Modeling and Numerical Simulation of Semitensioned Mooring Line under Taut-Slack State

. In this paper, considering the mooring system can be taut-slack in the motion, the dynamic equations of the mooring system are derived by using the theory of large deformation. Te nonlinear dynamic response of semitensioned mooring line is numerically simulated. Assuming the platform motion is known, the efect of platform on mooring line is simplifed as an end-point excitation. Directly using the fnite diference method for numerically solving partial diferential equations of mooring line, the dynamic responses can be obtained. Ten, the causes about the nonlinear state are analyzed, and the location where the taut-slack phenomenon occurs can be located by calculating the tension of mooring. Te results show that the mooring line is more likely to be taut-slack under tangential excitation with the tension change, while the mooring remains taut under normal excitation. Te taut-slack state of the mooring line is concentrated near the anchor point. Trough the amplitude-frequency curve and bifurcation point set, it is found that the taut-slack region is accompanied by multiperiod motion. And the taut-slack phenomenon will lead to the unstable motion.


Introduction
With the development of ocean engineering technology, ofshore oil production has gradually shifted from shallow sea to deep sea.As an indispensable part of the positioning of the deep-sea oil platform, the mooring system has become one of the key technologies that must be studied.Due to the structural characteristic which has large length-diameter ratio, mooring line's dynamic tension fuctuates greatly and results in the alternation of taut-slack state.Meanwhile, the constitutive relation of the mooring line also has nonlinear characteristics.Terefore, the analysis of the dynamics problem on mooring line under taut-slack state is always a concern in the feld of marine engineering.
Te numerical analysis of mooring line is basically realized by establishing equations and developing fnite element models.Perkins and Mote [1] derived the threedimensional dynamic equations of the elastic cable fxed at both ends by the Hamilton principle.Tangential, normal, and binormal motions along the cable are studied, respectively, and a theoretical model of the foating body was established for the analysis of cable vibration in water [2].According to this model, Tang et al. [3,4] studied the tension characteristic and impact tension under taut-slack condition.Ivan and Neven [5] developed a fnite element model considering the diameter and axial deformation of the mooring line, which can calculate hydrodynamic force more accurately when simulating the cable's large motion.Meng et al. [6] designed a mooring system of wave energy converter by using the second-order Stokes theory and verifed the stability of the mooring system and the whole device through model experiments.Behbahani-Nejad and Perkins [7] studied longitudinal-transverse coupled waves propagating along elastic cables.A mathematical model describing the three-dimensional nonlinear response of tensile elastic cables is presented.Te asymptotic form of the model is derived from the linear response of a cable with a small equilibrium curvature.Luo and Huang [8] studied a theoretical hydrodynamic model which is developed to describe the coupled dynamic response of a submerged foating tunnel (SFT) and mooring lines under regular waves.In that model, wave-induced hydrodynamic loads are estimated by the Morison equation for a moving object, and the simplifed governing diferential equation of the tunnel with mooring cables is solved using the fourth-order Runge-Kutta and Adams numerical method.
All the above scholars obtained the continuity equation of the system by establishing the continuity model, and the equation obtained thus is generally more practical but relatively complex and difcult to solve.Such as centralized mass method, the stifness matrix, and the recursive relation of the mooring can be easily written through the constitutive relation between elements, but the deformation of the element and the geometric nonlinearity of mooring are easily ignored.Terefore, in this research, the dynamics equations of the mooring system are deduced by the large deformation theory.Tis model can be solved by numerical method, and the geometric nonlinearity of the mooring is considered.
