research on statistical characteristics of vibration in...

Rev. Téc. Ing. Univ. Zulia. Vol. 38, Nº 1, 49 - 61, 2015

Research on Statistical Characteristics of Vibration in Centrifugal Pump

Yin Luo1*, Hui Sun1, Shouqi Yuan1, Jianping Yuan1 1 Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China

Abstract When a centrifugal pump operates in flow instability conditions, the dynamic characteristics of the

pump change, which is reflected by changes in the statistical properties of its vibration. This paper presents a

study on the application of vibration signals to detect the operating condition of centrifugal pumps, by using a

statistical analysis method. The statistical features of vibration from the time domain, particularly the rapid

increase in peak and root mean square (RMS) values, may indicate flow instability, with the peak indicator

being much more sensitive. Most of the extreme points of the crest and kurtosis have obvious physical

significance such as the onset and end of the flow instability area, the onset of cavitation, and the maximum

efficiency point. The probability density function (PDF) is a good indicator of the early stages of cavitation and

could be used to accurately detect the cavitation inception point. Moreover, the statistical features of vibration

from the frequency domain, namely the kurtosis, crest factor, and entropy, might be suitable for detecting

intensity changes in the broadband noise and discrete frequency peaks in the frequency domain, which are

caused by variations in the flow condition. Hence, the statistical analysis of vibration could be an effective

method of detecting unstable flow conditions.

Keywords centrifugal pump, condition monitoring, vibration, statistical characteristics, hydraulic instability.

1. INTRODUCTION

A centrifugal pump is designed to achieve its best performance at a specific combination of capacity, head,

and speed, known as the best efficiency point (BEP). At the best efficiency flow rate, the fluid motion is

compatible with the physical contours of the hydraulic passages. However, in industry, not all pumps function at

their optimal operating point (H.I, 2001). A pump that is working significantly below the best efficiency flow

rate is said to be operating at part load. Because of the considerably large blade inlet angles and channel cross

sections for the reduced flow rate, the flow patterns during part-load operation are fundamentally different from

those at the design point and introduce factors like hydraulic instability, which can cause damage to the

machines (Parrondo et al., 1998). By contrast, a pump working significantly above the best efficiency flow rate

risks operating under cavitation conditions, which can damage pump components and produce high levels of

vibration, noise, and additional energy. Hence, if BEP is not considered, the pump will be subject to increased

wear, and its operational life will be reduced (H.I, 2001). Figure 1 shows the detrimental effects of operating a

centrifugal pump away from the BEP.

Fig. 1. Adverse effects of operating away from the BEP

In traditional condition monitoring, many sensors are necessary, and most of them have installation,

implementation, and economic concerns. Because of such limitations and their invasive nature, sensors are

rarely deployed for efficiency checks. Therefore, a system that can identify the operating point of a centrifugal

pump needs to be developed (Rodriguez et al., 2014).

*Corresponding author.

E-mail address: [email protected]

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mailto:[email protected]


The basic principle of such a monitoring system is to analyze and subsequently classify the time signal

recorded by a single vibration accelerometer. This has the advantage of being easily achievable without

changing anything in the plant. Therefore, a method of extracting highly useful information from a simple

monitoring device needs to be studied (Al. 2011).

A number of authors have investigated and discussed the main features of the vibration signal from

centrifugal pumps, which are relevant for condition monitoring and fault diagnosis. In these studies, the

operating condition was detectedeither by tracking the variation in characteristic frequency or by computing the

change in energy content of the vibration within specific frequency bands or in the time domain. However, the

industry tends to prefer the simple approach: single-value indicators, obtained directly from the signal, which

can be easily compared against accepted standards (e.g., allowable RMS vibration levels). Single-value

measurements also have the advantage of easily identifiable trends, allowing for remedial action to be taken

when preset levels are reached. From this standpoint, the formal method is not the preferred in industry (Heng et

al., 1998).

The parameters PDF, peak value, RMS value, crest factor, kurtosis, and standard deviation are the common

statistical parameters that could reflect the variation in the statistical properties of a signal. For example, in the

case of fault development in a pump, changes in the time domain of the vibration signal, such as increase in the

RMS value or appearance of sharp peaks, are strongly suggestive of fault development and incipient damage.

