#### 4.1. Single-Value Indicators of Vibration from Time Domain

In terms of experimental analysis, accelerometers mounted on the centrifugal pump three direction as shown in

Figure 4, In order to simplify, the X direction data is mainly used in the indicator research

Figure 8 and

Figure 9 show the peak and

RMS trend of the time domain vibration signal with the flowrate from 0 to 1.6

Q_{d}.

Both peak and

RMS could reveal the change law of signal energy intensity, these two indicators share a similar tendency with the pump’s flowrate as shown in the figures above. Both indicators would rapidly increase when pump’s flowrate exceeds 60 m

^{3}/h, which may also be the inception cavitation point according to

Figure 3.

However, it was found that the change tendency of the RMS was much more flatter than that of the peak, especially during the flowrate range from 10 m^{3}/h to 18 m^{3}/h, which is the hump area of the pump performance curve. In this area, a relatively strong flow instability phenomenon exist in the pump, peak curve has a relatively steep rise trend, yet this trend is not observed in the RMS curve. Meanwhile, the peak value would become extreme low at 53 m^{3}/h, which indicates the highest efficiency operation point of the pump, and such tendency is also not shown in RMS curve. Furthermore, the start point of the dramatic rise of the peak occurs earlier than RMS. From these analyses, it might be concluded that the peak value is more sensitive in flow instability detection than RMS. By contrast, RMS performs better in predicting cavitation, it has strong anti-jamming capability contrast to low sensitivity.

Crest factor is described as the peak factor divided by the corresponding

RMS value. As the high detection sensitivity of peak and the strong anti-jamming capability of

RMS, the crest factor might be highly suitable for flow instability detection. As shown in

Figure 10, an interesting phenomenon could be found whereby the extreme points of the crest factor curve are the pump’s particular operating points, which show the flow instability boundary, cavitation inception and even the highest efficiency point.

Figure 11 shows the kurtosis curve for the time domain vibration signal, and two different definite increase could be easily found in the kurtosis curve. The first was observed at the operation range of 10–18 m

^{3}/h, the other was between 55–62 m

^{3}/h. The first increase indicates the hump area of the pump performance curve; in this area, the internal flow status of the centrifugal pump is rather unstable. The second increase locates at the onset of the cavitation, which also means stronger intensity of flow instability.

As can be seen from

Figure 7, the probability density function (

PDF) of the vibration signal is approximately Gaussian with the exact shape depends on its details. As the flowrate increases, the vibration would change and, obviously, the shape of the

PDF curve would also change correspondingly.

The shape of the PDF curve could be evaluated by standard deviation and variances, and as a result, both mathematical parameters could be used as the single value indicators.

The change tendency of the standard deviation and the variances for the time domain vibration signal with the pump’s flowrate are shown in

Figure 12 and

Figure 13. It could be found that the standard deviation curve and the variances curve have almost the same variation law with the

RMS curve, thus add nothing qualitatively new to the discussion.

Figure 14 shows the amplitude of the

PDF curve for the time domain vibration signal under different flowrates, which clearly shows that there is a broad maximum value of the

PDF exists at the flowrate range from 50 m

^{3}/h to 55 m

^{3}/h. When the pump’s flowrate exceeds this range, a definite decline trend occurs in the curve. Such a change law is confirmed as a universal phenomenon, and this turning point appears in the curve could detect the likely beginning of cavitation. However, the flow instability which should be reflected in the range of 10–17 m

^{3}/h is not shown, obviously.

The reason for such a pattern of the PDF is that the distribution of the vibration signal in terms of frequency would change with the pump’s operation condition. As the flowrate keeps increasing and the pump approaches cavitation, it could be seen that variations occur in the spectral structure of the vibration signal, where under the relatively low flowrates the vibration spectrum includes a few isolated low-level structural resonances. When centrifugal pump operates under the established cavitation detection point, a number of high peaks appear in the spectrum, where the PDF for the time domain vibration signal has its sharpest peak. When the centrifugal pump operates under the highest flowrate during the test, which means a severe cavitation condition, a large number of peaks appear across the spectrum, where the PDF for vibration is inclined to flatten out. Meanwhile, it should be noted that some other flow instability phenomena like backflow would cause stronger broadband noise in some areas, which would result in the change of the PDF pattern for the vibration signal. However, the degree of deformation of the PDF curve is weaker than the cavitation caused. Therefore, the PDF curve could be employed for cavitation diagnosis for centrifugal pump.

#### 4.2. Single-Value Indicators of Vibration from Frequency Domain

Previous investigations suggest that the vibration spectral of a centrifugal pump contains a broadband noise and a few discrete frequency peaks, whose intensity would be strongly affected by the flow status of the centrifugal pump. In the analysis of the frequency domain vibration signal, crest factor, kurtosis and entropy might possess the ability to evaluate the intensity change for the two components in the frequency domain.

As shown in

Figure 15 and

Figure 16, the change regularity of these two statistical parameters with the pump’s flowrate is similar, both parameters could clearly reflect the hump area and the cavitation onset by the curve drops, and indicates the cavitation process with a relatively low value.

Figure 17 shows the spectral entropy obtained from the frequency domain vibration signal as a function of the pump’s flowrate. The value of the spectral entropy for the frequency domain vibration signal would be expected to be relatively low in the area centered at the pump’s design flowrate, where the harmonic components of

BPF and rotational frequency stand out above the spectrum, and information uncertainty is lowest. As the pump’s flowrate deviates from its design, spectral entropy would increase correspondingly due to the stronger noise. The vibration signal would increasingly resemble broadband noise and spectral entropy would finally have its highest value. The curve obtained confirms this general analysis, and further indicates that the spectral entropy of the frequency domain vibration signal could detect cavitation onset and unstable flow area of centrifugal pump.