod and hexapod. The experimental model is mounted on the mobile platform of the hexapod.</figcaption></figure>

<table><caption>**Table 2**
The hexapod operational capabilities in six degrees of freedom.</caption><thead><tr><td>Motions</td><td>Displacement</td><td>Velocity</td><td>Acceleration</td><td>Frequency range</td><td>Precision</td></tr></thead><tbody><tr><td>Surge</td><td>±465 mm</td><td>±1000 mm/s</td><td>±20 000 mm/s<sup>2</sup></td><td>0–3 Hz</td><td>±1 mm</td></tr><tr><td>Sway</td><td>±465 mm</td><td>±1000 mm/s</td><td>±20 000 mm/s<sup>2</sup></td><td>0–3 Hz</td><td>±1 mm</td></tr><tr><td>Heave</td><td>±300 mm</td><td>±600 mm/s</td><td>±12 000 mm/s<sup>2</sup></td><td>0–3 Hz</td><td>±1 mm</td></tr><tr><td>Roll</td><td>±30°</td><td>±70°/s</td><td>±1200°/s<sup>2</sup></td><td>0–3 Hz</td><td>±0.1°</td></tr><tr><td>Pitch</td><td>±30°</td><td>±70°/s</td><td>±1200°/s<sup>2</sup></td><td>0–3 Hz</td><td>±0.1°</td></tr><tr><td>Yaw</td><td>±45°</td><td>±120°/s</td><td>±2000°/s<sup>2</sup></td><td>0–3 Hz</td><td>±0.1°</td></tr></tbody></table>

The free surface deformations caused by the body motion (radiated waves) are measured by 7 resistive wave gauges positioned in the tank. The wave gauges are distributed around the model at intervals of 120 degrees in the circumferential direction and 1.4 m and 2.1 m from the

center of the model at rest, $O_0$, in the radial direction as depicted in Fig. 1.

Fig. 4 presents the sensor arrangements inside the cylinder shell. A six-component force transducer (HBM:MSC10-BG2) has nominal forces
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** The captive model in the wave tank: a truncated cylinder with a heave plate mounted at the bottom. Five heave plates ranging $1.0 \le D_d/D_c \le 3.5$, including a perforated plate with a porous ratio, $\tau = 0.1$.</figcaption></figure>

<table><caption>**Table 3**
Dimensions of the experimental models: a truncated circular cylinder with and without heave plates.</caption><thead><tr><th>Parameters</th><th>Unit</th><th>Model A</th><th>Model B</th><th>Model C</th><th>Model D</th><th>Model E</th></tr></thead><tbody><tr><td>D<sub>c</sub></td><td>m</td><td>0.35</td><td>0.35</td><td>0.35</td><td>0.35</td><td>0.35</td></tr><tr><td>D<sub>d</sub></td><td>m</td><td>0.35</td><td>0.53</td><td>1.00</td><td>1.00</td><td>1.23</td></tr><tr><td>t<sub>d</sub></td><td>m</td><td>0.005</td><td>0.005</td><td>0.005</td><td>0.005</td><td>0.005</td></tr><tr><td>d</td><td>m</td><td>0.775, 0.250</td><td>0.775, 0.250</td><td>0.775<sup>a</sup>, 0.500<sup>b</sup> 0.250<sup>c</sup></td><td>0.775, 0.250</td><td>0.775, 0.250</td></tr><tr><td>Δ<sup>d</sup></td><td>kg</td><td>74.6, 24.1</td><td>75.2, 24.7</td><td>78.0, 51.6, 27.5</td><td>78.0, 27.5</td><td>80.0, 29.5</td></tr><tr><td>m<sup>e</sup></td><td>kg</td><td>36.9, 24.7</td><td>41.6, 29.5</td><td>64.2, 58.1, 52.1</td><td>60.9, 48.8</td><td>79.7, 67.6</td></tr><tr><td>τ</td><td>–</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.1</td><td>0.0</td></tr><tr><td>D<sub>d</sub>/D<sub>c</sub></td><td>–</td><td>1.0</td><td>1.5</td><td>2.9</td><td>2.9</td><td>3.5</td></tr><tr><td>d/D<sub>c</sub></td><td>–</td><td>2.2, 0.7</td><td>2.2, 0.7</td><td>2.2, 1.4, 0.7</td><td>2.2, 0.7</td><td>2.2, 0.7</td></tr><tr><td>Note</td><td>–</td><td>Truncated cylinder</td><td>Cylinder with heave plate</td><td></td><td></td><td></td></tr></tbody></table>

and moments of 2 kN, 10 kN, and 0.15 kN m for $F_x$, $F_z$, and $M_y$, respectively. Furthermore, two accelerometers installed on the base plate and beam structure are used to check each structural vibration. An inertial measurement unit (IMU) is also utilized to check the Euler angles and angular rate to verify the motion signal measured by Qualisys. The data are obtained using a sampling rate of 200 Hz.

Fig. 5 presents the reference frames and load conventions in the forced oscillation tests. A body-fixed coordinate system {$b$}, given by ($O_b$, $X_b$, $Y_b$, $Z_b$) is introduced with Tait-Bryan angles: Yaw $\psi$, Pitch $\theta$, and Roll $\phi$ based on the Galilean frame. The instantaneous state of the reference frame {$b$} is described based on the reference frame {$O$}, using the distance vector $\mathbf{r}_{b/0}^0$ and the Euler angles $\mathbf{\Theta}_{0b}$. In this test, $\mathbf{r}_{b/0}^0$ and $\mathbf{\Theta}_{0b}$ are imposed by the hexapod, while the Qualisys system measures these motions to verify that they correspond to the prescription.

The loads acting on the moving body are measured by the force transducer located at $O_t$. Since the transducer is fixed on the rigid body, the loads are expressed in the body-fixed reference frame {$b$}. Therefore, the measured loads are expressed at the origin of the body-fixed

reference frame, $O_b$, using the equation below:

$$ 
\begin{aligned}
\mathbf{F}_m^b &= \mathbf{F}_t^b \\
\mathbf{M}_m^b &= \mathbf{M}_t^b + \mathbf{r}_{t/b}^b \times \mathbf{F}_t^b
\end{aligned} 
\qquad (2) 
$$

where subscripts $t$ and $m$ represent the measured force and moment about $O_t$ and $O_b$ respectively. $\mathbf{r}_{t/b}^b$ is the position vector from $O_b$ to $O_t$ expressed in {$b$}.

## 2.4. Prescribed motions

In this study, prescribed motions are defined with a periodic function encompassing 30 motion periods $T$ over the total duration. The motion is characterized by a specific amplitude and frequency as follows,

$$ \left[ \begin{array}{c} \mathbf{r}_{b/0}^0(t) \\ \mathbf{\Theta}_{0b}(t) \end{array} \right] = \xi_j \cos(\omega t + \varepsilon), \quad j = 1 \sim 6 \qquad (3) $$

<footnote>a Deep draft ($d/D_c = 2.2$).</footnote>

<footnote>b Middle draft ($d/D_c = 1.4$).</footnote>

<footnote>c Shallow draft ($d/D_c = 0.7$).</footnote>

<footnote>d $\Delta$ is the mass of the fluid displaced.</footnote>

<footnote>e $m$ is the mass of the experimental model.</footnote>
<figure><img src="image_8.png" /><figcaption>Fig. 4. General arrangement of sensors inside the cylinder shell: the force transducer (blue), IMU (red), and accelerometers (green).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Load convention of the moving body in the wave tank: $F_{m,x}^b, F_{m,z}^b, M_{m,y}^b$ stand for the measured forces and moments at the origin of the body-fixed reference, $O_b$, converted from the force transducer located at $O_t$.</figcaption></figure>

where $\mathbf{r}_{b/0}^0(t) = [\mathbf{x}_{b/0}^0(t), \mathbf{y}_{b/0}^0(t), \mathbf{z}_{b/0}^0(t)]^T$ is the translational vector expressed in the global reference frame $\{0\}$. $\Theta_{0b}(t) = [\phi(t), \theta(t), \psi(t)]^T$ is the Euler angle by taking $\{0\}$ into $\{b\}$. $\xi_j$ is the motion amplitude for surge, sway, and heave for $j = 1 \sim 3$ in meters, and roll, pitch, and yaw for $j = 4 \sim 6$ in radians. $\omega = 2\pi/T$ is a circular frequency (rad/s), $t$ is time (s), and $\epsilon$ is a phase of the motion (rad). In the following prescribed motions, only one motion amplitude of the six is non-zero in each configuration.

As illustrated in Fig. 6, a ramp function is considered at the beginning and the end of the prescribed motions to minimize the transient behavior of the response. Before the beginning of the motions, the experimental model moves slowly to achieve the target submerged

depth with an initial offset and pauses for $t_{zero} = 50$ s to get the zeroing signals. The center of rotation is defined at $O_b$, which is on the free surface at rest.

## 2.5. Test parameters

A single vertical cylinder with heave plates oscillating in surge, heave, and pitch motions was tested without incident waves to obtain the hydrodynamic loads induced by the prescribed motion. Two distinct test campaigns were conducted with different flow parameters. First, the study considered various motion periods from 0.9 s to 3.0 s, represented by a non-dimensional frequency parameter ($0.16 \le \omega^2 D_c/g \le 1.74$). These motions had a constant amplitude of 0.05 m for surge and heave and 0.05 rad (3.0 deg) for pitch. These small amplitude motions are intended to minimize the nonlinearity due to the motion amplitude.