Thus, the extraction of useful information from the time domain can be achieved by calculating the statistical

parameters that are related to the changes in the signal induced by defects. The most obvious case is the use of

peak or RMS values to identify the severity of pump defects (Wu et al., 2013).

However, other statistical parameters of vibration, such as crest factor and kurtosis, have not been

considered for condition monitoring of centrifugal pumps before, as their variation laws have not been

systematically studied. As a result, this paper focuses on the statistical features of vibration from a centrifugal

pump within the whole range of operating conditions, in order to extract as much information as possible from

these statistical parameters.

2. THEORY AND METHODOLOGY

2.1 Characteristics of vibration from centrifugal pump

A centrifugal pump has two main parts: the rotating element, which consists of a shaft and an impeller, and

the stationary element, which consists of the casing, casing box, and bearing and the electrical motor with a

cooling fan (Nelson. 1992). Based on the working process of the pump, vibrations are understood to be

generated by both hydrodynamic and mechanical sources (Florjancic et al., 1993). The hydrodynamic sources

usually cause fluid-flow perturbations in the pump and facilitate the interaction of the rotor blades with nearby

stationary objects such as the volute tongue or the guide vanes. Meanwhile, the mechanical sources include

vibration of unbalanced rotating mass and friction in the bearing and seals. These mechanisms of generating

vibration cause the structure of the pump to vibrate. From this perspective, the basic generating mechanisms for

both structure-borne vibration and airborne noise are the same for a sealed pump system (Black, 1969).

Previous investigation showed that the vibration contains both broadband noise and a number of discrete

frequency peaks (Liu et al., 1994). The broadband content is the result of pressure fluctuations generated by

flow turbulence, viscous forces, boundary-layer vortex shedding, boundary-layer interaction between high-

velocity and low-velocity regions of the process fluid, and vortices generated in the clearances between the rotor

of the centrifugal pump and the adjacent stationary part of the casing. Mechanical sources also contribute noise

from the rotation of the pump shaft and bearings.

The amount of turbulence strongly depends on the flow conditions. If the pump operates at the design point

or the operating point, at which a maximum proportion of the energy is used to move the process fluid, a

minimum value is obtained. However, if the pump operates at less than the design flow rate condition,

additional hydraulic noise is created because of internal recirculation in the suction and discharge areas of the

pump impeller, increasing the overall noise of the pump. Moreover, if the pump operates at more than the design

flow rate condition, boundary-layer vortex shedding increases, flow turbulence increases, and additional

hydraulic noise is generated. The discrete component characteristics present in the overall spectrum are mainly

due to the interactions of the rotor blades, which have a discrete nature compared to the nearby stationary

objects, such as the volute tongue, and the periodicities in the flow. These two mechanisms generate discrete

components at the rotational frequency and/or the blade passage frequency of the pump and at their higher

harmonics.

Given that the turbulence at or near the BEP is at a minimum, the discrete components tend to dominate the

measured spectrum, and the harmonics lower than three are more distinctive. If BEP is not considered during

operation, the turbulent disturbance increases and may even exceed the tonal noise (Oh.et al., 2007).

50


2.2 Statistical analysis theory and methodology

Statistical analysis is the science of collecting, exploring, and presenting large amounts of data to discover

underlying patterns and trends. When the dynamic characteristics of a system change, the statistical

characteristics also change, and their trends can be extracted through statistical analysis.

