Mechanical Systems and Signal Processing 114 (2019) 275–289

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Mechanical Systems and Signal Processing

journal homepage: www.elsevier .com/locate /ymssp

Cyclostationary approach to detect flow-induced effects onvibration signals from centrifugal pumps

https://doi.org/10.1016/j.ymssp.2018.05.0270888-3270/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: chuning@zju.edu.cn (N. Chu).

Shiyang Li a,b, Ning Chu a,⇑, Peng Yan a, Dazhuan Wu a,b, Jérôme Antoni c

a Institute of Process Equipment, College of Energy Engineering, Zhejiang University, Hangzhou 310027, Chinab The State Key Laboratory of Fluid Power Transmission and Control, Hangzhou 310027, Chinac Laboratoire Vibrations Acoustique, Univ Lyon, INSA-Lyon, LVA EA677, F-69621 Villeurbanne, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 September 2017Received in revised form 2 May 2018Accepted 14 May 2018

Keywords:Flow-induced effectPump vibration modelCyclostationarityComputational fluid dynamics (CFD)Signal demodulation

This study aims to investigate the mechanism of flow-induced effects on vibration signalsfrom centrifugal pumps by combining computational fluid dynamics (CFD) and signalcyclostationarity. A pump vibration model is established as an amplitude-modulated(AM) model, and the modulation mechanism is elaborated in detail. A general AM modelis mathematically analyzed with spectral correlation density and spectral coherence, andanalytic solutions of modulation intensity are derived. Transient CFD simulations underdesign and off-design conditions are conducted to obtain the pressure pulsation in thevolute and the radial fluid forces on the impeller. Then, vibration signals from the pumpand motor feet with two acceleration transducers are processed by spectral coherence.The signature of second-order cyclostationarity is detected at the shaft rotating frequencyand blade passing frequency (BPF). The mean spectral coherence is used to evaluate theintensity of the modulating signals produced by the flow-induced effects. Finally, the signalprocessing results are compared with the unsteady CFD results under design and off-designconditions. These comparisons show a good agreement. Therefore, this study confirms thatthe flow-induced signals calculated by CFD can be considered as modulating componentsfor pump vibration signals. The results provide solid supporting theory for designing lowvibration pumps.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

The vibration performance of turbomachines such as pumps has caught the attention of engineers. In some specialsituations, vibration performance is prioritized over pump efficiency. For fluid engineers, the main method to reduce pumpvibration is to optimize the pressure pulsation in the pump and the radial fluid force fluctuation exerted on the impeller andvolute. This optimization is achieved by computational fluid dynamics (CFD) simulations. A common approach is to primarilyadopt empirical specifications for low-vibration designs instead of CFD simulations because the latter requires massive com-putational resources and careful model treatments. However, the direct relation between fluid-induced effects and vibrationsignals has yet to be elaborated clearly using the common approach. Many engineers consider the fluid-induced effects ofblade passing frequency (BPF) as the direct excitation source of vibration signals. However, they may fail to detect BPF in

276 S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289

the Fourier analysis of vibration signals. In fact, pump vibration models based on fluid-induced effects comprise a modula-tion mechanism that is generally ignored by researchers.

A number of researchers have recently used CFD methods to design low-vibration pumps with low-pressure pulsations.Spence et al. [1] studied the geometrical variations of the pressure pulsations of a centrifugal pump with a CFD parametricmethod. They concluded that the cutwater gap and vane arrangement would exert the greatest influence across variousmonitored locations and the flow range; in addition, they offered some recommendations that would enable pump designersto reduce vibrations. Pressure pulsation in pumps is a significant factor in evaluating stability under operating conditions. Toinvestigate pressure pulsation, González et al. [2,3] conducted a 3D unsteady calculation with a sliding mesh technique andinvestigated the dynamic characteristics inside a low specific-speed pump; they successfully correlated the load and globalparameters for the flow in the pump and verified the results by experimental data. Barrio et al. [4] studied radial load andunsteady pressure distribution around the impeller and found that unsteady components could account for a large propor-tion of the average magnitude during operation under off-design conditions. Yan et al. [5,6] used the standard deviation ofthe pressure pulsation around the volute to visualize pressure pulsation intensity with contour plots and thereby provide aclear understanding of the distribution of pressure pulsation intensity in the volute.