More and more software simulations can also be used to study mooring systems, such as AQWA [9], FhSim [10], and OrcaFlex [11].It is difcult to obtain the taut-slack phenomenon by software simulations.
Problems of the taut-slack efect and impact tension have been studied by scholars.Huang and Vassalos [12] used the spring-mass model to solve the impact tension of the cable from taut to slack state.Qiao et al. [13] analyzed the impact tension in the taut-slack state of the tensioned mooring by using the fnite element method.Wang et al. [14] studied the infuence of a submerged buoy on the tautslack state of the mooring line by using the concentrated mass method.Zhang et al. [15] studied the impact tension of taut mooring cable by using the fnite element method.Touzon et al. [16] used the catenary model to study the mooring system based on a wave energy converter and found that the sudden force generated by the foating body has a signifcant infuence on cable tension and resistance.Hsu et al. [17] studied the damping and impact load of anchor chain in a shallow water environment by experimental method.Xu and Guedes Soares [18][19][20] studied the hydrodynamic response of the point absorber and the dynamic response of the mooring system through a series of regular and irregular wave model tests on the buoy wave energy converter.Subsequently, based on the model experiment results, a new Markov chain by Bayesian inference method was proposed to study the short-term limit mooring tension.A mixed distribution extreme anchorage tension and fatigue damage analysis model was proposed, and the parameters of the mixed model were estimated by the expectation-maximization algorithm, and the limit tension analysis was carried out by depolymerization method to investigate the infuence of the polymerization process setting on the prediction of limit tension.Low et al. [21] studied the efect of line dispersion and the seafoor model on tension fuctuation by using the spring-pad method and improved seafoor reaction model.Hsu et al. [22] proposed a comprehensive probability distribution model to predict the impact load on the mooring line when the fan experiences large wave and wind-induced motion.Zhang et al. [23] used the hyperbolic tangent (tanh) method to transform the nonlinear partial diferential equation of the taut-slack mooring system into a nonlinear algebraic equation for solving and studied the tension changes of the mooring line in the process from slack to taut.
Tese above scholars have studied the taut-slack state of the mooring and found that this state is accompanied by impact load.Te impact load is fatal to the mooring in engineering, so it is very necessary to study the mechanism of the taut-slack state of the cable.
In addition, some scholars also study the impact tension of mooring system by experimental methods.Gomes et al. [24] conducted an experimental study on the buoy-swinging wave energy converter of the fve-device array in the wave pool with diferent confgurations, focusing on the analysis of the device's motion and mooring tension.Liang et al. [25] evaluated the feasibility of the simplifed mooring system by model experiment on a very large foating structure (VLFS) mooring system which is composed of 20 mooring chains.Gao et al. [26] conducted a series of tests in a tank with a model scale of 1 : 70 to study the dynamic response of mooring line based on catenary structure when wave and heave motions were applied.Te dynamic tension and trajectory of mooring lines were measured.Furthermore, the numerical results of this paper have the same trend as the experimental results by Gao et al. [26].
In this paper, the dynamics equations of the mooring system are deduced by the large deformation theory.Te nonlinear dynamic response of the semitensioned mooring line is numerically simulated.Assuming the platform motion is known, the efect of the platform on mooring line is simplifed as an end-point excitation.Directly using the fnite diference method to numerical solving partial differential equations of mooring line, the dynamic responses can be obtained.Ten, the causes about the nonlinear state are analyzed, and the location where the taut-slack phenomenon occurs can be located by calculating the tension of mooring.Tis research is based on the assumption that the model ignores the wave load generated by fuid motion in quasi-static fuid.And the diferential scheme of the fnite diference method has been verifed in previous research.

