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The experiment was carried out in a laboratory wave flume facility in Kyushu University, Japan. The wave flume having the size of 20 m long, 0.8 m wide, and 1.0 m deep, which has equipped with two vertical motion wave generators in the both ends of the flume as shown in Fig. 1. Two wave generators operated by a computer are mainly used to make a generation of regular waves and at the same time to break up the propagating wave by active absorbing mechanism. Therefore, the designated amplitude and wave period of regular waves could be generated by this active wave control system. In order to make a variety of model experiment, the circular cylinder model is placed in the middle of the wave flume and it is connected with a designated mooring system as shown in Fig. 2.

In order to observe the synchronous wave height, a wave height gauge is installed at 75 cm further upstream of the cylinder model. The measured voltage signals are recorded in a 32 multi-channel data acquisition system (NEC OMNIACE RA2800). In addition, to make an accurate measurement of the movement of cylinder, a high density (HD) video camera was used in the wave flume. The HD camera was set to be the frame resolution of 1,920×1,080 pixel², data transfer rate 16,885 kbps, and the frame rate 30 fps (frames per second), etc. In order to find the exact position of floating body from the acquired images, two black points marked at 5 cm vertical distance on the cylinder surface were used as reference points. In order to find the movement of the cylinder, a video camera is located at 3 m apart from the side of wave flume with the aim to record the model movement in 30 fps. After obtaining the video files, based on two black points can be analyzed by using a image processor (Dipp-motion PRO, Kato Koken) so that the heave and pitch motions were acquired.

Fig. 1. 
Schematic diagram of wave flume experiment

Fig. 2. 
Schematic diagram for top view of mooring system

In this study, the basic model (model 0) is a single cylinder connected with a ballast bolt. Other various models depending on heave plates are orderly numbered from 1 to 4. The detailed configurations are shown in Tables 1 and 2.

The experimental model is moored at the level of center of gravity (CG) by long taut elastic lines in the both sides of model. The angle between the taut line and free surface is around 6 degs. In the experiment, we assume the mooring system has a negligible effect on heave and pitch motion of model.

In principle, the heave plate could enhance the heave added mass and viscous damping. Based on the dynamics theory, the heave and pitch natural period of a floating cylinder is given as:

where M , a₃₃, A and GML¯ are mass and added mass in heave, the area of water plane and longitudinal metacenter height, respectively. In addition, ρ and g are the density of water (the working fluid where the model is submerged) and acceleration of gravity.

The governing equation of heave motion is given as:

Equation (3) could be rearranged as an ordinary differential equation as follows:

In Eqn (4), the damping ratio ζ and the natural frequency w are defined as

For the second order ordinary differential equation, the general solution of Eqn (4) can be written as:

Finally, the motion of cylinder could be expressed in Eqn (6) and added mass a₃₃ can be obtained through the curve fitting once the motion of cylinder is known.

The added mass coefficient (C_a) is defined and calculated as follows:

However, when attaching the heave plate on the bottom of the spar, the viscous damping, which is caused by vortex shedding around the plate, becomes to the dominant damping resources. Therefore, the damping ratio could be approximately calculated by

where the logarithmic decrement, δ is defined as the natural log of the amplitude of adjacent peak.

The free decay experiment is designed to investigate the damped heave and pitch period of the models, and these values can be used for make a proper wave period for the further detailed experiment. The model is firstly placed in initial vertical displacement and pitch angle for finding the characteristic curve of the heave and pitch free decay experiment, and then the subsequent motions were recorded and stored (see Figs. 3-6).

As shown in figures, the heave and pitch damped period can be calculated by fitting the curves. (i.e., Model 0: 0.78s and 1.12s, Model 1: 0.89s and 1.1s, Model 2 : 0.90s and 1.08s, and Model 4: 0.86s and 1.08s). In Figs. 5 and 6, the heave plate has little effect on the pitch added mass, whereas the heave added mass varies significantly depending on different heave plates (see Figs. 3 and 4). In order to quantify the added mass and damping ratio, the added mass coefficients of natural frequency and damping ratio needs to be estimated and summarized in Table 3.

Fig. 3. 
Heave motion for models with various diameter

Fig. 4. 
Heave motion for models with various thickness

Fig. 5. 
Pitch motion for models with various diameter

Fig. 6. 
Pitch motion for models with various thickness

Phillip et al.^[6] conducted free decay test using different spar-buoy structure. The measured added mass coefficient and damping ratio are shown in Table 4.

As shown in Tables 3 and 4, the (spar-buoy) cylinder has relatively low added mass coefficient, whereas the added mass coefficient increases up to around 0.2 when having the heave plate on the bottom of the cylinder. In addition, since the heave plate is installed apart from the bottom of the cylinder by the thin rod, the separated heave plate has more efficient performance to increase the heave added mass compared to those of Phillip et al.^[6]

In terms of damping effect, heave plate shows better effect on enhancing the damping ratio of spar-buoy structure. The damping ratio for the Models 1 and 2, which installed with a larger diameter heave plate, increases by 212% and 258%, respectively, when compared with the single spar, Model 0. To give a summary of the Table 3, the larger diameter and thinner heave plate indicates relatively better damping effect on suppress the heave motion of models.

The results from Phillip et al.^[6] also shows the similar tendency of the effects of heave plate on damping ratio.

In this study, the wave amplitude, heave and pitch motion responses were also calculated by applying the LSM (Least Square Method) curve-fitting and Taylor series expansion. As shown in Figs. 7-10, the Response Amplitude Operator (RAO) for the heave and pitch motion with a variety of parameters such as diameter, thickness of heave plate were acquired and analyzed in our floating models. Depending on the parametric cases, the comparison was made with different group. The abscissa of figures denotes the non-dimensional wave period λ/L .

Fig. 7. 
Heave motion response for models with various diameter

Fig. 8. 
Heave motion response for models with various thickness

Fig. 9. 
Pitch motion response for models with various diametera

Fig. 10. 
Pitch motion response for models with various thickness

A series of regular waves were generated for all the models over a range of wave periods from 0.51s to 1.24s. The dispersion relation is given by:

The corresponding wave period and the nondimensional quantity λ/L , where L is equal to diameter of cylinder for this study, are also given in Table 5.

In Figs. 7 and 8, the model 0 (base model) shows a dominant peak RAO at λ/L≅17, but for the model 1, 2, and 4 with heave plate the peak heave RAO value changes to λ/L≅25 due to the increased added mass. In addition, the peaks in heave RAO distribution are also suppressed because of the extra heave damping in model 1, 2, and 4. Regarding the damping effect, the larger diameter of heave plate shows relatively better in terms of efficiency, whereas the thickness of heave plate is little effective.

Figures 9 and 10 show the pitch RAO distribution against non-dimensional quantity λ/L , and the heave plate effect seems mostly negligible. Interestingly, it is found that a local peak in λ/L≅17 is due to the large heave motion so that the heave motion could enhance the pitch motion in the heave resonant region.