Spoke Dimension on the Motion Performance of a Floating Wind Turbine with Tension-Leg Platform

The tension-leg platform (TLP) supporting structure is a good choice for floating offshore wind turbines because TLP has superior motion dynamics. This study investigates the effects of TLP spoke dimensions on the motion of a floating offshore wind turbine system (FOWT). Spoke dimension and offshore floating TLP were subjected to irregular wave and wind excitation to evaluate the motion of the FOWT. This research has been divided into two parts: (1) Five models were designed based on different spoke dimensions, and aerohydroservo-elastic coupled analyses were conducted on the models using the finite element method. (2) Considering the coupled effects of the dynamic response of a top wind turbine, a supporting-tower structure, a mooring system, and two models on a reduced scale of 1 : 80 were constructed and experimentally tested under different conditions. Numerical and experimental results demonstrate that the spoke dimensions have a significant effect on the motion of FOWT and the experimental result that spoke dimension can reduce surge platform movement to improve turbine performance.


Introduction
Wind energy is one of the most environmentally friendly renewable energy sources that are used to generate power.Global interest in wind energy has increased because of abundant global offshore wind resources.Offshore wind resources are distributed in areas within 5-50 km from coastlines at water depths greater than 30 m.Studies show that fixed foundations, such as monopolies, jackets, and gravity, are not economic for offshore wind turbines with depths greater than 50 m.To solve this issue, floating foundation concepts, such as spar, semisubmersible, and tension-leg platform (TLP) supporting structures, have been used [1,2].
As an active research direction, the TLP supporting structure concept for offshore wind turbines is an appropriate choice for intermediate water depths because of its superior motion and dynamic features compared with other floating concepts [2][3][4][5][6].Withee [7] developed the first TLP wind turbine supporting structure in 2004.Wayman et al. [8] performed fully coupled time-domain simulations of system responses for a 1.5 MW wind turbine mounted on a TLP floater subject to wind and wave forces.Bachynski and Moan [9] analyzed diverse parametric single-column TLP designs and found that a larger spoke radius had the potential to display the best overall behavior under special environmental conditions; however, they did not verify the hypothesis.Crozier studied the effects of design parameters on performance based on a 10 MW wind turbine [10].Another study [11] proposed a new TLP concept by combining the traditional TLP and spar structure; this concept is called the MIT-TLP (Massachusetts Institute of Technology tensionleg platform) model.The advantage of this model is that, without the moorings, this design is stable in calm sea states when the turbine is not operating.Matha [12] modified the spoke length of the MIT-TLP model and corrected the faults to improve motion using the FAST software program; this modified model is called the NREL-TLP (National Renewable Energy Laboratory Tension Leg Platform) model.On the basis of a popular 5 MW wind turbine designed by NREL, Martin built a wind-and-wave-based basin-wide numerical model of a 5 MW wind turbine on a scale of 1 : 50; this model was intended to assess a commercially viable floating wind turbine structure [13].However, in the NREL-TLP model, the effect of the spoke on the motion was not determined.Therefore, in this work, we investigated the spoke effect in detail for the first time.
For the model test, Goupee et al. [14] investigated the unique behavior of various floating wind turbine platforms and conducted model tests at three different floating-type wind turbines on a scale of 1 : 50 using a wind-and-wavebased basin-wide model at the Maritime Research Institute of the Netherlands.Coulling conducted model tests on a semisubmersible floating wind turbine system on a scale of 1 : 50.They verified the effectiveness of the experimental data for a floating wind turbine model, which was constructed by NREL [15].Ren et al. performed a model test on a floatingtype wind turbine system on a scale of 1 : 60 to investigate the influence of wind-wave coupling effects on its performance.They also conducted a numerical simulation and compared the results with that of the model output [16].Next, they proposed another TLP system with mooring cables and compared the model results of both TPL systems [17].
In this research, the spoke dimension effect will be investigated through numerical simulations and modeling tests.Six numerical models were designed with consideration of different spoke dimensions; then the added mass, damping, exciting force, and response amplitude operators (RAO) were compared.Afterwards, two models considered for the spoke dimension experiment on a reduced scale of 1 : 80 were demonstrated in the Harbin Institute of Technology Joint Laboratory of Wind Tunnel and Water Flume.The displacements of these two floating wind turbines in two different directions were compared and analyzed.

