# 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
#### 2.1.1 Environmental Load of Paving System  
Wind loads on marine structures can generally be calculated by empirical formulas derived from wind tunnel tests on hull models. In this paper, the wind loads of pipe-laying vessels are calculated according to the empirical formulas of wind loads provided by the Oil Companies International Marine Forum (OCIMF) [19]. Current loads can generally be considered as steady-state planar flows, producing only drag forces on the hull. In order to analyze the force of ocean current on the hull, Equation (1) needs to be used to calculate the superposition of horizontal velocity and wave particle velocity. \[\left\{ \begin{array}{l}F_{\text{XC}}=0.5 dC_{\text{XC}}(\varphi_{cR})\rho_{c}V _{cR}{}^{2}L_{BP}\\ F_{\text{YC}}=0.5 dC_{\text{YC}}(\varphi_{cR})\rho_{c}V_{cR}{}^{2}L_{BP}\\ M_{\text{XYC}}=0.5 dC_{\text{XYC}}(\varphi_{cR})\rho_{c}V_{cR}{}^{2}L_{BP}{}^{2 }\end{array} \right. \tag{1}\]  
where \(\varphi_{cR}\) is the relative angle, \(\rho_{c}\) is the density of the seawater in units kg/m\({}^{3}\). \(F_{\text{XC}}\), \(F_{\text{YC}}\), and \(M_{\text{XYC}}\) are X-direction, Y-direction, and moment of flow around the vertical axis of the ship, respectively. \(d\) is the average draft depth of the ship, unit m. \(v_{c}\) is the current velocity  
Figure 1: Overall structural diagram of the flexible pipeline-laying system. per m/s, which varies with the depth of the seawater. \(V_{CR}\) is the relative velocity of the ship and the current; \(L_{BP}\) is the vessel vertical length. \(C_{XC}\), \(C_{YC}\), and \(C_{XYC}\) are the drag force coefficients. Their values depend on the relationship between the current and the ship, which can also be found from the map data. Irregular waves are used for the calculation of wave loads. The wave elements are random variables and are usually expressed in terms of their statistical characteristics. According to the principle of linear waves, irregular waves are superimposed by waves with different heights, frequencies, and random phases, which can be defined by energy spectrum. The spectrum selected in this paper is the JONSWAP wave [20]. JONSWAP spectrum is obtained by [PERSON] through a large number of data taken in the North Sea. The expression is Equation (2):  
\[S_{\zeta}(\omega)=\alpha g^{2}\omega_{p}{}^{4}\omega^{-5}\exp\left[-1.25\left( \frac{\omega_{p}}{\omega}\right)^{4}\right]\gamma^{\exp\left[-\frac{(\omega- \omega_{p})^{2}}{2(\omega_{p})^{2}}\right]} \tag{2}\]  
where \(\alpha\) is the generalized Philips coefficient, \(\alpha=0.0076(gx/U^{2})^{-0.22}\). \(\omega_{p}\) is the peak frequency of the spectrum, \(\omega_{p}=22(g/U)(gx/U^{2})^{-0.33}\); \(\sigma\) is the peak parameter, \(\sigma=0.07\) when \(\omega<\omega_{p}\); When \(\omega>\omega_{p}\), \(\sigma=0.09\). \(\gamma\) is the peak enhancement factor with an average value of 3.3. U is the average wind speed, and \(x\) is the length of the wind zone. \(S_{\zeta}(\omega)\) is a function of the energy density of waves. The expression of wave surface height and time of irregular waves is Equation (3). \[\eta(t)=\sum\limits_{n-1}^{N}\sqrt{2S(\omega_{i})\Delta\omega}\cos(k_{n}x- \omega_{n}t-\varepsilon_{n}) \tag{3}\]  
where \(N\) represents the number of regular waves and \(\Delta\omega\) is the interval frequency. The first-order wave force and the second-order wave force are solved by deriving the transformation and Fourier transformation. The first-order wave force and the second-order wave force can be solved by Equation (4) and Equation (5) respectively. \[ \begin{array}{ll}F_{\rm wave}^{(1)}(t)&=\sum\limits_{n-1}^{N}\frac{1}{2\pi}f_ {-\infty}^{t}h(\tau)\zeta_{n}\cos[\omega_{n}(t-\tau+\varepsilon_{n})]d\tau\\ &=\sum\limits_{n-1}^{N}\zeta_{n}f_{ni}(\omega)\cos(\omega_{n}t+\varepsilon_{n }+\varphi_{n})\end{array}  \tag{4}\]  
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