If the probability density of distribution of a data sample is expressed as

Prob 𝑥 ≤ 𝑥 𝑡 ≤ 𝑥 + d𝑥 = 𝑝 x dx (1)

Then the expectation (mean) of a random function of time, x(t), is

E x = 𝑥𝑝 𝑥 𝑑𝑥+∞

−∞ , 𝑥𝑖𝑝𝑖

∞𝑖=1 (2)

Next, the rth-order moment about the mean 𝑥is given by

𝐸 𝑥 − 𝐸(𝑥) 𝑟 = (𝑥 − 𝑥)𝑟𝑝(𝑥)𝑑𝑥+∞

−∞ (3)

From Eq. (3), the mean 𝑥 or E(x) of the random variable is the first-order moment, the RMS value is the

square root of the second-order moment, and the variance σ2 is the second-order central moment. The mean, the

RMS value, and the variance are measures of the variable’s average value, intensity, and deviation from the

mean, respectively. If the data is in discrete form, Eq. (3) can be written as

𝑀𝑟 =1

𝑁 (𝑥𝑘 − 𝑥)𝑟𝑁

𝑘=1 (4)

where N is the number of data points, and r is the order of the moment. The following equations are used for the

calculation of the other variables from continuous or discrete data:

𝑅𝑀𝑆 = 𝑥2𝑝 𝑥 𝑑𝑥∞

−∞=

1

𝑛 𝑥𝑘

2𝑛𝑘=1 (5)

Standard deviation:

σ = 𝑥 − 𝑥 2∞

−∞𝑝 𝑥 𝑑𝑥 =

1

𝑛 (𝑥𝑘 − 𝑥)2𝑛

𝑘=1 (6)

Complex signals, which are composed of a large number of sinusoids whose relative amplitudes and phases

are not constant, require additional descriptors such as crest factor (Cf) and kurtosis (K). The crest factor is given

by Eq. (7), where Vp and VRMS are the peak and RMS values of the signal, respectively. It is a measure of the

number and sharpness of the peaks in the signal and may be used to determine whether a signal contains

repeated impulses.

𝐶𝑓 =𝑉𝑝

𝑉𝑅𝑀𝑆 (7)

The kurtosis of the signal is given by Eq. (8). The subtraction of three is included to normalize the

expression so that the Gaussian/Normal distribution has zero kurtosis, i.e., the natural non-normalized kurtosis

of a Gaussian distribution is three. When the peaks in the spectrum are sharper or more pointed than the

Gaussian distribution, the kurtosis is greater than zero. A high value of kurtosis usually means infrequent

extreme deviations rather than frequent modestly sized deviations. If the value of K is negative, it implies that

the distribution is flatter than the Normal distribution.

𝐾 =𝑀4

𝜎4− 3 (8)

The probability density function (PDF) of a continuous spectrum is an expression that describes the

probable value of the spectral energy between any two frequencies. The normalized Gaussian distribution curve

is a probability density function given by

PDF =1

2𝜋𝜎2𝑒𝑥𝑝 −

𝑥−𝜇 2

2𝜎2 (9)

3. EXPERIMENTAL SETUP

3.1 Test bed

To validate the theoretical predictions and to conduct diagnostic studies, a pump test rig was constructed to

simulate pump operation at different conditions. The rig consists of an integrated centrifugal pump, a water tank,

a suction line, and a discharge line (Fig. 2). This construction forms a closed loop for water circulation.

51


1, 3, 13. Ball valves 2. Vacuum pump 4. Cavitation tank 5, 10. Butterfly valves 6. Gate valve 7. Electromagnetic

flow meter 8. Pressure transmitter 9. Model pump 11. Motor 12. Stable-flow tank

Fig. 2. Schematic of pump test system

Fig. 3. Photo of pump test system

The left tank, called the cavitation tank, works with a vacuum pump and is used for operation under

cavitation conditions, whereas the right tank, called the stable-flow tank, is used to stabilize the flow. The

centrifugal pump is mounted in the cavitation tank with its impeller in a horizontal plane and its outlet flange 2

m below the water surface. The stainless-steel pipes that carry the upstream and downstream flows of the pump

have diameters of 60 mm and 50 mm to match the suction and discharge flanges of the pump, respectively. The

capacity of the tanks is based on the maximum flow rate. The water temperature in the system can be

maintained at 1 ºC for more than one hour, during which full measurements can be completed at different flow

rates. The centrifugal pump data are shown in Table 1, and the pump characteristic curves are shown in Fig. 4,

where H is the head, Q is the flow rate and η is the efficiency of the pump.

