# 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  
Consider the path as composed of multiple polylines, and the intersection of the polylines is the control node of the path. Arrange the control nodes of the path in the order of connection as follows:  
\[\{(x_{s},y_{s},z_{s}),\ldots,(x_{i},y_{i},z_{i}),\ldots,(x_{t},y_{t},z_{t})\} \tag{2}\]  
The coordinates of the starting and ending points are \((x_{s},y_{s},z_{s})\) and \((x_{t},y_{t},z_{t})\). Constructing a suitable fitness function is essential in industries such as aircraft and ships. The primary purpose of this article is to propose a routing algorithm that considers natural frequency to solve the design problems that require the maximum natural frequency. It is necessary to minimize the pipeline's total length, maximize the natural frequency, and ensure that the spacing between adjacent supports meets the minimum distance for the installation of the pipeline and other requirements. For these pipe design problems, pipes are typically short (less than 1000 mm) but require high natural frequencies (often exceeding 10,000 Hz). Simply assigning weights to different objectives may not satisfy constraints. Therefore, this article normalizes the length and natural frequency to obtain the optimal path for the selected pipe for an evaluation function, which includes the total path length and the natural frequency of the maximum length segment, as follows:  
\[ \begin{array}{l}\min\limits_{i=1}^{P}{f(\{(x_{s},y_{s},z_{s}),\ldots,(x_{t},y _{t},z_{t})\})}=\\ \beta_{1}\sum\limits_{i=1}^{\text{m}}{\sqrt{(x_{i}-x_{i-1})^{2}+(y_{i}-y_{i-1 })^{2}+(z_{i}-z_{i-1})^{2}}}/{L_{re}}\\ +\beta_{2}*F_{re}/F(L_{\text{max}})+\beta_{3}*\left(D_{subject}-Distance_{ min}\right)/D_{subject}+\\ \beta_{4}*Count(\text{N}_{obstacles})\end{array}  \tag{3}\]  
In the proposed method, the weight values \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), and \(\beta_{4}\) represent the relative importance of length, natural frequency, and the minimum distance between two adjacent supports. \(L_{max}\) represents the maximum length of the straight-line segment. \(L_{re}\) represents the path length when the starting and ending points are directly connected. The nature frequency of the maximum length segment is denoted as \(F(L_{max})\), while \(F_{re}\) represents the natural frequency of the pipe when the starting and ending points are directly connected. The variable \(m\) represents the number of segments in the pipe path. \(Count(N_{obstacles})\) is the number of nodes intersecting obstacles or other pipes. To ensure that the path can be processed, it is necessary to incorporate additional constraints based on engineering rules. One important constraint is \(D_{subject}\), which limits the minimum allowable length for the pipe. \(D_{min}\) is the shortest distance between any two supports in the path. When the shortest distance is greater than \(D_{subject}\), \(\beta_{3}\) is set to 0. A Natural Frequency Surrogate Model for Design Phase  
There are several methods available to solve for the natural frequency of a pipe, such as the transfer-matrix method [29], the nonlinear vibration method [30], and the finite element method. However, these methods involve intricate numerical calculations, including mesh division and inverse matrix operations, which can be time-consuming. They become unacceptable during the iterative design process when the pipe path, shape, and support positions are not yet determined. To address the challenge of time-consuming natural frequency calculations during the iterative design process, this paper proposes a simplified approach. It treats the pipe support as a fixed constraint and introduces a surrogate model to establish a numerical replacement model for natural frequency. This surrogate model aims to expedite the calculation of the natural frequency, enabling consideration of the natural frequency and other dynamic characteristics in the pipe routing process.