1 Introduction

The tension leg platform (TLP) system is one of the best design for oil production in deep water, because the construction cost does not increase dramatically with increasing depth of water, and the resonant motions are detuned from the frequencies of dominant wave energy spectrum (Zou, 1997). The floating system connected to the sea floor by a series of pretensioned tendons, which provide horizontal and vertical stiffness to ensure the stability of platform in a wide range of sea conditions.

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Various methods for the hydrodynamic analysis of TLPs at wave frequencies have been presented in the recent technical literature. Coupled nonlinear dynamic analysis of the SeaStar mini TLP in the time domain using the finite element method with wave loading by the linear diffraction-radiation theory is carried out and compared with the calculations based upon the Morison equation for wave forces as well as with the experiments (Bhattacharyya et al., 2003). A theoretical model was established for analyzing the nonlinear dynamic behavior of a tension leg platform with finite displacement by considering the instantaneous position and instantaneous wet surface (Zeng et al., 2007). Coupled response of a 4-column TLP for random waves was investigated by using modified Morison equation (Gu et al., 2012). Coupled effect of the horizontal motions (surge, sway and yaw) and the set-down motion is considered, and nonlinear dynamic response of TLP with six degrees of freedom on the wave height is acquired in both time and frequency domains (Wang et al.,2015). However, the second-order wave force were not considered in the above-mentioned literature. A new consistent restoring stiffness of a 4-column TLP considering the motion of set-down was presented, and a set of simulations considering difference-frequency wave forces were carried out (Senjanović et al., 2013). A simple method for incorporating set-down of a TLP in the extreme response prediction of the air-gap was developed by considering wave-frequency and low-frequency responses (Low, 2010),but the high-frequency motion was not discussed. The application of a time-domain second-order method in the numerical simulation of the nonlinear wave interaction with surface-piercing fixed and floating circular cylinders was described (Bai and Teng, 2013). A method to assess the effect of second-order hydrodynamics on a TLP floating offshore wind turbine was presented by using combi-nation of the frequency-domain tool WAMIT and the time domain tool FAST (Roald et al., 2013; Gueydon et al.,2015). Experimental modeling of a moored TLP structure interacting with a plunging breaker was carried out in a laboratory (Chuang et al., 2015). The resonance of a floating TLP excited by the third-harmonic force of a regular wave is investigated based on fully nonlinear theory with a higher order boundary element method (Zhou and Wu, 2015). Therefore, there are few papers mentioned the effect of second-order wave forces which includes the both diff- and sum-frequency forces on the TLP considering the nonlinear set-down motion.

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In this study, in order to investigate the effect of the second-order wave force on the 6-DOF motions of TLP considering nonlinear set-down motion, the second-order wave forces which include the mean-drift part, sum- and difference-frequency parts, are calculated using the fullfield quadratic transfer function (QTF), and then the lowand high-frequency resonant responses of the platform due to the effects of difference- and sum-frequency forces are analyzed. A set of time-domain simulation is conducted using a code, which integrating the nonlinear restoring force, first- and second-order wave force, and then the nonlinear dynamic responses of TLP in 6-DOFs are acquired.

2 Equation of Motion

The equation of motion of the moored offshore floating foundation can be written as follows (MARINTEK, 2013)

where M is the body mass matrix, A∞ is the frequencydependent added-mass matrix when the motional frequency closes to infinity ω=∞, D is the nonlinear damping matrix, f is the vector function where each element is given byis the nonlinear restoring matrix,h is the retardation function,represent the position, velocity and acceleration vectors of platform, q is the exciting force vector including first- and second-order wave forces, t is time.

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2.1 Mass and Damping Matrix

The body mass matrix is shown as

where M is the total mass of TLP; Jx, Jy and Jz are the moments of inertia of roll, pitch and yaw.

In order to decouple the damping term of the oscillation equation, Rayleigh damping is adopted. Suppose that the Rayleigh damping matrix is

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where α and β are constants referred to as the Rayleigh damping coefficient.