Table 1. The parameter of the test pump

Rated flow Qd 50 m3/h Impeller inlet diameter D1 75 mm

Rated head Hd 32 m Impeller outlet diameter D2 174 mm

Rated speed n 2900 r/min Blade width b2 12 mm

Efficiency η 72% Blade number Z 6

Specific speed ns 101 Volute base diameter D3 184 mm

Pump inlet diameter Ds 60mm Pump outlet diameter Dd 50 mm

52

13121110987654321


0

10

20

30

40

50

60

70

80

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90

ɳ(%

)

H(m

)

Q (m3/h)

H-Q Q-ɳ

instable area

Fig. 4. Pump performance characteristic curves

A pump characteristic is regarded as stable if dH/dQ is negative, i.e., if the head drops as the flow rate

increases. Meanwhile, positive dH/dQ means that the pump is operating in an unstable range, usually caused by

unreasonable design parameters. Operating under this unstable range can enhance vibration. In Fig. 4, the

gradient dH/dQ is positive in the area between 5 m3/h and 20 m

3/h, demonstrating the phenomenon of

instability.

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70 80 90

H (

m)

Q (m3/h)

NPSHa

NPSHr

Cavitation start

Fig. 5. Pump performance characteristic curves for cavitation

Following ISO 3555 , the experimental characteristics between the net positive suction head available

(NPSHA) and the net positive suction head required (NPSHR) for the pump system in this study were obtained by

progressively throttling the valve in the discharge line, with the pump speed maintained at 2900 rpm and the

valve in the suction line fully open (100%).

Figure 5 shows that pump cavitation fully occurred at a flow of approximately 80 m3/h, when the head was

18.7 m and NPSHR was higher than NPSHA. Meanwhile, a flow rate of more than 83 m3/h can cause full

cavitation, as suggested in ISO 3555. Moreover, during the test study, the cavitation performance rapidly

deteriorated beyond 65 m3/h, as observed in Fig. 5, while the vibration significantly changed in amplitude and

frequency. Thus, the flow range from 65 m3/h to 83 m

3/h is considered to be the cavitation progression phase, in

which the cavitation becomes increasingly severe as the flow rate increases.

3.2 Data acquisition test system

To monitor the pump performance, an induction coil, which can detect the leakage flux of the motor, is

used to measure pump speed. A flow sensor is installed in the discharge line, and two pressure sensors are

installed in the suction and discharge lines to measure the pump delivery head.

The vibration of the pump was measured using four identical accelerometers with a flat frequency band of

20 Hz to 2 kHz and a resonant frequency of about 40 kHz. The wide frequency band allowed for the

measurement of high-frequency pump vibrations. The four sensors were placed on the pump case as shown in

Fig. 7. 53


Fig. 6. Performance-monitoring sensor installation

Fig. 7. Vibration sensor installation

Taking into account the characteristics of the vibration, the test system should have a highly accurate audio

frequency, simultaneous sampling, wide dynamic range, low noise, and low distortion. At the same time, the

vibration sensor must work with the integrated electronic piezoelectric (IEPE) signal conditioning. Hence, PXI-

4472 dynamic signal acquisition board, whose parameters are shown in Table 2, was adopted to measure the

vibration. This multi-function data acquisition board was used because of the large number of measured

parameters and various outputs from the sensors.

Table 2. Data acquisition board parameters

PXI-4472 - dynamic signal acquisition PXI-6251 - High-Speed M Series Multifunction

DAQ

Resolution 24-bit Resolution 16-bit

Dynamic range 110 dB Analog inputs

32 channel with Data

Transfer Rate 1.25

MS/s

Sampling rate 102.4 kS/s

maximum Analog outputs

2 channel with Data

Transfer Rate 2.5Ms/s

Range ±10 V Range

7 programmable input

ranges (±100 mV to

±10 V) per channel

IEPE

conditioning

Software

configurable Synchronization Multiple-device

Synchronization Multiple-device

3.3 Experimental process

The vibration was measured when the pump was operated at different flow rates with a fixed speed of 2900

rpm. The flow rate was adjusted step by step using a throttling valve in the discharge line.

