# Design and Dynamic Stability Analysis of a Submersible Ocean Current Generator Platform Mooring System under Typhoon Irregular Wave

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{C}of rope C. On one side of the floating platform, rope B is used to pull the ocean current generator, and the other side of the floating platform is pulled down and anchored on the deep seabed. The buoyancy of the floating platform can be adjusted to be smaller than that of static balance so that, when ropes A and B are pulled, the floating platform has negative buoyancy and pontoon 3 has positive buoyancy, and rope C is used to connect the floating platform and pontoon 3 to achieve a balance of positive and negative buoyancy. In this way, the depth of the floating platform can be calculated by the length L

_{C}of rope C.

- -
- The current flow is steady;
- -
- The masses of the turbine, floating platform and the pontoon are concentrated;
- -
- Lightweight and high-strength PE mooring ropes are used;
- -
- Under the ocean velocity, the deformed configuration of PE rope is nearly straight;
- -
- The elongation strain of the ropes is small;
- -
- The tension of the rope is considered uniform.

_{i}, y

_{i}) for the i-th element shown in Figure 1 and Figure 2 are the sum of two parts: (1) the static one subjected to the steady current and (2) the dynamic one subjected to the wave, as follows:

#### 2.1. Static Displacements and Equilibrium under the Steady Current and without the Wave Effect

_{A}can be expressed as:

_{B}can be expressed as:

_{1}are the static tensions of ropes A and B, the buoyancy of the floating platform and the weight of the floating platform, respectively. The steady drag of the floating platform under current ${F}_{DFs}=\frac{1}{2}{C}_{DFy}\rho {A}_{FY}{V}^{2}$.

_{2}are the static tensions of rope D, the static buoyancy and the weight of the turbine, respectively. The static vertical equilibrium of pontoon 3 is expressed as:

_{B3s}and W

_{3}are the static buoyancy and the weight of pontoon 3, respectively. The static vertical equilibrium of the pontoon 4 is expressed as:

_{B}

_{4s}and W

_{4}are the static buoyancy and the weight of pontoon 4, respectively.

#### 2.2. Simulation of Irregular Wave

_{p}is the peak frequency and H

_{s}is the significant wave height.

_{s}during the 50-year regression period H

_{s}=15.4 m, and the peak period P

_{w}= 16.5 s.

_{s}and the peak period P

_{w}into Equations (12) and (13), the Jonswap wave spectrum is determined, as shown in Figure 3.

#### 2.3. Dynamic Equilibrium with the Effects of the Steady Current and Irregular Wave

_{3}is the mass of pontoon 3. T

_{C}is the tension of rope C. Substituting Equations (2) and (10) into Equation (17), one obtains:

_{Cd}is the effective spring constant. ${x}_{3d}-{x}_{1d}$ is the dynamic elongation between floating platform 1 and pontoon 3. Considering the safety of the rope, some buffer springs are used to serially connect the rope between elements 1 and 3. The effective spring constant of the rope–buffer spring connection is obtained:

_{C}are the Young’s modulus, cross-sectional area and length of rope C, respectively.

**Figure 4.**Relation among phase f

_{i}, wave length λ

_{i}and relative direction α of wave and current and distance L

_{E}between the two pontoons.

_{E}can be changed to obtain the desired phase angle ${\varphi}_{i}$.

_{4}is the mass of the pontoon 4, and T

_{D}is the tension of the rope D. Substituting Equations (2) and (11) into Equation (29), one obtains:

_{Dd}is the effective spring constant. ${x}_{4d}-{x}_{2d}$ is the dynamic elongation between floating platform 2 and pontoon 4. Considering the safety of the rope, some buffer springs are used to serially connect the rope between elements 2 and 4. The effective spring constant of the rope–buffer spring connection is obtained:

_{D}are the Young’s modulus, cross-sectional area and length of the rope D, respectively.

_{1}is the mass of the platform. The dynamic, effective mass of rope 1 in the x-direction, ${m}_{eff,x}=\frac{4{f}_{g}{L}_{As}\mathrm{sin}{\theta}_{1}}{{\pi}^{2}}$, was derived by Lin and Chen [15]. Substituting Equations (2) and (7) into Equation (33), one obtains:

_{A}is the dynamic tension of rope A.

