Coupled Translational–Rotational Stability Analysis of a Submersible Ocean Current Converter Platform Mooring System under Typhoon Wave

: This study proposes a mathematical model for the coupled translational–rotational motions of a mooring system for an ocean energy converter working under a typhoon wave impact. The ocean energy convertor comprises two turbine generators and an integration structure. The conﬁguration of the turbine blade and the ﬂoating platform is designed. The two turbine blades rotate reversely at the same rotating speed for rotational balance. If the current velocity is 1.6 m/s and the tip speed ratio is 3.5, the power generation is approximately 400 kW. In the translational and rotational motions of elements under ocean velocity, the hydrodynamic parameters in the ﬂuid–structure interaction are studied. Initially, the hydrodynamic forces and moments on the converter and the platform are calculated and further utilized in obtaining the hydrodynamic damping and stiffness parameters. The 18 degrees of freedom governing equations of the mooring system are derived. The solution method of the governing equations is utilized to determine the component’s motion and the ropes’ dynamic tensions. In the mooring system, the converter is mounted under a water surface at some safe depth so that it can remain undamaged and stably generate electricity under typhoon wave impact and water pressure. It is theoretically veriﬁed that the translational and angular displacements of the converter can be kept small under the large wave impact. In other words, the water pressure on the converter cannot exceed the predicted value. The relative ﬂow velocity of the convertor to the current is kept ﬁxed such that the power efﬁciency of convertor can be maintained as high. In addition, the dynamic tension of the rope is far less than its breaking strength.


Introduction
Ocean current power generation is a potential renewable energy technology. The Taiwan Kuroshio current has a potential capacity of over 4 GW [1]. However, the seabed beneath the Kuroshio current is almost over 1000 m in the area mentioned above. The deep mooring technology is essential for harnessing that energy. Additionally, the typhoon wave impact affects the operation of the ocean power generation system. Hence, there is a need to model and develop a technology overcoming these limitations.
So far, Chen et al. [1] successfully tested the 50-kW ocean current convertor mooring to the 850 m deep seabed in the Taiwan Pingtung sea area. IHI and NEDO [2] tested a 100 kW ocean current convertor mooring to the 100 m deep seabed beneath the Japan Kuroshio current. The current converter generated approximately 30 kW under the current speed of 1.0 m/s. These two experiments were conducted under the small excitation of wave and in a few weeks. The goal of these experiments was to test the power performance of the developed convertors. However, it is for a commercial power farm that the convertor will be safe working for a long time and under different wave impact. Therefore, the deep 2 of 34 mooring theory and technology for the ocean current convertor system are in great need of development. Nobel et al. [3] presented the standards and guidance for the development and testing of the devices for marine renewable energy.
Lin et al. [4] investigated the dynamic stability of the mooring system under regular wave and ocean current. The significant effects of some parameters on the dynamical stability of the mooring system were detected. The lightweight high-strength PE (HSPE) mooring rope was determined to be suitable for the deep mooring system. C'atipovic et al. [5] investigated the hydrodynamic damping force of fiber mooring lines taking longitudinal deformation by the finite element method. Lin and Chen [6] developed the linear elastic model for the mooring system with PE mooring rope. They proposed a methodology to protect the convertor from the typhoon wave-current impact. The protection function of the proposed methodology under Typhoon wave impact was theoretically verified. Lin et al. [7] investigated the dynamic stability of the mooring system for surfaced convertor under the regular wave during non-typhoon periods and steady ocean current. Lin et al. [8] proposed a mooring system that enabled the energy convertor to work under typhoon wave impact. The plane translational motion of the mooring system was simulated in the linear elastic model. The concentrated mass assumption was made. Meanwhile, only the hydrodynamic forces of the convertor and platform were considered in the surge motion.
The mathematical model of the mooring system is also important for wave energy converters (WEC). Davidson and Ringwood [9] reviewed the mathematical modeling of mooring systems for wave energy converters. Chen et al. [10] investigated the waveinduced motions of a floating WEC with mooring lines by using the Smoothed Particle Hydrodynamics (SPH) method. Xiang et al. [11] proposed the finite element cable model to study the performance of a buoy mooring system. Paduano et al. [12] validated the quasi-static and dynamic lumped-mass models. Touzon et al. [13] compared a linearized frequency domain model, a non-linear quasistatic time domain model, and a non-linear dynamic model for WEC. Xiang et al. [14] investigated the dynamic response of a floating wind turbine foundation with a Taut Mooring System.
Anagnostopoulos [15] studied the dynamic performance of offshore platforms under wave loadings in the Morison model. It was determined that the effect of hydrodynamic damping on the resonant response of the structure is significant. Bose et al. [16] studied the dynamic stability of an airfoil supported by a spring. The problem of fluid-structure interaction is usually solved by using numerical methods such as the boundary element method [17], the finite volume method [18], the Lagrangian-Eulerian Method [19], the particle-based method [20], and the hybrid methods [21].
Lin et al. [5][6][7] investigated the plane translational motion of the mooring system in the linear elastic model. The concentrated mass assumption was made but the fluidstructure interaction (FSI) was not completely considered. In this study, a mooring system for an ocean energy convertor that is working under the typhoon wave impact is proposed. The mathematical model of the coupled translational-rotational motions of the system is derived. The configuration of the turbine blade and the floating platform is designed. The hydrodynamic forces and moments on the operational convertor and the platform in motion are determined by using the finite volume method. The damping effect of the fluid-structure interaction on the stability of the mooring system under typhoon wave is investigated.

