# Dynamically Scaled Model Experiment of a Mooring Cable

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Similitude

#### 2.1. Governing Equations

_{1}x

_{2}x

_{3}]

^{T}is the position vector of the cable in a Cartesian coordinate system with the origin at the anchor point, the coordinate x

_{1}directed vertically upwards perpendicular to the sea bottom and the coordinate x

_{2}horizontally in the plane of the resting cable. The sea bottom is assumed to be flat, horizontal and parallel to the mean sea surface. See Figure 1. The terms in Equation (1) represent forces per length unit of unstretched cable. The first term is the mass inertia force on the cable. The second and third terms are forces perpendicular to the cable that arise due to the acceleration in the water. The forth term is the reaction force of the cable. Finally, the fifth term, f, represents the external forces. The hydrodynamic forces due to the acceleration in the water (second and third term) and the drag forces contained in the external forces (fifth term) due to the relative velocity in the water are derived from the common Morison equation, and follows ideas in [27].

_{1}x

_{2}x

_{3}]

^{T}is the position vector of the cable and γ

_{o}the cable mass per length unit of unstretched cable:

_{o}: characteristic diameter of the cable; ε, $\tilde{\mathsf{\epsilon}}$: longitudinal strains of the cable; K: the cable stiffness; t: time; s

_{o}: curvilinear coordinate from the origin along the unstretched cable to a material point P; $\dot{\phantom{x}}=\frac{\partial}{\partial t}$; $\ddot{\phantom{x}}=\frac{{\partial}^{2}}{\partial {t}^{2}}$; $\stackrel{\prime}{\phantom{x}}=\frac{\partial}{\partial {s}_{o}}$.

_{2}= (1/2) C

_{DT}d

_{o}ρ

_{w}

_{DT}tangential drag force coefficient

_{3}= (1/2) C

_{DN}d

_{o}ρ

_{w}

_{DN}normal drag force coefficient

#### 2.2. Dimensionless Variables and Dynamic Similarity

_{o}/L

_{o}L/T

^{2}we then get the following dimensionless equation:

_{1}, α

_{2}, α

_{3}, α

_{4}, and α

_{5}are dimensionless parameters:

_{v}is a volume coefficient. For a wire rope or a circular solid steel rod, C

_{v}= 1. Then, with the help of Equations (2), (8), (10), (29), and (30), we can interpret the dimensionless parameters ${\mathsf{\alpha}}_{1}$ to ${\mathsf{\alpha}}_{5}$ as

#### 2.3. Scale Factors

_{w}, and ρ

_{c}, and the earth acceleration, g, are the same in the model and prototype, which is very common. Equation (31a) then indicates that the ratio C

_{MN}/C

_{v}shall be equal in model and prototype. We then introduce the length scale λ:

_{3}to be equal:

_{o}are characteristic values of, respectively, the relative velocity and the cable diameter, and ν is the kinematic viscosity. The scale of the relative velocity is

_{v}= 1 above. The stiffness scale Equation (2.36) was introduced by [13]. In [1] it is proposed that Equations (38) and (40) can be utilised with experimental data of the drag coefficients as functions of the Reynolds number to get drag-force similarity. In [1], this is shown for a cable with a circular cross section using different scales, λ and β. Similarity of both tangential and transverse drag forces normally cannot be achieved.

## 3. Physical Experiments

#### 3.1. Choice of Chain

_{m}= 10,000 ± 500 N, which is a low value caused by the links being open. Again, the index m denotes model values and the index p prototype values.

_{om}= 0.0818 ± 0.0005 kg/m by weighing a long length of the chain. The longitudinal wave speed was then calculated as ${c}_{m}=\sqrt{{K}_{m}/{\mathsf{\gamma}}_{om}}\approx 350\text{}\mathrm{m}/\mathrm{s}$. The density was assumed to be 7800 kg/m

^{3}, neglecting the thin lacquer.

_{om}= 2.2 mm. A prototype chain in a mooring system can have a link diameter of d

_{op}= 76 mm. Such a chain has the stiffness of K

_{m}= 5.8 × 10

^{8}N [29] and the mass γ

_{op}= 126.5 kg/m [29,30] but the same density as the model chain, because both are made from steel. The longitudinal wave speed can then be calculated as ${c}_{p}=\sqrt{{K}_{p}/{\mathsf{\gamma}}_{pm}}\approx 2143\text{}\mathrm{m}/\mathrm{s}$.

_{m}= 33 ± 0.005 m for practical reasons and to get a somewhat taut cable. The vertical span from the concrete floor to the mean elevation of the upper attachment point was 3.3 m. The chain length also included a shackle and a force probe in the upper end with the length 0.159 m and a shackle in the other, lower end with a length of 0.026 m. These are neglected in the numerical simulations in Section 4. With λ = 0.027 and L

_{m}= 33 m the length of the prototype chain would become L

_{p}$\approx $ 1240 m.

