Abstract
The continuous model is introduced, and the nonlinear partial differential equations of taut-slack mooring line system are transferred to nonlinear algebraic equations through tanh method, and four solitary solutions are obtained further. At the same time, to express the results clearly, the curve surfaces of strains, displacements, and tension are plotted. The results show that there are four different solutions in the system. With the pretension increasing, the tension changes from one solitary solution to snap tension, and when the pretension is increased further, the curve converts to continuous line, until straight line, which is corresponding to the taut mooring line. In the process of increasing of pretension, the mooring line transfers from slack to taut, accompanied with tension skipping, which is reduced by the system parameters, and different combination of parameters may introduce different tension in line, and the uncertainty may cause the breakage of mooring system. The results have an agreement with experiment, which shows that the calculating method in this paper may be believable and feasible. This work may provide reference for design of mooring system.
1. Introduction
With the development of exploitation of deep-water resources, TLP and SPAR platforms are designed and used in some areas and are considered as the excellent platforms in deep water, in which mooring line is one of the most important parts of platforms.
Mooring lines are flexible structures and have unidirectional stiffness; that is, they can only bear tension and cannot bear compressive loading. When the platform moves with large amplitude, the mooring line will transfer from taut to slack, accompanied with snap tension. The research showed that the snap tension was usually several times to dozen times more than the mean tension [1], which may be the real reason causing breakage of mooring line.
Many works have been done by the scholars to reveal the mechanism of breakage of mooring line. Niedzwecki and Thampi [2] investigated the snap load behavior of marine cable systems in regular seas. Huang and Vassalos studied the impulse loads under alternative taut-slack conditions by using the model of bilinear string [3, 4]. They neglected the effect of floating structure, the cable was simplified into the model subjected to periodic sinusoidal excitation at the top end of cable, and the snap tension of the mooring cable was calculated [5]. The results show that the effect of tension magnification is due to the shock. Experiments show that, under sinusoidal excitation and quasi-static condition, the result calculated by the quasi-static method is near the experiment result. But under the impact condition, the impact tension is much larger than the result calculated by the quasi-static method [6]. The equivalent force model is used to analyze the effect of mooring systems on the horizontal motions [7]. Zhang et al. studied the analytical solution of taut-slack mooring line with tanh method and obtained snap tension curves in four cases [8]. Until now, most of the solutions to the equations are numerical simulation, and the analytical solution is not fully developed.
At the same time, different methods solving the nonlinear evolution equations have been developed. The tanh method, developed for years, is one of the most direct and effective algebraic methods for finding exact solutions of nonlinear diffusion equations. Recently, much work has been concentrated on the various extension and applications of the method [9]. Huibin and Kelin [10] introduced a power series in tanh as a possible solution and substituted this expansion directly into a higher-order KdV equation. Wazwaz [11] employed the tanh method for traveling wave solutions of nonlinear equations and the work was extended to equations that do not have tanh polynomial solutions. Abdou and Soliman [12] presented and implemented the solutions of nonlinear physical equations for constructing multiple traveling wave in a computer algebraic system by means of computerized symbolic computation and a modified extended tanh-function method. Zheng et al. [13] presented the generalized extended tanh-function method for constructing the exact solutions of nonlinear partial differential equations (NPDEs) in a unified way making use of a new generalized analysis. As a result, the solitary wave solutions and other new and more general solutions were obtained. Based on the symbolic computation, Zhi and Zhang [14] combined the tanh-function method with the symmetry method to construct new type of solutions of the nonlinear evolution equations for the first time. With the combined method, some new types of solutions of the coupled (2 + 1)-dimensional nonlinear system of Schrodinger equations were obtained.
In this paper, to reveal the mechanism of snap tension in mooring line while transforming from taut to slack or from slack to taut, the continuous model is introduced and tanh method is used to resolve the nonlinear equations of taut-slack mooring line. Four types of analytical solutions of taut-slack mooring line are obtained, and the displacement and tension curve surfaces in different cases are presented, and the effects of pretension will be discussed in Section 4. The snap tension is obtained, which may be the reason causing the breakage of mooring line.
2. The tanh Method
Consider a given evolution, say in two variables [15],where and is a polynomial about and its derivatives.
The fact that the solutions of many nonlinear equations can be expressed as a finite series of tanh function motivates us to seek for the solutions of (1) in the formwhere . Notice that the highest order of isSo can be obtained by balancing the derivative term of the highest order with the nonlinear term in (1). are parameters to be determined. Substituting (2) into (1) will yield a set of algebraic equations for because all coefficients of have to vanish. Using these relations, can be obtained. Therefore, the traveling solitary wave solutions are obtained. Usually is a positive integer; however, once in a while, the value of is a negative or a fraction. In these cases, we can introduce a transformation , where is the denominator of , and transform (1) into another equation for , whose balancing number will be a positive integer. Then it can be dealt with by the above method. Therefore the final solution may be a rational or radical function about .
3. Equations of Motion
The model of mooring line is shown in Figure 1. The horizontal uniform mooring line is considered, and the shear deformation is neglected. The left end of the line is fixed, and the right end is attached to the buoy; here the condition is simplified by assuming harmonic excitation . and are the unit vector in tangential and normal direction, respectively, and is the binormal unit vector. and represent the static and dynamic configuration, and is the displacement relative to the static configuration.