Each test covered several operating conditions. To obtain reliable results, each test was repeated at least

three times, with a sufficient time interval in between. The water temperature for each test remained the same.

The data records from each test were processed using MATLAB to characterize the signals and to identify a set

of consistent parameters.

The data acquisition had a sampling rate of 20 kHz, with a built-in anti-aliasing filter, and the sample time

was set at 500 rotation cycles of the pump. Sample signals at different flow rates are shown in Fig. 10. 54


a. 0.2*Qd

b. 1*Qd

c. 1.6*Qd

Fig. 10. Raw time-domain vibration signals at different flow conditions

The data sets were also explored in the frequency domain. Figure 11 shows the spectra at typical flow rates

for five types of measurement conditions, namely, part-load operation, efficient operation, design operation,

progressive cavitation, and full cavitation.

Fig. 11. Vibrationspectra at different flow conditions

4. ANALYSIS OF THE STATISTICAL FEATURES OF CENTRIFUGAL PUMP OPERATION

4.1 Statistical features of vibration from time domain

4.1.1 Peak value, RMS, and crest factor

Figures 12 and 13 show the peak and RMS values of the time-averaged time-domain vibration signal for

different flow rates between zero and 1.6*Qd (rated flow 50 m3/h).

55

0500

10001500

20002500

30003500

40004500

5000

10 m3/h

30 m3/h

50 m3/h

65 m3/h

75m3/h0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

f (Hz)

Am

plit

ude (

g)


0

2

4

6

8

10

12

0 10 20 30 40 50 60 70 80

Vpe

ak (

g)

Q (m3/h)

Peak

Fig. 12. Peak value of time-domain vibration signal against flow rate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80

RM

S (

g)

Q (m3/h)

Root mean square

Fig. 13. RMS value of time-domain vibration signal against flow rate

The peak and RMS values both reflect the signal energy intensity index, and the trends of these parameters

against the flow rate are consistent with the data in Fig. 4 and Fig. 5. Both of them increase rapidly when the

flow rate exceeds 60 m3/h, which is the cavitation inception point as shown in Fig. 5.

However, the variation of the RMS is much flatter than that of the peak, especially for the flow rates

between 10 m3/h and 18 m

3/h. This range corresponds with the hump of the pump characteristic curve and has

relatively strong flow instabilities. The peak value shows a rising trend in this range and also an extreme low at

53 m3/h, the maximum efficiency point, both of which are absent in the RMS curve. Moreover, the starting point

of the dramatic rise is earlier for the peak than for the RMS. These factors show that the peak value is much

more sensitive than the RMS value as a flow instability indicator. On the other hand, the RMS value has the

ability to predict cavitation as well as strong anti-jamming capabilities.

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50 60 70 80

Cres

t fa

ctor

Q (m3/h)

Crest Factor

Fig. 14. Crest factor of time-domain vibration signal against flow rate

The crest factor is defined as the peak value divided by the corresponding RMS value. Because of the high

sensitivity of the peak value and low sensitivity of the RMS value to flow instability, the crest factor would be

particularly sensitive to instability. As seen in Fig. 14, the extreme points of the crest factor curve correspond to

particular operating conditions, namely the flow instability boundary, cavitation inception, and maximum

efficiency point. 56


4.1.2 Probability density function

Figure 15 shows the probability density of the time-domain vibration signal at different flow rates. It can be

seen that the PDF of such signals is approximately Gaussian, with the specific shape depending upon the details

of the signal. As the flow rate increases, the flow conditions and the vibration change, causing the shape of the

PDF curve to change also.