The nonlinearity of the loads was then addressed with different motion amplitudes at a single motion frequency ($\omega^2 D_c/g = 0.35, T = 2.0$ s). For the translational motions, the amplitudes ranged from 0.0125 m to 0.20 m, which corresponds to the translational $KC_c$ number

$$KC_c = 2\pi\xi_{1,3}/D_c \quad (4)$$

which ranges from 0.2 to 3.6.

The pitch amplitudes ranged from 0.02 rad (1.0 deg) to 0.21 rad (12.0 deg). The rotational $KC_c$ number considering the tangential velocity was introduced as,

$$KC_c = 2\pi\xi_5 d/D_c \quad (5)$$

which differs depending on the submerged depth.

Furthermore, deep (0.775 m), middle (0.500 m), and shallow (0.250 m) drafts were considered to investigate the influence of the free surface's proximity. The middle draft was only tested for Model C (solid plate with $D_d/D_c = 2.9$), which is the reference model. In addition, two more submerged drafts were added for the heave motion: the intermediate (0.375 m) and very shallow (0.125 m) drafts. All the heave plate configurations were tested with surge and heave motions. For the pitch motion, only the effect of the presence of the heave plate, including the perforated plate, was studied using Models A, C and D. The test condition parameters are listed in Table 4.

Fig. 7 presents examples of disturbed free surface in forced heave oscillation with the largest heave plate ($D_d/D_c = 3.5$) at the shallow draft. Strong interactions between the free surface and heave plate were observed during the experiments.

## 3. Data analysis

The general mathematical description to obtain the hydrodynamic load from the measurements is presented.

### 3.1. Derivation of the fluid forces from the measurement

The main objective of these experiments is to investigate the hydrodynamic load on an oscillating body. Since the force measurement contains the inertia, gravity and fluid loads, the first two terms need to be handled in order to obtain the fluid loads using the Newton–Euler equation of motion at the given point $O_b$ shown as:

$$ 
\begin{aligned}
m \left( \frac{d\mathbf{V}_{b/0}^b}{dt} + \omega_{b/0}^b \times \mathbf{V}_{b/0}^b + \frac{d\omega_{b/0}^b}{dt} \times \mathbf{r}_{G/b}^b + \omega_{b/0}^b \times (\omega_{b/0}^b \times \mathbf{r}_{G/b}^b) \right) \\
= \mathbf{F}_g^b + \mathbf{F}_f^b + \mathbf{F}_c^b \\
I_b \frac{d\omega_{b/0}^b}{dt} + \omega_{b/0}^b \times I_b \omega_{b/0}^b + m \mathbf{r}_{G/b}^b \times \left( \frac{d\mathbf{V}_{b/0}^b}{dt} + \omega_{b/0}^b \times \mathbf{V}_{b/0}^b \right) \\
= \mathbf{M}_g^b + \mathbf{M}_f^b + \mathbf{M}_c^b
\end{aligned}
\quad (6) $$

where $m$ is the mass of the model, $I_b$ is the moment of inertia at position $O_b$, $\mathbf{r}_{G/b}^b$ is the vector from $O_b$ to the center of gravity, $\mathbf{V}_{b/0}^b$ is the
<figure><img src="image_10.png" /><figcaption>Fig. 6. Prescribed motions including times for zeroing, ramp up and down.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. Snapshots of the radiated waves around the truncated cylinder with heave plate ($D_d/D_c = 3.5$) at the shallow submerged depth ($d/D_c = 0.7$, $d = 0.775$ m). A regular heave motion ($K C_c = 1.8$, $\omega^2 D_c / g = 1.41$) is imposed by the hexapod.</figcaption></figure>

body velocity and $\omega_{b/0}^b$ is the angular velocity, following the notation of Fossen (2011). The superscript b means the body-fixed reference frame, and the subscript g, f, and c stand for the gravity, fluid, and constraint loads.

The LHSs of Eq. (6) are defined as the inertia forces and moments: $\mathbf{F}_i^b$, $\mathbf{M}_i^b$. On the other hand, the constraint loads on the RHSs are treated as the measured loads as $\mathbf{F}_c^b = -\mathbf{F}_m^b$ and $\mathbf{M}_c^b = -\mathbf{M}_m^b$. Then, the equation of motion is simplified as follows,

$$ \mathbf{F}_f^b(t) = \mathbf{F}_m^b(t) + \mathbf{F}_i^b(t) - \mathbf{F}_g^b(t) \quad (7) $$

$$ \mathbf{M}_f^b(t) = \mathbf{M}_m^b(t) + \mathbf{M}_i^b(t) - \mathbf{M}_g^b(t) $$

where the subscript m stands for the measured loads.

From these last equations, the fluid loads acting on the body are obtained by subtracting the inertial and gravity load contributions from the measurements. The general expressions of the inertia, gravity and hydrostatic loads are summarized in Appendix A.

## 3.2. Hydrodynamic loads on the vertical circular cylinder with the heave plate

When a free surface piercing body oscillates in calm water, the fluid loads involve dynamic and static contributions as discussed in Appendix B. Therefore, hydrodynamic loads are considered the dynamic contributions to fluid loads and are obtained by subtracting the static contribution.

$$ \mathbf{F}_{hd}^b(t) = \mathbf{F}_f^b(t) - \mathbf{F}_{hs}^b(t) \quad (8) $$

$$ \mathbf{M}_{hd}^b(t) = \mathbf{M}_f^b(t) - \mathbf{M}_{hs}^b(t) $$

where the subscripts, hs and hd, denote the hydrostatic and hydrodynamic loads, respectively.

In the case of the vertical circular cylinder, the linear hydrostatic loads are appropriate for modeling hydrostatic loads due to the simplicity of the geometry. Thus, the equations for hydrodynamic loads can be simplified for the pure surge, heave, and pitch motions as below.

• Horizontal hydrodynamic force in pure surge motion:

$$ F_{hd,x}^b(t) = F_{m,x}^b(t) + m\dot{\mathbf{x}}_{b/0}^b(t) \quad (9) $$

where $\dot{\mathbf{x}}_{b/0}^b$ is the x-acceleration (surge) vector of $O_b$ from $O_0$ expressed in the body-fixed reference frame. For the surge motion, there is no hydrostatic contribution to the fluid forces.

• Vertical hydrodynamic force in pure heave motion:

$$ F_{hd,z}^b(t) = F_{m,z}^b(t) + m\dot{\mathbf{z}}_{b/0}^b(t) + mg - \rho g \nabla + \rho g A_w \mathbf{z}_{b/0}^b(t) \quad (10) $$

where $\dot{\mathbf{z}}_{b/0}^b$ is the z-acceleration (heave) vector of $O_b$ from $O_0$ expressed in the body-fixed frame, $\nabla$ is a submerged volume under the free surface at rest and $A_w$ is the water plane area of the cylinder.

• Hydrodynamic moment in pure pitch motion at the center of rotation:

$$ \begin{aligned} M_{hd,y}^b(t) = & M_{m,y}^b(t) + I_b \ddot{\theta}(t) \\ & - mg \left( \mathbf{x}_{G/b}^b \cos \theta(t) + \mathbf{z}_{G/b}^b \sin \theta(t) \right) \\ & + \rho g \left( \nabla \mathbf{z}_{B/b}^b + \frac{\pi}{64} D_c^4 \right) \sin \theta(t) \end{aligned} \quad (11) $$

where $\ddot{\theta}(t)$ is an angular acceleration of pitch motion and $\mathbf{z}_{B/b}^b$ is the vertical position of the center of buoyancy at rest in the body-fixed reference frame.

The mass m, center of gravity $\mathbf{r}_{G/b}^b = [\mathbf{x}_{G/b}^b, \mathbf{y}_{G/b}^b, \mathbf{z}_{G/b}^b]^T$, and moment of inertia of each configuration were obtained from the forced oscillation tests in air with the quasi-static and dynamic motions. Detailed procedures and the results are given in Appendix C.