For signal processing, cyclostationarity is a powerful tool to deal with random signals that are mixed by a hidden periodicphenomenon; these signals are categorized as cyclostationary signals [7]. Classical methods that model signals as stationarycannot effectively deal with cyclostationary signals because they neglect periodic or almost-periodic time variability in sta-tistical moments. Gardner et al. [8,9] developed the theory of cyclostationary processes, which has been widely used in com-munication fields such as radar, sonar, and telemetry. Recently, researchers [10–14] have started to apply cyclostationarityto mechanics for fault detection, especially those for rotating machines. Antoni et al. [15,16] established a cyclostationarymodel for the vibration signals of rotating machines, proposed a general methodology for analyzing other types of rotatingmachine signals, and then introduced spectral correlation density to successfully detect BPF in pump vibration signals.Botero et al. [17] recently adopted cyclostationary analysis to study the rotating stall in a pump turbine with tuft visualiza-tion and developed a non-intrusive method for detecting rotating stall instability and number of stall cells. Napolitano[18,19] reviewed the application of cyclostationarity in recent years and identified new trends and limits to provide a com-prehensive understanding of cyclostationarity.

The present study investigates the flow-induced effects in centrifugal pumps under design and off-design conditions onthe basis of CFD simulations. The results are compared with vibration signals processed with second-order cyclostationarytools. The widely used CFD package Fluent and the mesh generator ICEM are used for all the 3D numerical computations toobtain the pressure pulsation and radial fluid force fluctuation. The mean spectral coherence is used to evaluate the intensityof the modulating signals and prove their effectiveness. The main objective of this study is to elaborate the modulationmechanism of the flow-induced effects on vibration signals and offer guidance for designing low-vibration pumps.

2. Numerical model and method

2.1. Model descriptions

The centrifugal pump under study is a canned motor pump. The pump impeller is mounted at the overhung end of ashared pump-and-motor shaft. The main parameters of the centrifugal pump are presented in Table 1. Fig. 1(a) shows theconfiguration of the pump on the test rig that is used to test the pump characteristics and vibration performance. Two accel-eration transducers are set on the pump and motor feet for vibration data acquisition, and corresponding positions areshown in Fig. 1(a). For the CFD simulations, the calculation model of the centrifugal pump consists of six main domains,which are displayed in Fig. 1(b). Aside from the impeller and volute, the front/rear sidewall gap, balance holes, and inletguide vanes are considered in the CFD simulations. This consideration provides an accurate result, given that the back flowin the balance holes can deteriorate the flow fields in the impeller domain. Furthermore, the increment of the shaft power bysidewall friction cannot be ignored in the efficiency calculation. The impeller consists of seven main blades and seven splitterblades. Such a combination can produce two types of BPFs, which correspond to the number of total blades and main blades.

Table 1Main parameters of centrifugal pump.

Parameters Values

Qd (design volume flow rate, m3/h) 550Hd (design pump head, m) 25n (rotation speed, rpm) 1470D1 (impeller suction diameter, mm) 117.5D2 (impeller outlet diameter, mm) 310b2 (impeller outlet width, mm) 55Z1 (number of main blades) 7Z2 (number of splitter blades) 7b3 (volute inlet width, mm) 55D3 (volute tongue diameter, mm) 454

Fig. 1. (a) Test rig and (b) diagram of centrifugal pump.

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For this specific pump with a rotation speed equal to 1470 rpm, BPFs can be predicted as 171.5 and 343 Hz. However, theamplitudes of radial fluid forces and the pressure pulsation at the BPFs cannot be predicted accurately without transientCFD simulations.

2.2. CFD methods and boundary conditions

Steady simulations using a multiple reference frame approach are first performed to determine the pump performancecharacteristics under full operating conditions. The boundary conditions of the velocity inlet and pressure outlet are setfor the steady and transient CFD simulations. No-slip boundary conditions are adopted for all walls. Velocity–pressure cou-pling is addressed with the well-known SIMPLEC strategy. Two-equation models are widely used in industry applicationsdue to their relatively low calculation cost and reasonable accuracy. Hence, the Realizable k-e model is adopted in thisresearch. The second-order upwind scheme is applied to discretize the convection terms in the momentum, turbulent kineticenergy, and turbulent dissipation rate equations. The sizes of the front and rear sidewall gaps are extremely small, and eachgap is divided into five layers of meshes. Thus, the y+ value is around 3.5, whereas the y+ values in the other regions are morethan 30. Thus, the enhanced wall treatment that implements hybrid wall functions for varying y+ values in the CFD model isapplied to simulate the flow in the boundary layer for the entire computational domain.

This study focuses on the low-frequency range caused mainly by rotor–stator interactions; it does not cover the high-frequency range caused by turbulence. The unsteady Reynolds-averaged Navier–Stoke (URANS) method is adopted in suchapplication [20-22] to achieve efficient simulations and obtain the radial fluid force and pressure pulsation under design con-ditions (550 m3/h) and off-design conditions (220 m3/h). The second-order implicit spectral scheme is adopted for transientformulation. The sliding mesh technique, which considers the effects of relative motion between rotating and stationarydomains, is also applied. The time step should be lower than the value of the minimum cell size over the maximum velocityin the impeller domain. Thus, it is set to 1.13 � 10�4 s, which is equal to the time interval in which the impeller rotates forone degree. The total mesh models contain 4.5 million cells, and the grid quality is rigorously checked. The number ofmeshes is already numerous that the results barely change as this number increases. Local mesh refinement is applied nearthe blades. The hexahedral meshes of the impeller and the sidewall gap are shown in Fig. 2.