2.2 Restoring Stiffness Matrix

Each component of the nonlinear restoring stiffness KP with respect to the reference point P, which is the center point of the hull bottom, is shown as follows (Senjanović et al., 2013)

where K11, K22, K31, K32, K33, K36, K44, K55 and K66 are shown as follows

where T denotes the pretension of the tension leg, Lz is the mean of the vertical distance from the upper endpoints to the seabed, ρ is the density of water, g is the acceleration of gravity, Aw is the area of water plane of the column, δs is the increment of the set-down motion, E is the elasticity modulus of tension leg, A is the area of tension leg sec-tion, L is the length of the tension leg, δx, δy and ψ are the displacement of surge, sway and the angle of yaw, r is the horizontal distance between the center of the bottom of the platform and the center of the column section, Ix and Iy are the inertia moments of water plane about x-axis and y-axis, Ixx and Iyy are the inertia moments of all tension leg about x-axis and y-axis, φ and θ are the rotation angles of roll and pitch, zB, zT and zG are the vertical coordinates of the buoyancy center, the upper endpoint of tension leg,and the center of gravity of platform, respectively.

Nonlinear restoring stiffness of the platform with respect to the center of gravity G is shown as follows

where T is a displacement transformation matrix in Eq.(15).

发电机高压油顶起系统由高压油顶起油泵、逆止阀、过滤器、高压油管、管件、压力开关和压力表等部件组成，集成在2 250 mm×1 200 mm×1 800 mm的框内，布置在机坑外。常压油取自推力油槽内，高压油通过高压油泵加压后输送至推力轴瓦，每个高压油供油支管设置有一个单向阀，其系统原理见图1。

Expressed in a complex form, the wave elevation at origin changes into the following equation

Thus, the restoring stiffness matrix with respect to G is shown in as follows

2.3 Hydrodynamic Forces

2.3.1 Random waves

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The random wave model (Faltinsen, 1993) is based on Longuet-Higgins model. It is the random wave surface elevation propagating along the positive x-axis; The surface elevation η(t) can be decomposed into the sum of a number of regular wave units in the following form

For each regular wave component, an denotes the wave amplitude, ωn is the circular frequency, kn is the wave

where is the conjugate of the complex wave amplitude Am, the QTF is denoted as the functions H(2+)(ωn, ωm)number, εn is the random phase angle, N is the number of the wave components. The wave amplitude an can be expressed by the wave energy spectrum S(ω) in the form

From Fig.5, it can be found that both the set-down and vertical motions are different from those in first- and second-order wave force models. In the second-order model,the amplitude of set-down motion is much larger than the result of first-order model. For example, at t=590 s, the set-down motion reaches the maximum amplitude of−0.011 m under first-order wave force, but the maximum of set-down motion becomes −0.047 m under both firstand second-order forces at the same time. That is to say,the latter maximum is 3.3 times larger than the former one. Besides, the equilibrium of set-down under both first- and second-order forces is also even lower than that under first-order force only. Furthermore, the vertical motion under both forces is slightly larger than that under first-order force only.

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However, it is shown that the simulated random wave generated by equal frequency spacing is pseudo-random,and the extreme crest will repeat with a period of 2π/∆ω,which results in an inaccurate random wave. For this reason, the random frequency spacing can be used to solve this problem.

where zGT is the vertical distance between the center of gravity of the platform and the top of the tension leg.

2.3.2 First- and second-order wave forces

To obtain the random wave force (Faltinsen, 1993) on the platform in time domain, firstly, the wave force transfer functions are calculated based on potential theory in frequency domain, according to DNVGL software WADAM.Then, based on these hydrodynamic parameters and specified wave spectrum, the random wave forces series can be generated in time domain.

Wave forces can be expressed as follows

The time-dependent second-order wave forcesincluding the mean-drift, difference- and sum-frequency forces, are obtained by using full-field quadratic transfer functions (QTF). The calculation method of QTF is based on Kim and Yue (1989, 1990), and the calculation is also performed by WADAM. The second-order wave forces can be expressed as

and which are the second-order sum- and difference-frequency forces transfer functions.

The time-dependent first-order wave forcesis described in the frequency domain by a transfer function between wave elevation and force

where H(1)(ωn) is the first-order transfer function computed by WADAM.

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whereis time-dependent first-order wave force,is time-dependent second-order wave force.

For this platform, the second-order sum- and differencefrequency force QTFs H(2+)(ωn, ωm) and H(2−)(ωn, ωm) are calculated in all DOFs. The QTFs with respect to the incident angles of 30˚ are shown in Fig.1. In Fig.1, the x-axis and y-axis represent the two different incident wave frequencies, respectively, and the z-axis represents the value of QTF with respect to the corresponding incident wave frequencies.

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Fig.1 2nd SUM- and DIFF-Freq force QTF in 6-DOF.