NREL 5 MW Offshore Wind Turbine and Environmental Conditions.
In 2006, the National Renewable Energy Laboratory (NREL, US) provided a detailed design of a 5 MW offshore wind turbine for the preliminary design of support structures based on a Repower wind turbine which is produced by Repower System AG in Germany.The rotor and hub diameters are 126 and 3 m, respectively, while the hub height is 90 m.The cut-in, rated, and cut-out wind speeds are 3, 11.4, and 25 m/s, respectively.The maximum rotor and generator speed are 12.1 rpm.The 1 and 3 p frequency values were between 2.7 and 3.5 s [18].This wind turbine was chosen because the required parameters are easily available and because many studies have already used it as a base wind turbine.
The environmental conditions considered in this study are shown in Table 1.These values were adopted from previous studies [6].Similar to other studies, the water depth for this study was fixed at 200 m [9,12,14,19].The energy density of waves was defined by Hua et al. [20] in the Pierson-Moskowitz wave spectrum: where  is the wave frequency,   is the significant wave height, and   is the zero-crossing period.[11], Zambrano et al. [21], and Wayman et al. [8] reported the requirements for conceptual design; however, they did not consider the entire lifecycle, from installation scenarios to extreme situations.Wang [6] proposed a new design requirement for TLP based on the NREL-TLP model.In this study, we considered the new design requirement, which we call the modified NREL-TLP model, as the basic requirement for conceptual design.
Table 2 shows the four designs used in this study, which were selected based on the displacement and mass considering cost and research.In this study, two other models were considered, namely, TLP-0 and TLP-5; the only difference between TLP-1/TLP-4 and TLP-0/TLP-5 is that the spoke effect was not considered in the calculations.
Computer algorithms were used to simulate the TLP supporting structure.CATIA software was used to generate a geometric model, and HyperMesh was used to create a mesh model.Ansys AQWA software was used to calculate hydrodynamic characteristics.An aerohydroservo-elastic coupled analysis was conducted in the time domain [13,22].This analysis was performed by using the numerical tools of FAST, which is a publicly available software.The matrices for the wind turbines and platform were transferred to the dynamic analysis module, where they were combined to represent the coupled system.To study the spoke dimension effect, TLP-0 and TLP-5 models and specially designed TLP-2 and TLP-3 models were chosen as examples.TLP-0 (TLP-5) has the same dimensions as TLP-1 (TLP-4).The mass (displacement) of TLP-0 was 1279.943T (7418.25 M 3 ) and that of TLP-5 was 2531.154T (6181.88M 3 ).Compared with TLP-1 and TLP-4, the mass increased by 12.8% (18.3% displacement) and 4% (19.6% displacement), respectively.TLP-2 and TLP-3 were designed so that the spoke displacement constitutes up to 37% of the entire displacement.Furthermore, the entire displacement of TLP-2 was only 83% of the displacement of TLP-3.Next, the differences in detail are analyzed.Figure 2 shows the FEM mesh for the different models.

Response Amplitude Operators (RAO).
The equation of motion that governs the rigid body motions of a floating structure consists of standard Newtonian equations of motion and is summarized in a matrix form below, describing the six modes of motion: TLP-1 [6]   (a)  where  added is the added mass matrix,  WT is the mass matrix of the wind turbine at a constant wind speed,  structure is the mass matrix of the platform,  structure is the damping matrix of the platform,  WT is the damping matrix of the wind turbine,  WT is the stiffness matrix of the wind turbine,  structure is the stiffness matrix of the platform,  mooring is the stiffness matrix of the mooring system, and ζ , ζ , and  are the acceleration, velocity, and displacement of the system.
In this study, RAO represents the nondimensional response of a system to a unit-amplitude incident wave in a direction along the  coordinate, that is, the zero incident angle [13].The motion equations that control the system's linear dynamic motion are summarized in (3) [10] as follows: For the translational modes of motion, RAO is expressed by [10] RAO trans modes () =       ()      wave .
For the rotational modes of motion, RAO is expressed by [10] RAO rot modes () =       () where  wave represents the wave amplitude and  is the cylinder radius.Although RAO are not based on sea state, the damping and stiffness of the wind turbine are based on wind speed.were used: a high-frequency force balance, data acquisition, and its analysis system for receiving dynamic signals, a highprecision laser displacement meter, an acceleration sensor, a current meter, and a floating body instrument that comprises six components.Figure 4(a) shows the high-frequency force balance.The forces along the two lateral directions, which are perpendicular to the axial direction of the balance, have ranged between 0 and 660 N. Similarly, the lateral bending moments along the two directions (i.e.,   and   ) are ranged between 0 and 60 Nm, and the torque (i.e.,   ) is ranged between 0 and 60 Nm.The measuring error is, usually, less than 1%.length, which is ranged between 300 and 500 mm. Figure 4(c) shows the system for dynamic signal data acquisition and analysis.A system of data acquisition and its analysis of a dynamic signal (i.e., NI-PXI, National Instruments, America) have been used during the experiments of this paper, which has (24 + 12) dynamic acquisition channels.It can meet the requirements of a dynamic-response data acquisition of a large and complicated structure, which may be subject to the action of the wind and the wind-wave; it can be shown in Figure 3.