_{A}and L

_{Ad}are the static and dynamic lengths of rope A, respectively. The effective spring constant of the rope–buffer spring connection is:

_{Bs}and L

_{Bd}are the static and dynamic lengths of rope B. The effective spring constant of the rope–buffer spring connection is:

_{1d}is the dynamic horizontal displacement of the floating platform. The dynamic effective mass of rope A in the y-direction is ${m}_{eff,y}=\frac{4{f}_{g}{L}_{A}\mathrm{cos}{\theta}_{A}}{{\pi}^{2}}$ [26]. The horizontal force on the platform due to the current velocity V and the horizontal velocity ${\dot{y}}_{1d}$ of the platform are expressed as [34]:

_{BY}and (3) the current velocity V.

_{2d}is the dynamic, horizontal displacement of the turbine. The horizontal force on the platform caused by the current velocity V and the horizontal velocity ${\dot{y}}_{2d}$ of the turbine is expressed as [34]:

_{Ty}is the effective operating area of the turbine. The theoretical effective drag coefficient of optimum efficiency is C

_{DTy}= 8/9, [27]. Considering ${\dot{y}}_{2d}<<V$, the term ${\dot{y}}_{2d}^{2}$ is negligible. Substituting Equations (2), (8), (39), (42) and (51) into Equation (50), one obtains:

_{DTy}, (2) the damping area A

_{TY}and (3) the current velocity V.

#### 2.4. Solution Method

#### 2.4.1. Free Vibration

#### 2.4.2. Forced Vibration

_{m}t and integrating it from 0 to the period T

_{m}, 2π/Ω

_{m}, Equation (66) becomes:

_{m}t and integrating it from 0 to the period T

_{m}, 2π/Ω

_{m}, Equation (66) becomes:

## 3. Numerical Results and Discussion

_{D}= 60 m, the diving depth of the floating platform L

_{C}$\ge $ 60 m and (2) the diving depth of the floating platform L

_{C}= 60 m, the diving depth of the turbine L

_{D}$\ge $ 60 m. Meanwhile, the effects of several parameters on the dynamic response are investigated.

_{bed}= 1300 m, (2) the cross-sectional area of pontoon 3 connecting to floating platform A

_{BX}= 2.12 m

^{2}, (3) the cross-sectional area of pontoon 4 connecting to turbine A

_{BT}= 2.12 m

^{2}, (4) no buffer spring, (5) the ropes A, B, C and D are made of some commercial, high-strength PE dyneema; Young’s modulus E

_{PE}= 100 GPa, weight per unit length f

_{g,PE}= 16.22 kg/m, diameter D

_{PE}= 154 mm, cross-sectional area A

_{PE}= 0.0186 m

^{2}, fracture strength T

_{fracture}= 759 tons, (6) the static diving depth of the turbine L

_{D}= 60 m, (7) the horizontal distance between the turbine and floating platform L

_{E}= 100 m, (8) the inclined angle of the rope A, ${\theta}_{\mathrm{A}}={30}^{\xb0}$, (9) the current velocity V = 1 m/s, (10) the irregular wave is simulated by six regular waves which are listed in Table 1, (11) the wave phase angles ${\phi}_{i},i=1,2,\dots ,6$ are assumed as $\left\{{30}^{\xb0},{60}^{\xb0},{90}^{\xb0},{120}^{\xb0},{170}^{\xb0},{270}^{\xb0}\right\}$, (12) the masses of turbine, floating platform and pontoons ${M}_{1}=300tons$, ${M}_{2}=838tons$, ${M}_{3}={M}_{4}=250tons$, (13) the cross-sectional area of the floating platform and turbine ${A}_{FY}={23\mathrm{m}}^{2}$ and ${A}_{TY}={500\mathrm{m}}^{2}$, (14) the effective damping coefficients ${C}_{DFy}=0.3$ and ${C}_{DTy}=8/9$, (15) the static axial force to turbine F

_{DTs}= 180 tons and (16) the relative orientation between current and wave α = 60°.

