1. Introduction
A multi-branch pipeline is a typical structure, consisting of a straight pipe, nut, rear ferrule, front ferrule and four parts of the joint body. It is widely used in aerospace, shipbuilding and other fields, with the advantages of small size and convenient operation. The pipeline suffers serious vibration from fluid-structure interaction, and long-term vibration will reduce the life of the pipeline and hydraulic components, and in serious cases may even cause wall rupture and damage to the pipeline support structure. It causes loss of support stiffness, piping system failure, and joint leakage [
1,
2,
3]. Therefore, it is urgent to study the vibration behavior of pipeline to reveal the influence mechanisms of various parameters on its vibration behavior.
The fluid-structure interaction mechanism is defined as three kinds of coupling [
4,
5,
6]. Poisson coupling is due to the Poisson effect, in which the oscillating pressure causes the radial pipe wall to expand, resulting in axial strain and movement. Due to the unbalanced pressure acting on a certain area, the coupling of the joint occurs at the boundary of change, such as elbow, valve, joint and pipe end. Friction coupling is caused by the shear stress on the pipe wall and is generally considered less important than the other two couplings. To describe the vibration characteristics of pipelines, several models are proposed according to basic equations. Ouyang et al. [
7,
8] considered the stress distribution around the section perpendicular to the axis of the hydraulic pipe according to the two-dimensional model, taking into account the effects of curvature and stress concentration. On the basis of the characteristic line method, the vibration characteristics of a simple straight pipe were accurately predicted by using the eight-equation model, and on this basis, the effects of curvature and friction on the resonance frequency of the pipe section were analyzed. Andrade et al. [
9] developed a quasi-two-dimensional flow model for fluid transport in pipes, analyzed the energy transfer and dissipation effects in fluid piping systems, effectively described frictional coupling, and accurately captured energy dissipation in fluids. Zhang, Quan et al. [
10,
11] used the 14-equation fluid-structure interaction vibration model of hydraulic pipe. The accuracy of the FSI frequency domain solution at high Reynolds number is improved and the accuracy of the friction coupling model analysis is verified. Bai et al. [
12] established a new dynamic model for the cantilever pipe conveying variable density fluid in combination with the Bernoulli-Euler beam model. The results showed that the pipeline system is prone to losing stability when the fluid density increases. Gao et al. [
13,
14,
15] analyzed the research status of aviation hydraulic pipeline vibration and control technology in the field. A model reduction method was developed, and artificial springs were introduced to simulate boundary conditions, which significantly reduced the computational cost of the vibration analysis of long-distance multi-elastically supported hydraulic pipelines. Tan, Chai et al. [
16,
17,
18] established a nonlinear coupling dynamic model of the flow conveying pipeline based on the Timoshenko beam and verified the accuracy of the model through the finite difference method (FDM). The results showed that the Timoshenko model was more accurate in analyzing large flow rate, amplitude or short pipeline length. Yuan et al. [
19,
20] developed a novel finite element model based on the absolute nodal coordinate formula (ANCF) and a dynamic mechanical model based on the Timoshenko beam theory for curved pipes conveying fluid, and analyzed the vibration characteristics of curved pipes under the influence of the fluid flow. Then, through the finite element numerical simulation, they verified the accuracy of the model.
On the basis of the model equations, the following researchers have investigated the fluid–structure interaction of pipelines using different calculation methods. Wiggert et al. [
21,
22,
23] used the characteristic line method to carry out research on pipeline system vibration for many years. They then put forward a mathematical model describing the fluid–structure interaction of a liquid-filled pipeline and carried out modal analysis of a liquid-filled pipeline system through the transfer matrix method. Dai et al. [
24] studied the influence of the combined force caused by fluid flow on the natural frequency of straight pipes and elbows by using the transfer matrix method. When analyzing the vibration of pipes containing flowing fluid, the stable resultant force generated by fluid flow must be considered. Kalliontzis [
25] realized accurate FSI calculation under the structural response dominated by axial, bending and shear motion through the ALE formula, and verified the accuracy of the method using existing commercial software. Huang et al. [
26] used the element-free Galerkin method to calculate the natural frequencies of the two ends of the straight pipe under different boundary conditions and obtained the natural frequency equations of the fluid transmission pipeline under different boundary conditions. Zhu et al. [
27] used fluid dynamics (CFD) and fluid–structure interaction calculation methods to obtain the flow characteristics of the oil flow, and analyzed the effect of the diameter ratio of the branch pipe to the straight pipe in the T-pipe and the merging angle of the three branches on the flow and deformation of the pipe. Francis [
28] proposed a complete Lagrangian finite element method (FEM) with node integration to simulate the fluid–structure interaction problem. The accuracy of the method was proved through experimental tests, providing a reference for solving complex FSI problems. Lavooij, Wang et al. [
29,
30] used the finite element method combined with the characteristic line method (MOC-FEM) to solve the pipe equations based on the characteristic line method. The transient analysis of the fluid-filled pipeline was completed, which significantly improved the calculation efficiency of the coupled vibration of the hydraulic pipeline system. Liu et al. [
31] developed the transfer matrix method for vibration analysis of branch pipes to compare the calculated pressure frequency response of rigid pipes and flexible pipes. This method was easier to use and can be used to calculate branch pipes with different shapes. Based on the fluid-structure interaction, Wu et al. [
32,
33] used the structural and fluid-dynamic models to analyze the fluid-structure coupling vibration of the pipeline, as well as analyzing the pipeline vibration response under different working conditions, providing a reference for practical engineering application.
