## 4 Results and Discussion  
To verify the accuracy of the proposed prediction method, the numerical transfer function for finite element model was used. The pipe was assumed to be made of aluminium (\(D=6241\) Nm\({}^{2}\), \(m_{s}=0.5174\) kg/m, \(L=1\) m, \(E=70\) GPa, \(\rho_{s}=2700\) kg/m\({}^{3}\), \(\
u=0.39\)). The fluid in the pipe was assumed to be water at 4 \({}^{\circ}\)C (\(m_{f}=2.827\) kg/m, \(\rho_{f}=1000\) kg/m\({}^{3}\), \(c=1492\) m/s). The amplitude of force \(F\) was 1 N, and the position of the excitation force \(x_{e}=0\) m. The positions of measurement points were \(x_{1}=-0.62\) m, \(x_{2}=0.07\) m, and \(x_{3}=0.5\) m.  
The transfer functions of the pipe with varying flow velocities were calculated from Equation (7), as shown in Figure 4. As the flow velocity increased, the frequencies of all the mode tended to shift toward zero. The structural response at the mode frequencies increased as the flow velocity increased. Although the force excitation was applied at the centre point, the predicted transfer function had even modes of the pipe and increased as the flow velocity increased. To investigate the vibrations of the even modes, the mode shapes of the pipe were predicted as shown in Figure 5. Analytical mode shapes were calculated from Equation (4) at the resonance frequency of the piping system. At zero flow velocity, the mode shapes were symmetrical, and the even modes at \(x=0\) were minuscule. With increasing flow velocity, mode shapes were skewed along the flow direction and the number of modes at  
Figure 4: Analytical transfer functions for fluid-conveying pipe at (**a**) \(x=x_{1}\), (**b**) \(x_{2}\), and (**c**) \(x_{3}\) with varying flow velocities. \(x=0\) increased. Thus, the nodal points of the even modes were shifted in the direction of flow. Therefore, the force excitation at the centre of the pipe generated the even modes not at zero velocity but as the flow velocity increased. The flow velocity influences the mode shape owing to the increased tension force applied to the pipe. Because the flow velocity increases, the wavenumbers could be determined by using the transfer function. To consider the shifted nodal points of the even modes, the measurement point at \(x_{2}\) was located near the centre of the pipe. The locations of other acceleration signals at \(x_{1}\), \(x_{3}\) were specified by considering the difference in accelerations along each mode shape. The elements of the solid and fluid were generated to simulate the vibration of the straight fluid-conveying pipe using the finite element model, as shown in Figure 6. The solid contained 151,910 elements, and the fluid 354,322. Both ends of the pipe imposed the fixed condition to model the clamped-clamped pipe. To validate the analytical mode shapes of the fluid-conveying pipe, numerical mode analysis was performed as shown in Figure 7. The shapes of the numerical modes of the  
Figure 5: Analytical mode shapes of the fluid pipe when the flow velocity increased: (**a**) 1 st, (**b**) 2 nd, (**c**) 3 rd, and (**d**) 4 th mode shape. Figure 6: Finite element model for the vibration of clamped-clamped fluid-conveying pipe. system only have minor differences when compared to those of analytical modes because of insignificant torsional displacement of the finite element model in three dimensions. In the actual pipe system, the torsional displacements will cause considerable error to the predicted fluid flow. Additionally, the shifted nodal points were described while the flow velocity was increased. The vibration of the fluid-conveying pipe had symmetric shapes at the 1 st, 2 nd, 3 rd, and 4 th mode frequencies. As the flow velocity increased, the mode shape changed asymmetrically. Moreover, the frequency response at \(x_{2}\) was not zero because of the asymmetric even-mode shape. Other points at \(x_{1}\), \(x_{3}\) had several changed values along the mode number. Figure 8 compares the analytical and numerical transfer functions.