3 Analysis of Examples

In this section, for a working water depth 450 m, the calculations and analysis are based on the ISSC TLP model. The schematic of the platform is shown in Fig.2,and the platform’s main parameters are shown in Table 1(Senjanović et al., 2013)

3.1 6-DOF Motions Under First- and Second-Order Wave Force

Four types of wave force models are adopted for simulation, including first-order model, second-order (d) model,second-order (s) model and second-order model. Details of four models are shown as Table 2.

In order to study the impact of second-order wave force,the responses of the platform subjected to the wave forces are simulated. The simulation is carried out by using the first- and second-order models respectively. The chosen operating wave is based on JONSWAP spectrum with a significant wave height of Hs=10.4 m and a peak-spectral period of Tp=12.5 s, which is the 50-yr random sea state in the South China Sea. The incident angle of the random waves is 30 degree. The 6-DOF responses of the platform in time domain are shown in Fig.3 for the range of time from 500 s to 2500 s.

Fig.2 Schematic of the TLP platform.

Table 1 Main parameters of ISSC TLP

Parameter Value Column diameter (m) 16.88 Column spacing (m) 86.25 Pontoon width (m) 7.5 Pontoon height (m) 10.5 Initial draft (m) 35.0 Displacement (kg) 5.45×107 Mass (kg) 4.05×107 Roll mass moment of inertia (kg m2) 82.37×109 Pitch mass moment of inertia (kg m2) 82.37×109 Yaw mass moment of inertia (kg m2) 98.07×109 Length of tension leg (m) 415.0 Inner diameter of tension leg (m) 0.3464 Outer diameter of tension leg (m) 0.8 Elastic modulus of tension leg (N m−2) 2.04×1011(EIxx/L), (EIyy/L) (Nm rad−1) 1.501×1012 Vertical position of COG above keel (m) 38.0 Vertical position of COB above keel (m) 22.3

Fig.3 Comparison of 6-DOF motions under first- and second-order wave forces.

Table 2 Details of four wave force models

Note: The symbol ‘○’ represents the corresponding wave force included in model.

Wave force Models First-order Mean-drift Difference-frequency Sum-frequency Second-order First-order ○Second-order (d) ○ ○ ○Second-order (s) ○ ○Second-order ○ ○ ○ ○

From Fig.3, it can be seen that there is a large difference between the results of two the models (first- and second-order models) only except for the yaw motion. In the surge and sway motions, both amplitudes and equilibrium positions in second-order model are larger than those in first-order wave force model. The amplified surge motion indicates that the second-order wave force can excites a large surge motion. Furthermore, the heave motions under two models also have a large difference which is reflected on two aspects: the amplitude and the equilibrium position. Besides, it is found that for the simulation using second-order wave force, the roll motion and pitch motions are obviously larger than the results of the firstorder model. However, there is little difference between the results of the yaw motion in the two models. In other words, the second-order wave force have little impact on the yaw motion of the platform.

In addition, the power spectral densities of platform motions in first-order, second-order (d) and second-order models are shown in Fig.4, respectively.

From Fig.4, it can be found that, in models of secondorder (d) and second-order, a peak of the surge spectrum appears at a low-frequency which is the natural frequency of surge/sway motion, but at this low-frequency there is no peak existing in the first-order model. The other peak appears at the wave characteristic frequency. However, in the first-order model, the peak of the surge spectrum only focuses on the characteristic wave-frequency, and there is no power in the low-frequency domain. Therefore, it can be concluded that the second-order difference-frequency wave force induces a large low- frequency motion of surge,and the wave-frequency motion of surge is usually caused by the first-order wave force. The spectral characteristics of sway motion are identical to the surge motion. From the heave spectrum, it can be seen that the second-order difference-frequency force increases both low-frequency and wave-frequency motions, and the wave-frequency motion obviously dominates the heave motions. A lot of high-frequency motion components appear in the roll and pitch spectrums when the platform under the second-order wave force, which includes sum- and difference-frequency force. From the spectrums, we can see that the high-frequency motion is not induced by the differencefrequency force. In other words, the sum-frequency force will induce a set of high-frequency motions of roll and pitch. However, the wave-frequency motions also dominate the roll and pitch motions. Eventually, it can be seen that the power spectral densities of the yaw motions in the three models are almost overlapping due to the slight impact of the second-order wave force on the yaw motion.