The research on the fluid-structure interaction vibration characteristics of pipelines has already yielded a wealth of results. Different mathematical theoretical models have been established for straight pipes, elbows, T-shaped pipes, etc. The pipeline vibration has been analyzed using the characteristic line method, transfer matrix method, and finite element method, etc. However, there are few literature reports on the multi-branch pipeline, so it is necessary to explain the law of fluid in the multi-branch pipeline. Compared with other types of pipes, the fluid changes acutely when flowing in the multi-branch pipeline, especially at the junction. Where the fluid collides with the wall to generate eddy current, the influence of the fluid-structure interaction on the pipeline is significantly increased. Therefore, the finite element model of a multi-branch pipeline is established, and the influence of different parameters on the vibration characteristics of the pipeline is analyzed, providing a reference for the subsequent structural optimization design.
4. Influence of Different Parameters on Pipeline Vibration Characteristics
For analyzing the influence of different parameters on the dynamic characteristics, a simulation method is applied. The fluid pressures (7, 14, 21, 28, 35, 42 MPa), flow rates (0, 4, 8, 12, 16, 20 m/s), pipe diameters (6, 8, 10, 12, 14, 16, 18, 20 mm), fluids (air, kerosene, hydraulic oil, lubricating oil, water), and elastic support stiffness (103–1010 N/m), straight pipe lengths (100, 150, 200, 250, 300, 350 mm) and pipe wall thicknesses (0.5, 1, 1.5, 2, 2.5, 3 mm) are investigated by analyzing the fluid–structure interaction effect of the pipeline.
4.1. Analysis of the Flow Field in the Multi-Branch Pipeline
Figure 9 shows the flow rate, pressure and particle trajectories extracted along the central section of the pipeline. The inlet flow rate is 5 m/s, the pressure is 7 MPa, and the fluid is 46# hydraulic oil. Five lines are extracted along the pipeline section, and the schematic diagram of the extraction position is shown in
Figure 10.
Figure 9 shows that the fluid flow changes abruptly after passing through the joint. In the two curved parts, eddy currents are generated due to the collision between the fluid and the wall. Due to the shunting of the branch pipes, the flow rate decreases sharply after passing through the inner wall of the connection, and the flow direction of the fluid changes.
Figure 11 shows that the maximum flow rate in the pipeline before entering the joint reaches 6.84 m/s. After passing the joint, the flow rate decreases rapidly, and the minimum flow rate is 0.34 m/s Then, the pressure decreases continuously with the fluid flow path. There is an obvious low-pressure area downstream near the connection, vortices are observed in this area. The particle trajectory reveals that the fluid generates vortices due to the effects of high and low pressure. The fluid fluctuation at this location is the main cause of fluid-induced pipeline vibration.
4.2. The Joint Effect on Fluid-Structure Interaction
To analyze the joint effect on the fluid-structure interaction, multi-branch pipelines with and without a joint are analyzed as shown in
Figure 12. The calculation iteration step is set as 0.001 s, and each step is iterated 10 times. The two-way fluid–structure interaction simulation analysis is carried out.
Figure 13 shows the deformation and equivalent stress cloud diagram of the pipeline. The difference value of the maximum deformation and equivalent stress due to the joint effect are 6.02% and 8.22%, respectively. The joint reduces the deformation of the pipeline and the value of the equivalent stress.
Figure 14 shows the frequency response function of pipelines with and without a joint. Due to the mass effect of the joint, the natural frequencies of the pipeline shift to the low-frequency domain, and the magnitudes of FRF are significantly reduced with the addition of the joint. For engineering applications, the joint effect on the pipeline system cannot be ignored.
4.3. Influence of Flow Rate on the Dynamic Characteristics of the Multi-Branch Pipeline
When the fluid pressure is set at 2 MPa, the flow rate is set at 0, 4, 8, 12, 16, and 20 m/s for analysis.
Table 5 shows the first three-order natural frequencies of the pipeline at different flow rate.