## 3.3. Harmonics of hydrodynamic loads and hydrodynamic coefficients

An example of the vertical hydrodynamic forces acting on the cylinder with the heave plate ($D_d/D_c = 2.9$, Model C) in the forced heave oscillations is shown in Fig. 8. A low-pass filter with a cut-off
<table><caption>**Table 4**
Test parameters for the forced oscillation in surge, heave, and pitch motions.</caption><thead><tr><th>Motions</th><th>Test type</th><th>T [s]</th><th>ω<sup>2</sup>D<sub>c</sub>/g</th><th>ξ [m or deg]</th><th>KC<sub>c</sub></th><th>D<sub>g</sub>/D<sub>c</sub><br>(model)</th><th>d/D<sub>c</sub> (draft)</th></tr></thead><tbody><tr><td rowspan="12">Heave</td><td rowspan="7">freq.</td><td>3.0</td><td>0.16</td><td rowspan="7">0.050</td><td rowspan="7">0.9</td><td>1.0 (A)</td><td>2.2 (deep)</td></tr><tr><td>2.0</td><td>0.35</td><td>1.5 (B)</td><td>1.4 (middle)</td></tr><tr><td>1.5</td><td>0.63</td><td>2.9 (C)</td><td>1.1 (intermediate)</td></tr><tr><td>1.3</td><td>0.83</td><td>2.9τ (D)</td><td>0.7 (shallow)</td></tr><tr><td>1.2</td><td>0.98</td><td rowspan="5">3.5 (E)</td><td rowspan="5">0.4 (very shallow)</td></tr><tr><td>1.1</td><td>1.16</td></tr><tr><td>1.0</td><td>1.41</td></tr><tr><td>0.9</td><td>1.74</td></tr><tr><td rowspan="12">amp. &amp; freq.</td><td rowspan="6">amp.</td><td rowspan="6">2.0</td><td rowspan="6">0.35</td><td>0.0125</td><td>0.22</td><td>1.0</td><td>2.2</td></tr><tr><td>0.025</td><td>0.45</td><td>1.5</td><td>1.4</td></tr><tr><td>0.050</td><td>0.90</td><td>2.9</td><td>1.1</td></tr><tr><td>0.075</td><td>1.35</td><td rowspan="5">3.5</td><td rowspan="5">0.4</td></tr><tr><td>0.100</td><td>1.80</td></tr><tr><td>0.125</td><td>2.24</td></tr><tr><td>0.150</td><td>2.69</td></tr><tr><td>0.175</td><td>3.14</td></tr><tr><td>0.190</td><td>3.41</td><td>2.9</td><td>1.4</td></tr><tr><td rowspan="12">Surge</td><td rowspan="6">freq.</td><td>3.0</td><td>0.16</td><td rowspan="6">0.050</td><td rowspan="6">0.9</td><td>1.0</td><td>2.2</td></tr><tr><td>2.0</td><td>0.35</td><td>1.5</td><td>1.4</td></tr><tr><td>1.5</td><td>0.63</td><td>2.9</td><td>0.7</td></tr><tr><td>1.3</td><td>0.83</td><td rowspan="5">2.9τ</td><td rowspan="5">3.5</td></tr><tr><td>1.2</td><td>0.98</td></tr><tr><td>1.1</td><td>1.16</td></tr><tr><td>1.0</td><td>1.41</td></tr><tr><td>0.9</td><td>1.74</td><td>2.9τ</td><td>1.4</td></tr><tr><td rowspan="12">amp.</td><td rowspan="6">freq.</td><td>3.0</td><td>0.16</td><td rowspan="6">0.050</td><td rowspan="6">0.45</td><td>1.0</td><td>2.2</td></tr><tr><td>2.0</td><td>0.35</td><td>1.5</td><td>1.4</td></tr><tr><td>1.5</td><td>0.63</td><td>2.9</td><td>0.7</td></tr><tr><td>1.3</td><td>0.83</td><td rowspan="5">2.9τ</td><td rowspan="5">0.7</td></tr><tr><td>1.2</td><td>0.98</td></tr><tr><td>1.1</td><td>1.16</td></tr><tr><td>1.0</td><td>1.41</td></tr><tr><td>0.9</td><td>1.74</td><td>2.9τ</td><td>1.4</td></tr><tr><td rowspan="12">Pitch</td><td rowspan="6">freq.</td><td>3.0</td><td>0.16</td><td rowspan="6">0.050</td><td rowspan="6">0.73/0.47/0.23<br>(deep/middle/shallow)</td><td>1.0</td><td>2.2</td></tr><tr><td>2.0</td><td>0.35</td><td>1.5</td><td>1.4</td></tr><tr><td>1.5</td><td>0.63</td><td rowspan="5">2.9τ</td><td rowspan="5">0.7</td></tr><tr><td>1.3</td><td>0.83</td></tr><tr><td>1.2</td><td>0.98</td></tr><tr><td>1.1</td><td>1.16</td></tr><tr><td>1.0</td><td>1.41</td></tr><tr><td>0.9</td><td>1.74</td><td>2.9τ</td><td>1.4</td></tr><tr><td rowspan="6">amp.</td><td rowspan="6">2.0</td><td rowspan="6">0.35</td><td>1.0</td><td>0.24/0.16/0.08</td><td>1.0</td><td>2.2</td></tr><tr><td>3.0</td><td>0.73/0.47/0.23</td><td>2.9</td><td>1.4</td></tr><tr><td>5.0</td><td>1.21/0.78/0.39</td><td rowspan="5">2.9τ</td><td rowspan="5">0.7</td></tr><tr><td>7.0</td><td>1.70/1.10/0.55</td></tr><tr><td>9.0</td><td>2.19/1.41/0.70</td></tr><tr><td>12.0</td><td>2.91/1.88/0.94</td></tr></tbody></table>

frequency of 4.5 Hz is applied to the signal to avoid interference from the natural frequency of the plate. The hydrodynamic loads induced by sinusoidal motions show a periodic oscillatory behavior over time.

The harmonic content is determined using Fourier analysis within the integer periods to minimize the energy losses. The time window is selected after the transient period and is limited to 10T in order to avoid the effect of the reflection on the side wall. It was confirmed that the effect is negligibly small in the selected time window since the radiated wave propagates away from the body in three dimensions, leading to a decrease in the amplitude. The Fourier analysis clearly demonstrates that higher harmonics are visible up to the 10th harmonic, even with linear imposed motion. The first harmonic is the most dominant, but higher harmonics contribute significantly to the overall load.

The contribution of the first harmonic depends on the motion amplitude. As shown in Fig. 9, the first-harmonic contribution accounts for over 90% of the harmonics up to the fifth ($\sum_{n=0}^{5} F_{hd,z}^{h,(no)}$) when the $KC_c$ number is less than 0.5. However, as the $KC_c$ number increases, the relative contribution of higher harmonics gradually increases, resulting in a decrease in the contribution of the first harmonic; this is important to bear in mind when analyzing the first harmonic only. Additionally, the submerged depth also affects the first harmonic contribution. For the drafts deeper than ($d/D_c$ > 1.1), the contribution of the first harmonic shows a similar trend in general. Interestingly, for shallow drafts close to the free surface, the contribution of the first harmonic rapidly decreases to approximately 55% at $KC_c$ = 2.7 at the draft $d/D_c$ = 0.7, for example.
<figure><img src="image_12.png" /><figcaption>Fig. 8. Time histories of vertical hydrodynamic forces on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$, Model C) and its harmonic content. The model oscillates in heave motion with a single amplitude $KC_c = 2.7$ ($\xi_3 = 0.15$ m) and frequency $\omega^2 D_c / g = 0.35$ ($T = 2.0$ s).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. Contribution of the first harmonic to vertical hydrodynamic loads acting on the vertical cylinder with the solid plate ($D_d/D_c = 2.9$) in the forced heave oscillations with a single frequency ($\omega^2 D_c / g = 0.35$) and various amplitudes, varying the proximity to the free surface.</figcaption></figure>

With the periodic behavior hypothesis, the loads can be modeled with Eq. (12), where $f_{hd}$ is a general expression of the hydrodynamic loads expressed in the body-fixed reference frame. The least-square method is used to obtain the amplitudes of each harmonic.

$$f_{hd}(t) \approx \bar{f}_{hd} + \mathfrak{R}\mathfrak{e}\{f_{hd}^{(\omega)}e^{-i(\omega t+\epsilon)}\} + \mathfrak{R}\mathfrak{e}\{f_{hd}^{(2\omega)}e^{-2i(\omega t+\epsilon)}\} \\ + \mathfrak{R}\mathfrak{e}\{f_{hd}^{(3\omega)}e^{-3i(\omega t+\epsilon)}\} \\ + \mathfrak{R}\mathfrak{e}\{f_{hd}^{(4\omega)}e^{-4i(\omega t+\epsilon)}\} + \mathfrak{R}\mathfrak{e}\{f_{hd}^{(5\omega)}e^{-5i(\omega t+\epsilon)}\} \quad (12)$$

where the superscript indicates the corresponding harmonic component. For example, $f_{hd}^{(n\omega)}$ is a complex amplitude of the nth harmonic, where $f_{hd}^{(n\omega)} = f_{hd}^{c,(n\omega)} + f_{hd}^{s,(n\omega)}$ with $i = \sqrt{-1}$.

In marine hydrodynamics, the first harmonic component of the oscillating body is often considered the only contribution and is further interpreted as in-phase and out-of-phase components with the body motion. The in-phase component leads to the added mass, and the out-of-phase component represents damping.

$$A_{ij} = \frac{1}{\xi_j \omega^2} f_{hd,i}^{c,(\omega)}, \quad B_{ij} = \frac{1}{\xi_j \omega} f_{hd,i}^{s,(\omega)} \quad (13)$$

where $A_{ij}$ and $B_{ij}$ are the added mass (kg or kg m²) and the total (or equivalent linear) damping (kg/s or kg m²/s) from the first harmonic load in the direction i induced by the sinusoidal motion in the direction j. This study presents only the diagonal components of the added mass and damping where $i = j$.

# 4. Results I: forced heave oscillations

This section shows the vertical hydrodynamic force time series and its harmonic content. Firstly, the frequency effect is examined, and then the influence of $KC_c$ is presented with a solid plate ($D_d/D_c = 2.9$, Model C). The harmonic content for the porous plate (Model D) with the same diameter ratio is compared to that for the solid plate. Finally, the behavior of hydrodynamic coefficients is described in terms of frequency, $KC_c$, plate diameter, and submerged drafts.

## 4.1. Time series and harmonic content of vertical hydrodynamic forces

A single configuration with a solid heave plate ($D_d/D_c = 2.9$) is examined to address motion amplitude and frequency effects on the hydrodynamic forces. The deep ($d/D_c = 2.2$) and shallow ($d/D_c = 0.7$) drafts are selected to highlight the influence of the free surface's proximity. The time histories are displayed in 5 cycles, and the harmonic content is given up to the fifth harmonics obtained from 10 cycles. The hydrodynamic forces are nondimensionalized as $F_{hd,z}^b / \rho g D_c^3$. The results cover the magnitude of the forces as well as in- and out-of-phase components, which are physically interpreted as the added mass and damping components in the following.

