Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 8519523, 13 pages

https://doi.org/10.1155/2017/8519523

## The Snap Tension Analysis of Taut-Slack Mooring Line with tanh Method

School of Mechanical Engineering, Tianjin Key Laboratory of Nonlinear Dynamics & Chaos Control, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Su-xia Zhang

Received 14 March 2017; Revised 6 June 2017; Accepted 24 July 2017; Published 29 August 2017

Academic Editor: Fazal M. Mahomed

Copyright © 2017 Su-xia Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.