_{C}on the maximum dynamic tensions of the four ropes, T

_{A,max}, T

_{B,max}, T

_{C,max}and T

_{D,max}under the typhoon irregular wave when the diving depth of the turbine L

_{D}= 60 m. The irregular wave is simulated by six regular waves which are listed in Table 1. When the depth L

_{C}increases from 60 m, the dynamic tensions of the ropes increase significantly. If L

_{C,res}= 66 m, the resonance happens, and the maximum dynamic tensions T

_{A,max}= 2422 tons, T

_{C,max}= 7329 tons and T

_{D,max}= 18,835 tons, which is over that of the fracture strength of rope, T

_{fracture}= 759 tons. Figure 5b demonstrates the vibration mode at the resonance. It is found that the displacements x

_{2d}and x

_{4d}of turbine 2 and pontoon 4 are largest. Therefore, the maximum dynamic tension is that of rope D, T

_{D,max}.

_{C}increases further, the dynamic tension decreases sharply. If L

_{C}> 80 m, all the dynamic tensions are significantly less than the fracture strength of rope, T

_{fracture}= 759 tons. If L

_{C}= 80 m, T

_{A,max}= 442 tons, T

_{B,max}= 27 tons, T

_{C,max}= 187 tons, then T

_{D,max}= 478 tons. The maximum one among the four dynamic tensions is T

_{max}= T

_{D,max}= 478 tons. If L

_{C}= 150 m, the maximum dynamic tension T

_{max}= T

_{A,max}= 367 tons. This is because the natural frequency changes with the length L

_{C}. The excitation frequencies of the irregular wave are different to the natural frequency of the mooring system. Therefore, the resonance does not exist. It is found that the greater the diving depth of the floating platform L

_{C}, the smaller the maximum dynamic tension. In other words, the mooring system of the diving depth of the floating platform L

_{C}= 150 m is better than that of L

_{C}= 80 m. Because the diving depth of the floating platform is different to that of turbine, the water flowing through the floating platform does not interfere with the flow field of the turbine. Moreover, for L

_{C}> 80 m, the dynamic tension T

_{A,max}of rope A decreases with the diving depth L

_{C}. This is because the angle θ

_{A}of rope A decreases with the diving depth L

_{C}. The towing force is horizontal due to the ocean velocity. Meanwhile, the dynamic tension T

_{B,max}of rope B increases with the diving depth L

_{C}. It is because the angle θ

_{B}of rope B increases with the diving depth L

_{C}.

_{C}of the floating platform and the dynamic tensions of ropes under the typhoon irregular wave for the distance L

_{E}= 200 m. Aside from the distance L

_{E}=200 m, all other parameters are the same as those of Figure 5. It can be observed in Figure 6 that, when L

_{E}= 200 m, the maximum resonant position L

_{C,res}= 80 m is different to L

_{C,res}= 66 m for L

_{E}= 100 m in Figure 5. The effect of the horizontal distance between the turbine and floating platform L

_{E}on the dynamic tension with L

_{C}= 150 m is negligible.

_{C}and the mass of pontoons M

_{3}and M

_{4}on the dynamic tensions of the four ropes, T

_{A,max}, T

_{B,max}, T

_{C,max}and T

_{D,max}under the typhoon irregular wave. In this case, the mass of pontoons M

_{3}= M

_{4}= 150 tons; other parameters are the same as those of Figure 6. It is found that, if the mass of pontoons M

_{3}= M

_{4}= 150 tons, the resonance occurs at several diving depths of the floating platform L

_{C}, and the maximum dynamic tensions are over that of the fracture strength of rope, T

_{fracture}= 759 tons. In other words, if the weight of the pontoon is too low, the dynamic displacement of the system is too intense, resulting in the excessive dynamic tension of the rope.

_{D}and the mass of pontoons M

_{3}and M

_{4}on the maximum dynamic tensions of the four ropes, T

_{A,max}, T

_{B,max}, T

_{C,max}and T

_{D,max}under the typhoon irregular wave when the diving depth of the turbine L

_{C}= 60 m and the horizontal distance between the turbine and floating platform L

_{E}= 100 m. All the other parameters are the same as those of Figure 5. It is found that there is no resonance. The dynamic tension increases with the diving depth of the floating platform L

_{D}, especially in the case where M

_{3}= M

_{4}= 150 tons. The maximum tension is that of rope A, T

_{A,max}, which is close or over that of the fracture strength of rope, T

_{fracture}= 759 tons. It is concluded that this mooring system should not be proposed.

