Dynamic response analysis of the equivalent water depth truncated point of the catenary mooring line

The real-time computer-controlled actuators are used to connect the truncated parts of moorings and risers in the active hybrid model testing system. This must be able to work in model-scale real time, based on feedback input from the floater motions. Thus, mooring line dynamics and damping effects are artificially simulated in real time, based on a computer-based model of the problem. In consideration of the nonlinear characteristics of the sea platform catenary mooring line, the equations of the mooring line motion are formulated by using the lumped-mass method and the dynamic response of several points on the mooring line is investigated by the time and frequency domain analysis method. The dynamic response of the representative point on the mooring line is analyzed under the condition of two different corresponding upper endpoint movements namely sine wave excitation and random wave excitation. The corresponding laws of the dynamic response between the equivalent water depth truncated points at different locations and the upper endpoint are obtained, which can provide technical support for further study of the active hybrid model test.


Introduction
Ocean accounts for about 71% of the surface of the earth, and is rich in resources, such as oil, gas and biological resources. By the end of 2009, it is estimated that there were 185.504 billion tons of oil reserves and 187.16 trillion cubic meters of natural gas reserves, but 44% of the oil and gas resources are distributed in over 300 m deep sea (Li, 2006). Besides, in the past 10 years, 40% to 45% of the world's major oil and gas fields have been discovered in the deep sea. Owing to the international energy shortage, in the countries with ocean, especially with the field of deep sea, the competition is increasingly fierce.
With the advancement of marine oil and gas exploration to the deep-sea, the working depth of deep-sea platform is getting deeper and deeper. Since the existing scale of ocean engineering experimental basin is not large enough, the entire platform and its mooring system cannot conduct the model test by using conventional scale proportion, and thus the hybrid model test method is needed (Stansberg et al., 2002). This method mainly has two styles: passive and active. By the passive style hybrid model testing technique, the motion and mechanical characteristics of the full depth deep-sea platform system is obtained through indirect numerical method. To obtain satisfactory results, rich experience and reliable numerical software support are essential, yet inadequate especially for innovative deep-sea platform. By active style hybrid model testing technique, results are obtained directly through the model test, but the motion and mechanical characteristics in truncated point shall be acquired in advance. So for now, a control device of six-de-grees of freedom motion simulation is a promising design that can be used as an approximated alternative for the part below the truncated point in the mooring line to solve all deep-sea platform system model test problems once for all.
The basic principle of the active style hybrid model testing technique is shown in Fig. 1. According to the simulation depth of the pool, the truncated part of each mooring line/riser at the bottom of the pool is replaced by a computer-controlled real time servo, to simulate the motion and force of the mooring line/riser at truncated point actively. Thus the actual dynamic characteristics of the system can be directly obtained by the test, which is also an advantage of this method.
To realize the active style hybrid model test method, the primary job is to precisely calculate the motion and stress conditions at the truncated point. A great deal of research has been carried out by the corresponding scholars at home and abroad. Shen et al. (2015) studied the mooring types and dynamic analysis properties blow underwater 1000 m. In their study, the Deep-C of Sesam software was used to calculate the effect of dynamic analysis by changing the mooring length and cable angle. When the equivalent truncation of the mooring line is designed, if the degree of truncation is sizeable, it will be very difficult to design the mooring line. To solve this problem, Wang et al. (2015) proposed a method which uses a spring system to replace the truncated mooring line and to accurately simulate its stiffness characteristics. The accuracy and rationality of their method were proved by static analysis and dynamic analysis in the time domain. In addition, Fan et al. (2015) used the numerical solution of piecewise extrapolation and made static characteristics analysis on the top angle of the mooring line, and an optimal design program was developed based on the genetic algorithm.
To calculate the motions and stress conditions of the truncated points, lumped mass method has been a commonly used method in recent years. For example, Chai et al. (2002) raised a three-dimensional lump-mass formulation of the catenary risers, which can analyze the irregular seabed interaction, and the method can be widely used in the static and dynamic analysis of the mooring lines, the vertical pipe and the submarine tunnel. In addition, Tang et al. (2009) adopted the three-dimensional lumped mass model to analyze the cable configuration and tension, and the dynamic response of the cable was analyzed with different excitation amplitudes and frequencies. Besides, the occurrence condition of the taught-slag state of the mooring line was predicted, and the snap tension of the mooring line was calculated. This paper ignores some small factors and mainly considers nonlinear factors like fluid resistance and geometric characteristics, adopts the lumped mass method to analyze the dynamic characteristics of nonlinear, multidegree freedom of a mooring line, and provides technical support for the active style hybrid model testing technique.