Mathematical Model
To avoid the typhoon wave impact, the energy convertor and the floating platform were submerged to a depth of approximately 60 m, as shown in Figure 1. Therefore, the direct impact of the typhoon wave is almost negligible. In this study, the translationalrotational response of the mooring system under coupled wave-ocean effect is investigated. The translational motions include 'heave', 'surge', and 'sway'. The rotational motions include 'pitch', 'roll', and 'yaw'. The ocean energy convertor is composed of two turbine generators and an integration structure. When ocean currents flow through the energy convertor, the turbine blade rotates and drives the power generator to generate electricity. Meanwhile, the convertor and the floating platform are subjected to the hydrodynamic force and moment due to the ocean current-structure interaction. Lin and Chen [3] determined that the HSPE rope could be assumed as a straight line over a certain amount of ocean current drag force because the force deformation of the HSPE rope was negligible. The linear elastic model proposed by Lin and Chen [3] is used to analyze the motion equation of the overall mooring system.
To avoid the typhoon wave impact, the energy convertor and the floating platform were submerged to a depth of approximately 60 m, as shown in Figure 1. Therefore, the direct impact of the typhoon wave is almost negligible. In this study, the translationalrotational response of the mooring system under coupled wave-ocean effect is investigated. The translational motions include 'heave', 'surge', and 'sway'. The rotational motions include 'pitch', 'roll', and 'yaw'. The ocean energy convertor is composed of two turbine generators and an integration structure. When ocean currents flow through the energy convertor, the turbine blade rotates and drives the power generator to generate electricity. Meanwhile, the convertor and the floating platform are subjected to the hydrodynamic force and moment due to the ocean current-structure interaction. Lin and Chen [3] determined that the HSPE rope could be assumed as a straight line over a certain amount of ocean current drag force because the force deformation of the HSPE rope was negligible. The linear elastic model proposed by Lin and Chen [3] is used to analyze the motion equation of the overall mooring system. Based on the facts for ocean current energy converters (OCEC), the following assumptions are made:

−
The current flow is steady. − The HSPE mooring ropes are used. − Under the ocean velocity, the deformed configuration of the HSPE rope is nearly straight. − The elongation strain of the ropes is small. − The translational and rotational displacements of the components are small. − The tension of the rope is considered uniform.
These displacements of the component and tensions of ropes include (1) the static one under the steady current only, (2) the dynamic one due to the wave impact and current. The global translational and the rotational displacement of the component are expressed as The total tensions of the ropes are expressed as Based on the facts for ocean current energy converters (OCEC), the following assumptions are made: − The current flow is steady. − The HSPE mooring ropes are used. − Under the ocean velocity, the deformed configuration of the HSPE rope is nearly straight. − The elongation strain of the ropes is small. − The translational and rotational displacements of the components are small. − The tension of the rope is considered uniform.
These displacements of the component and tensions of ropes include (1) the static one under the steady current only, (2) the dynamic one due to the wave impact and current. The global translational and the rotational displacement of the component are expressed as The total tensions of the ropes are expressed as

