## 4 Results Analysis
### Numerical Model Validation
#### 4.1.2 Comparison of Vibration Characteristics of Main Shaft  
The variation curves of vibration accelerations along X and Y directions versus time were gained from numerical calculation and vibration measurement. In the numerical model, a node of the main shaft which connects with the bearing was chosen for study and comparison. It was found that spectral distribution characters of vibration acceleration under different flow rates are similar in some aspects, so the \(Q_{\text{des}}\) condition was chosen for the comparisons of the spectral distribution of the vibration accelerations, as shown in Figure 12.  
In Figure 12, distribution laws can be seen between computation and measurement results. Specifically, the vibration acceleration magnitude reaches the peak at \(f_{\text{BPF}}\) (148 Hz), which is the dominant frequency. Meanwhile, the magnitudes at \(f_{\text{n}}\) (24.7 Hz), \(f_{\text{BPF}}\) and their harmonics. Moreover, the acceleration magnitude along X direction is generally higher than that along the Y direction. To sum up, the numerical calculation of vibration information of the rotor system is reliable. However, the calculated results are different from measurement results to some extent with respect to specific vibration acceleration magnitude and distribution features. On one hand, the calculated acceleration magnitudes under dominant frequency is higher than the measurement results. For example, the calculated vibration acceleration magnitudes along X and Y directions under the \(f_{\text{BPF}}\) are 0.88 m s\({}^{-2}\) and 0.26 m s\({}^{-2}\) respectively, which are higher than the corresponding  
Figure 12: Comparisons of the frequency domains of the vibration acceleration under \(Q_{\text{des}}\): (**a**) Calculated values in X direction; (**b**) Calculated values in Y direction; (**c**) Test values in X direction; (**d**) Test values in Y direction. measurement results (0.46 m s\({}^{-2}\) and 0.19 m s\({}^{-2}\)). On the other hand, the measured vibration acceleration magnitudes at some other multiple shaft frequencies (e.g., 98.7 Hz and 296 Hz) are significantly higher than the calculated results. Differences between calculated results and measurements are mainly caused by the following three aspects. Firstly, the unsteady CFD model applies Reynolds-average method to average the instantaneous values of turbulence phenomenon, thus ignoring the effects of high-frequency random disturbance. Secondly, vibration of the rotor system in the numerical model is only attributed to hydraulic exciting. However, due to mechanical processing and assembly, it is inevitable to have some other vibration sources in practical operation, such as external forces caused by mass imbalance of impeller, rotor asymmetry, and electrical motor. Thirdly, the computation model cannot provide vibration information of bearing directly due to the limitation of element types in the rotordynamics finite element model. Instead, bearing vibration is approximately reflected by information on the shaft node connected with the bearing. Considering the high stiffness of the bearing, the vibration acceleration measurements are smaller than the calculated values, and there are multiple harmonic components in measurement results. In this study, a quantitative evaluation on vibration energy of a centrifugal pump was carried out by using the root-mean-square (RMS) value of vibration velocity. At the first step, curves of vibration velocity versus time were gained through time integral of vibration acceleration series. Then, RMS of vibration velocities under different flow rates were calculated and shown in Figure 13. It is easy to see that the RMS velocity distribution feature of computation results is highly similar to measurements. Generally speaking, vibration velocity level is relatively low around \(Q_{des}\), and the RMS ranges are between 0.4 m s\({}^{-1}\)-1.0 m s\({}^{-1}\). And RMS velocity along the X direction is generally higher than that along the Y direction.