Figure 15 shows that the first three-order natural frequencies of the pipeline at different flow rates. The first three-order natural frequencies of the pipeline decrease continuously with the increase in flow rate, but the frequencies of each order under different flow rate are relatively close, and the change is not obvious. The first three-order natural frequencies have changed by 0.26%, 0.4%, 0.43%, respectively. The effect of the flow rate on the pipeline is due to the pressure difference caused by the variation in flow rate.
4.4. Influence of Fluid Pressure on Natural Frequency of the Multi-Branch Pipeline
When the flow rate is set at 5 m/s, the fluid pressures are set at 7, 14, 21, 28, 35, and 42 MPa for analysis.
Table 6 shows the first three-order natural frequencies of the pipeline at different pressures.
Figure 16 shows that the first three natural frequencies of the pipeline gradually decrease with the increase in the fluid pressure. The first three-order natural frequencies are changed by 2.85%, 1.9%, and 0.62%, respectively. The influence of pressure on pipelines cannot be ignored in practical engineering analysis.
4.5. Influence of Fluid Medium on Natural Frequency of the Multi-Branch Pipeline
Table 7 shows the parameters of different fluids. In order to explore the effect of different fluids on the natural frequency of the pipeline, different fluids (air, aviation fuel, hydraulic oil, lubricating oil, water) are investigated.
Table 8 shows the first three-order natural frequencies of the pipeline with different fluids.
Figure 17 shows that with the change in fluid, the first three-order natural frequencies of the pipeline gradually decrease. This change is caused by the modification of density. In different industrial fields, attention should be paid to the effect of fluid density variations on the natural frequency.
4.6. Influence of Pipe Diameter on Natural Frequency of the Multi-Branch Pipeline
Table 9 shows the First three-order natural frequencies of the pipeline at different pipe diameters.
Figure 18 shows the first three natural frequencies of the pipeline gradually increase with the increase in pipe diameter. The first three-order natural frequencies change by 261.18%, 206.68%, and 174.58%, respectively. The size of the pipe diameter has a significant impact on the vibration characteristics of the pipeline, and the natural frequency corresponding to different pipe diameters changes obviously. The size of the pipe diameter is an important factor when it comes to the actual design. The appropriate pipe diameter size should be selected according to the actual needs to reduce costs. Avoid the resonance caused by the similar frequency of the pipeline, causing the failure of the pipeline.
4.7. Influence of Constraint Stiffness on Natural Frequency of the Multi-Branch Pipeline
The fixed constraint of the pipeline is changed to elastic constraint. Different restraint stiffnesses (10
3, 10
4, 10
5, 10
6, 10
7, 10
8, 10
9, 10
10 N/m) are applied.
Table 10 shows the first three natural frequencies of the pipeline at different constraint stiffness.
Figure 19 shows that with the increase in restraint stiffness, the first three-order natural frequencies of the pipeline gradually increase. When the stiffness is less than 10
5 N/m, the pipeline is less constrained, resulting in a slow increase in the natural frequency. The elastic constraint gradually approaches the fixed constraint when the stiffness increases to 10
9–10
10 N/m. At this point, the increase in the natural frequency becomes slow and gradually approaches the value of the natural frequency at the fixed constraint. Within this constraint stiffness range, The first three-order natural frequencies change by 2803.88%, 3374.03%, and 556.29%, respectively. When elastic restraint is applied to the pipeline, appropriate restraint stiffness shall be selected on the basis of avoiding resonance.
4.8. Influence of Pipe Length on Natural Frequency of the Multi-Branch Pipeline
The length of different straight pipes is set at 100, 150, 200, 250, 300, and 350 mm.
Table 11 shows the first three-order natural frequencies of pipelines at different pipe lengths.
Figure 20 shows that with the increase in the straight pipe length, the first three-order natural frequencies of the pipeline gradually decrease. The first three-order natural frequencies change by 793.83%, 553.61%, and 850.55%, respectively. With the increase in the pipeline length, the natural frequency path of the pipeline decreases rapidly, especially the high order mode.
4.9. Influence of Wall Thickness on Natural Frequency of the Multi-Branch Pipeline
The tube inner diameter is set at 10 mm. The wall thickness of the pipe is adjusted by changing the outer diameter (0.5, 1, 1.5, 2, 2.5, 3 mm).
Table 12 shows the first three natural frequencies of the pipeline at different wall thicknesses.
Figure 21 shows that the natural frequency of the pipeline increases with the increase in wall thickness. When the wall thickness increases, the equivalent stiffness and equivalent mass of the piping system increase. If the increasing trend of the equivalent stiffness is greater than that of the equivalent mass, this makes the natural frequency of the whole piping system increase. The first three-order natural frequencies change by 110.31%, 51.44%, and 24.35%, respectively. The selection of the pipeline should be based on the objective situation to select the appropriate wall thickness of the pipeline.