_{D}and the mass of pontoons M

_{3}and M

_{4}on the maximum dynamic tensions of the four ropes, T

_{A,max}, T

_{B,max}, T

_{C,max}and T

_{D,max}under the typhoon irregular wave when the diving depth of the floating platform L

_{C}= 60 m and the horizontal distance between the turbine and floating platform L

_{E}= 200 m. All the other parameters are the same as those in Figure 8. It is found that the maximum tension of the four ropes is the dynamic tension of rope A, T

_{A,max}. If the mass of pontoons M

_{3}= M

_{4}=150 tons, the maximum tension T

_{A,max}decreases with the diving depth of the turbine L

_{D}. However, it is the reverse for the case of the mass of pontoons M

_{3}= M

_{4}= 250 tons. Moreover, the dynamic tension T

_{A,max}, with the mass of pontoons M

_{3}= M

_{4}= 150 tons, is obviously less than that of the mass of pontoons M

_{3}= M

_{4}= 250 tons.

_{D}= 30 m to that of L

_{D}= 150 m is about 4.85. In other words, the deeper the diving depth of the turbine L

_{D}, the smaller the power generation.

_{C}= 60 m, the diving depth of the turbine L

_{D}= 70 m and the mass of pontoons M

_{3}= M

_{4}= 150 tons. The other parameters are the same as those in Figure 9. Dynamic displacements are multi-frequency coupled. The horizontal displacements of the turbine and the floating platform y

_{1d}and y

_{2d}are very small, the amplitude is about 0.30 m, the vertical displacements x

_{1d}and x

_{3d}are large and the amplitude is about 15.5 m, which is close to the significant wave H

_{S}= 15.4 m. The amplitudes of vertical displacements x

_{2d}and x

_{4d}are about 15.5 m. The amplitudes of vertical displacements x

_{1d}and x

_{3d}are about 9.69 m. The vertical displacements of pontoon 3 and the floating platform directly connected by using rope C are synchronized and similar. The vertical displacements of pontoon 4 and the turbine directly connected by using rope D are synchronized and similar.

_{A,max}of rope A connecting the floating platform and the mooring foundation is about 589 tons. The maximum dynamic tension T

_{B,max}of rope B connecting the turbine and the floating platform is about 38 tons. The maximum dynamic tension T

_{C,max}of rope C connecting pontoon 3 and the floating platform is about 322 tons. The maximum dynamic tension T

_{D,max}of rope D connecting pontoon 4 and the turbine is about 75 tons.

_{C}= 150 m, the diving depth of the turbine L

_{D}= 60 m. The other parameters are the same as those in Figure 6. Dynamic displacements are multi-frequency coupled. The horizontal displacements of the turbine and the floating platform y

_{1d}and y

_{2d}are very small, the amplitude is about 0.14 m, the amplitudes of vertical displacements x

_{1d}and x

_{3d}are about 8.6 m and the amplitudes of vertical displacements x

_{2d}and x

_{4d}are about 9.6 m, which are significantly lower than the significant wave H

_{S}= 15.4 m. The vertical displacements of pontoon 3 and the floating platform directly connected by using rope C are synchronized and similar. The vertical displacements of pontoon 4 and the turbine directly connected by using rope D are synchronized and similar.

_{A,max}of rope A connecting the floating platform and the mooring foundation is about 375 tons. The maximum dynamic tension T

_{B,max}of rope B connecting the turbine and the floating platform is about 58 tons. The maximum dynamic tension T

_{C,max}of rope C connecting pontoon 3 and the floating platform is about 143 tons. The maximum dynamic tension T

_{D,max}of rope D connecting pontoon 4 and the turbine is about 131 tons.

_{C}and the buffer spring connected in series with ropes C and D on the dynamic tension of the rope. The diving depth of the turbine L

_{D}= 60 m. The effective spring constants of the two buffer springs are K

_{C,spring}= K

_{D,spring}= K

_{rope A}. The other parameters are the same as those in Figure 5. Compared with Figure 5, it is found that the dynamic tensions T

_{A,max}, T

_{B,max}and T

_{C,max}of the ropes A, B and C are significantly reduced at the resonance point, but the effect on T

_{D,max}is not obvious and is still over the fracture strength T

_{fracture}. If the diving depth of the floating platform L

_{C}> 72 m, the effect of the buffer springs on the dynamic tensions is negligible. It is concluded that the effect of the buffer springs on the dynamic tensions of this mooring system is slight.