_{1}should be the same in the model as in the prototype

_{vm}$\approx $ 2.76 and that of the prototype chain to be C

_{vp}$\approx $ 3.58. Then, the volume coefficient scale is φ = C

_{vm}/C

_{pm}$\approx $ 0.74. See Equation (36). The ratio between the added mass coefficients in the model and the prototype can then be solved from Equation (42):

_{om}/d

_{op}= 2.2/76 = 0.029 and λ = 0.027 inserted in Equation (38), based on ${\alpha}_{5}$ being equal, gives the ratio between the normal drag coefficients in the model and the prototype as:

#### 3.2. Choice of Other Conditions

_{m}, to its centre of rotation and the electrical motor at various rotational speeds to produce various periods, T

_{m}. Table 1 and Table 2 show the radii of excitation and the period times respectively, for both model and prototype scale.

Model | 0.075 | 0.100 | 0.125 | 0.150 | 0.200 |
---|---|---|---|---|---|

Prototype | 2.82 | 3.75 | 4.69 | 5.63 | 7.51 |

Periods, T_{p} (s) | 1.25 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 |
---|---|---|---|---|---|---|

Periods, T_{p} (s) | 7.7 | 9.2 | 12.3 | 15.3 | 18.4 | 21.5 |

#### 3.3. Alternative Dimensional Analyses

Quantity | Model | Prototype | |||
---|---|---|---|---|---|

L | 33 | m | 1240 | m | Length of cable |

V | 3.3 | m | 124.0 | m | Vertical span |

d_{o} | 0.0022 | m | 0.076 | m | Material diameter |

d_{e} | 0.00365 | m | 0.144 | m | Equivalent solid-rod diameter (reference diameter) |

$w=(1-{\rho}_{w}/{\rho}_{c})g{\gamma}_{o}$ | 0.699 | N/m | 1078 | N/m | Weight in water |

T_{s} | 22.68 | N | 1.314 × 10^{6} | N | Static tension |

K | 10 | kN | 5.8 × 10^{5} | kN | Stiffness |

u_{max} | 0.135–1.0 | m/s | 0.825–6.16 | m/s | Excitation speed |

ν | 10^{−6} | m^{2}/s | 10^{−6} | m^{2}/s | Kinematic viscosity |

${\rho}_{c}$ | 7800 | kg/m^{3} | 7800 | kg/m^{3} | Cable density |

${\rho}_{w}$ | 1000 | kg/m^{3} | 1024 | kg/m^{3} | Water density |

r | 0.075–0.2 | m | 2.82–7.51 | m | Diameter of motion |

ψ | 0.455 | rad | 0.454 | rad | Angle between motion and the horizontal |

${C}_{v}$ | 2.76 | - | 3.58 | - | Volume coefficient |

_{D}and C

_{m}—essentially depending on the Reynolds number—and one containing the steady current drag over the weight of the mooring line. In our physical model there is no bending stiffness, there is no current and the drag and added-mass parameters may be combined into a Reynolds number. This leaves us with nine parameters. These are checked below for the presented model. Papazoglou et al. [2] and Mavrakos et al. [3] also present nine non-dimensional parameters for similitude, but not in the same combinations.

**Table 4.**The nondimensional parameters (groups) used by Webster [9] and relevant for this problem with Webster’s explanations of their physical meanings. Notations as in the present article.

Parameter | |
---|---|

t/T | is the time relative to the period of the sinusoidal excitation. If a steady state is achieved, the state of the mooring line repeats for each unit increment in this parameter; This parameter scales with the square root of the length scale and is not listed explicitly below in Table 5 and Table 6; |

L/V | is the ratio of the mooring line length to the vertical span and is commonly called the “scope” of the mooring line; |

T_{s}/(wV) | is the ratio of the static pretension of the mooring line to the weight in water of a length of mooring line equal to the vertical span. This parameter, together with the scope, governs the geometry of the mooring line when no motions are imposed; |

r/V | is the ratio of the amplitude of motion to the water depth; |

${C}_{D}\sqrt{{C}_{v}}$ | is the ratio of the hydrodynamic, cross-flow drag forces acting on the real mooring line to those which would act on the reference mooring line if exposed to the same flow situation; |

${C}_{m}{C}_{v}$ | is the ratio of the added mass loads to those which would act on the reference mooring line exposed to the same flow situation; |

$\frac{T}{2\pi}\sqrt{\frac{g}{V}}$ | is the ratio of the period of the excitation to the period of a pendulum of length V; |

${\rho}_{w}/{\rho}_{c}$ | is the ratio of the water mass density to the mass density of the mooring line material. This parameter measures the relative importance of the hydrodynamic loads to the internal mechanical loads; |

K/wL | is the inverse of the strain at the top of the cable resulting from suspending the mooring line vertically in water of unrestricted depth. This parameter measures the relative “stiffness” of the mooring line; |

d_{o}/V | is the ratio of the diameter of the reference mooring line to the vertical span. Since almost all mooring lines are exceptionally thin compared to the vertical span, this parameter approaches zero. |

**Table 5.**Nondimensional parameters according to [9] calculated for the present experimental data with the notations used in the present article.