(a)
(b)
Only the in-plane motions of mooring line are considered, and the equations are as follows [16]: where is the dynamic strain, and the expression is as follows:EA represents the cable section modulus, represents the cable mass/length, and are components of an arbitrary external excitation, including the hydrodynamic loading and exciting from the buoy attached at the end, mainly the end exciting. In other words, the transforming from taut to slack of mooring line is mainly caused by the large amplitude motion of attached buoy. The hydrodynamic loading is calculated with Morison’s equation as follows: and are the added mass coefficients in tangential and normal direction, respectively, and are fluid drag coefficients in tangential and normal direction, respectively, is the velocity of fluid particle, and the subscripts and denote the projection in tangential and normal directions, respectively. and are the displacements in tangential and normal directions, subscript denotes partial differentiation with respect to arc lengths, and denotes partial differentiation with respect to time. Moreover, and are the equilibrium tension and curvature distributions describing a catenary as given by where denotes the horizontal component of the equilibrium cable tension.
Now considering the effects of curvature and tension in line, substitute (6) and (8) into (4) and (5), and the equations can be reduced aswhere . The dynamic tension is that In the quasi-static fluid, and . Then the initial equations may be simplified and rewritten as follows:where and are coefficients, shown in the Appendix.
4. Results and Discussions
Assuming that , are the polynomials of with and ranks, respectively, the nonlinear term in (11) is -order polynomials of , and is -order polynomials of , while the highest order linear term is , which is polynomials of . Similarly, in (12), the nonlinear terms and are and polynomials of , respectively, and the highest order linear term . To look for the traveling wave solution of (11) and (12), the parameters and are first determined by balancing the linear term of highest order with the nonlinear term:It is solved that
For cases (), (), and (), there is no nontrivial solution in terms of tanh function. For case (), the following transformation is introduced:where , and are undetermined coefficients.
Submitting the transformation into (9), collecting the coefficients of with the same orders, the nonlinear algebraic equations can be obtained:
Solving them with RATH package [17, 18], the four solutions are obtained.
Case 1. The parameters are thatThe constraint parameters are thatAnd other parameters are arbitrary.
Case 2. The parameters are thatThe constraint parameters are thatAnd other parameters are arbitrary.
Case 3. The parameters are thatThe constraint parameters are thatAnd other parameters are arbitrary.
Case 4. The parameters are thatThe constraint parameters are thatAnd other parameters are arbitrary.
To express the solutions clearly, the plots of the solutions are drawn, as shown in Figures 2–5, and the parameters used are shown in Table 1. The four solitary solutions are 2 left traveling waves and 2 right traveling waves according to the expressions of . From the figures, it can be seen that, in the four solitary solutions, two of them introduce solitary tension in the system, and the tension does not change with the time delay. The other solitary solutions in and induce solitary wave tension, but with the time, the tension attenuates and disappears, and the energy is absorbed by the line and depleted by the damping from structure and external medium. The solitary tension wave in line may induce the breakage of line structure.
(a) Displacement of
(b) Displacement of
(c) Tension surface
(a) Displacement of
(b) Displacement of
(c) Tension surface
(a) Displacement of
(b) Displacement of
(c) Tension surface
(a) Displacement of
(b) Displacement of
(c) Tension surface
In the problem of taut-slack mooring line, the pretension is an important factor which relates to the line station. At the same exciting loading, when the pretension is altered, the tension curves are different, as shown in Figure 6. It can be seen that when the pretension is 10000 N, the solution is 2-solitary solution, and with the pretension increasing, the tension changes from solitary solution to snap tension, and with the continuous increasing, the curve converts to continuous line, until straight line, which is corresponding to the taut mooring line. In the process of increasing the pretension, the mooring line transfers from slack to taut, accompanied with tension skip.
(a) N
(b) N
(c) N
(d) N
(e) N
(f) N
Given as 26000 N, the tension at m is calculated as shown in Figure 7. The results show that there is a tension skip at about s, and the tension decreases. When the parameters are given the same as those in [19], where the length of cable is 30 m, excitation frequency is 1.2 Hz, and amplitude is 8 cm, the calculated result is shown in Figure 8, and the experimental result is shown in Figure 9. The calculated maximum tension is 945.5 N, and it is 790.8 N in the experiment. This is because the cable is elastic, and the pretension is not so accurate in the experiment. Excluding the effect, the theoretical result is consistent with the experimental one. It indicates that the calculating method in this paper may be believable and feasible.
The results show that the skip tension is reduced by the system parameters, and different combination of parameters may introduce different tension in line, and the uncertainty may cause the breakage of mooring system.
5. Conclusions
In this paper, the continuous model is introduced, and the nonlinear partial differential equations of taut-slack mooring line system are transferred to nonlinear algebraic equations through tanh method, and four solitary solutions are obtained further. To express the results clearly, the curve surfaces of displacement , and tension are plotted. The results show that there are four different solutions in the system. With the pretension increasing, the tension changes from solitary solution to snap tension, and when the pretension increased further, the curve converts to continuous line, until straight line, which is corresponding to the taut mooring line. In the process of increasing the pretension, the mooring line transfers from slack to taut, accompanied with tension skip, which is reduced by the system parameters, and different combination of parameters may introduce different tension in line, and the uncertainty may cause the breakage of mooring system. The results have an agreement with experiment, which shows that the calculating method in this paper may be believable and feasible.
Appendix
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Grateful thanks are due to the National Natural Science Foundation of China (Grant no. 51479136, no. 11502161, and no. 11302145) and Tianjin Natural Science Foundation (no. 17JCYBJC18700).