3. Cyclostationary analysis methods

3.1. Second-order cyclostationary tool

For rotating machines, second-order cyclostationarity is widely used by researchers [23–25]. Therefore, it is mainly intro-duced in this section. Before processing the second-order cyclostationarity, the first-order cyclostationarity processed viamean value calculations should be subtracted from the signals if it exists [6]. The second-order cyclostationarity indicatesa periodically time-varying autocorrelation function, which is defined in Eq. (1).

Rxðt; sÞ ¼ limN!11

2N þ 1

XNn¼�N

x t þ nT0 þ s2

� �x� t þ nT0 � s

2

� �¼ Rxðt þ T0; sÞ ð1Þ

Fig. 2. Mesh model of flow domains.

278 S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289

where T0 and s denote the period and time lag, respectively. s should be less than T0. The cyclic autocorrelation function isdefined as the extracted coefficients by expanding the autocorrelation function into a Fourier series, as shown in Eq. (2).

Rax ðsÞ ¼ limT!11T

ZTx t þ s

2

� �x� t � s

2

� �e�j2patdt ð2Þ

where a is the cyclic frequency of the signal and its inverse is the cycle, and j is the imaginary unit. The spectral correlationdensity function is obtained by performing a Fourier transform of the cyclic autocorrelation function.

SCax ðf Þ ¼

Z 1

�1Rax ðsÞe�j2pfsds ð3Þ

This definition of spectral correlation density has been introduced by many researchers [2,19]. Nevertheless, another def-inition developed by Antoni [7] is briefly introduced to improve the understanding of its physical meanings. These two def-initions are essentially identical.

Eq. (4) is introduced as a pure second-order cyclostationary signal, which considers a random stationary carrier mðtÞ .

xðtÞ ¼ cosð2pa0tÞ � mðtÞ ð4Þ

In detecting its hidden periodicity T0 ¼ 1=a0, a spectral correlation that measures the interaction between two spectralcomponents at frequencies f 1 and f 2 is introduced as

corrxðf 1; f 2Þ ¼ limDf!01Df

limT!11T

ZTxDf ðt; f 1Þx�Df ðt; f 2Þe�j2pðf 1�f 2Þtdt ð5Þ

where xDf ðt; f Þ is the filtered version of signal xðtÞ through a frequency band of width Df centered on frequency f ; the order ofthe two limits cannot be interchanged. By changing the variables f ¼ ðf 1 þ f 2Þ=2 and a ¼ f 1 � f 2, Eq. (5) finally turns to thesecond definition of spectral correlation density.

SCax ðf Þ ¼ limDf!0

1Df

limT!11T

ZTxDf t; f þ a

2

� �x�Df t; f � a

2

� �e�j2patdt ð6Þ

Eq. (6) is notably intuitive, and it represents the density of the correlation of two spectral components spaced apart by a.However, some small vibration signatures are masked in the spectral correlation density due to the spectral scaling effect.The degree of cyclostationarity, which is a relative measure for spectral correlation, generally applies to the detection ofvibration signatures. Thus, spectral coherence is defined as follows [7,14], where Pxðf Þ is the power spectral density andSC0

x ðf Þ � Pxðf Þ.

cax ðf Þ ¼corrxðf þ a

2 ; f � a2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pxðf þ a2ÞPxðf � a

2Þq ¼ SCa

x ðf ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSC0

x ðf þ a2ÞSC0

xðf � a2Þ

q : ð7Þ

Considering the effect of impulsive response for yðtÞ ¼ hðtÞ � xðtÞ the spectral correlation density and spectral coherence ofyðtÞ can be derived as follows:

S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289 279

SCayðf Þ ¼ H f þ a

2

� �H� f � a

2

� �SCa

x ðf Þ: ð8Þ

cayðf Þ ¼Hðf þ a

2 ÞH�ðf � a2 Þ

jHðf þ a2 ÞjjHðf � a

2 Þjcax ðf Þ ð9Þ

3.2. Pump vibration model

The vibration model for pumps can be derived from the vibration model for rolling element bearings with inner-racefaults, and it is widely used to provide an intuitive understanding of bearing fault signals [24,26]. A simplified version thatdoes not consider random slips is as follows:

xðtÞ ¼ hðtÞ �XNn¼1

dðt � nTÞ½1þ qðtÞ�( )

þ nðtÞ ð10Þ

where hðtÞ denotes the impulse response to a single impact as measured by the sensor and T is the inter-arrival time betweentwo consecutive impacts on the fault. qðtÞ ¼ qðt þ PÞ is the periodic modulation of period P due to the load distributionimparted by gravity, given that the inner-race defect moves in and out of the bearing load zone. The vibration is strongestwhen the defect is in the load zone, and weakest when the defect is outside the load zone. Therefore, the reciprocal of periodP is the shaft rotating frequency for a simple rolling-element bearing. nðtÞ accounts for the additive background noise thatincludes all other vibration sources.

