# Transient Translational–Rotational Motion of an Ocean Current Converter Mooring System with Initial Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Dynamic Governing Equations

#### 2.2. Integration of Two Ropes with Different Lengths in Parallel

#### 2.2.1. Relation between Tension and Elongation

_{iα}and L

_{iβ}, where L

_{iβ}> L

_{iα}. The effective spring constants of the two ropes are ${\overline{K}}_{ij}={E}_{ij}{A}_{ij}/{L}_{ij0},i=B,C,D;j=\alpha ,\beta $. When the elongation of the integrated rope ${\delta}_{i}$ < 0, in stage 0, its tension is zero, i.e., the integrated effective spring constant ${K}_{i}={K}_{i0}=0$. When the elongation of the integrated rope is in the interval $0<{\delta}_{i}<{L}_{i\beta}-{L}_{i\alpha}$ in stage 1, the integrated effective spring constant is ${K}_{i}={K}_{i1}={\overline{K}}_{i\alpha}$. When the elongation of the integrated rope ${\delta}_{i}$ is greater than the critical one ${\delta}_{ic1}={L}_{i\beta}-{L}_{i\alpha}$, in stage 2, the integrated effective spring constant is ${K}_{i}={K}_{i2}={\overline{K}}_{i\alpha}+{\overline{K}}_{i\beta}$, as shown in Figure 2. The critical tension was ${\overline{T}}_{ic}={\overline{K}}_{i\alpha}\left({\delta}_{ic1}+{\delta}_{i\beta}\right)$, which is lower than the fracture strength of rope iα. The effective spring constant of rope i is

#### 2.2.2. Strain Energy, Effective Spring Constant, and Fracture Strength

#### 2.3. Examples of Integrated Ropes

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

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

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

_{PE}= 0.0186 m

^{2}, length L

_{PE}= 300 m, fracture strength T

_{frac,PE}= 759 tons. The fracture strain ${\epsilon}_{\mathrm{fract},\mathrm{PE}}={T}_{\mathrm{frac},\mathrm{PE}}/{E}_{\mathrm{PE}}{A}_{\mathrm{PE}}=$ 0.004. The effective spring constant ${K}_{PE}={E}_{PE}{A}_{PE}/{L}_{PE}=632\left(\text{tons/m}\right)$.

_{frac,i}=750 tons and L

_{i}= 300 m, respectively.

_{iα}of the fracture strength of the rope iα to that of the integrated rope on the effective spring constants of the integrated rope $\left\{{K}_{i1},{K}_{i2}\right\}$, the critical elongations $\left\{{\delta}_{ic1},{\delta}_{ic2}\right\}$, the transformed tension T

_{trans}, and the fracture energy of the integrated rope ${e}_{\mathrm{frac},\mathrm{i}}$. It was found that if the elongation ${\delta}_{i}<{\delta}_{ic1}=0.597\mathrm{m}$, only rope iα will be working, and the effective spring constant K

_{i1}increases with fracture strength ratio γ

_{iα}. If the elongation ${\delta}_{ic1}<{\delta}_{i}<{\delta}_{ic2}=1.196\mathrm{m}$, ropes iα and iβ will be simultaneously subjected to loads. If the elongation ${\delta}_{i}\ge {\delta}_{ic2}$, the rope iα will be broken but not rope iβ. The higher the fracture strength ratio γ

_{iα}, the larger the fracture energy ${e}_{\mathrm{frac},\mathrm{i}}$. If the fracture strength ratio ${\gamma}_{i\alpha}=0.8$, the fracture energy ${e}_{\mathrm{frac},\mathrm{i}}=403.8(\text{tons-m})$. The higher the fracture strength ratio γ

_{iα}, the larger the effective spring constant of rope iα. In other words, the buffering effect decreases with decreasing fracture strength ratio γ

_{iα}. If the elongation ${\delta}_{i}<{\delta}_{ic1}=0.597\mathrm{m}$, only rope iα will be working, and the effective spring constant K

_{i1}will be significantly lower than that of the commercial HSPE dyneema, ${K}_{PE}=632\left(\text{tons/m}\right)$. Therefore, a significant application of the buffering effect of impact was achieved by the integration of parallel ropes.

_{frac,PE}= 759 tons, L

_{P}

_{E}= 100 m, its effective spring constant ${K}_{PE}=$1896 (tons/m).