Static Displacements and Equilibrium under the Steady Current Only
Under the steady current only, the static displacements of the components are obtained: x 1s = H bed − L C = L A sin θ As , y 1s = L A cos θ As , z 1s = 0, x 2s = H bed − L D = x 1s − L B sin θ Bs , y 2s = y 1s + L B cos θ Bs , z 2s = 0, The global setting angle θ As of rope A is The global setting angle θ Bs of rope B is Under the steady current only, the static equilibrium of the platform in the y-direction is The static equilibrium of the platform in the x-direction is T Cs + F B1s = T As sin θ As + T Bs sin θ Bs + W 1 .
The static equilibrium of the energy convertor in the y-direction is The static equilibrium of the energy convertor in the x-direction is The static equilibrium of the pontoon 3 in the x-direction is The static equilibrium of the pontoon 4 in the x-direction is The damping force on the pontoon is negligible. Because the length of the rope connecting the platform and pontoon 3 is long and the connecting point runs through the mass center of the platform, the rotational motion of pontoon 3 is not affected by the rotational motion of the platform. The dynamic equilibrium of the pontoon 3 in the heaving motion is M 3
Considering the linear elastic model, the dynamic tension of the rope C is [5] where K Cd and δ Cd are the effective spring constant and the dynamic elongation of rope C, respectively. Dynamic elongation is the difference between the dynamic and static lengths of rope C. Further, by using the Taylor formula, the following is obtained: where Assume the coordinates of the pontoons 3 and 4 as shown in Figure 2: where Cd K and Cd  are the effective spring constant and the dynamic elongation of rope C, respectively. Dynamic elongation is the difference between the dynamic and static lengths of rope C. Further, by using the Taylor formula, the following is obtained: where ( ) ( ) (  )   2  2  2  3  1  3  1  3  1  Cs  s  s  s  s  s  s   L  x  x  y  y  z Assume the coordinates of the pontoons 3 and 4 as shown in Figure 2: The wave elevations at the pontoons 3 and 4 are  The wave elevations at the pontoons 3 and 4 are x w,pontoon4 = H w0 sin(Ωt + ϕ).
The corresponding dynamic buoyance of the pontoon 3 due to the difference in wave elevation and the vertical dynamic displacement is where f Bs = A Bx ρgH w0 . Substituting Equations (14) and (20) into Equation (13), one obtains ..
where the third term is the restoring force. The last term is the wave exciting force.

Equation of Heaving Motion for Pontoon 4
The dynamic equilibrium of the pontoon 4 in the heaving motion is ..
Considering the linear elastic model, the dynamic tension of the rope D is [5] where K Dd and δ Dd are the effective spring constant and the dynamic elongation of the rope D, respectively. The dynamic elongation is the difference between the dynamic and static lengths of the rope D. Further, by using the Taylor formula, the following is obtained: where The corresponding dynamic buoyance of the pontoon 4 due to the difference in wave elevation and the vertical dynamic displacement is  ..
where the third term is the restoring force. The last two terms are the wave exciting force.

Equation of Heaving Motion of the Platform
The dynamic equilibrium of the floating platform in the heaving motion is .. where Substituting Equations (7) and (30) into Equation (29), one obtains ..
where the hydrodynamic force on the floating platform due to the fluid-structure interaction is expressed in Taylor series as follows: ϕ kz ≡ (s k1 , s k2 , s k3 , s k4 , s k5 , s k6 , s k7 , s k8 , s k9 ), k = 1, 2. When the symmetry configuration of the platform is considered, the hydrodynamic force on the platform in the x-direction under the current only is f Px (V, 0, 0, 0, 0, 0, 0, 0, 0, 0) = 0. Considering small oscillation, the higher-order terms are neglected later. The right-handed side second term of Equation (32) is the hydrodynamic force due to the fluid-structure interaction.
The dynamic tensions of ropes A and B are The dynamic elongation is the difference between the dynamic and static lengths, δ βd = L βd − L βs , β = A, B. Using the Tylor formula, the dynamic elongations are derived, x 1d + f damp,Px + f sti f ,Px where the dynamic effective masses of the rope A, M e f f ,j , j = x, y, z are listed in Appendix A.
The hydrodynamic damping force f damp, s 1j and the hydro- s 1j on the platform about the x-axis due to the FSI.