### 4.1.1. Influence of the motion frequency

In Fig. 10, time histories of vertical hydrodynamic forces are presented for several frequencies with a single amplitude ($KC_c = 0.9$) at the deep and shallow drafts. Generally, the hydrodynamic forces are shifted compared to the imposed heave motion given as a black dashed line. The measured force amplitudes increase with increasing motion frequency, which can be observed in both time series and harmonics. The first harmonic is dominant in the harmonic content, and the amplitudes of the zeroth and higher harmonics are negligibly small compared to the first harmonic.

The evolution of the first harmonic with the motion frequency is investigated in Fig. 11. The magnitude, in-phase and out-of-phase components are presented at five different submerged depths for the solid and perforated plates with the same diameter ratio ($D_d/D_c = 2.9$).

The magnitude of the first harmonic linearly increases with the motion frequency at the deep draft $d/D_c = 2.2$. The in- and out-of-phase components follow the tendency of the magnitude, although the increasing rates are different. By decreasing the submerged depth, the linearity of the first harmonics is broken. Interestingly, $\omega^2 D_c / g = 0.6$ appears to be a critical frequency, and the general trend of the magnitude of the first harmonic is distinct before and after this value. At lower frequencies, the magnitude increases with decreasing draft, while at higher frequencies, it decreases.

The in-phase component of forces shows similarity at low frequencies up to the third draft, but differences become larger as the frequency increases. As the submerged depth becomes shallower, the
<figure><img src="image_14.png" /><figcaption>Fig. 10. Time series and harmonic content of vertical hydrodynamic forces on the vertical circular cylinder with the solid heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts under the forced heave oscillations with a single amplitude ($K_c = 0.9$) and various frequencies ($\omega^2 D_c/g = 0.2 - 1.7$).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11. Amplitude of the first harmonic component of vertical hydrodynamic forces on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the various drafts under the forced heave oscillations with a single amplitude ($K_c = 0.9$) and various frequencies ($\omega^2 D_c/g = 0.2 - 1.7$). Filled maker: solid plate. Unfilled markers: perforated plate. Note: the perforated plate was not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>

increase rate with frequency decreases, but it can still be seen that the component increases with frequency. The out-of-phase component increases linearly at deep drafts, while the slope begins to flatten at low drafts. At the shallow draft, $d/D_c = 0.7$, there is no change in frequency above 0.8, and it tends to decrease at the lowest draft, $d/D_c = 0.4$. The general trend of the out-of-phase component is distinct at the frequency $\omega^2 D_c/g = 1.41$. The value increases due to the proximity of the free surface below this frequency but decreases at the highest motion frequency.

The perforated plate shows a similar tendency compared to the solid plate. At the low motion frequency ($\omega^2 D_c/g \le 0.35$), the first harmonics of solid and porous plates have similar magnitudes. However, as the frequency increases, the difference between the two becomes more apparent. Generally, the in-phase component of the porous plate is lower than that of the solid plate at all submersed depths, whereas

the out-of-phase component is larger at deeper drafts and smaller at shallower drafts.

### 4.1.2. Influence of motion amplitude, $K_c$

Fig. 12 presents the effects of motion amplitudes given in $K_c$ on the force time series. At the deep draft, the amplitude of the forces increases with the motion amplitude, which is similar to what is observed when varying the motion frequency. Notably, the positive and negative peaks are sharpened on increasing the motion amplitude. The linear increase of all harmonics can be clearly seen in the zeroth, first, and higher harmonics according to the $K_c$ number. A similar trend is observed at the shallow draft up to $K_c = 2.2$. It is worth noting that the positive peaks of the time series are flattened at the $K_c = 2.7$ compared to that of the deep draft.
<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Time series and harmonic content of vertical hydrodynamic forces on the vertical circular cylinder with the solid heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts under the forced heave oscillations with a single frequency ($\omega^2 D_c / g = 0.35$) and various amplitudes ($KC_c = 0.2 - 3.4$).</figcaption></figure>

**Fig. 13** presents harmonic content up to the third harmonic in magnitude, in- and out-of-phase components on the solid and perforated plates. The magnitude of the first harmonics increases with an increase of the $KC_c$ number and slightly increases with a decrease in the submerged depth. The in-phase components of the first harmonic show a similar trend up to the draft ratio $d/D_c = 1.1$. However, differences appear when the draft ratio falls below 0.7 and $KC_c \le 1.8$, corresponding to $\xi_3/d = 0.4$. Similarly, the difference is clearly visible at the very shallow draft ($d/D_c = 0.4$) after $KC_c$ number 0.9, which corresponds also to $\xi_3/d = 0.4$. The value starts to decrease after this $KC_c$ number.

The out-of-phase components of the first harmonic force show a distinct effect of submerged depth with respect to the $KC_c$. The growth rate increases with the vicinity of the free surface. Interestingly, there is no difference at very low $KC_c$ numbers, but for higher $KC_c$, large differences appear depending on the submerged depth. Furthermore, the magnitude, in-phase, and out-of-phase components generally increase monotonically for the higher harmonics at the deep draft. It can be seen that the in-phase and out-of-phase components of the second and third harmonics decrease with a decrease in the submergence. In general, the higher harmonic values are negative when the draft ratio is $d/D_c \le 0.7$, except in the in-phase component of the third harmonic, where the values are mostly negative regardless of the depth.

## 4.2. Hydrodynamic coefficients for the heave motion

The first harmonics examined in the previous section are now used to compute the added mass and damping coefficients, which are in- and out-of-phase components as given in Eq. (13). An investigation is carried out on the hydrodynamic coefficients of the heave plates ($D_d/D_c = 2.9$) with and without porosity. The motion frequency and amplitude effects are studied by varying submerged depth. Lastly, the effect of the diameter ratio is presented.

### 4.2.1. Influence of the motion frequency for the solid and perforated plates

**Fig. 14** presents the non-dimensional added mass and damping of the heave plate ($D_d/D_c = 2.9$) with and without the porosity

for various motion frequencies in the range $0.16 \le \omega^2 D_c / g \le 1.74$. The experimental measurements are compared with a linear potential flow model given in continuous shaded lines. The model is based on the matched eigenfunction expansion method (MEEM), which solves a linear radiation problem for a vertical circular cylinder with a solid plate attached at the bottom, oscillating in the surge, heave, and pitch motions (Han et al., 2024). This model will also be used in the following sections.

At the deep draft ($d/D_c = 2.2$), the motion frequency does not affect the added mass and damping as reported in Lopez-Pavon and Souto-Iglesias (2015). The added mass from the experiment is higher than the results of the potential flow theory model (MEEM), which is close to a theoretical value ($A_{33}/\rho D_c^3 = 7.1$ for Model C) computed using a formulation proposed by Tao et al. (2007) under the unbounded fluid assumption.

At high frequencies ($\omega^2 D_c / g > 0.8$), the added mass decreases significantly with the draft. This reduction is observed to have almost no frequency dependence, even at shallower drafts. However, at low frequencies, the added mass shows a notable dependence on frequency. Particularly at the lowest frequency ($\omega^2 D_c / g = 0.16$), the added mass increases as the draft decreases, which is contrary to the behavior at high frequencies. In comparison to the theoretical formula in the unbounded fluid, the potential flow theory model (MEEM) is able to capture the general trend of the added mass in shallower drafts by considering the linearized free surface boundary condition. This shows increasingly sharp maxima and minima, as described in Martin and Farina (1997).

The decrement of the added mass was also reported in McIver and Evans (1984) and Greenhow and Ahn (1988) with a submerged body. In particular, according to McIver and Evans (1984) based on the potential flow, the added mass is related to both the kinetic and potential energies of the fluid motion. The kinetic energy decreases with the reduction of submergence, while the potential energy is only affected by the deformation of the free surface. This results in the reduction of the added mass due to the proximity of the free surface.

Meanwhile, the non-dimensional damping increases as the draft decreases, particularly at frequencies below 1.4. Notably, at the highest
<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Amplitude of the first three harmonic components of vertical hydrodynamic forces on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the various drafts under the forced heave oscillation with a single frequency ($\omega^2 D_c/g = 0.35$) and various amplitudes ($KC_c = 0.2-3.4$). Filled makers: solid plate. Unfilled markers: perforated plate. Note: the perforated plate was not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>

frequency $\omega^2 D_c/g = 1.74$, the damping decreases when the draft decreases. Furthermore, the motion frequency strongly affects the damping with the proximity of the free surface, and the maximum damping appears around $\omega^2 D_c/g \approx 0.35$. Similar to the findings of Molin et al. (2008), the peak of the experimental results occurs at a lower frequency than the theoretical one, and the potential flow theory model agrees well at the very shallow draft in the high-frequency region.

During the experiments, it was observed that the heave motion of the plate at the shallow drafts generated significant radiated waves, while there were almost no radiated waves at the deep draft. Potential flow theory is consistent with the findings, demonstrating that there is no wave radiation damping at the deep draft, but significant wave radiation damping when the submerged depth decreases. This suggests that in the case of deep drafts, the damping is dominated by viscous effects, in the form of vortices shed from the edges of the plates and dissipating energy. Previous research on a heaving cylinder with a heave plate already emphasized that the free surface had a limited effect on the

dissipation of energy in deep draft configuration, see Garrido-Mendoza et al. (2014, 2015). However, in the case of shallow drafts, the damping is also given by the radiated waves.

The added mass for the perforated plate is less than for the solid plate with the tested motion frequency range and shows a similar trend according to the frequency for the solid plate. The perforated plate produces more damping than the solid plate at the draft ratios of 2.2 and 1.1. These findings align with those presented in Tao and Dray (2008), Li et al. (2013) and Tian et al. (2017). However, below the draft ratio of 0.7, the damping of the porous plate is lower than that of the solid plate. Interestingly, we observed that the amplitude of the waves radiated by the perforated plate is lower than that of the solid plate at the shallow draft.