When the impeller is implemented in a centrifugal pump, steady and unsteady radial fluid forces are imposed on thebearing as the pump operates. The steady radial fluid force could impart a much larger load than the original steady loadimparted by gravity. By contrast, the unsteady radial fluid force can generate oscillatory bearing load at BPF. Thus, even with-out any incipient fault in the rolling element bearing, the random vibration is modulated by the steady and unsteady loadzones. The following equation represents the vibration model for a centrifugal pump without any bearing faults:

xðtÞ ¼ hðtÞ � fvðtÞ½1þ qðtÞ þ qf ðtÞ�g þ nðtÞ ð11Þ

where mðtÞ represents the random vibration and qf ðtÞ denotes the periodic modulation due to the unsteady bearing load. Forsimplicity, the next section mainly discusses the cyclostationary analyses of the amplitude-modulated (AM) models withoutimpulse response. Nevertheless, the effect of impulse response can still be considered with Eqs. (8) and (9) thereafter.

3.3. Cyclostationary analyses of AM models

An AM signal with infinite modulating components is studied with spectral correlation density. Its model is shown asfollows:

xðtÞ ¼ 1þXNi¼1

Ai cosð2paitÞ" #

� mðtÞ ð12Þ

where Ai is the amplitude of the modulating signal and ai is the modulating frequencies in the condition of aiþ1 > ai > 0. TheAM signal is filtered by a narrow frequency band ½f � Df=2; f þ Df=2�, and the results are shown as follows:

xDf ðt; f 1Þ ¼ vDf ðt; f 1Þ þXNi¼1

Ai

2½vDf ðt; f 1 � aiÞ þ vDf ðt; f 1 þ aiÞ� ð13Þ

xDf ðt; f 2Þ ¼ vDf ðt; f 2Þ þXNk¼1

Ak

2½vDf ðt; f 2 � akÞ þ vDf ðt; f 2 þ akÞ� ð14Þ

Substituting Eqs. (13) and (14) into Eq. (5) yields the following:

corrxðf 1; f 2Þ ¼ corrvðf 1; f 2Þ þXNi¼1

Ai

2½corrvðf 1 � ai; f 2Þ þ corrvðf 1 þ ai; f 2Þ�

þXNk¼1

Ak

2½corrvðf 1; f 2 � akÞ þ corrvðf 1; f 2 þ akÞ� þ

XNi¼1

XNk¼1

AiAk

4½corrvðf 1 � ai; f 2 � akÞ

þ corrvðf 1 � ai; f 2 þ akÞ þ corrvðf 1 þ ai; f 2 � akÞ þ corrvðf 1 þ ai; f 2 þ akÞ� ð15Þ

In this equation, mðtÞ is a random stationary carrier, which means that any spectral correlation in mðtÞ equates to zero whenthe two frequencies involved are not identical. In other words, non-zero values exist only when the frequencies are identicalfor at least one item in Eq. (15). Furthermore, the frequencies corresponding to the non-zero values can be predicted, and the

280 S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289

values can be obtained. Changing the variables a ¼ f 1 � f 2 and f ¼ ðf 1 þ f 2Þ=2 yields the following equation, which only con-siders the condition in which a is greater than or equal to zero:

SCax ðf Þ ¼

1þXNi

A2i2

!Pvðf Þ; if a ¼ 0 ðIÞ

AiPvðf Þ; if a ¼ ai ðIIÞAiAk4 Pvðf Þ; if a ¼ 2ai ðIIIÞ

AiAk2 Pvðf Þ; if a ¼ ai ak; i > k ðIVÞ

8>>>>>>><>>>>>>>:

ð16Þ

where Pmðf Þ is the power spectral density of carrier mðtÞ. The results can be categorized into four main items: (I) boiling downto the power spectral density, (II) modulating frequencies, (III) harmonics, and (IV) interference term. Thus, when an AM sig-nal with multiple modulating frequencies is processed by the spectral correlation density, the modulating frequencies can bedetected, and the interference frequencies and harmonics from items (III–IV) are obtained. The corresponding values of thespectral correlation density can be calculated with Eq. (16), and they are indeed affected by the amplitudes of the modulatingsignals. The corresponding results of spectral coherence are shown in the following equation:

Fig. 3. (a) Spectral coherence and (b) mean spectral coherence of first synthetic signal with a1 ¼ 7 Hz and a2 ¼ 14 Hz.

