## 4 Results and Discussion
### Hydraulic Radial Force on Each Blade  
In this study, the centrifugal pump impeller has six blades that exhibit a cyclic symmetry relationship, which are named Blade 1 to Blade 6, respectively. The hydraulic radial force on all the blades together is regarded as the vector sum of that acting on each of these six blades. According to this concept, under different flow conditions, the curves of the components of the hydraulic radial force in the X and Y directions of Blade 1 to Blade 6 changing with time are plotted, as shown in Figures 16 and 17. As can be seen from Figures 16 and 17, under each flow condition, the curves of the components of the hydraulic radial force in the X and Y directions of each blade changing with time form a curve cluster with exactly the same shape and size but with the same time interval. Each curve is very close to a sine curve, and the period is basically equal to the impeller rotation period \(T.\) Meanwhile, except for the local abrupt changes in the Y-direction components of the hydraulic radial force under the two low flow conditions of \(0.8Q_{d}\) and \(0.9Q_{d}\), all the curves are relatively smooth. Figure 14: Time-varying curves of angle of hydraulic radial force on all blades. Figure 15: Time-varying curves of hydraulic radial force components on all blades: (**a**) X direction; (**b**) Y direction. In brief, by observing Figures 16 and 17, it can be initially speculated that the curves of components of the hydraulic radial force on each blade in the X and Y directions changing with time are approximately a cluster of sine curves with equal amplitudes and frequencies but with initial phases that change successively at the same interval. To verify this hypothesis, taking the flow condition of \(0.8Q_{d}\) as an example, the time varying curves of the hydraulic radial force on each blade in the X and Y directions are fitted according to the following formula:  
\[f(t)=a\times\sin\left(\omega\times t+\Phi\right)+b \tag{7}\]  
where \(f\) reflects the hydraulic radial force component in X or Y direction, and \(t\) is time; \(a\), \(\omega\), \(\Phi\), and \(b\) are the amplitude, angular frequency, initial phase, and constant terms in the trigonometric function expression, respectively. Figure 16: Curves of hydraulic radial force component in the X direction on each blade under different flow conditions: (**a**) \(0.8Q_{d}\); (**b**) \(0.9Q_{d}\); (**c**) \(1.0Q_{d}\); (**d**) \(1.1Q_{d}\); and (**e**) \(1.2Q_{d}\). The fitted trigonometric function curves are plotted as dashed lines in Figure 18. For the convenience of observation and comparison, the original curves are also plotted as solid lines. The fitting residuals, that is, the differences between the original values and the fitted values, are plotted in Figure 19. It is easy to see from Figures 18 and 19 that for both the hydraulic radial force components in the X and Y directions, the original curves are very close to the sine function curves obtained by fitting, with only some differences at the peaks or troughs of each curve. Figure 17: Curves of hydraulic radial force component in the Y direction on each blade under different flow conditions: (**a**) 0.8\(Q_{d}\); (**b**) 0.9\(Q_{d}\); (**c**) 1.0\(Q_{d}\); (**d**) 1.1\(Q_{d}\); and (**e**) 1.2\(Q_{d}\). The fitting parameters are listed in Table 3. As can be seen from the table, for each curve of the hydraulic radial force component, the goodness of fit, that is, the value of \(R^{2}\), is not lower than 0.986. This indicates an extremely high fitting accuracy and a satisfactory fitting effect. Figure 19: Preliminary residual curves of radial hydraulic force components on each blade under 0.80\({}_{d}\): (**a**) X direction; (**b**) Y direction.