# Scaled Model Simulation and Experimental Verification of Submarine Flexible Pipeline Laying System
## 2 Similarity Theory and Scaled Model of Flexible Pipeline-Laying System
### Similarity Principle and Derivation of Similar Norms  
When testing a designed structure, the prototype design is usually uneconomical and difficult to meet the requirements of the experiment. Therefore, we usually choose a  
Figure 3: Finite element meshing of the hull. sample similar to the original sample for the experiment. This is called a model experiment. When the scaled model is similar to the prototype, it must be geometrically similar to the prototype and similar in strength [25]. Since the model is designed in a four-level sea environment, the deformation errors caused by the environment on the prototype and the model can be neglected [26]. When the prototype and model of flexible pipe-laying system are subjected to corresponding loads, the geometrical and static effects of the variables that can be compared with each other are size \(l\), external force \(F\), external moment \(M\), elastic modulus \(E\), strain \(\varepsilon\), density \(\rho\), and stress \(\sigma\). According to these variables, the corresponding relationship between prototype and model can be obtained. The relationship between the main physical parameters can be expressed as Equation (10). \[f(F,M,E,\rho,l,\varepsilon,\sigma) \tag{10}\]  
For general elastic questions, the characteristics of introducing physical constants are analyzed, and three basic force scales are adopted, namely mass \(m\), length \(L\), and time \(T\). Each variable can be expressed in units of the basic dimension, whose dimension matrix is as Equation (11). \[ \begin{array}{cccccccc}&F&M&E&\rho&l&\varepsilon&\sigma\\ \par m&\left(\begin{array}{cccccccc}1&1&1&1&0&0&1\\ 1&0&-1&-3&1&0&-1\\ -2&0&-2&0&0&0&-2\end{array}\right)\end{array}  \tag{11}\]  
Represent physical parameters from \(F\), \(M\), \(E\), \(\rho\), \(l\), \(\varepsilon\), \(\sigma\) with \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\), \(a_{6}\), \(a_{7}\), respectively. Equation (12) can be simplified by Equation (11). \[\left\{ \begin{array}{ll}m:a_{1}+a_{2}+a_{3}+a_{4}+a_{7}=0\\ \par L:a_{1}-a_{3}-3a_{4}+a_{5}-a_{7}=0\\ \par T:-2a_{1}-2a_{3}-2a_{7}=0\end{array} \right. \tag{12}\]  
Equation (13) can be obtained by using \(a_{3}\), \(a_{4}\), \(a_{5}\) as independent variables. \[\left\{ \begin{array}{ll}a_{3}=-a_{1}-a_{7}\\ \par a_{4}=-a_{2}\\ \par a_{5}=-2a_{1}-3a_{2}\end{array} \right. \tag{13}\]  
Equation (13) is a statically indeterminate equation with several parameters, including three basic dimensions. The number of independent \(\pi\) can be known by the \(\pi\) Theorem to be four (\(7-3=4\)). The Similarly \(\pi\) Theorem is the Similarly Second Theorem expressed as if a phenomenon can be described by Equation (14). \[D_{l}(x_{1},x_{2},\cdots,x_{k},x_{k+1},\cdots x_{n})=0 \tag{14}\]  
The equation includes a total of n physical quantities. If \(K\) physical quantities have independent dimensions, the dimension of the last \(n-k\) physical quantities can be derived from the previous independent dimensions. By means of transformation, the above formula can be transformed into a functional relation between similar criteria \(\pi_{1}\), \(\pi_{2}\),... \(\pi_{n-k}\), that is, Equation (15). \[F_{l}(\pi_{1},\pi_{2},\cdots\pi_{n-K})=0 \tag{15}\]  
Therefore, Equation (13) can be rewritten as Equation (16). \[f(\pi_{1},\pi_{2},\pi_{3},\pi_{4})=0 \tag{16}\]Then the similarity matrix containing each variable can be obtained as Equation (17). \[ \begin{array}{cccccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}&a_{7}\\ F&M&E&\rho&l&\varepsilon&\sigma\\ \pi_{1}(a_{1})&\pi_{2}(a_{2})&\left(\begin{array}{cccccccc}1&0&-1&0&-2&0&0\\ 0&1&0&-1&-3&0&0\\ 0&0&0&0&0&1&0\\ 0&0&-1&0&0&0&1\end{array}\right)\\ \pi_{3}(a_{6})&\pi_{4}(a_{7})&\left(\begin{array}{cccccccc}a_{1}&a_{2}&a_{3} &a_{4}&a_{5}&a_{6}&a_{7}\\ F&M&E&\rho&l&\varepsilon&\sigma\\ \end{array}\right)\\ \end{array}  \tag{17}\]  
Then the scale of model scale meets the requirements as Equation (18). \[\left\{ \begin{array}{l}C_{F}=C_{E}{C_{l}}^{2}\\ C_{M}=C_{\rho}{C_{l}}^{3}\\ C_{e}=1\\ C_{\sigma}=C_{E}\end{array} \right.