Introduction of Experimental Models.
In order to compare the result for two models, parameters have been designed in detail.Table 3 shows the model parameter, and A tower was installed at the top of both models.The blade of a wind turbine was prepared with a ratio of 1 : 80.The rotating speed of a wind turbine was approximately 8.3 rev/min, and the facial area of the blade was approximately 1.766 m 2 .Under extreme wind speed weather conditions, the The added mass matrices in different directions are shown in Figure 6.The difference is evident if spoke dimension is not considered.In A11 (surge-surge direction), TLP-1 and TLP-4's values were larger than those of TLP-0 and TLP-5; however, this finding is not evident in other directions.If we consider displacement only, then TLP-3 is larger than TLP-2, and all the values are larger for TLP-3.However, displacement does not only affect A11 because TLP-0 displacement was lower than TLP-4, while the TLP-0 A11 value was larger than TLP-4 value.In A33 (heave-heave direction) and A55 (pitchpitch direction), the values increase when we considered the spoke dimension effect.A comparison between the TLP-2 and TLP-3 results showed that displacement and values for A33 and A55 were larger.In A66 (yaw-yaw direction), TLP-0 and TLP-5 results were zero.However, the magnitude of A66 was very large in TLP-1 and TLP-4 and cannot be ignored.Moreover, TLP-1's result was four times larger than TLP-4.When the spoke dimension effect was considered, the added mass matrix always increased.This effect was most evident in A66 because the value increased from zero to 1E5 and 4E5 in A42 (roll-sway direction) and A15, respectively, because the unit was E5; designers should consider this result in the near future.For the same spoke dimension ratio, when the total displacement increases, the added mass matrix also increases.In the surge-pitch (A15) component, the absolute value was larger than the others.In total, the coefficient of the added mass matrices increases when the spoke dimension effect is considered, thereby being useful for damping and motion.Martin [13] assumed that all offdiagonal translational coefficients are zero.The calculations in this section show that the coefficient of mass matrices is not zero because of the spoke effect.The size effect on the overall motion will be analyzed in the next section.
The damping effect (Figure 7) approached zero at high and low frequencies; however, the four models clearly differ at intermediate frequencies.TLP-4 had the maximum damping matrix coefficients in the surge direction.The spoke size effect ensures that a larger damping coefficient can be obtained, particularly in B66 (yaw-yaw) where the value for TLP-0 and TLP-5 became zero.Thus, yaw instability may be severe in the calculation stage.A comparison between Figures 9(a), 9(b), 9(d), and 9(e) for TLP-0/TLP-1 shows that the damping coefficient decreased by 1%, 10%, 10%, and 24%, respectively.The decreases were 0.9%, 28%, 25%, and 31% for TLP-4 and TLP-5.These results indicate that the B55 value (pitch-pitch) was more sensitive to the spoke size effect.Heave direction displacement was restricted because of tension leg; therefore, B33 (heave-heave direction) was not considered in the damping matrix, and its value was assumed large enough in damping.In B55 (pitch-pitch direction), the magnitude was E5; therefore, the spoke dimension effect should be considered.In fact, the spoke dimension effect increased.At the same displacement ratio in TLP-2 and TLP-3, the change was smaller in TLP-2, but TLP-3 was larger than TLP-2, which indicates that larger displacement leads to larger damping.B66 (yaw-yaw direction) had the same situation as A66; the damping matrix was limited to zero if the spoke dimension effect was not considered.In the off-diagonal translational matrices B42 and B15, the effect was clearly enhanced damping.Interestingly, in TLP-2 and TLP-3, the off-diagonal matrices did not change.When the spoke dimension effect was considered, the damping did not always increase.In B66, the spoke dimension effect increased damping from zero to a larger value.In the offdiagonal translational matrices, the spoke dimension effect was enhanced damping.At the same displacement ratio, the off-diagonal coefficient did not change at B42 and B51.
Obviously, the exciting force reduced when the spoke effect was not considered.In this section, TLP-2 and TLP-3 had the minimum exciting force.For the surge exciting force, TLP-0 and TLP-1 had similar curves, thereby indicating that the effect was limited.The same situation occurred in TLP-4 and TLP-5.For the pitch exciting force, the trend was opposite, as shown in Figure 8(c).The exciting force of TLP-1 and TLP-4 was larger than that of TLP-0 and TLP-5, respectively.For TLP-2 and TLP-3, large displacement indicates a large pitch exciting force.The yaw exciting force exhibited the same patterns as the pitch exciting force; however, the values were the same for the yaw exciting force.Therefore, this result can be ignored.
Overall, the exciting force reduced when we did not consider the spoke effect in the translation direction, which is the opposite for rotation.The same situation occurred in TLP-2 and TLP-3.In this wave direction, the pitch exciting force was the largest, whereas the other forces were so small that they could be neglected.
For B66 (yaw-yaw), TLP-2 was the largest in all the modes and could be used to improve yaw damping.For the exciting force matrices, the surge exciting force was similar; clearly, the pitch and yaw exciting force of TLP-3 did not change.Martin [13] assumed yaw instability in his model because the spoke size effect on the yaw-yaw damping is larger and improves motion performance.