Model
When the ratio of the sag to the length of the mooring is greater than 1/8, the mooring is generally considered to be in a slack state and cannot be regarded as a taut string, so that the static stretching theory is no longer applicable [27].As the mooring is forced to vibrate at a large amplitude and low frequency by the end excitation, the local position of the mooring will show two alternating states of taut-slack.For the elastic elongated structure, this is a form of local geometric large deformation, so it is necessary to introduce the theory of large deformation to describe the dynamics model of the mooring system.
Considering the infnitesimal body with large deformation of mooring structure, the coordinate system is established as shown in Figure 1, where Te generalized Serret-Frenet formula is introduced [28]: where r is the radius of curvature, τ is the radius of torsion, s is the natural coordinate of the mooring, and the relation is Figure 1 shows the force analysis of the infnitesimal body.R → is the space vector describing position of the element, T → is the tension, F → is the external load acting on the mooring, G → is gravity, ρ is the mooring line density, and A is the efective cross-sectional area.For the momentum theorem, Decomposed in three directions: tangential, normal, and binormal.
Substitute equations ( 1)-( 3) into (6a), equation (6b) is obtained: Ten, three-dimensional dynamic equations of the infnitesimal body can be obtained as follows: where U 1 , U 2 , andU 3 are used to represent the displacement of the cable along the tangential, normal, and binormal directions relative to the equilibrium position, respectively, and l t and l n are the directional cosine of the body coordinate in the absolute coordinate system [29].Te absolute coordinate system is a Cartesian coordinate fxed on the seabed.
In the study of the mooring system, the mooring and the excitation load are placed in the same plane, so that the binormal motion is smaller than that in other directions and can be ignored, so the in-plane motion is mainly considered [27].And the mooring system can be shown in Figure 2.
By ignoring the spatial state of the mooring, 1/τ � 0 [28] so that equations ( 7) and ( 8) can be written as follows: where P is the static equilibrium tension, E is the elastic modulus, A is the efective cross-sectional area, κ is the curvature function of the mooring line in static state, ε is the tensile dynamic strain of the mooring line along the neutral layer in Lagrangian coordinates, s is the body coordinates, ρ w is the fuid density, and g is the gravitational acceleration.
Te obtained dynamic equations ( 10) and ( 11) are consistent with those established by the method of the Hamiltonian principle in literature [2].It means that the mooring's motion conforms to the conservative system, but the boundary excitation needs to be considered separately.
Since it is assumed that the mooring line is in a quasistatic fuid, the diameter is smaller than the wavelength.Te hydrodynamic force comes from the motion of the mooring structure, and its hydrodynamic force can be approximately obtained by the Morrison formula [5].Ten, the tangential and normal hydrodynamic forces acting on the unit mooring length can be expressed as F 1 and F 2 , respectively: where C at and C dt are inertia coefcient and drag force coefcient in the tangential direction, C an and C dn are inertia coefcient and drag force coefcient in the normal direction, and d is the efective diameter of the mooring.
Te boundary condition at the anchor point of the mooring line is Te upper point U D (t) gives diferent displacement functions according to the working conditions.U D (t) � U Dτ (t) + U Dn (t), where U Dτ (t) is tangential displacement and U Dn (t) is normal displacement.

Selection of Parameters and Difference Scheme
Te structural and environmental parameters related to mooring lines are shown in Table 1.
Due to the forced vibration of the mooring system under the given excitation of the platform, the two states of tautslack will occur repeatedly and alternately in the motion.During the slack state, the tension tends to 0, and the impact caused by tension will suddenly increase the tension several times [5], which makes the vibration state of the mooring system very complex.PDE (partial diferential equations), equations ( 10) and ( 11), belong to the wave equation form of hyperbolic function, and it is a dynamic boundary problem.In addition, the fuid drag force is also a nonlinear term.Terefore, the fnite diference method is the most suitable for direct numerical solution [30].
During the motion of the mooring system, there will be local taut-slack phenomenon, so there are high accuracy requirements for numerical calculation.Te diference scheme adopted is as follows [31].Some scholars have used this scheme to solve similar equations of fexible bodies and obtained more accurate results [29,32].
Te spatial diference scheme is as follows: Te time diference scheme is as follows: Shock and Vibration 5 where subscript j represents spatial coordinates and superscript k represents time coordinates, u is the displacement, v is the velocity, a is the acceleration, ∆t is the step length of time, and α 1 , α 2 , β 1 , and β 2 are the weights of acceleration and velocity at adjacent time step, respectively, and the values are {1, 1, 1, 2} [29].

Dynamic Response Analysis of Mooring Line with
Excitation.In the process of establishing the mathematical model, the body coordinate has been used to divide the mooring into tangential and normal motion.And there is a nonlinear coupling relationship among dynamic strains which generated by the mooring motion in diferent directions.For study patterns of the nonlinear phenomena between tangential and normal motion, the normal motion caused by tangential excitation and the tangential motion caused by normal excitation is calculated so that it is assumed that the excitation subjected on the upper point of the mooring replaces the large motion of the foating structure.