S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289 281

cax ðf Þ ¼

1; if a ¼ 0 ðIÞ

Ai= 1þXNi

A2i2

!; if a ¼ ai ðIIÞ

AiAk4 = 1þ

XNi

A2i2

!; if a ¼ 2ai ðIIIÞ

AiAk2 = 1þ

XNi

A2i2

!; if a ¼ ai ak; i > k ðIVÞ

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð17Þ

The spectral coherence shows a degree of modulation intensity, and the power spectral density of the carrier is ignored. Thus,in the next sections, only the spectral coherence is shown.

3.4. Synthetic signal simulation

Before cyclostationarity analyses are performed on the pump vibration signals, two synthetic signals are adopted to verifythe derived equations. For simplicity, both synthetic signals are designed to comprise two modulating components, whichhave the same formula as Eq. (18).

xðtÞ ¼ ½1þ cosð2pa1tÞ þ cosð2pa2tÞ� � mðtÞ: ð18Þ

Fig. 4. (a) Spectral coherence and (b) mean spectral coherence of second synthetic signal with a1 ¼ 10 Hz and a2 ¼ 14 Hz.

Fig. 5. Mean spectral coherence of a synthetic signal with varied SNR.

Fig. 6. Comparison of steady CFD simulation results with experimental data: (a) pump head and (b) pump efficiency.

282 S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289

For the first synthetic signal, the modulating frequencies are set as a1 ¼ 7 Hz and a2 ¼ 14 Hz, where a2 is a multiple of a1.This case is similar to that of most rotating machinery vibrations. For the second synthetic signal, the modulating frequenciesare set as a1 ¼ 10 Hz and a2 ¼ 14 Hz, where the two modulating components are independent and considered as two irrel-evant modulating components. For the first synthetic signal, interference items and harmonics mix with each other. There-fore, only four cyclic frequency lines that display non-zero values exist in the spectral coherence image. For the secondsynthetic signal, the harmonics and interference items do not mix, and a total of six spectral lines are predicted to appear.

The spectral coherences of the two synthetic signals are computed by a fast algorithm developed in Ref. [27]. The sam-pling frequency is set to 1000 Hz for both synthetic signals. As shown in Figs. 3(a) and 4(a), the spectral lines can be clearlydistinguished in the gray-level images, and the numbers of the spectral lines representing non-zero values are consistentwith the theoretical prediction in Eq. (17). As white Gaussian noise is used for the random stationary signal mðtÞ, spectrallines are uniformly distributed along the frequency domain. Figs. 3(b) and 4(b) display the average of the spectral coherencesalong the f frequency axis. This value is defined as the mean spectral coherence, which can be interpreted as the intensity ofthe modulating components. Note that the results of the simulation algorithm slightly underestimate the mean spectralcoherence with respect to the theoretical values.

The signal-noise-ratio (SNR) is a serious concern in industrial applications. Therefore, the second-order cyclostationarytool is tested with a synthetic signal for different SNR values. This synthetic signal is also an AM signal, but it has one mod-ulating component, and the modulating frequency is set to 7.5 Hz. According to the theoretical prediction in Eq. (17), thissynthetic signal produces two significant amplitudes at the modulating frequency and its harmonics in the mean spectralcoherence image. As observed in Fig. 5, the mean spectral coherence declines as the value of SNR decreases. But its valueis still distinct even when the SNR is as small as �9 dB.

Fig. 7. Pressure pulsation in volute under design and off-design conditions.

Fig. 8. Radial fluid force fluctuation under design conditions.

S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289 283

Eq. (9) and Fig. 5 indicate that both the frequency response function and the background noise can affect the mean spec-tral coherence. However, this fact does not impede the application to the evaluation of the modulation intensity of differentimpellers with the same blade numbers. For example, when the optimized impeller and the original impeller are tested fortheir vibration performances, the mean spectral coherence can still be effective in evaluating the relative modulation inten-sities of different impellers with the same blade numbers. As the modulating frequencies in the centrifugal pump can be con-sidered as a narrow band compared with the large band carrier signal, then the effects of the frequency response function onthe mean spectral coherence are reasonably assumed as invariant.

4. Results and discussions

4.1. CFD simulation results

The steady CFD simulations are conducted under full operating conditions to determine the pump performance. Theresults are then compared with the experimental data. Flow rate is measured by an electromagnetic flowmeter, and the suc-tion and discharge pressure are measured by two pressure sensors. The test rig and experiment methods are stated in detail

Fig. 9. Radial fluid force fluctuation under off-design conditions.