0

0.5

1

1.5

2

2.5

3

3.5

-4 -3 -2 -1 0 1 2 3 4

Pro

bai

lity

Den

sity

(%

)

Vibration Amplitude (g)

PDF

0 10 16 20 30 40 50 55 67 72 Fig. 15. Probability densities of time-domain vibration signal for different flow rates

The values of standard deviation and variance for the PDF curves are shown in Fig. 16 and Fig. 17,

respectively. The standard deviation and variance follow the same trend as the RMS value, and hence do not add

anything new to the discussion qualitatively.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80

Sd

Q (m3/h)

standard deviation

Fig. 16. Standard deviation of time-domain vibration signal against flow rate

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 10 20 30 40 50 60 70 80

Var

iance

Q (m3/h)

Variance

Fig. 17. Variance of time-domain vibration signal against flow rate

57


0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80

Pea

k o

f p

df

Q (m3/h)

Peak of PDF

Fig. 18. Amplitude of PDF curve of vibration signal against flow rate

Fig.18 clearly shows that there is a broad maximum value of the PDF centered at a flow rate of about 50–

55 m3/h. After this maximum, there is a definite decline in the PDF value, which accelerates as the flow

approaches 58 m3/h. If this is confirmed as a general phenomenon, then the turning point in the PDF-peak curve

can be taken as a definite indicator of the likely onset of cavitation and can be used as such. Moreover, the PDF-

peak curve takes a slight dip between 10 m3/h and 18 m

3/h, but it is not conspicuous enough for detecting flow

instability.

The shape, or pattern, of the PDF curve changes with flow rate because the distribution of frequencies in

the vibration signal is different at each flow rate, as seen in Fig. 11. At low flow rates, the spectrum contains a

few isolated low-level structural resonances, but at flow rates approaching established cavitation, there are a

number of distinct large-amplitude peaks (PDF has its sharpest peak), and at established cavitation, there are a

large number of peaks across the spectrum (PDF curve flattens out). Other flow instability phenomena such as

backflow could also cause some increase in broadband noise in some areas, producing changes in the PDF.

However, the degree of deformation of the PDF curve due to these factors is less than that caused by cavitation.

As a result, the PDF curve is an effective indicator of cavitation.

4.1.3 Kurtosis

The kurtosis is usually normalized so that it is zero for a Gaussian distribution, as done in Eq. (8).

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70 80

Kurt

osi

s

Q (m3/h)

Kurtosis

Fig. 19. Kurtosis of time-domain vibration signal against flow rate

Fig.19 suggests that, in the time-domain vibration signal, there are two different sharp increases in kurtosis,

one at the 10–18 m3/h area and the other at the 55–62 m

3/h range. The first increase correlates with the hump of

the pump characteristic curve, which is an area of relatively strong flow instabilities, whereas the second

increase corresponds to the onset of cavitation, an area of even higher flow instabilities.

4.2 Statistical features of vibration from frequency domain

The vibration of the centrifugal pump contains both broadband noise and a number of discrete frequency

peaks. The broadband content is the result of pressure fluctuations generated by flow turbulence, viscous forces,

boundary-layer vortex shedding, boundary-layer interaction between high-velocity and low-velocity regions of

the process fluid, and vortices created in the clearances between the rotor of the centrifugal pump and the

adjacent stationary part of the casing. 58


The amount of turbulence strongly depends on the flow conditions. If the pump operates at the design point

or the operating point, at which a maximum proportion of the energy is used to move the process fluid, a

minimum value is obtained. However, if the pump operates at less than the design flow rate condition,

additional hydraulic noise is created because of internal recirculation in the suction and discharge areas of the

pump impeller, increasing the overall noise of the pump. Moreover, if the pump operates at more than the design

flow rate condition, boundary-layer vortex shedding increases, flow turbulence increases, and additional

hydraulic noise is generated. The discrete component characteristics present in the overall spectrum are mainly

due to the interactions of the rotor blades, which have a discrete nature compared to the nearby stationary

objects, such as the volute tongue, and the periodicities in the flow. In the analysis of this characteristic, the

kurtosis, crest factor, and the entropy might be suitable for detecting the intensity changes in the two

components of the frequency domain.

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80

Cre

at f

acto

r

Q (m3/h)

Creat factor for frequency domain

Fig. 20. Crest factor of vibration spectra against flow rate

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80

ku

rto

sis

Q (m3/h)

Kurtosis for frequency domain

Fig. 21. Kurtosis of vibration spectra against flow rate

As shown in Fig. 20 and Fig. 21, the variation trends of these two statistical parameters at different

operation conditions are similar. Both of them could detect the hump and the onset of cavitation by sharp drops

in their curves and show the cavitation by relatively low values.