_{BX}, A

_{PX}and the diving depth of the floating platform L

_{C}on the dynamic tensions of the four ropes. The cross-sectional area of the two pontoons is A

_{BX}= A

_{PX}= 4 m

^{2}. The other parameters are the same as those in Figure 5. Compared with Figure 5, it is found that the dynamic tensions are significantly increased. At the resonance point, the dynamic tension is over the fracture strength T

_{fracture}. If the diving depth of the floating platform L

_{C}> 85 m, the dynamic tension is close to the fracture strength T

_{fracture}. It is concluded that the larger the cross-sectional area of the pontoon, the larger the dynamic tension.

_{s}and the peak period P

_{w}on the dynamic tension T

_{A,max}. Based on Equations (14)–(16), the irregular wave is simulated by six regular waves, i.e., n = 6. The six regular waves share according to the ratio of energy {2,35,8,4,3,1}. The amplitude a

_{i}, frequency f

_{i}, the wave number k

_{i}and wave length l

_{i}can be determined. The diving depths L

_{C}= 60 m and L

_{D}=150 m. The horizontal distance between the turbine and floating platform L

_{E}= 200 m. Two buffer springs are connected in series with ropes C and D. The effective spring constants of the two buffer springs are K

_{C,spring}= K

_{D,spring}= K

_{rope A}. The other parameters are the same as those of Figure 5. It is found that the more the significant wave height H

_{s}, the larger the dynamic tension T

_{A,max}. For the peak period T

_{p}= 13.5 s, the dynamic tension T

_{A,max}increases dramatically with the significant wave height Hs. With the increase of the peak period T

_{p}, the increase rate of the dynamic tension T

_{A,max}becomes low.

_{s}= 15 m, and the peak period P

_{w}= 16.5 s. The other parameters are the same as those of Figure 14. It is observed that the effect of the relative angle a of the wave and ocean current on the dynamic tension is significant. T

_{A,max}(α = 180°) and T

_{C,max}(α = 180°) are much larger than T

_{A,max}(α = 0°) and T

_{C,max}(α = 0°), respectively. Moreover, T

_{D,max}(α = 90°) is significantly larger than T

_{A,max}(α = 0°) and T

_{A,max}(α = 180°).

_{3}, M

_{4}and M

_{2}and the distance L

_{E}and the areas on the natural frequencies are investigated and listed in Table 2. It is found that the larger the cross-sectional areas of pontoon A

_{Bx}and A

_{BT}, the higher the natural frequencies of the system. The larger the masses of pontoon M

_{3}and M

_{4}, the lower the natural frequencies of the system. The larger the mass of turbine M

_{2}, the lower the first natural frequency of the system. However, the effect of the mass of turbine M

_{2}on the second natural frequency of the system is negligible. The larger the distance between the turbine and the floating platform L

_{E}, the higher the second natural frequency of the system. However, the effect of the distance L

_{E}on the first natural frequency of the system is negligible.

## 4. Conclusions

- (1)
- Considering the first mooring configuration, the diving depth L
_{D}of the turbine is fixed at 60 m. When the diving depth of the floating platform L_{C}=150 m, the dynamic tension is significantly less than the fracture strength T_{Fracture}of rope, and it is far from the resonance. Moreover, because the diving depth L_{D}of the turbine is far from the depth L_{C}of the floating platform, the floating platform does not interrupt the turbine water flow. Because the floating platform is a structure without a rotating mechanism in it, such as the rotating blade of a turbine, the water-proof at the depth of 150 m under sea surface is easily constructed. Therefore, this mooring configuration is safe and feasible; - (2)
- Considering the second mooring configuration, the diving depth L
_{C}of the floating platform is fixed at 60 m. When the diving depth L_{D}of the floating platform is larger than the diving depth L_{C}, there is no resonance point, but the dynamic tension ${T}_{Ad,max}$ of rope A is obviously larger than that of the first method and close to the fracture strength T_{Fracture}. It is found [32] that, for the Kuroshio current on the eastern coast of Taiwan, the greater the depth under the sea surface, the lower the current flow rate. The ratio of the potential power generation of the diving depth of the turbine L_{D}= 30 m to that of L_{D}= 150 m is about 4.85. Moreover, because there are the rotating blades of the turbine, the water-proof at the higher pressure under sea surface is difficult to construct. Therefore, the second mooring configuration is not recommended; - (3)
- The larger the area of pontoons A
_{BX}and A_{TX}, the larger the maximum dynamic tensions, especially for ${T}_{\mathrm{A}d,max}$; - (4)
- For the first mooring configuration, if the weight of the pontoon is too low, the dynamic displacement of the system is too intense, resulting in the excessive dynamic tension of the rope;
- (5)
- The effect of the buffer springs on the dynamic tensions of the first mooring configuration is slight.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{BX}, A_{BT} | cross-sectional area of pontoons 3 and 4, respectively |