_{frac,i}= 550 tons and L

_{i}= 100 m, respectively. Figure 4 shows the critical elongations ${\delta}_{ic1}=$0.199 m, ${\delta}_{ic2}=$0.399 m, which are less than those in Figure 3. If the fracture strength ratio ${\gamma}_{i\alpha}=$ 0.8, the fracture energy ${e}_{\mathrm{frac},\mathrm{i}}=$ 98.7 (tons-m), which is smaller than that in Figure 3 due to the fracture strength and length of the rope. If the elongation ${\delta}_{i}<{\delta}_{ic1}=$ 0.199 m, only rope a will be working, and the effective spring constant K

_{1}will be significantly smaller than that of the commercial HSPE dyneema, ${K}_{PE}=$ 1896 (tons/m). Therefore, the significant application of a buffering effect of impact was achieved through the integration of parallel ropes. When considering the double-rope parallel mode, the stiffness coefficients K

_{ij}depend on the dynamic tension of ropes, i.e., the stiffness coefficients are time-varying, as shown in Equation (2). In this study, the solution method derived for solving the mooring system with time-varying coefficients will be presented later.

## 3. Solution Method

#### 3.1. Transient Response

_{i}, i = 1, 2, …, 36.

#### 3.2. Derivation of a Fundamental Solution

_{f}) is divided into m small subintervals, $\left({t}_{0},{t}_{1}\right),\left({t}_{1},{t}_{2}\right),\cdots ,\left({t}_{m-1},{t}_{m}={t}_{f}\right)$. If the number of subinterval m is large enough, the elements of the stiffness matrix are close to constant in each subinterval. Furthermore, one can derive the independent and normalized fundamental solution by using the modified Frobenius method.

## 4. Numerical Results and Discussion

#### 4.1. Effects of Initial Displacements, Effective Spring Constant, and Double-Rope Parallel Mode

_{frac}= 759 tons. The hydrodynamic damping coefficients of platform 1, inverter 2, and pontoons 3 and 4 are listed in Appendix B. The hydrodynamic damping coefficients ${C}_{3,3}=5756\left(\text{N-s/m}\right)$, ${C}_{4,4}=1.465\times {10}^{6}\left(\text{N-s/m}\right)$, ${C}_{7,7}={C}_{10,10}=300\left(\text{N-s/m}\right)$.

_{A}and T

_{B}, are very high. Furthermore, these tensions gradually converge to low values. However, the tension of rope C, T

_{C}, increases gradually. This is because the effective spring constants of ropes B, C, and D are high, and their buffet effects are weak. It is demonstrated in Figure 5c that the displacements of the invertor and the pontoon 4 in the x-direction decay. The displacements of the platform and pontoon 3 in the x-direction oscillate and finally converge to about 1 m. It is demonstrated in Figure 5d that the yaw and roll angles of the invertor are close to zero. The pitch angle of the invertor is smaller than one degree at a time of 40 s. The pitch, yaw, and roll angles of the platform are higher than those of the invertor. Therefore, it is verified that, because the effective spring constants of ropes C and D are very high, their buffering feature is weak, and their instability will be easily obtained.

_{A}= A

_{B}= 0.0186 m

^{2}. Their effective spring constants are the same as those in Figure 5. Ropes C and D are made of a single HSPE rope, the same as ropes A and B. However, their cross-sectional area was A

_{C}= A

_{D}= $0.004{\mathrm{m}}^{2}$. Based on the linear elastic theory, the effective spring constant is reduced to ${K}_{Cd}={K}_{Dd}=412.2\left(\text{tons/m}\right)$. Their fracture strengths decrease to ${T}_{frac,j}=\left({A}_{j}/{A}_{i}\right){T}_{frac,i}$ $=162.54\mathrm{tons};i=A,B,j=C,D$. Figure 6a shows that the tension of rope C converges to a lower value due to the lowering of the effective spring constants of ropes C and D. The maximum momentary tension of rope B is T

_{Bmax}= 464 tons. The maximum momentary tension of rope C is T

_{Cmax}= 125.8 tons, which is close to the fracture strength ${T}_{frac,C}=$162.54 tons. Figure 6b shows that the transient displacements are stable and converge to the same value, as shown in Figure 5c. Therefore, it was verified that the lower the effective spring constants of ropes C and D are, the greater the stability of the mooring system. However, this design decreases the safety factor of the rope.

_{Bmax}= 465 tons, which is close to that of Figure 6. It can be seen from Figure 7b that the displacements are stable and convergent. It was verified that the double-rope parallel mode is helpful for achieving system stability and high fracture strength.