Fig. 10. Static pressure distributions at varied time frames under (a) design conditions and (b) off-design conditions.

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in our previous work [5,6]. The comparison of the CFD-calculated performance curve with the experimental data is displayedin Fig. 6(a). Under design condition, the pump head calculated by the CFD is larger than the design pump head in Table 1, thisis because design margin has been considered. The comparison results show a good agreement. However, the CFD resultsoverestimate the pump head in high flow rate conditions and underestimate it in low flow rate conditions. Fig. 6(b) showsthe CFD-calculated pump efficiency and the experimental data, and the experimental pump efficiency indicates the hydrau-lic efficiency which has deducted the motor loss already. The comparison also shows good consistency. Under low flow ratecondition, both pump head and shaft power are underestimated by the CFD. However, the underestimated shaft power isstill larger than the pump head, and this results in larger efficiency calculated by the CFD compared with experimental data.As described in Section 2.1, the front/rear sidewall gap, balance holes, and inlet guide vanes are modeled in the 3D

Fig. 11. (a) Spectral coherence and (b) mean spectral coherence of vibration signal from pump feet under design conditions.

S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289 285

computation model. Thus, the volumetric loss and disk friction loss are both considered. These efforts lead to a good agree-ment between the CFD results and the experimental data.

Unsteady CFD computations are performed to obtain the pressure pulsation in the volute and the radial fluid forcefluctuation on the impeller and volute. The steady CFD results are used to initialize the unsteady CFD simulations. The cal-culating process of unsteady CFD computations is designed to involve 20 rotations of the impeller. However, the calculationresults of the first three rotations are not used for further analysis because of the convergence problem. Fig. 7 shows the pres-sure pulsation in the volute under design conditions (550 m3/h) and off-design conditions (220 m3/h). The low flow rate of220 m3/h is selected as an example, in which condition low-frequency modulating components will take place. This reveals adifferent feature from that in design flow rate condition. Significant amplitudes occur at 171.5 and 343 Hz, which correspondto the BPFs caused by the number of main blades and total blades, respectively. The harmonics of the BPFs are not observed.Thus, the frequency range is set to 500 Hz for improved observation. For the design conditions, the pressure pulsations at171.5 and 343 Hz are dominant, and the amplitudes are quite close to each other. In the low-frequency range, no significantamplitudes occur; hence, the pressure pulsation under the design conditions is mainly affected by the BPFs. For the off-design conditions, the amplitude at 343 Hz is approximately two times greater than that at 171.5 Hz, and both values aregreater than those under the design conditions. In addition, the pressure pulsations in the low-frequency range below150 Hz are dominant due to the intense non-uniformity of the flow field under the off-design conditions.

The frequency spectrum of the radial fluid force fluctuation on the impeller and volute under the design conditions isshown in Fig. 8. The x and y radial directions are orthogonal, and the y direction is parallel to the outlet tube of the pump.The amplitude at 171.5 Hz is dominant, whereas the amplitude at 343 Hz declines to a small value for all the CFD-calculatedforce signals. This characteristic is considerably different from that of the pressure pulsation in the volute. Fig. 9 shows thefrequency spectrum of the radial fluid force fluctuation on the impeller and volute under the off-design conditions. The

Fig. 12. (a) Spectral coherence and (b) mean spectral coherence of vibration signal from motor feet under design conditions.

286 S. Li et al. /Mechanical Systems and Signal Processing 114 (2019) 275–289

spectrum exhibits characteristics that differ from those under the design conditions. The radial fluid force fluctuations shar-ply increase in the whole frequency range in comparison with those under the design conditions. Furthermore, the radialfluid force fluctuations in the low-frequency range become dominant, a property that is consistent with the pressure pulsa-tion in the volute. The amplitudes at 171.5 Hz can still be recognized while the amplitudes at 343 Hz can be ignored.

The difference of the pressure pulsation and radial fluid force fluctuation between the design conditions and off-designconditions can be interpreted by the static pressure contour plots extracted from the CFD data. Fig. 10 shows the pressuredistribution under the design and off-design conditions at three different time frames in the pump operation. The time inter-val is equal to the time of two rotations of the impeller. The pressure distributions under the design conditions are moreuniform than those under the off-design conditions. For example, under the design conditions, the pressure distributionsin the outlet of the impeller are circumferentially uniform. As the single blade passes through the pressure monitoring point,pressure pulsation occurs. Thus, the pressure pulsations in the impeller and the volute are mainly affected by the BPFs. Underthe off-design conditions, the circumferential distribution is non-uniform, and such a pressure distribution leads to a direc-tive pattern, which generates a large pressure pulsation in the low-frequency range. This low-frequency pulsation results indifferent static pressure distributions at different time frames, as observed in Fig. 10(b) in contrast to Fig. 10(a). The maxi-mum and minimum values of pressure contour under both conditions are set the same to provide intuitive comparisons.

