# Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation
## 4 Analysis and Methodology  
On the foundation of Step 1, the above data in ever flow rate point would be turned into the reference sequence (RS) and comparative sequence (CS) as the following matrix expresses:  
\[\text{RS}=\left\{ \begin{array}{cccc}\text{\emph{NPSH}}_{r}(p_{1}),\text{ \emph{NPSH}}_{r}(p_{2}),\text{\emph{NPSH}}_{r}(p_{3})\cdots\text{\emph{NPSH}}_{ r}(p_{n-1}),\text{\emph{NPSH}}_{r}(p_{n})\end{array} \right\}\]  
\[\text{CS}=\left\{ \begin{array}{cccc}\text{\emph{Maximum}}(p_{1})&\text{ \emph{Maximum}}(p_{2})&\cdots&\text{\emph{Maximum}}(p_{n-1})&\text{\emph{Maximum}}(p_{n}) \\ \text{\emph{Minimum}}(p_{1})&\text{\emph{Minimum}}(p_{2})&\cdots&\text{ \emph{Minimum}}(p_{n-1})&\text{\emph{Minimum}}(p_{n})\\ \vdots&\vdots&&\vdots&&\vdots\\ \text{\emph{Impulse factor}}(p_{1})&\text{\emph{Impulse factor}}(p_{2})&\cdots& \text{\emph{Impulse factor}}(p_{n-1})&\text{\emph{Impulse factor}}(p_{n})\\ \text{\emph{Margin}}(p_{1})&\text{\emph{Margin}}(p_{2})&\cdots&\text{ \emph{Margin}}(p_{n-1})&\text{\emph{Margin}}(p_{n})\end{array} \right\}\]  
For the normalization processing of the data from a matrix by the maximum way according to Step 2, the tackled data are drawn on Figure 4.  
Figure 3: Instrument layout: (a) test site; (b) monitor points. From Figure 4, in the NPSH decreasing process, the corresponding fifteen (15) vibration feature value in the test interval coexist in the situation of increase and decrease instead of monotonous relations. Meanwhile, as the vacuum pump starts working, positive and negative relevance coexists between the NPSH and fifteen (15) vibration feature. On the other hand, some values might be abnormal since the negative values exist in the original signal and the potential unknown factors are distributing. For instance, Figure 4d shows the mean value in 40 m\({}^{3}\)/h. However, as weight entropy states, the rationale and credible value can be acquired based on the calculated value of grey relation and entropy weight. In this way, the objective relation between NPSH and feature parameter can be decided whether it is related or not. Furthermore, the relevance matrix \(\theta\) can be acquired with the data in Figure 4b-p through the Step 3 calculation, the consequence of which can be seen in Figure 5.  
In Figure 5, n denotes the numbers of the calculated slope, and \(\zeta\) expresses the grey slope coefficient of the corresponding feature target in different stages. From Figure 5, the trend of all targets except the Kurtosis factor basically considered has a positive or negative relevance with NPSH but the grey coefficient tends to 0 in the terminal. In mathematical terms, these parameters do not have strong relevance with NPSH in the terminal. However, from a physics perspective, this kind of description cannot satisfy common sense. According to the definition of grey slope correlation, using the slope in different stages reflects the relevance between the vibration feature target and NPSH. Mirrored in Figure 5, in the cavitation stage, the slope value of the feature target and vibration has a big difference. The physics states in the pump are changed and the corresponding physics meaning is the minimum variation in NPSH which would cause logarithmic leaps among the feature targets. Thus, these descriptions correspond to the fact of phase-change vibration caused by bubble burst. Figure 4: Normalization value: (**a**) NPSH, (**b**) Maximum, (**c**) Minimum, (**d**) Mean, (**e**) Peak, (**f**) Absolute mean, (**g**) Variance, (**h**) Standard deviation, (**i**) Kurtosis, (**j**) Skewness, (**k**) Root mean square, (**l**) Shape factor, (**m**) Crest factor, (**n**) Kurtosis factor, (**o**) Impulse factor, (**p**) Margin factor. Due to the existence of positive and negative value in the feature target, the relevance of the feature target cannot be judged directly. Therefore, transforming the negative and positive value into the same positive interval by Step 4 as Figure 6 depicted.