#### 4.2. Effects of the Length of Rope and Hydrodynamic Heaving Damping

#### 4.2.1. Transient Response and Improvement of Stability and Safety, L_{C} = 100 m, L_{D} = 110 m

_{frac,A}= T

_{frac,B}= 756 (tons). The fracture strengths of ropes C, D, T

_{frac,C}= T

_{frac,D}= 162.54 tons. The hydrodynamic damping coefficients ${C}_{3,3}=$5756 (N-s/m), ${C}_{4,4}=1.465\times {10}^{6}$ (N-s/m), ${C}_{7,7}={C}_{10,10}=$ 300 (N-s/m). It can be seen from Figure 8a that the tension of rope B is the greatest among ropes A, B, C, and D. The maximum momentary tension of rope B is T

_{Bmax}= 617 tons, which is significantly larger than the T

_{Bmax}= 464 tons shown in Figure 6, with L

_{C}= L

_{D}= 100 m. It can be concluded that the effect of the lengths of ropes C and D on the maximum tension of rope B is significant. Figure 8b shows that the angular and translational displacements of the invertor are small. However, the swaying displacement z

_{1d}of the platform increases significantly.

_{frac,A}= T

_{frac,B}= 750 tons. The fracture strengths of ropes C, D, T

_{frac,C}= T

_{frac,D}= 550 tons. The other parameters are the same as those in Figure 8. It can be observed from Figure 9a that the maximum momentary tension values of ropes A, B are T

_{Amax}= 455 tons and T

_{Bmax}= 444 tons. It can be seen from Figure 9b that the heaving displacement of platform 1, invertor 2, and pontoons 3 and 4 converge to near zero. The swaying displacement z

_{1d}of the platform oscillates.

_{1d}of the platform significantly decreased. The mooring system therefore became more stable.

#### 4.2.2. Transient Response and Improvement of Stability and Safety, L_{C} = 110 m, L_{D} = 100 m

_{frac,A}= T

_{frac,B}= 750 tons. The fracture strengths of ropes C, D, T

_{frac,C}= T

_{frac,D}= 162.5 tons. It can be observed from Figure 8a that the tension of rope A is the greatest among ropes A, B, C, and D. The maximum momentary tensions of ropes A and B are T

_{Amax}=461 tons and T

_{Bmax}= 294 tons, respectively, which is significantly lower than the T

_{Bmax}= 464 tons shown in Figure 6, with ${L}_{\mathrm{C}}={L}_{D}=100\mathrm{m}$. It can be concluded that the effect of the lengths of ropes C and D on the maximum tensions of ropes A and B is significant. However, it is shown in Figure 8b that the heaving displacements of the invertor and pontoon 4 are significantly increased. Finally, the invertor is lifted to the water surface. The swaying displacement of the platform increases significantly. This means that the mooring system is unstable.

_{frac,A}= T

_{frac,B}= 756 tons. The fracture strengths of ropes C, D, T

_{frac,C}= T

_{frac,D}= 550 tons. The hydrodynamic heaving damping coefficients are increased as follows: ${C}_{3,3}=1.15\times {10}^{6}$(N-s/m), ${C}_{4,4}=2.96\times {10}^{8}$(N-s/m), ${C}_{7,7}={C}_{10,10}=6\times {10}^{4}$(N-s/m). The other parameters are the same as those in Figure 11. Figure 12a shows that the tensions of ropes are significantly decreased. Figure 12b demonstrates that the heaving displacements x

_{2d}, x

_{4d}are almost fixed. The swaying displacement z

_{1d}of the platform is close to zero. This means that increasing the hydrodynamic damping is greatly helpful for the stability of the system.

#### 4.3. Effect of Initial Velocities

_{frac,A}= T

_{frac,B}= 756 tons. The fracture strengths of ropes C, D, T

_{frac,C}= T

_{frac,D}= 550 tons. The other parameters are the same as those employed in Figure 12. It can be seen from Figure 13a that the tensions are significantly lower than the fracture strengths. It is demonstrated in Figure 13b that the mooring system is stable.

## 5. Conclusions

- The lower the effective spring constants of the ropes are, the higher the buffering feature of the mooring system will be.
- The higher the effective spring constants of the ropes are, the higher the momentary tension of the ropes will be.
- In traditional setups, the lower the effective spring constants of single-rope mode are, the lower the fracture strength of the rope will be. This disadvantage can be overcome by using the double-rope parallel mode.
- The effect of the lengths of ropes C and D on the transient response is significant.
- The larger the hydrodynamic damping coefficients are, the stabler the mooring system will be.

