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:
The coordinates of the starting and ending points are
and (
). 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:
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.
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 allowed for aero engines.
is the flow velocity inside the pipe and
is the change in flow velocity. And, respectively,
represent the axial force and
represents the pressure of fluid inside the pipe.
is the position coordinate along the length direction of the pipe;
represents the force.
represents the outer diameter of the pipe, and
represents the thickness of the pipe. The nonlinear vibration equation of the pipeline is derived in reference [
31], which is listed in Equations (4) and (5).
In Equations (4) and (5),
is the Young’s modulus of the material,
is the cross-sectional area of the pipe, and
is the interface moment of inertia, and
and
represents the density of the liquid and the pipe material. The following dimensionless parameters are introduced.
Thus, the nonlinear vibration equation of the fixed support is obtained as follows:
In order to solve the above partial differential equation, the galerkin method is used to discretize the equation. Therefore, the displacement function in the pipeline can be expressed as (9) and (10)
Among them,
and
satisfies Equations (11) and (12).
satisfies the following Equation (13)
satisfies the following Equation (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.
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.
is the dimensionless natural frequency.
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.
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,
,
,
is the fitting constant, and
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,
,
, and
are 7.08 × 10
4, 1.08 × 10
2 and 4.03 × 10
2 (
Table 3). Finally, a fast surrogate model for natural frequency has been obtained.