RAO Result.
The RAO of TLP models with varying dimensions were obtained from FAST (Figures 9, 10, and 11).We first considered the spoke dimension effect on TLP-0/TLP-1 and TLP-4/TLP-5.A comparison between the TLP-0 and TLP-1 results shows that the trend and curve are almost the same.The spoke dimension effect on the surge RAO at a high frequency is insignificant.In the lower frequency range, the results of TLP-0 and TLP-5 were larger than those of TLP-1 and TLP-4.In the higher frequency range, the trend was the opposite; in fact, we focused on the higher frequency range only.In this portion, TLP-0 had the maximum RAO value, and TLP-5 was larger than TLP-1 and TLP-4.This finding indicates that considering the spoke dimension effect could enhance RAO values.In the roll direction, the result of TLP-0 was smaller than that of TLP-1, thereby indicating that when the spoke dimension effect is not considered, the result is lower than the real value.In the lower frequency range, the sway RAO values of TLP-0 and TLP-5 were larger than those of TLP-1 and TLP-4.At high frequencies, the trend was the opposite.At the same time, the frequency range of TLP-5 and TLP-4 was smaller than that of TLP-0 and TLP-1; this result was possible because of the larger displacement for TLP-0 and TLP-1.In the higher frequency range, the maximum value for TLP-0 was 50% of TLP-1.For the pitch RAO value, TLP-1 and TLP-4 were larger than TLP-0 and TLP-5 at low frequencies.At higher frequencies, TLP-1 and TLP-4 were larger than TLP-0 and TLP-5.The same situation for frequency range occurred in the pitch RAO, where the maximum frequency points for TLP-1 and TLP-4 were larger than those for TLP-0 and TLP-5.In the heave RAO value, their trends were similar.The gradient for TLP-1 was larger than that for TLP-0, and the same situation occurred for the TLP-4 and TLP-5 models.A smaller displacement had a smaller yaw RAO value.The TLP-1 value was smaller than the TLP-0 value, thereby indicating that a considerable dimensional effect is better for yaw response.

Shock and Vibration
In a similar displacement ratio for TLP-2 and TLP-3, the displacement of TLP-3 was larger than that of TLP-2.The maximum surge RAO value for TLP-3 was larger than that for TLP-2.For the sway RAO values at higher frequencies, the value of TLP-3 was higher than that of TLP-2, reaching 60%.Heave RAO were similar, during 0.4 rads/s to 1.2 rad/s, the TLP-1's value is smallest and TLP-2's value is biggest, and TLP-2's displacement for spoke part is biggest.The roll RAO values were similar at lower frequency, but, at higher frequency, the maximum of TLP-2 was larger than that of TLP-3.For the pitch, the RAO of TLP-3 was larger than that of TLP-2 regardless of frequency.For sway and pitch RAO, the result of TLP-3 is larger than that for TLP-2 at any frequency.For the heave, the TLP-1's value is smallest and TLP-2's value is biggest; for the yaw RAO, TLP-4's value is smallest; for the roll RAO, the result of TLP-3 is smaller than that of TLP-2.