A_{BY}, A_{TY} | damping area of platform and turbine under current, respectively |

a_{i} | amplitude of the i-th regular wave |

${C}_{DFy},{C}_{DTy}$ | damping coefficient of floating platform and turbine |

F_{B} | buoyance |

F_{D} | drag under current |

f | wave frequency |

H_{bed} | depth of seabed |

H_{s} | significant wave height |

g | gravity |

K | effective spring constant |

${\overrightarrow{K}}_{i}$ | wave vector of the i-th regular wave |

${\tilde{k}}_{i}$ | wave number of the i-th regular wave |

L_{i}, i = A,B,C,D | length of rope i |

L_{i} | length of rope i |

M_{i} | mass of element i |

${m}_{eff,x},{m}_{eff,y}$ | vertical and horizontal effective mass of rope A, respectively |

P_{w} | peak period of wave |

$\overrightarrow{R}$ | coordinate |

T_{i} | tension force of rope i |

t | time variable |

V | ocean current velocity |

x_{i}, i = 1~4 | vertical displacements of the floating platform, the turbine and the pontoons, respectively |

x_{w} | sea surface elevation |

y_{1}, y_{2} | horizontal displacements of the floating platform and the turbine, respectively |

α | relative angle between the directions of wave and current |

ρ | density of sea water |

Ω_{i} | angular frequency of the i-th regular wave |

ω | angular frequency |

${\phi}_{i}$ | phase angle of the i-th regular wave |

${\varphi}_{i}$ | phase delay of the i-th regular wave |

θ_{i} | angles of rope i |

λ_{i} | length of the i-th regular wave |

δ_{i} | elongation of rope i |

## Subscript

0~4 | mooring foundation, floating platform, turbine and two pontoons, respectively |

A, B, C, D | Ropes A, B, C and D, respectively |

s, d | static and dynamic, respectively |

PE | high-strength PE dyneema rope |

p | peak |

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**Figure 1.**Coordinates of the current energy system composed of submarined ocean turbine, pontoons, floating platform, traction ropes and mooring foundation in the static state under steady ocean current.

**Figure 2.**Coordinates of the current energy system composed of submarined ocean turbine, pontoon, floating platform, traction rope and mooring foundation in the dynamic state under steady ocean current and wave.

**Figure 5.**Dynamic tensions of ropes and Dynamic displacements of elements under the typhoon irregular wave. (

**a**) Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length for L

_{E}= 100 m; (

**b**) Dynamic displacements of the four elements at resonance for ${L}_{C}=66\mathrm{m}$.

**Figure 6.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length L

_{C}and the horizontal distance between the turbine and floating platform L

_{E}.

**Figure 7.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length L

_{C}and the mass of pontoons M

_{3}and M

_{4}.

**Figure 8.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length L

_{D}and the mass of pontoons M

_{3}and M

_{4}for L

_{E}= 100 m.

**Figure 9.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length L

_{D}and the mass of pontoons M

_{3}and M

_{4}for L

_{E}= 200 m.

**Figure 10.**(

**a**) Dynamic displacements of the four elements and (

**b**) dynamic tensions of ropes under typhoon irregular wave for ${L}_{C}=60\mathrm{m},{L}_{D}=70\mathrm{m}$.

**Figure 11.**(

**a**) Dynamic displacements of the four elements and (

**b**) dynamic tensions of ropes under typhoon irregular wave for ${L}_{C}=150\mathrm{m},{L}_{D}=60\mathrm{m}$.

**Figure 12.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the diving length L

_{C}and the buffer springs K

_{C,spring}, K

_{D,spring}.

**Figure 13.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the cross-sectional area of the two pontoons A

_{BX}, A

_{PX}and the diving depth of the floating platform L

_{C}.