Mathematical model for dynamic analysis of a mooring line
The three main models of dynamic calculation of a mooring line are the lumped mass model, slender rod model, and catenary model. The catenary model is mainly used for shallow water and static calculation of a mooring line (Wang et al., 2012;Guo, 2012). Garrett (1982) proposed the slender rod model and established a three-dimensional computational model. Tang et al. (2010) used the slender rod model to calculate the mooring system modal by a finite element software ABAQUS. Based on the slender rod model, Xiao (2006) calculated the dynamic response of the mooring line by the Ne~kar method and mN-R iteration method. In this section, the lumped mass method is chosen to calculate and analyze the dynamic properties of the mooring line.

Establishment of motion equations
The model of the lumped mass analysis is shown in Fig. 2, where the two-dimensional condition is considered, namely, the deformation of a mooring line in the plane of xoz is considered only. Taking the seabed anchor as the coordinate origin o, the horizontal direction toward the appendage as the positive direction of the x-axis, and the upwardly perpendicular direction toward the sea level as the positive of the z-axis.
The mooring line is divided into N segments. Set the mass of each segment unit at the node, and the mooring line unit is regarded as the ideal spring. The mass of the nodes generally equals to half the total weight of each two adjacent mooring line units, except for the upper and lower endpoints. The stress analysis of a mooring line unit node is shown in Fig. 3.  According to Newton's Second Law, the motion equations of the j-th node are established: where, M j is the mass of the j-th node; and are the acceleration components in the x-axis and z-axis direction for the j-th node; A nj and A tj represent the vertical and tangential added mass, respectively; is the average value of the corresponding mooring line angle of the (j-1)-th node and the j node (γ j is the included angle of the jth node in the horizontal direction). The calculation methods of these parameters are based on Miao's presentation (Miao, 1995).
F xj and F zj are the aggregated external force components acted on the j-th node with the formulae as follows: (4) where, T j represents the tension of the mooring line unit between the j-th and the (j+1)-th nodes; δ j is the revised weight of the j-th node; f dxj and f dzj are the hydraulic resistances in the horizontal and vertical directions at the j-th node, respectively represented in Eq. (5) and Eq. (6).
where, ρ is the density of seawater; D c is the diameter of the mooring line unit; is the length of the mooring line unit; A rx and A rz represent the projected area of the buoy or the weight in the x-axis direction and z-axis direction, respectively; C dx and C dz are the resistance coefficients of the buoy or the weight. If there is no buoy or weight, the underlined part will be negligible; C dn and C dt are vertical and tangential resistance coefficients, respectively, expressed in Huang et al. (1992). By substituting Eqs.
(3)-(6) into Eqs. (1) and (2), the new motion equations are obtained: Finally, the new motion equations can be described by the Houbolt difference method, and these equations are solved by the Newmark method.

Treatment on boundary conditions 2.2.1 Weight adjustment near the point of tangency
Since the bottom of the mooring line lies on the seabed and part of it touched the seabed, it is necessary to amend the lumped mass point of this part to minimize the error of the calculation.
As shown in Fig. 4, set the weight of the mooring line lower than the bottom of the seabed as zero on which the parabola touch the seabed, and the intersection of the parabola and the seabed is marked as I-1, and then the revised method of the weight of the two units above the intersection point (δ I and δ I+1 ) is shown as follows: When 0≤∆l I-1 <l I-1 : where When ∆l I-1 <0