## 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 surfaced cylinder of pontoons 3 and 4, respectively |

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

C | matrix of damping |

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

E_{i} | Young’s modulus of rope i, i = A, B, C, D |

e_{frac} | fracture energy of rope |

F_{B} | buoyance |

${f}_{Pys}$$,{f}_{Tys}$ | the drag of the floating platform and the convertor under steady current |

H_{bed} | depth of seabed |

${I}_{Tj},{I}_{Pj}$ | mass moment of inertia of the convertor and the platform about the j-axis. |

g | gravity |

K | matrix of stiffness |

K_{id} | $\mathrm{effective}\mathrm{spring}\mathrm{constant}\mathrm{of}\mathrm{rope}i,{E}_{i}{A}_{i}/{L}_{i}$ |

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

L_{E,} | $\mathrm{horizontal}\mathrm{distance}\mathrm{between}\mathrm{the}\mathrm{convertor}\mathrm{and}\mathrm{platform},\sqrt{{L}_{B}^{2}-{\left({L}_{C}-{L}_{D}\right)}^{2}}$ |

M | matrix of mass |

M_{i} | mass of element i |

${M}_{eff,i}$ | effective mass of rope A in the i-direction |

${m}_{ki}$ | hydrodynamic moment of convertor or platform about the i-axis |

$\overrightarrow{R}$ | coordinate |

T_{i} | tension force of rope i |

t | time variable |

V | ocean current velocity |

W_{i} | weight of component i |

w_{PE} | weight per unit length of HSPE |

x_{i}, y_{i}, z_{i} | displacements of component i |

ε | strain |

ρ | density of sea water |

${\phi}_{kj}$ | angular displacement of convertor or platform about the j-axis |

θ_{i} | angles of rope i |

δ_{i} | elongation of rope i |

Subscript: | |

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

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

crit | critical |

iα, iβ | component α, β of rope i = A, B, C, and D |

frac | fracture |

s, d | static and dynamic, respectively |

PE | PE dyneema rope |

P | platform |

T | convertor |

## Appendix A. Elements of the Mass Matrix $M={\left[{\mathit{M}}_{\mathit{i},\mathit{j}}\right]}_{18\times 18}$

## Appendix B. Elements of the Hydrodynamic Damping Matrix $C={\left[{\mathit{C}}_{\mathit{i},\mathit{j}}\right]}_{18\times 18}$

## Appendix C. Elements of the Stiffness Matrix $K={\left[{\mathit{K}}_{\mathit{i},\mathit{j}}\right]}_{18\times 18}$

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**Figure 2.**Effective spring constant of the main rope integrated by two parallel smaller ropes with different lengths. (

**a**) Two integrated ropes in parallel; (

**b**) effective spring constant K

_{i}elongation relation δ

_{i}.

**Figure 3.**The effect of fracture strength ratio on the properties of integrated rope with T

_{frac,i}= 750 tons and L

_{i}= 300 m.

**Figure 4.**Effect of fracture strength ratio on the properties of the integrated rope with T

_{frac,i}= 550 tons and L

_{i}= 100 m.

**Figure 5.**Transient response of mooring system with ${L}_{\mathrm{C}}={L}_{D}=100\mathrm{m}$, ${K}_{Cd}={K}_{Dd}=1889(\text{tons/m})$. (

**a**) Total tension of ropes (I). (

**b**) Total tension of ropes (II). (

**c**) Displacements of pontoons, turbine, and platform. (

**d**) Angular displacements of turbine and platform.

**Figure 6.**Transient response of mooring system with ${L}_{\mathrm{C}}={L}_{D}=100\mathrm{m}$, ${K}_{Cd}={K}_{Dd}=412.2(\text{tons/m})$, ${T}_{frac,C}={T}_{frac,D}=162.54\mathrm{tons}$. (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbine, and platform.

**Figure 7.**Transient response of mooring system with ${L}_{\mathrm{C}}={L}_{D}=100\mathrm{m}$, ${T}_{frac,C}={T}_{frac,D}=550\mathrm{tons}$. (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbine, and platform.

**Figure 8.**Transient response of mooring system with smaller diameters of ropes C, D [${L}_{\mathrm{C}}=100\mathrm{m},{L}_{D}=110\mathrm{m}$, T

_{frac,C}= T

_{frac,D}= 162.54 tons]. (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbine, and platform.