**Figure 14.**Dynamic tension T

_{A,max}under the typhoon irregular wave as a function of the significant wave height H

_{s}and the peak period T

_{p}.

**Figure 15.**Dynamic tension of the four ropes under the typhoon irregular wave as a function of the relative angle α.

**Table 1.**Irregular wave simulated by regular waves [H

_{s}= 15.4 m, P

_{w}= 16.5 s, n = 6, H

_{bed}= 1300 m].

Case | Number of Regular Waves | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|---|

1 | 3 | a_{i} (m) | 2.603 | 2.603 | 2.603 | - | - | - |

f_{i} (Hz) | 0.0369 | 0.0390 | 0.1893 | - | - | - | ||

2 | 4 | a_{i} (m) | 2.255 | 2.255 | 2.255 | 2.255 | - | - |

f_{i} (Hz) | 0.0365 | 0.0382 | 0.0390 | 0.0398 | - | - | ||

3 | 5 | a_{i} (m) | 2.017 | 2.017 | 2.017 | 2.017 | 2.017 | - |

f_{i} (Hz) | 0.0365 | 0.0382 | 0.0390 | 0.0398 | 0.0406 | - | ||

4 | 6 | a_{i} (m) | 1.841 | 1.841 | 1.841 | 1.841 | 1.841 | 1.841 |

f_{i} (Hz) | 0.0365 | 0.0382 | 0.0390 | 0.0398 | 0.0406 | 0.1901 | ||

5 | 6 | a_{i} (m) | 1.142 | 4.208 | 2.630 | 1.364 | 0.843 | 0.605 |

f_{i} (Hz) | 0.0425 | 0.0600 | 0.0850 | 0.1150 | 0.1500 | 0.2664 | ||

${\tilde{k}}_{i}$(1/m) | 0.0073 | 0.0145 | 0.0291 | 0.0533 | 0.0906 | 0.2859 | ||

λ_{i} (m) | 861.5 | 433.3 | 215.9 | 117.9 | 69.3 | 22.0 |

**Table 2.**The first two natural frequencies f

_{n1}and f

_{n2}as a function of the masses M

_{3}, M

_{4}and M

_{2}, the distance L

_{E}and the areas A

_{Bx}, A

_{BT}for M

_{1}= 300 tons.

L_{E}(m) | M_{3}, M_{4}(tons) | M_{2}(tons) | A_{Bx} = A_{BT} = 2.12 m^{2} | A_{Bx} = A_{BT} = 4 m^{2} | ||
---|---|---|---|---|---|---|

f_{n}_{1} (Hz) | f_{n}_{2} (Hz) | f_{n}_{1} (Hz) | f_{n}_{2} (Hz) | |||

130 | 250 | 838 | 0.0220 | 0.0703 | 0.0302 | 0.0761 |

535 | 0.0258 | 0.0703 | 0.0355 | 0.0761 | ||

200 | 838 | 0.0220 | 0.0806 | 0.0302 | 0.0857 | |

535 | 0.0258 | 0.0806 | 0.0355 | 0.0857 | ||

130 | 150 | 535 | 0.0277 | 0.0776 | 0.0379 | 0.0839 |

200 | 0.0277 | 0.0890 | 0.0380 | 0.0946 |

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**MDPI and ACS Style**

Lin, S.-M.; Liauh, C.-T.; Utama, D.-W.
Design and Dynamic Stability Analysis of a Submersible Ocean Current Generator Platform Mooring System under Typhoon Irregular Wave. *J. Mar. Sci. Eng.* **2022**, *10*, 538.
https://doi.org/10.3390/jmse10040538

**AMA Style**

Lin S-M, Liauh C-T, Utama D-W.
Design and Dynamic Stability Analysis of a Submersible Ocean Current Generator Platform Mooring System under Typhoon Irregular Wave. *Journal of Marine Science and Engineering*. 2022; 10(4):538.
https://doi.org/10.3390/jmse10040538

**Chicago/Turabian Style**

Lin, Shueei-Muh, Chihng-Tsung Liauh, and Didi-Widya Utama.
2022. "Design and Dynamic Stability Analysis of a Submersible Ocean Current Generator Platform Mooring System under Typhoon Irregular Wave" *Journal of Marine Science and Engineering* 10, no. 4: 538.
https://doi.org/10.3390/jmse10040538