Treatment of the upper and lower endpoint
As the lower endpoint is fixed in the seabed, T 0 does not exist, and in the constraint function of does not exist either, i.e. does not exist. Thus, it can set . Meanwhile, since the displacement of the fixed point (the origin) is always zero, the equations can be written as follows: For the upper endpoint, in the constraint function of is only related to time, and is independent of the displacement of the upper endpoint, i.e.
, therefore, it can set . Similarly, the spherical coordinate transform of the upper endpoint is only related to time, and we obtain that: 2.3 Example Based on the above theory, the C++ program to calculate the mooring motion is wriiten, and the calculation of the following example is verified.
The given mooring line equivalent diameter is 0.00599 m, the length of the mooring line is 9 m, the depth of water is 3 m, the elastic modulus of the mooring line is 2.107×10 10 N/m 3 , the initial pretension is 28.4886 N, the density of seawater is 1030 kg/m 3 , the time step is 0.02 s, and the velocity of water is 0.5 m/s. When the given upper endpoint incentive is the simple harmonic motion in the direction of the x-axis, by comparing the program calculation results with the test results of Nakajima et al. (1982), it is found that the present results are slightly larger than the reference results, as shown in Fig. 5. In addition, the initial step problem of this example is also analyzed in this paper. The central difference method and Newmark method are used respectively to verify the initial calculation, and the result is shown in Fig. 6. It turns out that the Newmark method is superior to the center difference method in terms of the stability in numerical integration. Fig. 7 gives the simple harmonic motion of the upper endpoint incentive in the directions of the x-axis and z-axis.
The curves show that, for these two different initial algorithms, the movement of the upper endpoint by the Newmark method can reach a steady state more quickly compared with that by the central difference method.
As to initial problems, when the step length Δt is 0.01 s or 0.05 s, the tension oscillation is relatively large, especially when the step length Δt is 0.01 s. As shown in Fig. 7, the iterative process goes into a stable state after 3 s, and when Δt are 0.02 s, 0.03 s and 0.04 s, the tension time trace curves almost coincide. And when the step length Δt is 0.01 s or 0.05 s, the maximum amplitudes and oscillation degree of the tension are larger than those in other three cases.

Dynamic response analysis of the water depth truncated mooring line
In the previous section, two-dimensional motion equations of the mooring line are established based on lumped mass method. The dynamic response of mooring line nodes will be investigated using time domain and frequency domain analysis method in this section on the basis of the previous section. The dynamic response of the representative points on the mooring lines is analyzed by giving two different corresponding upper endpoint's movements which are sine waves and random waves. In this section, the corresponding laws of the dynamic response between the equivalent water depth truncated points at different locations and the upper endpoint are obtained, which can provide technical support for further study in the active hybrid model testing technique.

Transformation of time domain and frequency domain
Transformation from time domain to frequency domain was conducted according to the Fourier transform and inverse Fourier transform. Details of the time domain solution of the motion equations can be found in the last section. Here, at any moment the movement of the mooring line can be regarded as a series of time sequence of pulses movement superimposed. At the same time, the movement of the upper endpoint is also decomposed into a series of impulse response, so that the mooring motion in the time domain can be extended to the movement frequency domain for spectral analysis.

JONSWAP spectrum
From 1968 to 1969, "Joint North Sea Wave Project" (JONSWAP) was conducted by UK, the USA, France, Denmark, the Netherlands, Canada and Federal Germany cooperatively to observe waves with instruments by setting 13 observation stations in the northwest of the Sylt Island. The storms spectra of limited wind distance were exported by 2500 measured spectra (Yu, 2000). The specific formula is as follows: where, S(f) is the density of spectrum (m 2 .s); α is energy scale parameter; f is the frequency of wave (Hz); g is the acceleration of gravity (m 2 /s); T p is the period of spectral peak (s); γ is the rise factor of spectral peak; f p is the frequency of spectrum peak (Hz)= 1/T p ; and .

Generation of random wave
The movement of the upper endpoint of the floating structure can be simulated by the method similar to the random wave numerical method with the formula as follows: (17) where, m is the wavelet number; ω i is the frequency of wave component (rad/s); ∆t is time step (s); δ i is the random initial phase angle (rad); n is time series; and . JONSWAP spectrum was selected as the target spectrum S(ω i ). For the convenience of the calculation, the target spectrum is recorded as: ;H s is the significant wave height (m); m is 1000, n is 10000, and ∆t is 0.05 s. The time curve of the motion for the upper endpoint was obtained by writing C++ program shown in Fig. 8. Fig. 9 shows the corresponding density spectrum function curve.

Calculation of the dynamic response of the mooring line
An FPSO mooring line with the working depth of 120 m was taken as the research object in this section. The length ZHANG Huo-ming et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 37-47 41 of the given mooring line is 400 m. Specific parameters are shown in Table 1. Given that the initial pretension is 980000 N; the density of seawater is 1025 kg/m 3 , the flow velocity is 1 m/s. Set the mooring line into N = 20 segments equally, thus, there are totally 21 lumped mass points including the upper endpoint. And the mass of each segment is assumed to concentrate on its two endpoints. The length of each segment is 20 m, and the time step ∆t equals 0.05 s. Each segment of the mooring line is affected by some external forces like gravity, buoyancy and fluid resistance.