Dynamic Response of Mooring Line Endpoint
Only Subjected to Tangential Excitation.Generally, the period range of the frst-order simple harmonic wave of the ocean is 5 to 25 seconds.Te swing period of the semisubmersible platform is about greater than 100 seconds, and the heave period is generally greater than 20 seconds [27].Taking the South China Sea as an example, the average wave height can reach 6 m by typhoon.Terefore, it is assumed that there is a displacement excitation in the tangential direction at upper point of the mooring line, and the boundary condition can be expressed as follows: where A max � 6m is the tangential excitation amplitude, and the excitation frequency f � 0.2865Hz (circular frequency ω A � 1.8rad/s).Te dynamic response of the mooring system under tangential excitation can be obtained by using the numerical solution in equations ( 10) and (11).
In order to avoid the infuence of initial calculation error, the steady-state part of the system motion is selected for research, and the time is after t � 80 sec.Figure 3 shows the tension distribution curve of the mooring at diferent times from 83 sec to 84.7 sec.It can be seen that the tension of mooring within 280 m near the anchor point will appear 0 N during motion so that the taut-slack phenomena occur.Other parts of the mooring remain taut during motion.Meanwhile, the max tension change is at the anchor point.Te tension change along the mooring gradually decreases, and the min tension change of the mooring is near 1110 m from the anchor point.Ten, the tension change gradually increases until the excitation point.
According to the tension distribution on the mooring, Figure 3 is divided into four areas: I, II, III, and IV.I is an area where the taut-slack phenomena can occur.II is the area that the total tension can't be 0, and the tension change has the same order with the pretension.III is the min change of dynamic tension, and the amplitude of dynamic tension has one order of magnitude smaller than the pretension.IV is the area near the excitation point.Take four points A 1 , A 2 , A 3 , and A 4 from the four diferent areas on the mooring for analysis.A 1 is 130 m away from the anchor point, and it has the max total tension during the motion.A 2 is 650 m away from the anchor point, which has the max sag of the mooring static confguration.A 3 is the position which is 1110 m away from the anchor point, and it has smallest tension change during motion.A 4 is the excitation point.
Figure 4 shows that the max displacement amplitude of tangential motion distributes with the coordinate of mooring.Its tangential amplitude reaches the maximum near 1110 m, which is exactly the point that has the smallest tension change.It indicates that the motion of this region near the point is relatively synchronous, so that the dynamic strain is small, and the tension changes small.Figure 5 shows that the max displacement amplitude of normal motion distributes with the coordinate of mooring, and the max displacement appears near the anchor point.Since the mooring is subjected to tangential displacement excitation, its normal motion is caused by the geometric nonlinear coupling of the system structure.It indicates that the mooring near the upper point mainly moves by the displacement excitation, but the closer mooring is to the anchor point, the greater normal motion appears by coupling efect.
During the motion, dynamic responses of points A 1 , A 2 , A 3 , and A 4 on the mooring are shown in Figures 6 to 8. It can be seen from Figure 6 that the amplitude of the total tension of the system gradually decreases from the excitation point along the mooring line to the anchor point and reaches the minimum at point A 3 .Ten, the amplitude of total tension gradually increases and reaches the maximum at point A 1 .It means that tension near the anchor point changes drastically, and it is more likely to appear tautslack phenomenon.Shock and Vibration Mooring confguration can be described by calculating dynamic response.According to Figure 7, since the length of mooring is larger relative to the excitation amplitude, the whole confguration does not change signifcantly.Label x (m) and y (m) represent the horizontal and vertical span in an absolute coordinate system fxed on the seabed.In local magnifcation, it can be seen that mooring near the anchor point has unstable motion, while the motion of most other areas keeps stable.Terefore, the region of mooring near the anchor point is worth studying.
As shown in Figures 8 and 9, the period of tangential and normal motion is consistent with the excitation period.Tangential motion is forced vibration, while normal motion is caused by nonlinear coupling, so the period is the same as tangential motion.