### 4.2.2. Influence of the motion amplitude $KC_c$ for the solid and perforated plates

Fig. 15 presents non-dimensional heave added mass and damping of the heave plate ($D_d/D_c = 2.9$) with various $KC_c$ numbers at a single
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Non-dimensional heave added mass and damping coefficients of the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the various drafts under the forced heave oscillation with a single amplitude ($KC_c = 0.9$) and various frequencies ($\omega^2 D_c/g = 0.2-1.7$). Filled markers: solid plate. Unfilled markers: perforated plate. Shaded lines without markers: matched eigenfunction expansion method (MEEM) of the solid plate. Note: the perforated plate was not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>

motion frequency $\omega^2 D_c/g = 0.35$. At the deep draft ($d/D_c = 2.2$), the added mass and damping on the solid plate increase linearly on increasing the $KC_c$ number, in line with the findings of Thiagarajan and Troesch (1998).

The growth rate of the added mass decreases as the draft becomes shallower. Conversely, the damping increases as the draft decreases, and the slope of damping increases with respect to the $KC_c$ number. Furthermore, at the very shallow draft $d/D_c = 0.4$, the maximum peak of the damping occurs at $KC_c = 0.9$, followed by a decrease in the damping, which aligns with the findings of Wadhwa and Thiagarajan (2009) and Garrido-Mendoza et al. (2014).

The study by Ezoji et al. (2022) reported a similar tendency based on the experimental results conducted by Molin et al. (2008), Wadhwa and Thiagarajan (2009), Li et al. (2013) and Lopez-Pavon and Souto-Iglesias (2015). Their results highlight the significance of the draft ratio over the motion amplitude and frequency. The reason might be the relation between energy dissipation and entrophy associated with the damping, as reported by Garrido-Mendoza et al. (2014). According to their numerical study, the vortex field generated by the heaving plate is found to be stronger near the free surface compared to the deep draft, where these enhanced vortices lead to an increased entrophy term.

In general, the porous plate produces less added mass than the solid plate, as presented by Tao and Dray (2008), Li et al. (2013) and Tian et al. (2017). The slope of the porous plate is similar to that of the solid plate at the deep draft but differs when decreasing the draft ratio. However, the rate of change is less steep than that of the solid plate. The damping with the perforated plate is larger than with the solid plate at the deep draft in whole $KC_c$ numbers. On the other hand, at the draft ratios of 1.1 and 0.7, the damping of the perforated plate is larger at low $KC_c$ numbers but lower at the high $KC_c$ numbers. This result is consistent with the findings of Tian et al. (2017). Notably, at the very shallow draft, the damping of the solid plate is larger than that of the perforated plate, even at low $KC_c$. This aligns with the findings in the previous section.

### 4.2.3. Influence of motion frequency and amplitude on solid plate

Further investigation was conducted on a solid plate with $D_d/D_c = 2.9$ by considering three additional motion frequencies $\omega^2 D_c/g = 0.09$, 0.16, 0.23, to complete the study otherwise limited to the single motion

frequency ($\omega^2 D_c/g = 0.35$). The additional frequencies correspond to the motion period $T = 4.0, 3.0, 2.5$ s at the model scale.

Fig. 16 shows the non-dimensional added mass and damping with various motion amplitudes at each frequency. The added mass and damping at the middle draft ($d/D_c = 1.4$) linearly increases with an increase of the $KC_c$ number and present an identical slope when changing the motion frequencies. This observation aligns with the results reported by Tao and Dray (2008), Li et al. (2013) and Lopez-Pavon and Souto-Iglesias (2015), which noted a weaker frequency dependence on hydrodynamic coefficients than on the motion amplitude.

However, at the shallow draft ($d/D_c = 0.7$), the slope increases from a negative value to a positive one with decreasing motion frequency. Additionally, with the motion frequency $\omega^2 D_c/g = 0.09$, the slope of the added mass at the shallow draft is almost identical to that of the middle and deep drafts. This effect can be seen in the damping by changing the slope according to the $KC_c$ numbers.

Mentzoni and Kristiansen (2019) also observed a similar trend in their experiments, demonstrating that the added mass of a heaving perforated plate near the free surface in the channel showed a strong dependence on the oscillating frequency at various motion amplitudes. They mentioned that this dependence is likely due to the presence of the free surface, as it was not captured in their numerical simulations that disregarded the free surface.

### 4.3. Influence of heave plate diameter

This section presents the added mass and damping with various heave plate ratios ($D_d/D_c$) ranging from 1.0 to 3.5. Two submerged depths are examined: deep ($d/D_c = 2.2$) and shallow ($d/D_c = 0.7$) drafts.

Fig. 17 presents the non-dimensional hydrodynamic coefficients with various motion frequencies, comparing the results with the potential flow theory model (MEEM). The hydrodynamic coefficients are not influenced by the motion frequency at the deep draft for all plate configurations. For the solitary cylinder and the cylinder with the small heave plate ($D_d/D_c = 1.5$), the added mass and damping are almost identical at both deep and shallow drafts.

At the shallow draft, the coefficients of the largest heave plate ($D_d/D_c = 3.5$) are affected by the motion frequency in a similar way to the plate $D_d/D_c = 2.9$. At the peak frequency $\omega^2 D_c/g = 0.35$, the
<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Non-dimensional heave added mass and damping coefficients of the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at various drafts ($d/D_c = 0.4 - 2.2$) under the forced heave oscillation with a single frequency ($\omega^2 D_c/g = 0.35$) and various amplitudes ($KC_c = 0.2 - 3.4$). Filled markers: solid plate. Unfilled markers: perforated plates. Note: the porous plate at the draft, $d/D_c = 1.4$ (red), was not tested.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 16.** Non-dimensional heave added mass and damping coefficients of the vertical circular cylinder with the solid plate ($D_d/D_c = 2.9$) at the middle and shallow drafts under the forced heave oscillations with various amplitudes ($KC_c = 0.2 - 3.4$) at each frequency.</figcaption></figure>

damping of the plate with $D_d/D_c = 3.5$ is more strongly influenced by the motion frequency than the plate with $D_d/D_c = 2.9$.

The potential flow theory model predicts that the influence of frequency is negligible at the deep draft and significant at the shallow draft, similar to the experimental findings. The wave radiation damping and its dependency on the motion frequency are very significant at the shallow draft with the large heave plates (Models C and E).

Fig. 18 shows the $KC_c$ effect on the hydrodynamic coefficients with various plate diameter ratios. Similar to the frequency result, the influence of the submerged depth on the added mass and damping with respect to the $KC_c$ for the solitary cylinder and the small heave plate ($D_d/D_c = 1.5$) is negligible compared to that of the larger plates. The trend of the coefficients with respect to the diameter ratio differs based on the $KC_c$. Generally, larger plates provide more added mass and damping than smaller plates at both deep and shallow drafts.

# 5. Results II: forced surge oscillations

This section examines the hydrodynamic forces in forced surge oscillation.

## 5.1. Time series and harmonic content of horizontal hydrodynamic forces

The non-dimensional horizontal force time histories ($F_{hd,x}^b/\rho g D_c^3$) are presented for the circular cylinder with the heave plate ($D_d/D_c = 2.9$, Model C) at the deep ($d/D_c = 2.2$) and shallow ($d/D_c = 0.7$) drafts. The first harmonics observed at the three studied drafts are also presented. The harmonics for the solid plate are compared to those for the solitary cylinder and the perforated plate.
<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Non-dimensional heave added mass and damping coefficients of the vertical circular cylinder with various heave plates at the deep (top) and shallow (bottom) drafts with a single heave motion amplitude ($KC_c = 0.9$) and various frequencies. Shaded lines without markers are the results of the MEEM.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 18.** Non-dimensional heave added mass and damping coefficients of the vertical circular cylinder with various heave plates at the deep (top) and shallow (bottom) drafts with a single heave motion frequency ($\omega^2 D_c / g = 0.35$) and various amplitudes.</figcaption></figure>

### 5.1.1. Influence of the motion frequencies

The following observations are made from Fig. 19, which shows horizontal force time histories by changing the motion frequency at deep and shallow drafts. At low frequency ($\omega^2 D_c / g \le 0.4$), the force behavior follows the imposed surge motion depicted by a black dashed line. However, as the frequency increases, the forces tend to shift, and their amplitude increases. At the shallow draft, the maximum and minimum peaks of the time series are more significantly shifted. The dominant harmonic content is the first harmonic, while the amplitudes of the zeroth and higher harmonics are very small compared to the first harmonic.

The harmonic content at the deep, middle, and shallow drafts is summarized in Fig. 20. At deep and middle drafts, the magnitude of the first harmonic increases with frequency. Meanwhile, the tendency is flattened at high frequency at the shallow draft, which is also pronounced in the in-phase components. Both the magnitude and in-phase components decrease significantly on decreasing the submerged depth, but not for the out-of-phase components. The out-of-phase component gradually increases as foreseen in Fig. 19.

Similar trends are observed with the solitary cylinder (Model A) and the cylinder with the perforated plate (Model D). The magnitude of the first harmonics of the perforated plate is slightly lower than that of the solid plate. Interestingly, the out-of-phase components of each configuration are close to each other at the deep draft, but differences are noticeable at the shallow draft.