**Figure 9.**Transient response of mooring system with the double-rope parallel modes B, C, D (${L}_{\mathrm{C}}=100\mathrm{m},{L}_{D}=110\mathrm{m}$, T

_{frac,A}= T

_{frac,B}= 750 tons, T

_{frac,C}= T

_{frac,D}= 550 tons). (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbine, and platform.

**Figure 10.**Transient response of mooring system with the double-rope parallel modes B, C, D, and larger heaving hydrodynamic damping. (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbine, and platform.

**Figure 11.**Transient response of mooring system with ${L}_{\mathrm{C}}=110\mathrm{m},{L}_{D}=100\mathrm{m}$. (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbines, and platform.

**Figure 12.**Transient response of mooring system with integrated rope and larger heaving damping. (

**a**) Total tension of ropes. (

**b**) Displacements of pontoons, turbine, and platform.

**Figure 13.**Transient response of mooring system with different initial conditions. (

**a**) Total tension of ropes. (

**b**) Displacement of pontoons, turbines, and platform.

Parameter | Dimension | Parameter | Dimension | |
---|---|---|---|---|

depth of seabed H_{bed} | 1000 m | length of rope C, L_{C} | 100 m | |

length of rope D, L_{D} | 100 m | horizontal distance between the inverter and platform L_{E} | 300 m | |

inclined angle of rope A, ${\theta}_{\mathrm{A}}$ | $3{0}^{0}$ | mass moment of inertia of the convertor $\mathrm{about}\mathrm{the}x,,\text{-axis},{I}_{Tx}/{I}_{Ty}/{I}_{Tz}$ | $8.940\times {10}^{10}$/ $2.712\times {10}^{10}/$ $8.940\times {10}^{10}$(kg-m ^{2}) | |

cross-sectional area of surfaced cylinder of pontoon 3, A_{BX} | 4 m^{2} | mass moment of inertia of the platform $\mathrm{about}\mathrm{the}x,y,\mathrm{z-axis},{I}_{Px}/{I}_{Py}/{I}_{Pz}$ | $3.0\times {10}^{8}$$/5.0\times {10}^{6}/$$3.0\times {10}^{8}$(kg-m^{2}) | |

cross-sectional area of surfaced cylinder of pontoon 4, A_{BT} | 4 m^{2} | distance from the gravity of invertor to rope B, D, R_{TB}/R_{TD} | 16.5/12.82 m | |

HSPE rope | Young’s modulus E_{PE} | 100 GPa, | current velocity V | 1.6 m/s |

weight per unit length w_{PE} | 16.22 kg/m | mass of the platform M_{1} | 300 tons | |

diameter D_{PE} | 154 mm | mass of the invertor M_{2} | 538 tons | |

cross-sectional area A_{PE} | 0.0186 m^{2} | mass of the pontoon 3, M_{3} | 250 tons | |

fracture strength T_{frac} | 759 tons | mass of the pontoon 4, M_{4} | 250 tons | |

distance from the gravity of platform to ropes A, B, C, R_{PA}/R_{PB}/R_{PC} | 5/5.8/2.5 m | static tension of ropes A, B, C, D, T_{As}/T_{Bs}/T_{Cs}/T_{Ds} | 78.07/67.53/5/5 tons | |

static drag of the invertor F_{DT} | 67.53 tons | net buoyance of invertor and platform F_{BNT}/F_{BNB} | 533/320.77 tons |

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## Share and Cite

**MDPI and ACS Style**

Lin, S.-M.; Wang, W.-R.; Yuan, H.
Transient Translational–Rotational Motion of an Ocean Current Converter Mooring System with Initial Conditions. *J. Mar. Sci. Eng.* **2023**, *11*, 1533.
https://doi.org/10.3390/jmse11081533

**AMA Style**

Lin S-M, Wang W-R, Yuan H.
Transient Translational–Rotational Motion of an Ocean Current Converter Mooring System with Initial Conditions. *Journal of Marine Science and Engineering*. 2023; 11(8):1533.
https://doi.org/10.3390/jmse11081533

**Chicago/Turabian Style**

Lin, Shueei-Muh, Wen-Rong Wang, and Hsin Yuan.
2023. "Transient Translational–Rotational Motion of an Ocean Current Converter Mooring System with Initial Conditions" *Journal of Marine Science and Engineering* 11, no. 8: 1533.
https://doi.org/10.3390/jmse11081533