Model Test
Result.The mooring system is not the point of this research.The mooring system is assumed to be undamaged under various operating conditions and may be permanently fixed to a system.In reality, considering that a large external tension is applied on the tension legs, the floating structure can be fixed tightly in a floating sea.Thus, the vertical rigidity of a structure is approximately equal to infinity; that is, the experimental results of a vertical fixed system can be treated as acceptable; however, it can be movement in surge, sway, and rotation direction.Wind turbine was rotated at a preset speed, which remained unchanged just before the maximum wind speed.Figure 12 shows the process for the model test.

Combining a Typical Wind and a Regular
Wave.Three different wind speeds were used in the experiments: a rated wind speed, a maximum wind speed, and an extreme wind speed.The wind turbine operated normally on the first two wind speeds.However, the wind turbine ceased to operate at the extreme wind speed.Table 4 shows the used parameters of a wind speed and an external wave.Figure 13(a) shows the surge displacement result under the rated wind speed coupled wave loads.The maximum surge displacement results for model-TLP-1 and model-TLP-2 were 2.5 and 1 mm, respectively.Considering the whole  displacement response in 90 s for the two models, the result of model-TLP-1 was significantly higher than that of model-TLP-2.Figure 13(b) shows the surge displacement response under the maximum wind speed coupled wave loads; within 40 s, the result of model-TLP-2 was significantly lower than that of model-TLP-1, while, in the remaining 50 s interval, the results of model-TLP-2 were greater than those for model-TLP-1.In Figure 13(c), the results of model-TLP-2 were less than model-TLP-1 maximum displacement.The result in Figure 13 indicates that spoke dimension affects surge displacement in rated and extreme load conditions, and the maximum displacement of model-TLP-2 was less than the results of model-TLP-1.

Combining a Typical Wind and an Irregular Wave.
This section examines the dynamic response of a normal operating wind turbine under an irregular wave.An irregular wave referred to as a "Pierson-Moskowitz sea spectrum" (i.e., a fully developed spectrum, which is abbreviated as "PM spectra") was selected for these testing scenarios [22].PM spectrum was derived based on the measured data of the North Atlantic Ocean; the data can be applied to simulate     fully developed waves in an infinite-wave region of the sea.PM spectrum has been widely applied in oceanographic engineering because of several advantages, such as the empirical spectra, sufficient references, the method of reasonable analysis, and convenience.When we compared this situation with the aforementioned coupled operating conditions, the conditions of wind-wave coupled operation fit well with their practical conditions.Their related parameters are shown in Table 5.
The results after the use of the irregular wave PM spectrum are shown in Figure 14.For maximum surge displacement, the result of model-TLP-2 was less than that of model-TLP-1 under maximum wind speed coupling irregular wave conditions, as shown in Figure 14(a).In Figure 14(b), the extreme wind speed coupling results under irregular wave conditions in model-TLP-2 were significantly lower than the results of model-TLP-1.According to the previous model data, the spoke length of model-TLP-1 increased by 15% compared with that of model-TLP-2, but the spoke diameter model-TLP-2 was three times that of model-TLP-1.A comprehensive comparison of the surge displacement load combination for the two responses under typical wind conditions and regular wave coupling conditions showed that model-TLP-1 surge displacement was significantly higher than that of model-TLP-2.Under typical wind conditions and irregular wave coupling conditions, model-TLP-1 surge displacement was significantly higher than that of model-TLP-2 in rated wind speeds and extreme wind speeds.However, at maximum wind speed, the result of model-TLP-1 was less than that of model-TLP-2 at an interval.Data show that the scale effect of spoke helps to reduce surge displacement response, while surge displacement response is sensitive to the spoke diameter.