### 5.1.2. Influence of the motion amplitude

The effects of the motion amplitude on the horizontal forces are shown in Fig. 21. Depending on the motion amplitude, the increase in the forces is more pronounced than the phase shift. The dominant horizontal hydrodynamic forces are mainly from the first harmonic, but as the $KC_c$ number increases, the third harmonic becomes more prominent, with the other components remaining insignificant.

In Fig. 22, the first and third harmonics are provided. By increasing the $KC_c$, the magnitudes of the first and third harmonics show a monotonic increase. It is also noticeable that the first harmonic of the forces is due to in-phase components, and that they are clearly distinguished by the draft. In contrast, the out-of-phase components are
<figure><img src="image_23.png" /><figcaption>Fig. 19. Time series and harmonic content of horizontal hydrodynamic forces on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts under the forced surge oscillations with a single amplitude ($KC_c = 0.9$) and various frequencies ($\omega^2 D_c / g = 0.2 - 1.7$).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 20. Amplitude of the first harmonic component of horizontal hydrodynamic forces on the vertical circular cylinder with and without the heave plate ($D_d/D_c = 2.9$) at the deep, middle and shallow drafts under the forced surge oscillations with a single amplitude ($KC_c = 0.9$) and various frequencies ($\omega^2 D_c / g = 0.2 - 1.7$). Note: the solitary cylinder and the perforated plate were not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>

similar regardless of the draft. At low $KC_c$ numbers, the magnitude of the first harmonic component is almost identical for the single cylinder and the cylinder with the heave plate. However, as $KC_c$ increases, the presence of the heave plate becomes more pronounced.

## 5.2. Hydrodynamic coefficients for the surge motions

The surge added mass and damping of the first harmonics are non-dimensionalized as $A_{11}/\rho D_c^3$ and $B_{11}/\omega\rho D_c^3$, respectively. Fig. 23 shows the effects of motion frequency on the hydrodynamic coefficients for all plate configurations at two different drafts: deep (top) and shallow (bottom) drafts. The surge hydrodynamic coefficients strongly depend on the motion frequency and demonstrate good agreement with the matched eigenfunction expansion method (Han et al., 2024) given in shaded solid lines for the solid plates.

The effect of the heave plate on the coefficients is observed at both deep and shallow drafts. Similar to the findings for heave motions, the results for the perforated plate generally show less added mass than the solid plate, and more damping at low frequencies ($\omega^2 D_c / g \le 0.6$) at both drafts. However, in high frequencies ($\omega^2 D_c / g \ge 1$), the damping of the perforated plate is less than that of the solid plate at the shallow draft, while the damping remains the same at the deep draft.

Fig. 24 presents the effect of $KC_c$ on the coefficients for various heave plates, excluding the largest plate ($D_d/D_c = 3.5$), studied at the low frequency ($\omega^2 D_c / g = 0.35$). For both deep and shallow drafts, the added mass is clearly differentiated by plate diameter, but the difference with $KC_c$ is small. However, the damping is strongly influenced by $KC_c$, as its magnitude increases. At the deep draft, the damping is similar for all plate configurations, while at the shallow draft, the larger heave plate produces more damping, which is more pronounced
<figure><img src="image_25.png" /><figcaption>**Fig. 21.** Time series and harmonic content of horizontal hydrodynamic forces on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts under the forced surge oscillations with a single frequency ($\omega^2 D_c / g = 0.35$) and various amplitudes ($KC_c = 0.4 - 3.6$).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. 22.** Amplitude of the first (top) and third (bottom) harmonic components of horizontal hydrodynamic forces on the vertical circular cylinder with and without the heave plate ($D_d/D_c = 2.9$) at the deep, middle and shallow drafts under the forced surge oscillations with a single frequency ($\omega^2 D_c / g = 0.35$) and various amplitudes ($KC_c = 0.5 - 3.6$). Note: the solitary cylinder and the perforated plate were not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Fig. 23. Non-dimensional surge added mass and damping coefficients of the vertical circular cylinder with various heave plates at the deep (top) and shallow (bottom) drafts with a single surge motion amplitude ($KC_c = 0.9$) and various frequencies ($\omega^2 D_c / g = 0.2 - 1.7$). Shaded lines without markers are the results of the MEEM on the solid plates.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 24. Non-dimensional surge added mass and damping coefficients of the vertical circular cylinder with various heave plates at the deep (top) and shallow (bottom) drafts with a single surge motion frequency ($\omega^2 D_c / g = 0.35$) and various amplitudes ($KC_c = 0.5 - 3.6$).</figcaption></figure>

at the large $KC_c$. Interestingly, the porous plate provides slightly larger damping than the solid plate at both deep and shallow drafts.

## 6. Results III: forced pitch oscillations

The results of the non-dimensional hydrodynamic pitch moments ($M_{hd,y}^b / \rho g D_c^2$) are presented. Three plate configurations (Models A, C, and D) are examined to address the presence of the heave plate and porosity.

### 6.1. Time series and harmonic content of hydrodynamic moments

Fig. 25 presents the time histories of the moments on the vertical circular cylinder with the heave plate (Model C, $D_d / D_c = 2.9$) at the deep and shallow drafts. For the deep draft, it can be observed that an increase in frequency leads to an increase in the amplitude of the load and a change in phase. A similar trend can be seen at the shallow draft, but the forces do not increase above a certain frequency ($\omega^2 D_c / g = 1.0$). The first harmonic dominates and the contribution of other harmonics

is negligibly small, which aligns with the findings from the surge and heave motions.

The results for the first harmonic are summarized in Fig. 26. At deep and middle drafts, the magnitude of the harmonic increases linearly with frequency. However, at the shallow draft, it can be seen that the load magnitude no longer increases after a certain frequency.

The results of the perforated plate are similar to that of the solid plate but with smaller values for both deep and low drafts in terms of magnitude and the in-phase component. Interestingly, at the deep draft, the out-of-phase component for the perforated plate is higher than that for the solid plate, and vice versa at the shallow draft. It is also worth noting that the presence of the heave plate increases the pitch moment compared to the solitary cylinder.

Fig. 27 presents the hydrodynamic pitch moment time histories and their harmonics with different motion amplitudes in the range $1.0^\circ \le \xi_5 \le 12.0^\circ$. At both the deep and shallow drafts, the forces increase on increasing the motion amplitude, and the phase is shifted compared to the prescribed pitch motion given in the dashed line. The
<figure><img src="image_29.png" /><figcaption>Fig. 25. Time series and harmonic content of hydrodynamic moments on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts under the forced pitch oscillations with a single amplitude ($\xi_s = 3.0^\circ$) and various frequencies ($\omega^2 D_c/g = 0.2 - 1.7$).</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Fig. 26. Amplitude of the first harmonic component of the hydrodynamic moments on the vertical circular cylinder with and without the heave plate ($D_d/D_c = 2.9$) at the deep, middle, and shallow drafts under the forced pitch oscillations with a single amplitude ($\xi_s = 3.0^\circ$) and various frequencies ($\omega^2 D_c/g = 0.2 - 1.7$). Note: the solitary cylinder and the perforated plate were not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>

first harmonics contribute significantly, and the third harmonics are visible but negligibly small compared to the first harmonics.

The variation of the first harmonics in the in- and out-of-phase components is given in Fig. 28. At the deep draft, the magnitude of the first harmonic with the heave plate is linearly increased and distinctively higher than that of the solitary cylinder. On the other hand, the perforated plate follows a similar tendency but slightly lower moments than the solid plate. At the deep draft, the out-of-phase component for the perforated plate is higher than that for the solid plate, but there is no difference at the shallow draft.

## 6.2. Hydrodynamic coefficients for the pitch motions

Fig. 29 presents the frequency effect on the pitch added mass and damping, nondimensionalized with $A_{55}/\rho D_c^5$ and $B_{55}/\rho D_c^5$. The potential flow theory model (Han et al., 2024) shows good agreement for the

added mass and damping coefficients on the solitary cylinder at both deep and shallow drafts, with weak frequency dependence. Similarly, the hydrodynamic coefficients on the solid plate ($D_d/D_c = 2.9$) are less affected by the frequency at the deep draft, but the frequency dependence is more pronounced at the shallow draft. The theoretical model underpredicts the hydrodynamic coefficients, especially the pitch damping. This discrepancy may also be attributed to the vortex shed at the plate edge, as previously discussed for the heave motion.

The results for the perforated plate show less added mass than that for the solid plate, at both deep and shallow drafts. Conversely, the perforated plate induces higher damping than the solid plate at the deep draft. For the shallow draft, this difference in damping depends on the frequency. These conclusions found for the pitch motion are the same as that for the heave motion.

The effects of motion amplitude on the pitch hydrodynamic coefficients are presented in Fig. 30. For the solitary cylinder, the added
<figure><img src="image_31.png" /><figcaption>**Fig. 27.** Time series and harmonic components of hydrodynamic moments on the vertical circular cylinder with the heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts under the forced pitch oscillations with a single frequency ($\omega^2 D_c/g = 0.35$) and various amplitudes ($\xi_5 = 1.0^\circ - 12.0^\circ$).</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>**Fig. 28.** Amplitude of the first harmonic component of the hydrodynamic moments on the vertical circular cylinder with and without the heave plate ($D_d/D_c = 2.9$) at the deep, middle, and shallow drafts under the forced pitch oscillations with a single frequency ($\omega^2 D_c/g = 0.35$) and various amplitudes ($\xi_5 = 1.0^\circ - 12.0^\circ$). Note: the solitary cylinder and the perforated plate were not tested at the middle draft ($d/D_c = 1.4$, red).</figcaption></figure>

mass shows an almost constant value even with varying $KC_c$, whereas the added mass with the heave plate gradually increases as $KC_c$ is increased. Furthermore, the increase of $KC_c$ significantly affects the damping of the solid plate. The porosity produces less added mass than the solid plate at the deep and shallow drafts. Notably, the damping for the perforated plate is higher than that for the solid plate at the deep draft but the damping varies with $KC_c$ at shallow drafts.