Parameter | Model | Prototype | Ratio Prototype/Model | |
---|---|---|---|---|

1 | L/V | 10 | 10 | 1 |

2 | T_{s}/(wV) | 1.017 | 1.018 | 1.00 |

3 | r/V | 0.023–0.061 | 0.023–0.061 | 1.00 |

4 | ${C}_{D}\sqrt{{C}_{v}}$ | C_{Dm}∙1.66 | C_{Dp}∙1.95 | 1.14 $({C}_{Dp}/{C}_{Dm})$ |

5 | ${C}_{m}{C}_{v}$ | C_{mm}∙2.76 | C_{mp}∙3.82 | 1.30 $({C}_{mp}/{C}_{mm})$ |

6 | $\frac{T}{2\pi}\sqrt{\frac{g}{V}}$ | 0.343–0.960 | 0.343–0.960 | 1.00 |

7 | ${\rho}_{w}/{\rho}_{c}$ | 0.128 | 0.131 | 1.02 |

8 | K/wL | 435 | 435 | 1.00 |

9 | d_{o}/V | 0.67 × 10^{−3} | 0.61 × 10^{−3} | 0.92 |

**Table 6.**Non-dimensional parameters according to [2] calculated for the present experimental data with the notations used in the present article.

Parameter | Model | Prototype | Ratio Prototype/Model | |
---|---|---|---|---|

1 | L/V | 10 | 10 | 1 |

2 | d_{o}/L | 67 × 10^{−6} | 61 × 10^{−6} | 0.92 |

3 | wL/T_{s} | 1.018 | 1.016 | 1.00 |

4 | Re_{max} = u_{max}d_{o}/ν | 300–2200 | 6.27 × 10^{4}–4.7 × 10^{5} | 212 |

5 | ${\rho}_{c}/{\rho}_{w}$ | 7.80 | 7.62 | 0.98 |

6 | K/T_{s} | 441 | 442 | 1.00 |

7 | r/d_{o} | 34.1–90.9 | 37.1–98.8 | 1.09 |

8 | ${\omega}^{2}{L}^{2}/({T}_{s}/({\rho}_{c}\pi {d}_{e}^{2}/4))$ | 12.7–99.2 | 12.7–99.5 | 1.00 |

9 | ψ | 0.455 | 0.454 | 1.00 |

_{e}instead of the material diameter d

_{o}in Parameters 2 and 7, we would get prototype over model ratios of 1.05 and 0.96, respectively. As discussed above, the Reynolds number, Parameter 4, cannot be the same in the model and prototype. The modelling error in Section 5 is simply caused by the fact that we had fresh water in the model and assumed sea water in the prototype. As for Parameter 8, [2] does not discuss its physical significance, but again we may encounter a modelling error. If we use the diameter d

_{o}instead of d

_{e}the prototype over model ratio becomes 0.77.

#### 3.4. Experimental Equipment

#### 3.5. Experimental Procedure

#### 3.6. Experimental Results

_{m}= 3.50 s, and one shorter period test with T

_{m}= 1.25 s. In Figure 5 the tension response is slightly asymmetric, with a mild slope during the up-stroke and a sharper drop in the down-stroke. During approximately 1 s of the cycle the cable has virtually no stiffness as the major part is slack. When tension is regained, the response is still smooth because of the relatively long period oscillation. The same effects can be seen for the tension force record of the shorter period time in Figure 6, but here the re-tensioned cable gives rise to clearly visible transients in the tension force. This results in both a double peak appearance and a step-like behaviour of the tension force history during the upstroke motion.

**Figure 5.**Recorded upper-end force F

_{m}(N) for the period T

_{m}=3.50 s and radius r

_{m}= 0.2 m. (Digitised from [13]).

**Figure 6.**Recorded upper-end force F

_{m}(N) for the period T

_{m}=1.25 s and radius r

_{m}= 0.2 m. (Digitised from [13]).

