# Application of Improved MFDFA and D-S Evidence Theory in Fault Diagnosis
## 4 Test Analysis  
### Analysis of Fault Characteristic Data  
Through the above test, the signals of different working conditions of the centrifugal pump are obtained, and then the classical time-frequency analysis method HHT and the improved MFDFA are used to extract the features, and the characteristic values reflect the fault information, and the test data analysis process is shown in Figure 4.  
Figure 3: Fault diagnosis model. Figure 2: Centrifugal pump test device. Taking the vibration signal from the centrifugal pump as an example, the data of slight cavitation, severe cavitation, and loose foot bolts were selected and analyzed with respect to their time and frequency using HHT, as shown in Figure 5. Due to the different intrinsic dynamics and mechanisms, different fault signals have different frequency ranges and as well as energy distributions. From the time-frequency diagrams of Figure 5a,b, it can be seen that although slight cavitation and severe cavitation can be distinguished by signal frequency to some extent after HHT transformation and there is a partial difference in energy value, there is no particularly obvious overall difference in energy value between them. Moreover, from Figure 5c, it can be seen the signal frequency deviation by the loose foot bolt fault from that of severe cavitation is insignificant. Because of the phenomenon of modal mixing in HHT itself, which resulted in poor differentiation, it can be stated to be unconducive to the accurate diagnosis of faults. To improve on this, the improved MFDFA method was applied to each condition of the centrifugal pump vibration, torque, motor vibration, and oscillation signals. The data length was set as 1024, the parameter \(s\) taken from 4-256 (steps of 8), and the value of \(q\) was selected from the range-10-10 (steps of 0.5). The resulting distributions of the multiple fractal spectral features \(\Delta f\), \(\alpha_{0}\), \(\Delta\alpha\), \(\alpha_{\text{min}}\) and \(\alpha_{\text{max}}\) extracted by the improved MFDFA method are shown in Figure 6.  
Figure 6 shows the relationship between the quality index \(\tau(q)\) of vibration signals of four kinds of centrifugal pumps and the order \(q\) of wave function. When the centrifugal pump was in a normal state, the nonlinear relationship between the quality index \(\tau(q)\) and \(q\) was poor. While when the centrifugal pump was in a cavitation state or the anchor bolts became loose, there was an obvious turning point and a very distinct nonlinear relationship between the quality index \(\tau(q)\) and \(q\), highlighting the multifractal characteristics of each state of the centrifugal pump. The multifractal characteristics of all fault states were stronger than those of the normal state, while the multifractal characteristics under  
Figure 4: Analysis flow chart. Figure 5: HHT time–frequency diagram of centrifugal pump vibration signal under (**a**) slight cavitation, (**b**)severe cavitation, and (**c**) loose foot. looseness of anchor bolts were stronger than those of the cavitation state. There were also prominent differences among multifractal characteristics under different cavitation degrees. Therefore, EMD-LS-MFDFA method can directly distinguish the running states of the centrifugal pump. Figure 7a shows the distribution of the eigenvalues of the multifractal spectrum of the pump vibration signal in four different states. Therein, it can be observed the eigenparameter values of slight cavitation, severe cavitation, and loose foot bolt are larger than the normal state values. Moreover, except for the value of \(\alpha_{\text{min}}\) for severe cavitation which is larger than that of the loose bolt state, the eigenparameter values of the remaining loose bolt states are larger than the state values of different cavitation degrees, indicating the multifractal characteristics caused by loose foot bolt are the most obvious. A typical trend of the multifractal characteristics of cavitation increasing with increasing severity can be seen.