## 7. Conclusions

The hydrodynamic loads have been investigated for surge, heave and pitch oscillations of a vertical and surface-piercing circular cylinder with a heave plate mounted at the bottom. It was observed that the harmonically oscillating cylinder with the heave plate causes hydrodynamic loads with various harmonic content. The magnitude and phase of these harmonics were significantly affected by the amplitude and

frequency of the motion. These effects were more pronounced when the heave plate was close to the free surface.

The study demonstrated that the most significant component of the hydrodynamic loads is the first harmonic. This harmonic was further examined as added mass and damping coefficients, which are key elements in the design of floating platforms. The behavior of the hydrodynamic coefficients in the surge, heave and pitch was extensively described with respect to the motion amplitude and frequency, plate diameter ratio, porosity, and the proximity of the free surface.

The hydrodynamic coefficients associated with the first harmonic (added mass and damping) are used to represent the loads in engineering design tools based on flow and geometrical parameters. This study emphasized the strong influence of the free surface on the hydrodynamic coefficients in shallow draft configurations. Numerical tools such as linear potential flow theory solvers are able to capture these effects with reasonable accuracy for small amplitude motions. However, this
<figure><img src="image_33.png" /><figcaption>**Fig. 29.** Non-dimensional pitch added mass and damping coefficients of the vertical circular cylinder with and without the heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts with a single pitch amplitude ($\xi_5 = 3.0^\circ$) and various frequencies ($\omega^2 D_c/g = 0.2 - 1.7$). Shaded lines without markers are the results of the potential flow theory model (MEEM) given for the vertical circular cylinder with and without the solid heave plate.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>**Fig. 30.** Non-dimensional pitch added mass and damping coefficients of the vertical circular cylinder with and without the heave plate ($D_d/D_c = 2.9$) at the deep (top) and shallow (bottom) drafts with a single pitch motion frequency ($\omega^2 D_c/g = 0.35$) and various amplitudes ($\xi_5 = 1.0^\circ - 12.0^\circ$).</figcaption></figure>

may not be enough, as we have observed that higher harmonics become more noticeable as the motion's amplitude increases. Moreover, the relative contribution of the first harmonic decreases when the plate approaches the free surface. This emphasizes the significance of considering higher harmonic loads to accurately model the hydrodynamic loads of heave plates.

In future studies, it would be beneficial to determine the contribution of the wave, vorticity, and other energy dissipation terms to the damping by conducting a parametric numerical study (CFD) using a combination of motion amplitude, frequency, and submerged depth. Furthermore, it is important to examine the effects of heave plates in incident waves for both fixed and freely floating structures to achieve a comprehensive understanding of the physics at stake. Finally, expanding knowledge to the practical design of floating substructures is essential, as these systems may involve complex geometry, including multiple columns, pontoons, braces, and reinforcements such as stiffeners or webs.

CRediT authorship contribution statement

**Seung-Yoon Han:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Benjamin Bouscasse:** Writing – review & editing, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. **Vincent Leroy:** Writing – review & editing, Supervision, Software, Methodology, Investigation, Conceptualization. **Sylvain Delacroix:** Writing – review & editing, Software, Investigation, Formal analysis. **Jean-Christophe Gilloteaux:** Writing – review & editing, Supervision, Funding acquisition, Conceptualization. **Félicien Bonnefoy:** Writing – review & editing, Software, Investigation. **David Le Touzé:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization.
Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research is part of FLOAWER (FLOAting Wind Energy netwoRK) project. The project has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement N° 860879. This work was also carried out within the HP_Flow project in the WEAMEC framework, West Atlantic Marine Energy Community, and with funding from the Pays de la Loire Region (France) and European Regional Development Fund. The work presented in this paper was supported by Centrale Nantes - Bureau Veritas research chair. The authors acknowledge the tank testing team of LHEEA for their enthusiastic support of the experiment.

Appendix A. Inertia, gravitational, and hydrostatic loads

Inertia forces and moments.

$$ \mathbf{F}_i^b = m \left( \frac{d\mathbf{V}_{b/0}^b}{dt} + \omega_{b/0}^b \times \mathbf{V}_{b/0}^b + \frac{d\omega_{b/0}^b}{dt} \times \mathbf{r}_{G/b}^b + \omega_{b/0}^b \times (\omega_{b/0}^b \times \mathbf{r}_{G/b}^b) \right) \quad (A.1) $$

$$ \mathbf{M}_i^b = I_b \frac{d\omega_{b/0}^b}{dt} + \omega_{b/0}^b \times I_b \omega_{b/0}^b + m \mathbf{r}_{G/b}^b \times \left( \frac{d\mathbf{V}_{b/0}^b}{dt} + \omega_{b/0}^b \times \mathbf{V}_{b/0}^b \right) \quad (A.2) $$

Gravity forces and moments.

$$ \mathbf{F}_g^b(t) = (R_g^0(t))^{-1} \mathbf{F}_g^0 = -mg \begin{bmatrix} -\sin\theta(t) \\ \sin\phi(t) \cos\theta(t) \\ \cos\phi(t) \cos\theta(t) \end{bmatrix} \quad (A.3) $$

where m and g are the mass and gravitational accelerations (≈9.81 m/s²), $\mathbf{F}_g^0 = [0; 0; -mg]$ is the gravity forces in the inertial reference frame and $R_g^0$ is the rotational matrix as,

$$ R_g^0 = \begin{bmatrix} \cos \psi \cos \theta & -\sin \psi \cos \phi + \cos \psi \sin \theta \sin \phi & \sin \psi \sin \phi + \cos \psi \cos \phi \sin \theta \\ \sin \psi \cos \theta & \cos \psi \cos \phi + \sin \phi \sin \theta \sin \psi & -\cos \psi \sin \phi + \sin \phi \cos \phi \sin \theta \\ -\sin \theta & \cos \theta \sin \phi & \cos \theta \cos \phi \end{bmatrix} \quad (A.4) $$

$$ \mathbf{M}_g^b(t) = \mathbf{r}_{G/b}^b \times \mathbf{F}_g^b(t) = -mg \begin{bmatrix} \mathbf{y}_{G/b}^b \cos \theta(t) \cos \phi(t) - \mathbf{z}_{G/b}^b \cos \theta(t) \sin \phi(t) \\ -\mathbf{x}_{G/b}^b \cos \theta(t) \cos \phi(t) - \mathbf{z}_{G/b}^b \sin \theta(t) \\ \mathbf{x}_{G/b}^b \cos \theta(t) \sin \phi(t) + \mathbf{y}_{G/b}^b \sin \theta(t) \end{bmatrix} \quad (A.5) $$

where $\mathbf{r}_{G/b}^b$ is the position of the center of gravity. The body is assumed rigid, therefore this vector can be considered as a constant value.

Hydrostatic forces and moments.

$$ \mathbf{F}_{hs}^b(t) = (R_b^0(t))^{-1} \mathbf{F}_{hs}^b(t) = \rho g \nabla(t) \begin{bmatrix} -\sin\theta(t) \\ \sin\phi(t) \cos\theta(t) \\ \cos\phi(t) \cos\theta(t) \end{bmatrix} \quad (A.6) $$

where $\nabla(t)$ is the immersed volume under the mean water level varying in time.

$$ \mathbf{M}_{hs}^b(t) = \mathbf{r}_{B/b}^b(t) \times \mathbf{F}_{hs}^b(t) $$

<figure><img src="image_35.png" /><figcaption>Fig. B.31. Description of load conventions, wetted surfaces of moving body in calm water, $\tilde{S}_B$, the variation of wetted body surface, $\delta S_B$, and the radiated waves, $\eta_R$.</figcaption></figure>

$$ = \rho g \nabla(t) \begin{bmatrix} \mathbf{y}_{B/b}^b(t) \cos \theta(t) \cos \phi(t) - \mathbf{z}_{B/b}^b(t) \cos \theta(t) \sin \phi(t) \\ -\mathbf{x}_{B/b}^b(t) \cos \theta(t) \cos \phi(t) - \mathbf{z}_{B/b}^b(t) \sin \theta(t) \\ \mathbf{x}_{B/b}^b(t) \cos \theta(t) \sin \phi(t) + \mathbf{y}_{B/b}^b(t) \sin \theta(t) \end{bmatrix} \quad (A.7) $$

where $\mathbf{r}_{B/b}^b(t) = [\mathbf{x}_{B/b}^b(t), \mathbf{y}_{B/b}^b(t), \mathbf{z}_{B/b}^b(t)]^T$ is the position vector of the center of buoyancy.

Appendix B. Mathematical description of the hydrodynamic loads

The fluid forces and moments can be expressed by integrating the stress on the wetted body surface as,

$$ \begin{bmatrix} \mathbf{F}_f^b(t) \\ \mathbf{M}_f^b(t) \end{bmatrix} = - \iint_{S_B(t)} \sigma \cdot n_i dS, \quad i = 1 \sim 6 \quad (B.1) $$

where $\sigma$ is the stress tensor, $S_B(t)$ is the instantaneous wetted body surface, $n_i$ are the six components of the generalized unit surface normal vector pointing out of the fluid domain, $(n_1, n_2, n_3) = \mathbf{n}$ and $(n_4, n_5, n_6) = \mathbf{r} \times \mathbf{n}$. $\mathbf{r}$ is the position vector.