Equation of Heaving Motion for the Convertor
The dynamic equilibrium of the convertor in the heaving motion is where the hydrodynamic force due to the motion of the convertor is expressed in Taylor series as follows: f Tx (V, s 21 , s 22 , · · · , s 29 , TSR) = f Tx (V, 0, 0, · · · , 0, 0, TSR) When the symmetry configuration of the convertor is considered, the hydrodynamic force on the convertor in the x-direction under the current and operation of blades is f Tx (V, 0, 0, 0, 0, 0, 0, 0, 0, 0, TRS) = 0. Considering small oscillation, the higher-order terms are neglected later.
x 2d + f damp,Tx + f sti f ,Tx + T Bs cos θ Bs where the hydrodynamic damping force f damp,  .

Equation of Surging Motion of the Convertor in the y-Direction
The dynamic equilibrium of the convertor in the surging motion is The hydrodynamic force on the convertor is expressed as in which f Tys = f Ty (V, 0, 0, 0, 0, 0, 0, 0, 0, 0, TRS) = C DTy .
where the hydrodynamic damping force f damp, s 2j on the convertor about the y-axis due to the FSI.

Equation of Surging Motion of the Pontoon 3 in the y-Direction
The dynamic equilibrium of the pontoon 3 in the surging motion is where sin φ Cy = (y 3d − y 1d )/L C . T C = T Cs + T C d , in which T Cs and T Cd are the static and dynamic tensions. It is observed from Equation (7) that for static equilibrium of the platform, the static tension T Cs and the buoyancy of the platform F B1s are lift forces. If the designed buoyancy F B1s is not sufficient, the static tension T Cs must be increased. In this study, the static tension T Cs is considered to be significantly larger than the dynamic tension T Cd . The horizontal impact force of regular wave on the pontoon 3 is F 3wave = F wave,3 sin ωt, where F wave,3 = C wave,3 H wave , in which C wave,3 is the wave impact coefficient depending on the geometry of the pontoon 3, and H wave is the wave amplitude. The y-and z-direction components of wave force are where A 3y = F wave,3 cos α and A 3z = F wave,3 sin α.

Equation of Surging Motion of the Pontoon 4 in the y-Direction
The dynamic equilibrium of the pontoon 4 in the surging motion is where sin φ Dy = (y 4d − y 2d )/L D . T D = T Ds + T D d , in which T Ds and T Dd are the static and dynamic tensions. The horizontal impact force of regular wave on the pontoon 4 is 4 is the wave impact coefficient depending on the geometry of the pontoon 4. The y-and z-direction components of wave force are where A 4y = F wave,4 cos α and A 4z = F wave,4 sin α.

Equation of Swaying Motion of the Platform
The dynamic equilibrium of the floating platform in the swaying motion is .
The hydrodynamic force is Considering small displacements and based on Equations (2), (14), (33) and (51), one obtains where the hydrodynamic damping force f damp, s 1j on the platform about the z-axis due to the FSI.

Equation of Swaying Motion of the Convertor
The dynamic equilibrium of the convertor in the swaying motion is where the hydrodynamic force is Considering small displacements and based on Equations (2), (24), (33) and (54), one obtains where the hydrodynamic damping force f damp, s 2j on the convertor about the z-axis due to the FSI.

Equation of Swaying Motion for the Pontoon 3
The dynamic equilibrium of the pontoon 3 in the swaying motion is where sin φ Cy = (y 3d − y 1d )/L C . Considering small displacements and substituting Equations (2), (14) and (45) into Equation (56), one obtains

Equation of Swaying Motion of the Pontoon 4
The dynamic equilibrium of the pontoon 4 in the swaying motion is

Equation of Yawing Motion of the Convertor
The dynamic equilibrium of the convertor in the yawing motion is where R TBx is the distance between the center of gravity and the rope B about the x-axis.
Considering small displacement and substituting Equation (34) into Equation (60), one obtains where the hydrodynamic damping moment m damp,Tx = −

Equation of Rolling Motion of the Convertor
The dynamic equilibrium of the convertor in the rolling motion is where R TDy is the distance between the center of gravity G and the rope D about the y-axis. The dynamic angle between the rope D and the line from G to the rope D is θ y = ϕ Ty + ∆θ y , Considering small displacement and substituting Equation (63) into Equation (62), one obtains where the hydrodynamic damping moment m damp,Ty = −

Equation of Pitching Motion of the Convertor
The dynamic equilibrium of the convertor in the pitching motion is where the dynamic angle about the z-axis between the rope B and the line from the center of gravity to the rope B is θ The moment m Tz = m Tzs + m Tzd , where the moment in static equilibrium the hydrodynamic moment Considering small displacement and substituting Equations (66) and (67) into Equation (65), one obtains where the hydrodynamic damping moment m damp,Tz = −