Calculation of mooring rope static shape and tension
The given preliminary tension T is 980000 N. Assume that the seabed is flat, the shape and tension of the mooring line in static balance state can be calculated according to parameters in Table 1. The theory and program of calculation are seen in the above section. Figs. 10 and 11 are the shape and tension curves of the mooring line in static balance state, respectively. As shown in the figures, the mooring line is in the shape of catenary, and the tension increases with the increasing horizontal span.

Dynamic response analysis for the upper endpoint in
sine wave movement The incentive of the upper endpoint of the mooring line in this section is given by (A is amplitude). Lumped mass method is used to calculate the time history of the upper endpoint of the mooring line. Figs. 12-14 are the tension time history curves under the condition of     When A=12 m, the corresponding maximum tension reduces significantly on the low-frequency stage smaller than 0.6 rad/s, and when the frequency is larger than 0.6 rad/s, the corresponding maximum tension gradually increases.
Figs. 16, 18 and 20 are the tension power spectrum curves when A = 1 m, A = 5 m and A = 12 m, respectively. As it is shown, the power spectral densities under the three conditions mainly concentrated around 0.6 rad/s. When the amplitude is 1 m, a spectral peak appears near 8.2 rad/s,   Fig. 19. Tension curve changed with ω when amplitude is 12 m. Fig. 20. Power spectrum curve of above endpoint when amplitude is 12 m.
ZHANG Huo-ming et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 37-47 43 which is mainly the spectral density at inherent frequency of the mooring line. As shown in Fig. 16, the tension time history curve in Fig. 12 is rather rough.

Dynamic response analysis for the upper endpoint in random wave movement
The same example is chosen here to analyze the motion and stress of the 5th, 10th, 13th and 15th lumped mass points of the mooring line. These points are imaginary truncated points for the equivalent water depth, among which, the water depth of the 13th point is half of the total water depth. The motion of the upper endpoint of the floating body is generated by the numerical method of random wave, as Eq. (17) shows. The time history curve of corresponding motion for the upper endpoint is shown in Fig. 8. The tension time history curves of the upper endpoint, the 5th, 10th, 13th and 15th truncated point are shown in Fig.   21  show the corresponding tension power spectrum curves of the 5th, 10th, 13th and 15th truncated point, respectively. From the comparison of these power density spectrums, it can be seen that the corresponding mooring line's power density spectrums of different nodes are basically the same with the power density spectrum's trend of the upper endpoint of the mooring line horizontally, and the upper endpoint's spectral peak is larger than those of the other nodes. The closer the nodes are to the bottom, the smaller the spectral peaks and envelope areas will be. Figs. 30-33 give the motion trails of the nodes under different conditions. Through the given motion of the upper point is in a single horizontal direction, it can be seen that the motion of mooring line nodes underwater are activated in both vertical and horizontal directions, and the deeper the water is, the smaller the movement range of nodes will be.

X = Asin( t)
According to the response amplitude operator (RAO) in the calculation of the frequency domain (Yang et al., 2008), by giving the motion of the upper endpoint of the floating body as , and defining the i-th (i=1, 2, 3, …, N+1) node's corresponding tension frequency response function as RAO, we obtain:     ZHANG Huo-ming et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 37-47 45 ponding RAO curves of the 5th, 10th, 13th, 15th node have the same trends with the frequency, and the corresponding peak positions are also basically the same, but the peak values are quite discrepant.

Conclusion and prospects
Lumped mass method was adopted to analyze the dynamic characteristics of mooring line. The motion equation in the time domain is established and the corresponding computer program is developed by C++ language. Through transforming the time domain into the frequency domain, the dynamic response of the representative points on the mooring line is analyzed under two kinds of different movement conditions of the floating structures' upper endpoint. The corresponding relationship of the dynamic response between the equivalent water depth truncated points at different locations and the upper endpoint are obtained in the horizontal range, which can provide technical support for further study of the active hybrid model testing technique.
Meanwhile, we must point out that the mooring lines applied in deep-sea are mostly synthetic fiber materials which are zero gravity in water, and the physical properties of the materials are not linear. But the method used in this paper is still feasible and effective. We just need to modify the mathematical model of the mooring nodes, and take its dynamic stiffness and material nonlinearity into consideration. This problem will be solved in future research.