**Table 7.**Measured mean of maximum upper-end tension (N). (Digitised from [13]).

Period (s) | Radius (m) | ||||
---|---|---|---|---|---|

0.075 | 0.1 | 0.125 | 0.15 | 0.2 | |

1.25 | 42.5 | 46.8 | 54.1 | 60.4 | 70.3 |

1.50 | 41.0 | 45.3 | 51.5 | 59.0 | 68.0 |

2.00 | 36.0 | 39.5 | 47.5 | 54.1 | 62.3 |

2.50 | 31.1 | 35.4 | 42.5 | 49.0 | 57.3 |

3.00 | 29.5 | 33.1 | 39.3 | 45.8 | 54.0 |

3.50 | 27.8 | 31.5 | 37.5 | 42.5 | 50.1 |

## 4. Simulated Tension

Quantity | Measure | Unit | |
---|---|---|---|

Water density | 1000 | kg/m^{3} | ρ_{wm} |

Water depth | 3 | m | The coefficients of added mass and drag are as default set to zero above the water surface |

Bottom: | |||

Friction | 0.3 | - | The friction coefficient is increasing linearly to the given value up to the sliding speed and 0.01 m/s |

Coordinate | 0 | m | Vertical coordinate is x_{1} = 0 m at bottom and points upwards |

Elastic modulus | 3 × 10^{9} | Pa | For x_{1} < 0 m but 0 Pa for x_{1} ≥ 0 m |

Damping factor | 1 | - | For x_{1} < 0 m but 0 for x_{1} ≥ 0 m |

Cable: | |||

Unstretched length | 33 | m | L_{m} |

Horizontal span | 32.554 | m | From anchor point to centre of sheave |

Vertical span | 3.3 | m | From bottom to centre of sheave |

Density | 7800 | kg/m^{3} | ρ_{c} |

Stiffness | 10000 | N | K_{m} |

Mass per length unit | 0.0818 | kg/m | γ_{om} |

Steel diameter | 2.2 | mm | d_{om} |

Normal drag coefficient | 2.5 | - | Applied on the steel diameter d_{om} |

Tangential drag coefficient | 0.5 | - | Applied on the steel diameter d_{om} |

Added mass coefficient | 3.8 | - | Applied on the material area γ_{om}/ ρ_{c} |

_{m}(N) for the excitation radius r

_{m}= 0.2 m for the period T

_{m}=3.50 s are drawn in Figure 7 and for T

_{m}=1.25 s in Figure 8. Overall, the simulations match very well with the measured data, suggesting that all important dynamic effects are correctly accounted for in the numerical model. Some numerical oscillations are seen in the slack region of Figure 8. These are artefacts of the numerical settings used in MooDy (10 elements of polynomial order 7) and the ill-posed nature of the dynamic problem of cables under negative strain with negligible bending stiffness [35]. The results shown in Figure 7 and Figure 8 are unfiltered.

_{m}, T

_{m}) are plotted as a function of measured maximum tension. The agreement is shown to be very good, with a correlation coefficient of R

^{2}= 0.98 compared to full agreement.

**Figure 7.**Recorded (solid black) and simulated (dashed red) upper-end force F

_{m}(N) for the period T

_{m}=3.50 s and radius r

_{m}= 0.2 m.

**Figure 8.**Recorded (solid black) and simulated (dashed red) upper-end force F

_{m}(N) for the period T

_{m}=1.25 s and radius r

_{m}= 0.2 m.

**Figure 9.**Simulated versus measured maximum upper-end force F

_{max}(N) for various excitation periods T

_{m}(s) and radii r

_{m}(m).

## 5. Conclusions

^{2}= 0.98 of linear regression between simulated and measured force maxima are both strong indications that the physical model and the numerical model are of high quality. The comparison was made using the data of the physical model and thus involved no scaling errors, only the usual experimental inaccuracies, which are judged to be small. Using the present input in other codes should give a similarly good fit. The experimental data is herein made available for validation of other numerical codes through publishing digitised time series of upper-end force from two of the experiments.

## Supplementary Materials

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Bergdahl, L.; Palm, J.; Eskilsson, C.; Lindahl, J. Dynamically Scaled Model Experiment of a Mooring Cable. *J. Mar. Sci. Eng.* **2016**, *4*, 5.
https://doi.org/10.3390/jmse4010005

**AMA Style**

Bergdahl L, Palm J, Eskilsson C, Lindahl J. Dynamically Scaled Model Experiment of a Mooring Cable. *Journal of Marine Science and Engineering*. 2016; 4(1):5.
https://doi.org/10.3390/jmse4010005

**Chicago/Turabian Style**

Bergdahl, Lars, Johannes Palm, Claes Eskilsson, and Jan Lindahl. 2016. "Dynamically Scaled Model Experiment of a Mooring Cable" *Journal of Marine Science and Engineering* 4, no. 1: 5.
https://doi.org/10.3390/jmse4010005