Equation of Yawing Motion of the Platform
The dynamic equilibrium of the floating platform in the yawing motion is where R PAx and R PBx are the distance in the y-z plane from the center of gravity to the rope A and B, respectively. The angles of rope A and B in the x-y plane θ A = θ As + ∆θ Ad , θ B = θ Bs + ∆θ Bd , respectively. The relative angles between rope A and B and the longitudinal axis of the platform in the y-z plane The hydrodynamic moment on the floating platform due to the FSI is expressed in Taylor series as follows: where m Px (V, 0, 0, 0, 0, 0, 0, 0, 0, 0) = 0. Considering small oscillation, the higher order terms are neglected. Substituting Equations (34) and (70) into Equation (69), one obtains where the hydrodynamic damping moment m damp,Px = −

Equation of Rolling Motion of the Platform
The dynamic equilibrium of the floating platform in the rolling motion is where R PAy and R PCy are the distance in the x-z plane from the center of gravity to the rope A and C, respectively. The relative angles between rope A and C and the lateral axis of the platform in the x-z plane ϕ PyA = ϕ Py + ∆φ Ay , and ϕ PyC = ϕ Py + ∆φ Cy , respectively, in which ∆φ Ay = z 1d /L A sin θ As , ∆φ Cy = (z 1d − z 3d )/L C . The hydrodynamic moment Considering small displacement and substituting Equations (14), (33) and (73)  ϕ 1y + m damp,Py + m sti f ,Py + T As cos θ As R PAy + T Cs R PCy ϕ 1y + where the hydrodynamic damping moment m damp,Py = − The dynamic pitching equilibrium of the floating platform about the z-axis is where the angles of ropes A, B: θ A = θ As + ∆θ Ad , and θ B = θ Bs + ∆θ Bd . The relative angles between ropes A, B and C and the axis of the platform in the x-y plane Considering small displacement and substituting Equation (76)  ϕ 1z + m damp,Pz + m sti f ,Pz + (T As cos θ As R PAz + T Bs cos θ Bs R PBz + T Cs R PCz )ϕ 1z + T As cos θ As R PAz where the hydrodynamic damping moment m damp,Pz = −

Force Vibration Equation of System
The governing Equations (21) where the dynamic displacement vector The elements of the force vector F d = F dj 18×1 are in which The elements of the mass, damping and stiffness matrix M, C, and K are listed in Appendices D-F, respectively. To reduce hydrodynamic drag on the platform and to avoid disturbing the current through the turbine, the following oval configuration is designed as shown in Figure 3. To reduce hydrodynamic drag on the platform and to avoid disturbing the current through the turbine, the following oval configuration is designed as shown in Figure 3.

Hydrodynamic Damping and Stiffness Parameters of Platform
Because the hydrodynamic forces and moments on the floating platform due to the FSI are expressed in Taylor series, the hydrodynamic damping parameter of platform can be determined by the two methods: (1) determine these forces and moments by using the commercial STAR-CCM + software, (2) calculate the hydrodynamic parameter based on the determined forces and moments. Py py Py where the cross-sectional area of the platform . It is observed that the velocity around the platform is symmetrical. The current near the platform will be disturbed.
Secondly, considering the condition, ( ) , , 0, 0, 0, 0, 0, 0, 0, 0 , d Vx and given n sets of parameters these n sets of numerical hydrodynamic forces and moments are calculated by using the commercial STAR-CCM software. The flow field around the platform is shown in Figure   5 with   . It is observed that the velocity around the platform is

Hydrodynamic Damping and Stiffness Parameters of Platform
Because the hydrodynamic forces and moments on the floating platform due to the FSI are expressed in Taylor series, the hydrodynamic damping parameter of platform can be determined by the two methods: (1) determine these forces and moments by using the commercial STAR-CCM + software, (2) calculate the hydrodynamic parameter based on the determined forces and moments.
Firstly, given V, ϕ 1z = (V, 0, 0, 0, 0, 0, 0, 0, 0, 0), 0 < V < 2.5 m/s and by using the commercial STAR-CCM software, it is determined that the hydrodynamic forces and moments are f Px = f Pz = m Px = m Py = m Pz = 0, because of the symmetry of the platform. The hydrodynamic drag is where the cross-sectional area of the platform A Py = 19.635 m 2 . According to the numerical hydrodynamic drag with different current velocity V, the drag coefficient C py = 0.034. The flow field around the platform is shown in Figure 4 with V = 1 m/s . It is observed that the velocity around the platform is symmetrical. The current near the platform will be disturbed.