By assuming an incompressible Newtonian fluid, the stress tensor $\sigma$ is expressed as,

$$ \sigma = -pI + 2\mu \mathbf{D} \quad (B.2) $$

where $p$ is the pressure, $\mu$ is the dynamic viscosity, and $\mathbf{D}$ is the rate of strain tensor, i.e. $\mathbf{D} = (\nabla \mathbf{u} + \nabla \mathbf{u}^T)/2$.

To discuss the static contribution, a dynamic pressure is introduced in the following equation.

$$ p_d = p - \rho g \cdot \mathbf{x} \quad (B.3) $$

where $p_d$ is the dynamic pressure (Rusche, 2003) and the last term on RHS can be interpreted as a static pressure. $\mathbf{x}$ is the position vector, and $\mathbf{g} = [0, 0, -\mathbf{g}]$ is the gravity acceleration.

The fluid loads can be decomposed with the definitions of the wetted body surface illustrated in Fig. B.31.

$$ \begin{bmatrix} \mathbf{F}_f^b(t) \\ \mathbf{M}_f^b(t) \end{bmatrix} = \iint_{S_B} p_d n_i dS - \iint_{S_B} 2\mu \mathbf{D} \cdot n_i dS + \iint_{\delta S_B} \rho g \cdot \mathbf{x} n_i dS + \iint_{\tilde{S}_B} \rho g \cdot \mathbf{x} n_i dS \quad (B.4) $$

where $S_B$ is the wetted body surface naturally caused by fluid-structure interaction. $\tilde{S}_B$ is the wetted body surface of the moving body below the mean free surface, and $\delta S_B$ is the difference between $S_B$ and $\tilde{S}_B$.

The first three terms of Eq. (B.4) are the dynamic contributions to the fluid loads. The first and second terms are the hydrodynamic
<figure><img src="image_36.png" /><figcaption>Fig. C.32. Flow chart for obtaining mass, the center of gravity and moments of inertia of the model from the measurement $F_m$ and $M_m$ at $O_b$. Here, $m$ is the mass, $(\mathbf{x}_{G/b}^b, \mathbf{y}_{G/b}^b, \mathbf{z}_{G/b}^b)$ is the vector of the center of gravity from $O_b$ expressed in the body-fixed reference frame and $I_G$ is the moment of inertia at the center of gravity $G$.</figcaption></figure>

<caption>Table C.5</caption>

Estimation of the experimental models' mass and moment of inertia from the measurement in the forced oscillation tests in air.

<table><thead><tr><th>Model</th><th>Draft (d/D<sub>c</sub>)</th><th>m<br>[kg]</th><th>z<sup>b</sup><sub>G/b</sub><br>[m]</th><th>I<sub>G,xx</sub><br>[kg m<sup>2</sup>]</th><th>I<sub>G,yy</sub><br>[kg m<sup>2</sup>]</th><th>I<sub>G,zz</sub><br>[kg m<sup>2</sup>]</th><th>Difference (estimation/CAD-1) [%]</th></tr></thead><tbody><tr><td rowspan="2">Model A (D<sub>d</sub>/D<sub>c</sub> = 1.0)</td><td>Deep (2.2)</td><td>36.9</td><td>-0.27</td><td>8.7</td><td>8.8</td><td>0.9</td><td>0.2</td><td>3.8</td><td>-0.1</td><td>0.6</td><td>0.0</td></tr><tr><td>Shallow (0.7)</td><td>24.7</td><td>-0.02</td><td>2.3</td><td>2.3</td><td>0.6</td><td>1.9</td><td>0.0</td><td>0.9</td><td>0.4</td><td>0.0</td></tr><tr><td rowspan="2">Model B (D<sub>d</sub>/D<sub>c</sub> = 1.5)</td><td>Deep (2.2)</td><td>41.6</td><td>-0.33</td><td>10.0</td><td>10.0</td><td>1.1</td><td>0.0</td><td>3.1</td><td>0.0</td><td>0.6</td><td>0.0</td></tr><tr><td>Shallow (0.7)</td><td>29.5</td><td>-0.06</td><td>2.6</td><td>2.6</td><td>0.8</td><td>1.4</td><td>0.0</td><td>0.4</td><td>0.4</td><td>0.0</td></tr><tr><td rowspan="3">Model C (D<sub>d</sub>/D<sub>c</sub> = 2.9)</td><td>Deep (2.2)</td><td>64.2</td><td>-0.48</td><td>14.8</td><td>14.9</td><td>4.8</td><td>-0.2</td><td>0.0</td><td>-0.5</td><td>0.5</td><td>0.8</td></tr><tr><td>middle (1.4)</td><td>58.1</td><td>-0.31</td><td>8.8</td><td>8.8</td><td>4.6</td><td>0.3</td><td>3.3</td><td>-0.5</td><td>0.0</td><td>0.9</td></tr><tr><td>Shallow (0.7)</td><td>52.1</td><td>-0.14</td><td>4.9</td><td>4.9</td><td>4.5</td><td>0.7</td><td>0.0</td><td>0.6</td><td>1.4</td><td>0.9</td></tr><tr><td rowspan="2">Model D (D<sub>d</sub>/D<sub>c</sub> = 2.9τ)</td><td>Deep (2.2)</td><td>60.9</td><td>-0.47</td><td>14.3</td><td>14.4</td><td>4.4</td><td>-0.2</td><td>2.2</td><td>-0.6</td><td>0.3</td><td>0.9</td></tr><tr><td>Shallow (0.7)</td><td>48.8</td><td>-0.13</td><td>4.7</td><td>4.7</td><td>4.1</td><td>0.7</td><td>0.0</td><td>0.9</td><td>1.1</td><td>0.7</td></tr><tr><td rowspan="2">Model E (D<sub>d</sub>/D<sub>c</sub> = 3.5)</td><td>Deep (2.2)</td><td>79.7</td><td>-0.54</td><td>18.4</td><td>18.5</td><td>9.8</td><td>-0.3</td><td>0.0</td><td>-0.2</td><td>0.6</td><td>1.2</td></tr><tr><td>Shallow (0.7)</td><td>67.6</td><td>-0.17</td><td>7.6</td><td>7.6</td><td>9.4</td><td>0.3</td><td>0.0</td><td>1.6</td><td>1.7</td><td>1.4</td></tr></tbody></table>

Note 1. $\mathbf{x}_{G/b}^b$ and $\mathbf{y}_{G/b}^b$ < 0.

Note 2. $I_b = I_G - m(\mathbf{r}_{G/b}^b(\mathbf{r}_{G/b}^b)^T - (\mathbf{r}_{G/b}^b)^T \mathbf{r}_{G/b}^b I_{3 \times 3})$ where $I_{3 \times 3}$ is the identity matrix.

pressure forces and the viscous forces on $S_B$. The third term is the hydrodynamic contribution of the hydrostatic pressure on $\delta S_B$ due to the disturbed free surface. Therefore, the hydrodynamic loads can be defined as below.

$$ \begin{bmatrix} \mathbf{F}_{hd}^b \\ \mathbf{M}_{hd}^b \end{bmatrix} = \iint_{S_B} p_d n_i dS - \iint_{S_B} 2\mu \mathbf{D} \cdot n_i dS + \iint_{\delta S_B} \rho_g \cdot x n_i dS \quad (B.5) $$

The last term of Eq. (B.4) provides the hydrostatic force on $\tilde{S}_B$ which can be defined as:

$$ \begin{bmatrix} \mathbf{F}_{hs}^b \\ \mathbf{M}_{hs}^b \end{bmatrix} = \iint_{\tilde{S}_B} \rho_g \cdot x n_i dS \quad (B.6) $$

where the loads are the hydrostatic pressure contributions to the moving body in calm water.

## Appendix C. Mass and moments of inertia of the model

A procedure to obtain the inertia matrix of the model by imposing motions using the hexapod is presented. Assuming the body is moving in the air, the fluid load is negligible due to the low air density compared to the water density ($\rho_{air} \ll \rho_{water}$). Therefore, the measured load consists of inertia and gravity loads. The position of the center of gravity and the inertia matrix are estimated with two series of tests.
Firstly, in quasi-static motions of pure roll and pitch (very low-frequency motion, 0.02 Hz), the contributions of the velocity and acceleration components in the inertia loads can be neglected. This assumption leads to a simplified expression of the equation of motion. The mass *m* and position of the center of gravity (x<sub>G/b</sub><sup>b</sup>, y<sub>G/b</sub><sup>b</sup>, z<sub>G/b</sub><sup>b</sup>) can be obtained from the measured forces F<sub>m</sub><sup>b</sup> and moments M<sub>m</sub><sup>b</sup> from the load cell. Then, the moments of inertia at the center of gravity *I*<sub>G</sub> are deduced from pure dynamic motions of the roll, pitch, and yaw at higher frequencies (1.0 Hz). The procedure is given in Fig. C.32.

To implement this procedure, the experimental model configurations were tested in air before the wave tank testing. During the tests, the hexapod imposed two types of motions: quasi-static and dynamic. In the quasi-static tests, the hexapod applied a motion amplitude of 10 degrees for 60 s, resulting in negligible angular velocity and accelerations of the inertia components in the measurements. On the other hand, in the dynamic tests, the hexapod applied a 5-degree amplitude motion with a period of 5 s.

The estimations of the mass, vertical center of gravity, and moments of inertia are summarized in Table C.5 for all configurations. The results show less than 4 % difference compared to the CAD design tool (SolidWorks). These values were used to calculate the inertial and gravitational loads.

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