Table 3 shows the statistics of the 6-DOF responses under four types of wave force models (the details of wave force models are shown in Table 2). From Table 3,it can be seen that the surge, sway and heave motions are greatly influenced by the second-order difference-frequency wave force. Taking surge motion for example,when the platform subjected to second-order differencefrequency wave force, the mean and std. of surge motion are all larger than those in first-order model. It can be found that the difference-frequency force (including the mean-drift force) greatly influences the equilibrium position and amplitude of surge motion. Besides, the roll and pitch motions depend strongly on the second-order sumfrequency wave force. For instance, the difference between the maximum and minimum of roll motion is 0.109 deg in the second-order (s) model, which is about 1.3 times larger than that in the first-order model.

Table 3 Statistics of the 6-DOF responses in different models

Wave force Motion Mean Std. Min Max Second-order 0.848 1.115 −2.280 5.368 Second-order (d) 0.847 1.103 −2.163 5.270 Second-order (s) 0.002 0.866 −2.582 2.589 First-order Surge(m)0.002 0.851 −2.600 2.439 Second-order 0.454 0.671 −1.450 2.955 Second-order (d) 0.453 0.656 −1.475 2.561 Second-order (s) 0.001 0.540 −1.461 1.958 First-order Sway(m)0.001 0.523 −1.556 1.495 Second-order −0.004 0.008 −0.052 0.019 Second-order (d) −0.004 0.008 −0.052 0.017 Second-order (s) −0.002 0.007 −0.027 0.016 First-order Heave(m)−0.002 0.007 −0.025 0.017 Second-order 0.000 0.013 −0.058 0.050 Second-order (d) 0.000 0.008 −0.023 0.024 Second-order (s) 0.000 0.013 −0.057 0.052 First-order Roll(deg)0.000 0.008 −0.024 0.024 Second-order 0.000 0.023 −0.088 0.084 Second-order (d) 0.000 0.020 −0.062 0.066 Second-order (s) 0.000 0.023 −0.088 0.086 First-order Pitch(deg)0.000 0.020 −0.059 0.063 Second-order −0.002 0.371 −1.241 1.220 Second-order (d) −0.002 0.371 −1.228 1.218 Second-order (s) 0.000 0.371 −1.240 1.219 First-order Yaw(deg)0.000 0.371 −1.228 1.217

3.2 Set-Down Motion Under First- and Second-Order Wave Force

Fig.4 Comparison of spectral densities of 6-DOF motions under first- and second-order wave forces.

The heave motion of the platform is a vector sum of the set-down motion and vertical motion. The motion induced by horizontal motions (including surge, sway and yaw motions) is called set-down motion. When the tension legs are stretched, the axial offset will occur at the top of the tension legs, and its projection on the vertical direction is called vertical motion. Therefore, in order to study the effect of second-order wave force on the setdown and vertical motions, it is important to show the setdown motion and the vertical motion in both time and frequency domains respectively, as shown in Figs.5 and 6.

where ∆ωn is the frequency interval between two adjacent wave frequencies, Sη is the wave spectrum function.

Based on Fig.6, under the first-order wave force only,the low-frequency motion of set-down appears at the natural frequency of surge/sway motion, and the super-harmonic motion observably appears at the multiple frequencies. Besides, due to the nonlinear of set-down motion, the wave-frequency motion of set-down is close to zero.

Compared to the first-order model, the low-frequency motion of set-down in second-order (d) model substantially increases. A large wave-frequency motion of setdown obtained with respect to the second-order wave force is caused by the frequency coupling of nonlinear stiffness that is the coupled effect of the low- and wavefrequency components of the horizontal motions. On the other hand, the equilibrium positions of surge, sway and yaw motions greatly increase with respect to the second-order wave force. According to Eq.(23), it can be seen that the amplitude of set-down motion is related to the amplitudes of surge, sway and yaw motions. When the equilibrium positions of surge, sway and yaw increase,even if the standard deviations are constant, the amplitude of set-down motion will greatly increase. Therefore, a large wave-frequency motion of set-down obtained with respect to the second-order wave force is not only due to the frequency coupling of nonlinear stiffness, but also due to the increase of equilibrium positions of surge, sway and yaw motions, i.e.,

whereare the setdown motions caused by surge, sway and yaw motions,respectively.

Fig.5 Comparison of set-down and vertical motions between first- and second-order wave forces.