In signal processing, a spectral entropy is introduced to measure the distribution features of the spectrum

X(i), i = 1, 2, . . . , N of a signal x(i), i = 1, 2, . . . , N. (Luo et.al., 2014)

The spectrum is first normalized by

𝑝𝑖 =𝑋(𝑖)

𝑋(𝑗 )𝑁𝑗=1

(10)

where 𝑝𝑖𝑁𝑖 = 1 = 1, and N is the length of the spectral sequence. The spectrum can be considered as a

probability distribution, and its entropy, denoted by SE, can be obtained using the following equation:

𝑆𝐸 = − 𝑝𝑖 ∙ 𝑙𝑜𝑔2(𝑝𝑖)𝑁𝑖=1 (11)

59


The SE changes with the length of the spectral sequence. Therefore, for easier comparison, it is generally

normalized by the length to obtain the normalized spectral entropy SEn as follows:

𝑆𝐸𝑛 = − 𝑝𝑖 ∙𝑙𝑜𝑔2(𝑝𝑖)𝑁𝑖=1

𝑙𝑜𝑔2(𝑁) (12)

Thus, the value range of spectral entropy, as defined in Eq. (12), is zero to one. SEn is higher for vibration

spectra with flat amplitude distributions, reaching the limit of one when the amplitudes are equal for all the

frequency components. On the other hand, SEn is lower if the amplitudes concentrate only in a few frequency

components and has a value of zero when only one frequency component has a non-zero amplitude. With this

numerical property, the spectral entropy is capable of evaluating the spectral structure of the vibration signal.

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0 10 20 30 40 50 60 70 80 90

Entr

op

y

Q (m3/h)

Fig. 22. Spectral entropy of the vibration at the throttling control operation point

Fig.22 shows the spectral entropy obtained from the individual vibration spectra as a function of flow rate.

The spectral entropy would be expected to have a relatively low value at lower flow rates, where the harmonic

peaks due to the blade passing and rotational frequencies stand out above the background noise. The spectral

entropy will then increase with increasing flow rate, as the background noise increases relative to the peaks, and

reach a maximum value when the signal resembles the broadband noise. The results obtained conform to this

general analysis and indicate that spectral entropy could be used for detecting the onset of cavitation, the

unstable-flow area, and even the high-efficiency area.

5. CONCLUSIONS

In exploring the characteristics of the vibration signals from a centrifugal pump using the statistical

analysis technique, the following conclusions were drawn:

1. When the centrifugal pump operates in flow instability conditions, the dynamic characteristics of the

pump change, which is reflected by changes in the statistic properties of the vibration. Hence, the statistical

analysis of vibration could be an effective method of detecting unstable flow conditions.

2. The statistical features of vibration from the time domain, particularly the rapid increase in peak and

RMS values, may indicate flow instability, with the peak indicator being much more sensitive. Most of the

extreme points of the crest and kurtosis have obvious physical significance such as the onset and end of the flow

instability area, the onset of cavitation, and the maximum efficiency point. The probability density function is a

good indicator of the early stages of cavitation and could be used to accurately detect the inception of cavitation.

3. The statistical features of vibration from the frequency domain, namely the kurtosis, crest factor, and

entropy, might be suitable for detecting intensity changes in the broadband noise and discrete frequency peaks in

the frequency domain, which are caused by variations in the flow condition.

CONFLICT OF INTERESTS

The authors declare that there is no conflict of interests regarding the publication of this paper.

ACKNOWLEDGMENTS

This project is supported by The National Natural Science Fund, (No. 51409125 ), the China Postdoctoral

Science Foundation (No. 2014M551515), Open Research Fund of Key Laboratory of Fluid and dynamic

mechanical, Xihua University (Grant No. szjj2013-006), (No. 51349004), Priority Academic Program

Development of Jiangsu Higher Education Institutions, Jiangsu University fund assistance (No. 13JDG082),

Jiangsu postdoctoral research grants program (No. 1302026B) and University Natural Science Foundation of

Jiangsu Province (No. 14KJB470002). 60


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