The Turbine Blade and Its Performance
The ocean energy convertor is composed of two turbine generators and an integration structure, as shown in Figure 6. Its normal power generation is 400 kW. The blade shape is shown in Figure 7. Secondly, considering the condition, V, x 1d , 0, 0, 0, 0, 0, 0, 0, 0 , and given n sets of parameters 0 < V < 2.5 m/s,−1.5 < .
x 1d < 0, these n sets of numerical hydrodynamic forces and moments are calculated by using the commercial STAR-CCM software. The flow field around the platform is shown in Figure 5 with {V = 1 m/s, .

The Turbine Blade and Its Performance
The ocean energy convertor is composed of two turbine generators and an integration structure, as shown in Figure 6. Its normal power generation is 400 kW. The blade shape is shown in Figure 7.

The Turbine Blade and Its Performance
The ocean energy convertor is composed of two turbine generators and an integration structure, as shown in Figure 6. Its normal power generation is 400 kW. The blade shape is shown in Figure 7.

The Turbine Blade and Its Performance
The ocean energy convertor is composed of two turbine generators and an integration structure, as shown in Figure 6. Its normal power generation is 400 kW. The blade shape is shown in Figure 7. The two turbine blades rotate reversely at the same rotating speed for rotational balance. Under the current velocity V = 2 m/s, the velocity field around the fixed convertor with rotating blade at the tip speed ratio TSR = 3.5 is calculated by using Star CCM+ and shown in Figure 8. It is observed that the current flows through the turbine blade along the guide tunnel. It will increase the flow velocity through the blade and the power generation. Moreover, the flow field around the two turbine blades will not disturbs each other. Figure 9 shows the effect of the TSR on the power coefficient of the turbine, CP = power/ The two turbine blades rotate reversely at the same rotating speed for rotational balance. Under the current velocity V = 2 m/s, the velocity field around the fixed convertor with rotating blade at the tip speed ratio TSR = 3.5 is calculated by using Star CCM+ and shown in Figure 8. It is observed that the current flows through the turbine blade along the guide tunnel. It will increase the flow velocity through the blade and the power generation. Moreover, the flow field around the two turbine blades will not disturbs each other. Figure 9 shows the effect of the TSR on the power coefficient of the turbine, CP = power/ 1 2 ρAV 3 , at the current velocity V = 2 m/s. The maximum power coefficient CP of the proposed turbine is 0.43 at TSR = 3.5. The two turbine blades rotate reversely at the same rotating speed for rotational balance. Under the current velocity V = 2 m/s, the velocity field around the fixed convertor with rotating blade at the tip speed ratio TSR = 3.5 is calculated by using Star CCM+ and shown in Figure 8. It is observed that the current flows through the turbine blade along the guide tunnel. It will increase the flow velocity through the blade and the power generation. Moreover, the flow field around the two turbine blades will not disturbs each other. Figure 9 shows the effect of the TSR on the power coefficient of the turbine, CP = power/      Further, Figure 10 shows the relation between the current velocity V and the output power at TSR = 3.5. It is determined that when the current velocity V = 1.6 m/s, the power of each turbine P each = 197 kW and the total output power of the two turbines is 394 kW. It is close to the nominal power of 400 kW. Further, Figure 10 shows the relation between the current velocity V and the output power at TSR = 3.5. It is determined that when the current velocity V = 1.6 m/s, the power of each turbine Peach = 197 kW and the total output power of the two turbines is 394 kW. It is close to the nominal power of 400 kW.

Hydrodynamic Damping Parameter of Convertor
Because the hydrodynamic force and moment due to the motion of the convertor are expressed in Taylor series, its hydrodynamic damping parameters can be determined as follows: Firstly, given (V, where the cross-sectional area of the convertor A T f y = 1034 m 2 . According to the numerical hydrodynamic drag with different current velocity V, the drag coefficient C T f y = 0.50. The flow field around the convertor is shown in Figure 8 with V = 2 m/s. Secondly, considering the condition, V, x 2d , 0, 0, 0, 0, 0, 0, 0, 0, 3.5), 0 < V < 2.5 m/s, −1.5 < .
x 2d < 0, the numerical hydrodynamic forces and moments are calculated.
x 2d , j = x, y, z one can determine the hydrodynamic parameters ∂m Tj /∂ .
Similarly, other hydrodynamic parameters are obtained and listed in Appendix C.