Taking the motion of mooring 10 m away from the anchor point as an example, the time histories of tension are obtained in Figure 10.In Figure 10(a), when a point of  Shock and Vibration mooring becomes slack, the total tension of this point is 0, and the slack state will last for a while.As mentioned previously, the mooring will appear slack state within 280 m from anchor point.As shown in Figure 10(b), the closer to anchor point the slack state appears, the longer duration will be.Te relaxation state is closer to the fxed end.At 280 m from the anchor point, the duration of the slack state is 0.02 s in one period, while at 10 m point, the duration is 0.579 s.
Te response of the 10 m point motion is shown in Figure 11.In one period, the point moves rapidly to wave trough in the negative direction and then moves to the peak in the positive direction with a slower speed.Te reason is that the taut-slack phenomenon appears.
When the point moves towards the wave trough, the tension of this point is 0, and the mooring is in a slack state.As the point moves towards the peak, the mooring gradually tenses.When the mooring changes from slack to taut, a sudden increase in tension will lead to the mooring taut instantly.And the mooring will appear to fall and rise regularly in the normal direction because of elasticity.
Te Fourier transform of normal motion is shown in Figure 12, and the amplitude-frequency curve can be obtained.It can be seen from the fgure that the phenomenon of these regularly falling and rising in the normal motion is composed of multiple periodic solutions, and the motion frequency is times of the excitation frequency.
Furthermore, it can be seen from Figure 11 that the normal motion is in-phase with the tension, while the tangential motion is out-of-phase with the tension.Tis is because T � EA((zU 1 /zs) − κU 2 ), zU 1 /zs is in-phase with tension and normal motion.And the position 10 m from the anchor point tends to be fat, so the curvature is very small, κ � 0.0005.Terefore, the tension is in phase with zU 1 /zs and similar in shape.
Under this displacement excitation, the bifurcation diagram of the normal displacement of each point on the cable can be obtained along the cable coordinates.As shown in Figure 13, 0 is the anchor point.It can be found that points from the excitation point to 50 m of the mooring are in single-period motion, which is region II.And from 50 m to the anchor point, multiperiod motion appears in region I. Te closer it is to the anchor point, the more multiperiod solutions appear.Terefore, multiorder frequency occurs during the motion near the anchor point.Tis is because during the motion, when the mooring is slack, part of mooring close to the anchor point will recover original length with 0 tension, and stop at the seafoor.If the mooring becomes taut, this part of mooring will be lifted and strained again, so the phenomenon of multiperiod solution appears.
In addition, the maximum Lyapunov exponents of displacement at each point of mooring can be calculated by the wolf method [33].In Figure 14, within 280 m of mooring near the anchor point, the maximum Lyapunov exponent is greater than 0. It means that the closer to the anchor point tangential motion is, the more unstable motion will appear.Meanwhile, this part of mooring is also the area where the taut-slack phenomenon occurs, which also indicates that the occurrence of the taut-slack phenomenon will lead to the unstable motion.Te maximum exponent from 280 m to the excitation point is approximately 0 or less than 0, indicating that this part of mooring is in stable periodic motion.
As shown in Figure 15, the exponent of normal motion is mostly around 0. According to the previous calculation results, the normal motion is periodic motion mainly depends on the geometric nonlinear coupling of the mooring structure, so there will be no unstable motion state.
Moreover, not every excitation amplitude and frequency will lead to the taut-slack phenomenon.As shown in Figure 16, the amplitude of displacement excitation ranging from 1 to 12 m and excitation frequency ranging from 0 to 0.4456 Hz (circular frequency 2.8 rad/s) is divided into two parts.In region I, the mooring is always in a taut state, while region II (including line) is in a slack state where the local tension of the mooring is 0. From Figure 16, it can be known that when the excitation frequency is 0.1 Hz, amplitude of displacement excitation must reach 10 m and lead to a slack    Shock and Vibration state.As the amplitude decreases, a slack state occurs need an increasing frequency.Tat means the taut-slack phenomenon appears only when the energy in the system reaches to a certain level.