Fig.6 Comparison of spectral densities of set-down and vertical motions under first- and second-order wave forces.

There is more obvious high-frequency motion at the multiple frequencies induced by the second-order sumfrequency force. It can be concluded that the differencefrequency force will induce a set of large low- and wavefrequency of set-down motion. Moreover, the vertical motions in the three models are all dominated by the wave-frequency motions.

The statistics of the set-down and vertical motions for the four wave force models are shown in Table 4. From Table 4, it can be seen that the equilibrium positions of the vertical motions in different models are all near 0 m,and those of the set-down motions are all lower than 0 m,which is the center of gravity. The maximum amplitude of set-down motion in the second-order (d) model reaches 0.044 m, it is about 3.7 times of that in the first-order model. However, the maximum amplitude of the setdown motion in the second-order (s) model is almost the same as that in the first-order model. It can be concluded that the second-order difference-frequency wave force will induce a large set-down motion. Nonetheless, the second- order wave forces, whether difference- or sum-frequency, all have a slight influence on the vertical motion.

Table 4 Statistic results of the set-down and vertical motions

Motion Wave force Mean Std. Min Max Second-order −0.004 0.005 −0.047 0.000 Set-down motion (m)Second-order (d) −0.004 0.004 −0.044 0.000 Second-order (s) −0.002 0.002 −0.014 0.000 Vertical motion (m)First-order −0.002 0.002 −0.012 0.000 Second-order 0.000 0.006 −0.020 0.024 Second-order (d) 0.000 0.006 −0.019 0.023 Second-order (s) 0.000 0.006 −0.020 0.021 First-order 0.000 0.006 −0.018 0.019

4 Conclusions

This study mainly discusses the motion response of an ISSC TLP subjected to the first- and second-order wave forces in 50-yr random sea state. The numerical analysis focuses on impact of the second-order wave forces on the motion characteristics of TLP. From results of simulation,the second-order wave force greatly influences the 6-DOF motions of TLP.

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1) The second-order difference-frequency wave force,which includes the mean-drift part, has an obvious influence on the low-frequency motions of surge and sway of the platform. The impact is mainly manifested in the equilibrium positions and amplitudes of motions. Besides,the second-order sum-frequency force will induce a set of high-frequency motions of roll and pitch, and has a significant effect on the amplitudes of roll and pitch. However, little influence of second-order wave force is found in the yaw motion.

2) The low- and high-frequency motions of set-down appear under the first-order wave force only. However,due to the nonlinearity of set-down motion, the wavefrequency motion of set-down is close to zero. When the platform subjected to both first- and second-order difference-frequency force, a large wave-frequency motion of set-down obtained is not only due to the frequency coupling of nonlinear stiffness, but also due to the increase of equilibrium positions of surge, sway and yaw motions.

3) The second-order difference-frequency wave force will induce a large set-down motion in both low- and wave-frequency domains. Nonetheless, the second-order wave force, whether difference- or sum-frequency, all have a slight and not obvious influence on the vertical motion.When the platform under both first- and second-order forces, the heave motion, which consists of set-down and vertical motions, is mainly dominated by wave-frequency motion, but there is a significant increase in the amplitude and equilibrium position of heave motion as a result of a large set-down motion which is induced by second-order difference-frequency force.

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标注规范的制定,可以在一定程度上缩小不同标注者在标注时的差异,减少语料标注过程中的错误和不一致性,提高标注的效率.面向事件的中文文本指代标注与传统文本的指代标注是有差别的,对于缺省要素的标注在2.2节已作了说明,这里仅对已存在要素和事件的标注作简要说明.

Acknowledgements

The study is supported by the National Natural Science Foundation of China (Nos. 51239008 and 51279130).

由于直线电机牵引运载系统的特殊性，直线电机与感应板之间的垂直方向最大存在50 kN的相互吸力。在交变吸力的作用下，直线电机出现垂向吊杆橡胶关节老化，以及垂向吊杆断裂的现象，从而引起直线电机下沉，导致多起直线电机与感应板之间的接触碰撞发生，造成直线电机和感应板之间损伤，影响地铁线路的正常运营。在我国广州地铁4号线首次引进并成功运用了日本的直线电机牵引运载系统。为了实时在线监测气隙的变化，以及实现系统对超出限值自动报警，广州地铁集团有限公司与我国相关企业共同攻关，相继研发了广州地铁4、5、6号线直线电机气隙在线监测系统。

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