4.2. Experimental results

The experiments on vibration performance are conducted under the design and off-design conditions corresponding tothe unsteady CFD simulations. Two acceleration transducers are set on the pump and motor feet. The length of the vibration

Fig. 13. (a) Spectral coherence and (b) mean spectral coherence of vibration signal from pump feet under off-design conditions.

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signals is recorded as 1.63 s, which is equal to the time of 40 rotations of the impeller. Fig. 11 displays the spectral coherenceand mean spectral coherence of the vibration signals from the pump foot under the design conditions. Fig. 11(a) shows theshaft rotation frequency and its harmonics, which produce spectral lines extending up to 20 kHz in the f frequency. Asexplained in Section 3.2, the steady bearing load is increased by the steady radial fluid force on the impeller. Thus, the shaftrotation frequency modulation is significant.

The spectral lines produced by BPFs can be observed. Their amplitudes can be mixed up with the harmonics of the shaftrotation frequency. Fig. 11(b) shows the mean spectral coherence with the same characteristics as those in Fig. 11(a). Thesame second-order cyclostationary signature can be found in Fig. 12. This result indicates that the vibrations of the motorroot are also modulated by the shaft rotation frequency, its harmonics, and BPFs. For cyclic frequencies at 100, 200, and300 Hz, the spectral coherences show non-zero values in the f frequency below 2 kHz resulting from the electromagneticinterference. However, the effect of the electromagnetic interference is great because the acceleration transducer is closeto the alternating current coil for the vibration signals from the motor feet.

Fig. 13 presents the spectral coherence and mean spectral coherence of the vibration signals from the pump foot underthe off-design conditions. Under these conditions, the mean spectral coherence is more effective than the spectral coherencein detecting the modulating components. Detecting the dispersed spectral lines in Fig. 13(a) is also difficult. As shown inFig. 13(b), the amplitudes in the cyclic frequencies below 150 Hz are dominant, which indicates that the vibration signalsare mainly modulated by the low frequency modulating components. This result is consistent with the transient CFD sim-ulations under the off-design conditions. The amplitude at 171.5 Hz can be detected as well, although the radial fluid forcefluctuation is considerably smaller than that of the low-frequency components shown in Fig. 9. The cyclostationary analysis

Fig. 14. (a) Spectral coherence and (b) mean spectral coherence of vibration signal from motor feet under off-design conditions.

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of the vibration signals from the motor feet under the off-design conditions is shown in Fig. 14. Here, the effect of electro-magnetic interference increases.

5. Conclusions

This study mainly investigates the quantitative relationship between the effects of pressure pulsation and radial fluidforce fluctuation through the vibration signal analysis in a centrifugal pump. The results from cyclostationary analysisand CFD simulation are expected to aid the design of low-vibration pumps. To achieve this object, a modulation model ofpump vibration signals is established by using an AM model with multiple modulating components. In addition, this mod-ulation model incorporates the flow-induced effect by modeling the similarity of rolling element bearings with inner-racefaults. Then cyclostationary analysis is employed as a powerful tool to extract the modulation components from complicatedvibration signals which are non-stationary and broadband during pump running. Moreover, this paper present a mean spec-tral coherence to evaluate the intensity of the modulating components related to flow-induced effect. In addition, meanspectral coherence detects the electromagnetic interference, and reveals that electromagnetic influence is significant forthe acceleration transducer on the motor feet.

The cyclostationary signatures are consistent with those of the unsteady CFD simulation. To obtain the pressure pulsationand radial fluid force fluctuation, the CFD are conducted under design and off-design conditions. The comparison of cyclo-stationary analyses and CFD simulations confirms the direct connection between these two methods. The related resultsshow that the fluid-induced effects on pump vibration signals can be considered as modulating components. Indeed, CFD

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simulations and cyclostationarity are complementary that the combination of these methods is promising to design the low-vibration pump and characterize the vibration features.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (NFSC) No. 61701440, by the National KeyResearch and Develop Program of China 2016YFF0203304, and by the Fundamental Research Funds for the Central Univer-sities 2017QNA4015. These supports by the foundation are gratefully acknowledged by the authors.

References

[1] R. Spence, J. Amaral-Teixeira, A CFD parametric study of geometrical variations on the pressure pulsations and performance characteristics of acentrifugal pump, Comput. Fluids 38 (6) (2009) 1243–1257.