Model Test and Numerical Result Comparison.
As seen formerly in Table 3 and Figure 5, model-TLP-1 and model-TLP-2 have the same draft diameter and draft height; in the spoke distance model-TLP-1 is 1.17 times as model-TLP-2's date; in the spoke diameter model-TLP-2 is 3 times as model-TLP-1's date.To the spoke displacement, model-TLP-1 is 13% of model-TLP-2.Based on the model test result, in a typical wind and an irregular wave condition, regardless of extreme wind (wave period is 1.03 s) and maximum wind (wave period is 1.34 s), it is obvious that model-TLP-2 surge displacement is smaller than model-TLP-1's result.In particular, in extreme wind load case, model-TLP-2 surge displacement reduces 33% comparing with model-TLP-1.In a typical wind and a regular wave condition, model-TLP-1 surge displacement is smaller than model-TLP-2 in extreme wind (wave period is 2.672 s) and maximum wind (wave period is 2.5 s) condition, because, in the model test, spoke part cannot be deleted absolutely but only can be minimized.Back to the surge RAO result of simulation, in the lower frequency range, the result for TLP model considering spoke dimension is bigger than model result without considering spoke dimension.In the higher frequency range, the trend was the opposite.Comparing the simulation and model test result, this conclusion has been verified.And model test shows spoke dimension increase to reduce platform movement to improve turbine performance.

Conclusions
In this study, the spoke dimension effect in TLP models was evaluated and tested for the first time.Results indicate that dynamic characteristics improve when spoke dimension is considered.This finding verifies the predictions of Bachynski and Moan [9] and Matha [12], in which spokes or pontoons enhance motion behavior.The primary effect of spoke on the dynamic characteristics is that the spoke dimension effect increases the added mass matrices.This effect was most evident in the yaw-yaw direction, where the M66 value increased from zero to E5.Moreover, A42 and A15 values were not neglected.For the same spoke dimension ratio, when the total displacement increases, the added mass matrices also increase.Damping did not always increase and became constant at some point.For the off-diagonal translation matrices, the effect of the spoke dimension on damping was positive.At the same displacement ratio, the offdiagonal coefficient did not change at any point.The exciting force reduced when the spoke effect was not considered in the translation direction, and the trend was opposite to the rotation direction.When considering spoke dimension to the surge RAO and sway RAO, in the lower frequency range, the result for model considering spoken dimension is smaller than those model without consider spoken dimension, in the higher frequency range, the trend was the opposite, and the model test has been done to verify surge RAO conclusion.For the pitch and heave RAO value, at low frequencies the result for model considering spoke dimension is larger than model without considering spoke dimension; in the higher frequency range, the trend was the opposite.A smaller displacement had a smaller yaw RAO value, thereby indicating that a considerable dimensional effect is better for yaw response.
At the same displacement ratio, sway, pitch, and roll RAO were more sensitive to displacement.Model tests showed that the scale spoke increase helps reduce platform movement to improve turbine performance.In the specific conditions the surge displacement was more sensitive to the spoke diameter.

2. 2 . 1 .
Wind Tunnel and Water Flume.The Joint Laboratory of Wind Tunnel and Water Flume at the Harbin Institute of Technology[17, 23]  has one of the largest atmospheric boundary layer wind tunnels in China and is used to evaluate experiments of a water flume on the wind-wave coupling effect.The displacement and acceleration of a TLP-FOWT along two different directions (i.e., the surge and the sway) were measured and analyzed.The following experimental devices

Figure 2 :
Figure 2: Mesh result for different models.

Figure 5
Figure5shows the real view for the two models.Model-TLP-1 and model-TLP-2 have the same draft diameter and draft height; however, in the spoke distance model-TLP-1 is 1.17 times as model-TLP-2's date; in the spoke diameter model-TLP-2 is 3 times as model-TLP-1's date.To the spoke displacement, model-TLP-1 is 13% of model-TLP-2.The purpose of this design is to distinguish parameter spoke distance and spoke diameter and also is to easily compare the simulation result with model test result.A tower was installed at the top of both models.The blade of a wind turbine was prepared with a ratio of 1 : 80.The rotating speed of a wind turbine was approximately 8.3 rev/min, and the facial area of the blade was approximately 1.766 m 2 .Under extreme wind speed weather conditions, the
(2).1.Hydrodynamic Properties.The added mass, damping, and exciting force matrices are considered based on the motion and dynamic equations in(2).The calculated wave excitation force, added mass, and damping matrices are shown in Figures6, 7, and 8, respectively.A portion of the results have been shown because of symmetry characteristics.

Table 4 :
Combining a typical wind and a regular wave.

Table 5 :
Combination of a typical wind and an irregular wave.