Dynamic Displacement
Multiplying Equation (78) by the inverse matrix of mass M −1 , one obtains ..
By using the balanced method for Equation (87), one obtains and where A = M −1 K − Ω 2 I . Substituting Equation (88) into (89), one obtains Based on Equation (90), the frequency equation is obtained:

Dynamic Tensions of Ropes
Under regular wave, the dynamic tensions of Ropes A, B, C, D are where where T Bdc = K Bd . T Cd = T Cdc cos Ωt + T Cds sin Ωt, |T Cd | = T 2 where T Cdc = K Cd (x 3dc − x 1dc ), T Cds = K Cd (x 3ds − x 1ds ).

Numerical Results and Discussion
Consider According to Figure 11b,c, the displacement of the platform is obviously larger than that of the convertor. Because the translational and rotational displacements of the convertor are small under the wave impact, the efficiency of power generation of convertor can be maintained to be high.
Obviously, the hydrodynamic damping parameters of the convertor and platform According to Figure 11b,c, the displacement of the platform is obviously larger than that of the convertor. Because the translational and rotational displacements of the convertor are small under the wave impact, the efficiency of power generation of convertor can be maintained to be high.
Obviously, the hydrodynamic damping parameters of the convertor and platform significantly depend on their configuration design. The dynamic performance of the system is decided by the corresponding hydrodynamic damping parameters or the configuration design. For clarity, the relationship between the hydrodynamic damping and the rope tension is investigated here. The hydrodynamic damping and stiffness parameters of some convertor and platform different to the proposed ones are assumed to be where the parameters with subscript '0 are those presented in Section 5 and Figure 11. β k , k = P, T are the hydrodynamic parameter ratio of different convertors and platforms to those presented in Section 5.
In Figure 12, the hydrodynamic parameter ratios are assumed to be β P = β T = 0.1. Other parameters are the same as those in Figure 11. The effects of the small hydrodynamic parameters and the typhon wave frequency on the dynamic tensions of the ropes, T Ad , T Bd , T Cd , and T Dd , are studied. It is determined that the resonant frequencies are 0.032 and 0.160 Hz. The maximum resonant dynamic tension of ropes A, B, C, and D: T Ad = 294.4 tons, T Bd = 165.0 tons, T Cd = 113.9 tons, and T Dd = 48.9 tons. These are significantly larger than those in Figure 11a. Further, if the hydrodynamic damping and stiffness parameters of the convertor are neglected, i.e., β P = 0.1 and β T = 0. The dynamic tension spectrum is presented in Figure 13. It is observed from Figure 13 that without the hydrodynamic damping of the convertor, the resonant tensions are significantly increased. The resonant dynamic tensions are greatly larger than the fracture strength of rope T fracture = 759 tons.   Figure 14a demonstrates the dynamic tension spectrum with L C = 140 m and L D = 60 m. In Figure 11a, with the rope lengths L C = L D = 60 m, the maximum dynamic tension T Ad = 84.56 tons.
In Figure 14a, with the rope lengths L C = 140 m, L D = 60 m, the maximum dynamic tension T Cd = 171.8 tons. It is because the surge and heave displacements of the pontoon 3 and platform at the resonance in Figure 14b are significantly larger than those in Figure 11b. Moreover, the pitch angle of the platform in Figure 14c is significantly larger than that in Figure 11c.    Figure 11a, with the rope lengths LC = LD = 60 m, the maximum dynamic tension TAd = 84.56 tons. In Figure 14a, with the rope lengths LC = 140 m, LD = 60 m, the maximum dynamic tension TCd = 171.8 tons. It is because the surge and heave displacements of the pontoon 3 and platform at the resonance in Figure 14b are significantly larger than those in Figure 11b. Moreover, the pitch angle of the platform in Figure 14c is significantly larger than that in Figure 11c.  Figure 15a demonstrates the dynamic tension spectrum with LC = 60 m and LD = 140 m. In Figure 11a, with LC = LD = 60 m, the maximum dynamic tension was TAd = 84.56 tons. In Figure 15a, the maximum dynamic tension TAd = 64.06 tons and TCd = 60.06 tons. It is observed from Figure 15b that the maximum resonance displacement is the surge of the pontoon 4. However, it is observed from Figure 11b that the maximum resonance displacement is the surge of the pontoon 3. In Figure 15c, the maximum yaw, roll and pitch  Figure 15a demonstrates the dynamic tension spectrum with L C = 60 m and L D = 140 m. In Figure 11a, with L C = L D = 60 m, the maximum dynamic tension was T Ad = 84.56 tons. In Figure 15a, the maximum dynamic tension T Ad = 64.06 tons and T Cd = 60.06 tons. It is observed from Figure 15b that the maximum resonance displacement is the surge of the pontoon 4. However, it is observed from Figure 11b that the maximum resonance displacement is the surge of the pontoon 3. In Figure 15c, the maximum yaw, roll and pitch angles of the platform ϕ 1x , ϕ 1y , and ϕ 1z are 1.3 • , 2.5 • and 21 • , respectively. The maximum yaw, roll and pitch angles of the convertor ϕ 2x , ϕ 2y , and ϕ 2z are 0.01 • , 0.104 • and 0.027 • , respectively. The maximum pitch angle of the platform in Figure 15c is significantly smaller than that in Figure 11c.      Figure 18 demonstrates the relation among the rope angle θA, the wave frequency f and the total tensions of ropes. It is observed that the angle θA will increase the resonant frequency; this is because if the angle θA is increased, the stiffness of system is increased. Moreover, if the angle θA is over critical, the dynamic tension TA increases with the angle θA.    Figure 18 demonstrates the relation among the rope angle θA, the wave frequency f and the total tensions of ropes. It is observed that the angle θA will increase the resonant frequency; this is because if the angle θA is increased, the stiffness of system is increased. Moreover, if the angle θA is over critical, the dynamic tension TA increases with the angle θA. Figure 17. Effect of length of rope D on the dynamic tension (β P = β T = 1). Figure 18 demonstrates the relation among the rope angle θ A , the wave frequency f and the total tensions of ropes. It is observed that the angle θ A will increase the resonant frequency; this is because if the angle θ A is increased, the stiffness of system is increased. Moreover, if the angle θ A is over critical, the dynamic tension T A increases with the angle θ A .

