## 4 Analysis of Vibration Characteristics of Cross-Type Hydraulic Pipeline
### Effect of Fluid Pressure on the Vibration Characteristics of Cross-Type Pipelines  
Set the fluid pressure to 0 MPa, 1 MPa, 5 MPa, 10 MPa, 15 MPa, and 21 MPa, respectively, and perform modal analysis on the cross-type pipeline to obtain its modal frequency  
Figure 6: Changes in steady-state parameters of pipelines with pressure. (**a**) response amplitude curve; (**b**) maximum stress curve.  
Figure 7: The variation of steady-state parameters of pipelines with elastic modulus. (**a**) response amplitude curve; (**b**) maximum stress curve.  
and vibration mode. The first six natural frequency variation curves under different pressures are shown in Figure 8. Taking the pressure inside the pipe as an example of 21 MPa, harmonic response analysis was conducted by adding a simple harmonic force with an amplitude of 1 N in the X, Y, and Z directions at the connection positions of the pipe joint, and the frequency range was set to basically cover the first six modal natural frequencies of the cross-type pipe. For example, when the pressure inside the pipe is 21 MPa, the frequency range to be considered for harmonic response analysis is the same as the first six natural frequencies in modal analysis, ranging from 0-1200 Hz. Figure 9 shows the displacement response of a cross-type pipeline when a simple harmonic force is added in the X, Y, and Z directions.  
Figure 8: Natural frequency variation curve of pipeline under different pressures.  
Figure 9: A 21 MPa harmonic response analysis curve of cross-type pipe.  
From the above curve analysis, it can be concluded that the first six natural frequencies of the cross-type pipeline show an increasing trend with the increase in fluid pressure inside the pipeline. With the increase in fluid pressure, the pipeline's stiffness will go up. So the self-excited vibration period of the vibration system decreases, and the natural frequency increases. Compared with the results of fluid-structure interaction vibration characteristics of a single pipe in the literature [25], it is found that the natural frequency of a cross-type pipe changes in the opposite direction to that of a single pipe when the fluid pressure in the pipe increases. The response of the cross-type branch pipeline under the action of this harmonic force shows maximum values at frequencies of 336 Hz, 582 Hz, 1136 Hz, and 1156 HZ. Comparing the response of the cross-type branch pipeline in the X, Y, and Z directions, it is found that the displacement response of the cross-type pipeline in the X direction was the largest, with a value of \(8.73\times 10^{-3}\) mm. By comparing the modal analysis, it is found that the natural frequencies of the cross-type branch pipe were very close to the first-order natural frequency of 335.53 Hz, the second-order natural frequency of 581.36 Hz, the fourth-order natural frequency of 1136.4 Hz, and the fifth-order natural frequency of 1158.7 Hz, respectively. This indicates that under external excitation, the first, second, fourth, and fifth order modes of the cross-type pipe are more sensitive to external excitation forces, and the response value is the highest in the first order mode. When conducting modal analysis on pipelines, we will find that the amplitude of vibration varies at different orders of modes. Therefore, when subjected to external excitation, although the magnitude of the excitation force is the same, sensitivity may vary due to the different amplitudes of the modes in which the structure is located.