# Routing Design for Pipe System Considering Natural Frequency
## 4 A Novel Path Planning Method for Pipe Arrangement with Considering Natural Frequency
### Evaluation Function Design Considering Natural Frequency
The pipe for establishing a numerical surrogate model for the natural frequency is shown in Figure 11. In Figure 11, the boundary conditions are the fixed support at both ends. The reason for using fixed supports at both ends is that cantilever pipelines are not  
\[w(0,\tau)=w(1,\tau)=\frac{\partial w(0,\tau)}{\partial\xi}=\frac{\partial w(1,\tau) }{\partial\xi}=0 \tag{12}\]  
\(\phi_{r}(\xi)\) satisfies the following Equation (13)  
\[ \begin{array}{l}\phi_{r}(\xi)=cosh(\lambda_{r}\xi)-cos(\lambda_{r}\xi)-\\ \frac{cosh(\lambda_{r})-\cos(\lambda_{r})}{sinh(\lambda_{r})-sin(\lambda_{r}) }(sinh(\lambda_{r}\xi)-sin(\lambda_{r}\xi))\end{array}  \tag{13}\]  
\(\lambda_{r}\) satisfies the following Equation (14)  
\[cosh(\lambda_{r})cos(\lambda_{r})=1 \tag{14}\]  
Multiply both sides of the equation to the left and integrate in the interval \([0,1]\) to obtain a discrete equation. In order to obtain the natural frequency of the pipe, the nonlinear term related to time in the equation can be ignored, so that the natural frequency can be obtained. \[ \begin{array}{l}\mathbf{M}\mathbf{\ddot{T}}(\tau)+2\mu\mathbf{G}\mathbf{ \dot{T}}(\tau)+\big{(}\mathbf{K}+\mu^{2}\mathbf{H}+\dot{\mu}\mathbf{G}+ \mathbf{S}\big{)}\mathbf{T}(\tau)\\ +\mathbf{N}(\mathbf{T}(\tau))=\mathbf{F}(\tau)\end{array}  \tag{15}\]  
To verify the accuracy of the natural frequency calculation in this article, a curve depicting the variation in natural frequency with fluid flow velocity in the pipe is provided and compared with the results presented in article [31] (Figure 12). The material parameters are the same as in article [31], which are listed in Table 2. The curve depicting the variation in natural frequency with fluid flow velocity calculated in this article. It should be noted that natural frequencies are dimensionless in Figure 12. According to Equation (16), it can convert the dimensionless natural frequency into the actual value. \(\overline{\lambda}_{n}\) is the dimensionless natural frequency. \(\lambda_{n}\) is the actual value of natural frequency. The relationship between natural frequency, length, and outer diameter for a steel pipe is obtained in Figure 13. In Figure 13, different colors represent the different natural frequency curves for different diameters. \[\overline{\lambda}_{n}=\lambda_{n}L^{2}\sqrt{\frac{m_{p}+m_{f}}{EI}} \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Name** & **E (Pa)** & **Beta** & **L (mm)** & \(p\) & \(\mathbf{\Pi}_{0}\) & **D (mm)** & **H (mm)** & \(\mu\) \\ \hline Value & 1e10 & 0.1 & 2000 & 0 & 0 & 20 & 0.2 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The material parameters for natural frequency verification. Figure 12: The natural frequency verification in this article. The curve depicting the variation in natural frequency with fluid flow velocity calculated in this article. Through the above derivation, it can be found that the solution of natural frequency is accompanied by a large number of inversion operations, which is very time-consuming for the path planning algorithm. Therefore, this paper uses the surrogate model [32] technology to establish the numerical model of natural frequency along with the pipe length, outer diameter, and wall thickness, so as to speed up the calculation of natural frequency. The surrogate model for natural frequency is obtained by the Equation in (17). Frequency represents the natural frequency of the pipe, \(A_{0}\), \(y_{0}\), \(t\) is the fitting constant, and \(L\) is the length of the pipe. For example, for a steel pipe with a radius of 20 mm and a wall thickness of 0.2 mm, \(A_{0}\), \(y_{0}\), and \(t\) are 7.08 \(\times\) 10\({}^{4}\), 1.08 \(\times\) 10\({}^{2}\) and 4.03 \(\times\) 10\({}^{2}\) (Table 3). Finally, a fast surrogate model for natural frequency has been obtained. \[Frequency=A_{0}exp(-L/t)+y_{0} \tag{17}\]