# Scaled Model Simulation and Experimental Verification of Submarine Flexible Pipeline Laying System
## 2 Similarity Theory and Scaled Model of Flexible Pipeline-Laying System
### Ambient Load of Paving System and Ship Motion Response
#### 2.1.2 Research on Mechanical Model of Flexible Pipe  
In the analysis of the laying process of flexible deep-sea pipelines, the following four assumptions are required: \(a\). It is assumed that the working ship is stationary when laying the pipeline. \(b\). Current is linear based on the distribution on the longitudinal section of the hull, with zero submarine velocity and maximum surface velocity. \(c\). Integral flexible pipes are laid on the mid-longitudinal plane of the ship. \(d\). Because of the large depth of water laid, the flexible piping can be considered, and the influence of bending rigidity of flexible piping can be ignored.  
After laying the pipeline for a period of time, the flexible pipeline in the ocean taken on the form of a catenary, as shown in Figure 2. The point at which the pipeline passes through the sea level is regarded as origin \(O\), the x axis as horizontal direction, the y axis as vertical downward direction, the pipeline from contact bottom to \(O\) point at sea level as suspension section, and the bottom contact section on the seabed. When analyzing the suspension section, first intercept the microelement of the dl length of the previous section and analyze its stress, as shown in Figure 2.  
\(F_{n}\) and \(F_{\tau}\) are used to calculate the ocean load per unit length. \(\tau\) represents the drag force of the unit pipeline in the ocean by the current in the normal and tangential directions. It can be obtained by [PERSON]'s semi-empirical Equation (6).  
\[\left\{\begin{array}{l}F_{n}=0.5\times\rho_{W}\times C_{n}\times D_{1}\times (\upsilon\times\sin\theta)^{2}\\ F_{\tau}=0.5\times\rho_{W}\times C_{\tau}\times D_{1}\times(\upsilon\times \cos\theta)^{2}\end{array}\right. \tag{6}\]  
where \(C_{n}\) and \(C_{\tau}\) are the normal and tangential drag force coefficients, respectively; \(\upsilon\) represents the current velocity in the horizontal plane where the pipeline unit is located; \(\theta\) represents the included angle between the pipe unit and the horizontal direction.  
The coordinate axis is established according to the shape of the pipeline. The intersection point between the pipeline and horizontal plane is the origin. The complete pipeline is considered to be composed of a large number of micro-elements. Each microelement is \(dy\) in the \(y\)-direction length. If the depth of seawater is \(h\), the total microelement segment is \(m=h/dy\). For any pipe element \(i\) in the suspension section, the parameters can be calculated by Equation (7), and the parameters at point \(i+1\) can be calculated by the parameters at point \(i\) of the pipeline.  
\[\begin{array}{l}w\delta l_{i}=0.25(\rho_{p}D_{2}{}^{2}-\rho_{W}D_{1}{}^{2}) g\delta l_{i}\frac{nt}{r(n-r)!}\\ F_{ni}=0.5\rho_{W}C_{n}D_{1}(\upsilon\sin\theta_{i})^{2}\\ F_{\tau}=0.5\rho_{W}C_{\tau}D_{1}(\upsilon\cos\theta_{i})^{2}\end{array} \tag{7}\]  
Based on the above analysis, the position parameters, tension at each point, line coordinate parameters, and total length of the pipeline can be calculated by computer programming, which can be used as basic parameters for dynamic simulation of pipeline laying. The linearity of pipe laying and the variation of internal force of pipe are calculated by setting the angle, maximum depth, and shape parameters of pipe laying. Several different pipelines are selected for the study, and the parameters are given in Appendix A.