Dynamic Response of Mooring Line Endpoint Only under
Normal Excitation.When only the normal excitation is applied at upper point of the mooring, the boundary condition can be expressed as follows:  Shock and Vibration where B max � 6m is the excitation amplitude and the excitation frequency f � 0.2865Hz (ω B � 1.8rad/s).Figures 17 to 19, respectively, describe the time history curves of total tension, tangential motion, and normal motion at 440 m, 530 m, 820 m, 1020 m, and 1230 m away from the anchor point.Te same conclusion can be drawn from Figure 17 that the total tension at each point of the mooring does not change much.Te tangential motion of the mooring is caused by nonlinear coupling with the normal motion, but the amplitude is an order of magnitude smaller than that of the normal motion, and the amplitude of the normal motion does not change much.It shows that motion under normal displacement excitation cannot be transmitted far, and it mainly afects the motion of the mooring near the excitation point, while the closer to the anchor point, the smaller change of displacement amplitude.More importantly, there is no sudden change in motion, which means there is no taut-slack phenomenon.And the taut-slack phenomenon is mainly induced by tangential displacement excitation.

Conclusions
In this paper, the nonlinear dynamic response of semitensioned mooring line under quasi-static fuid is numerically simulated.Assuming that the motion of the platform is known, the efect of the platform on the mooring is simplifed as endpoint displacement excitation.Te dynamic response of mooring line is numerically calculated by the fnite diference method.All above calculation results are consistent with the experimental results in the literature [26], and the following conclusions are obtained: (1) Considering the mooring system can be taut-slack in the motion, the dynamic equations of the mooring system are derived by using the theory of large deformation.(2) Under the tangential and normal excitation, the conditions of taut-slack motion of mooring are analyzed.It is found that when the tangential excitation is given, the mooring will be in the taut-slack state, while the system will always remain in the taut state with the normal excitation.For the mooring lines with the parameters in this paper, the excitation amplitude and frequency region that will cause the taut-slack motion state are given.(3) Te taut-slack state of the mooring line is concentrated near the anchor point.During the process from slack to taut, due to the sudden increase of tension, it will lead to the instant tightening of the mooring line and make the mooring in normal motion to appear regular falling and rising Shock and Vibration 11 phenomenon.Trough the amplitude-frequency curve and bifurcation point set, it is found that the taut-slack region is accompanied by perioddoubling motion.(4) Te stability analysis of the motion time series under tangential excitation shows that the taut-slack phenomenon will lead to the unstable motion, and the normal motion mainly depends on the geometric nonlinear coupling of the mooring structure, so there will be no unstable motion state.
In conclusion, from the analysis of numerical simulation results, it can be known that the tangential excitation of mooring line can make the tension change greatly, so it is easy to produce the taut-slack state, and the large amplitude of normal motion can also be caused due to the coupling efect.When the system is excited in the normal direction, it is not easy to cause the tangential motion of the system.Te system is always in a taut state, and its tension basically fuctuates within a certain range.Terefore, in order to avoid the impact caused by taut-slack and sudden change of tension in the system under actual working conditions, we should try to avoid the excitation area where this can happen and mainly limit the motion of tangential excitation.

Figure 2 :
Figure 2: Schematic diagram of mooring line motion.

Figure 3 :
Figure 3: Tension distribution of mooring line at diferent times.

Figure 4 :
Figure 4: Amplitude distribution of tangential motion of mooring line.

Figure 9 :
Figure 9: Time history of normal motion at A 1 , A 2 , A 3 , and A 4 .

Figure 10 :
Figure 10: (a) Time history of total tension at 10 m from the anchor point and (b) time history of total tension at positions when the cable is relaxed.

Figure 11 :
Figure 11: Time history of normal motion at 10 m from anchor point.

Figure 12 :
Figure12: Amplitude frequency curve of normal motion at 10 m from anchor point.

Figure 13 :Figure 14 :Figure 15 :Figure 16 :
Figure 13: Bifurcation diagram of normal displacement at each position on the cable: (a) the whole mooring structure and (b) partial of mooring structure 0-200 m.