[2] J. González, C. Santolaria, Unsteady flow structure and global variables in a centrifugal pump, J. Fluids Eng. 128 (2006) 937.[3] J. Gonzalez, J. Fernández, E. Blanco, C. Santolaria, Numerical simulation of the dynamic effects due to impeller-volute interaction in a centrifugal pump,

J. Fluids Eng. 124 (2) (2002) 348–355.[4] R. Barrio, J. Fernández, E. Blanco, J. Parrondo, Estimation of radial load in centrifugal pumps using computational fluid dynamics, Euro. J. Mech. B Fluids

30 (3) (2011) 316–324.[5] P. Yan, N. Chu, D. Wu, L. Cao, S. Yang, P. Wu, Computational fluid dynamics-based pump redesign to improve efficiency and decrease unsteady radial

forces, J. Fluids Eng. 139 (1) (2017) 011101.[6] P. Yan, S. Li, S. Yang, P. Wu, D. Wu, Effect of stacking conditions on performance of a centrifugal pump, J. Mech. Spectral Coherenceience Technol. 31 (2)

(2017) 689–696.[7] J. Antoni, Cyclostationarity by examples, Mech. Syst. Sig. Process. 23 (4) (2009) 987–1036.[8] W.A. Gardner, Introduction to Random Processes: With Applications to Signals and Systems, Macmillan Publishing Company, 1986.[9] W. Gardner, Measurement of spectral correlation, IEEE Trans. Acoust. Speech Signal Process. 34 (5) (1986) 1111–1123.[10] C. Cheong, P. Joseph, Cyclostationary spectral analysis for the measurement and prediction of wind turbine swishing noise, J. Sound Vibrat. 333 (14)

(2014) 3153–3176.[11] R. Boustany, J. Antoni, A subspace method for the blind extraction of a cyclostationary source: application to rolling element bearing diagnostics, Mech.

Syst. Sig. Process. 19 (6) (2005) 1245–1259.[12] Q. Leclere, L. Pruvost, E. Parizet, Angular and temporal determinism of rotating machine signals: the diesel engine case, Mech. Syst. Sig. Process. 24 (7)

(2010) 2012–2020.[13] M. Poirier, M. Gagnon, A. Tahan, A. Coutu, J. Chamberland-lauzon, Extrapolation of dynamic load behaviour on hydroelectric turbine blades with

cyclostationary modelling, Mech. Syst. Sig. Process. 82 (2017) 193–205.[14] I. Antoniadis, G. Glossiotis, Cyclostationary analysis of rolling-element bearing vibration signals, J. Sound Vibrat. 248 (5) (2001) 829–845.[15] J. Antoni, F. Bonnardot, A. Raad, M. El Badaoui, Cyclostationary modelling of rotating machine vibration signals, Mech. Syst. Sig. Proc. 18 (6) (2004)

1285–1314.[16] J. Antoni, Cyclic spectral analysis in practice, Mech. Syst. Sig. Process. 21 (2) (2007) 597–630.[17] F. Botero, V. Hasmatuchi, S. Roth, M. Farhat, Non-intrusive detection of rotating stall in pump-turbines, Mech. Syst. Sig. Process. 48 (1) (2014) 162–173.[18] A. Napolitano, Cyclostationarity: Limits and generalizations, Signal Process. 120 (2016) 323–347.[19] A. Napolitano, Cyclostationarity: new trends and applications, Signal Process. 120 (2016) 385–408.[20] R. Barrio, J. Parrondo, E. Blanco, Numerical analysis of the unsteady flow in the near-tongue region in a volute-type centrifugal pump for different

operating points, Comput. Fluids 39 (5) (2010) 859–870.[21] R. Spence, J. Amaral-Teixeira, Investigation into pressure pulsations in a centrifugal pump using numerical methods supported by industrial tests,

Comput. Fluids 37 (6) (2008) 690–704.[22] J. GonzáLez, J. FernáNdez, E. Blanco, C. Santolaria, Numerical simulation of the dynamic effects due to impeller-volute interaction in a centrifugal

pump, J. Fluids Eng. 124 (2) (2002) 348–355.[23] J. Antoni, D. Hanson, Detection of surface ships from interception of cyclostationary signature with the cyclic modulation coherence, IEEE J. Oceanic

Eng. 37 (3) (2012) 478–493.[24] J. Antoni, Cyclic spectral analysis of rolling-element bearing signals: facts and fictions, J. Sound Vibrat. 304 (3) (2007) 497–529.[25] G. Dalpiaz, A. Rivola, R. Rubini, Effectiveness and sensitivity of vibration processing techniques for local fault detection in gears, Mech. Syst. Sig. Proc.

14 (3) (2000) 387–412.[26] R.B. Randall, J. Antoni, S. Chobsaard, The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other

cyclostationary machine signals, Mech. Syst. Sig. Process. 15 (5) (2001) 945–962.[27] J. Antoni, G. Xin, N. Hamzaoui, Fast computation of the spectral correlation, Mech. Syst. Sig. Process. 92 (2017) 248–277.