Conclusions
This paper presents the mathematical model of the coupled translational-rotational motions of the mooring system for an ocean energy convertor operating under the typhoon wave impact. The configurations of the convertor and the floating platform are designed. The hydrodynamic damping and stiffness parameters under the fluid-structure interaction are calculated. The performance of the mooring system under typhon wave impact and with different parameters is investigated and discovered as follows: (1) The translational displacements of pontoons 3 and 4 are more obvious than those of the platform and convertor. (2) The angular displacement in pitch motion of the platform is greatly larger than those of the yaw and roll motions.

Conclusions
This paper presents the mathematical model of the coupled translational-rotational motions of the mooring system for an ocean energy convertor operating under the typhoon wave impact. The configurations of the convertor and the floating platform are designed. The hydrodynamic damping and stiffness parameters under the fluid-structure interaction are calculated. The performance of the mooring system under typhon wave impact and with different parameters is investigated and discovered as follows: (1) The translational displacements of pontoons 3 and 4 are more obvious than those of the platform and convertor. (2) The angular displacement in pitch motion of the platform is greatly larger than those of the yaw and roll motions.
The transformed boundary conditions are: At s = 0, The solution of Equation (A5) is assumed: Substituting Equation (A8) into Equations (A5)-(A7), the mode shape and frequency are obtained [8]: For simplicity, the rope system is simulated by an effective mass-spring model. Its equation of motion is [22] M e f f ,s d 2 u Ls dt 2 + k e f f ,s u Ls = 0, where u Ls is the displacement at the free end. The effective spring constant k e f f ,s = EA L s . M eff,s is the effective mass. The natural frequency is The first natural frequency in the effective mass-spring model is the same as that in the distributed model. Equating Equations (A10)-(A12), the effective mass is obtained: where the mass per unit length of rope A f g = ρA. The component of rope A in the x, y, and z axis are L Ax = L A sin θ A , L Ay = L A cos θ A and L Az = 0. The corresponding effective masses are

Appendix B. Hydrodynamic Damping and Stiffness Parameters of Platform
Appendix B.1. Hydrodynamic Damping Parameters of Platform x 1d = 0.