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Petroleum Pumps’ Current and Vibration Signatures Analysis Using Wavelet Coherence Technique

Petroleum Pumps’ Current and Vibration Signatures Analysis Using Wavelet Coherence Technique Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 659650, 6 pages http://dx.doi.org/10.1155/2013/659650 Research Article Petroleum Pumps’ Current and Vibration Signatures Analysis Using Wavelet Coherence Technique Rmdan Shnibha and Alhussein Albarbar Advanced Industrial Diagnostic Centre, Digital Signal Processing Research Group, School of Engineering, Manchester Metropolitan University, Manchester M1 5GD, UK Correspondence should be addressed to Alhussein Albarbar; a.albarbar@mmu.ac.uk Received 28 February 2013; Revised 7 May 2013; Accepted 22 May 2013 Academic Editor: K. M. Liew Copyright © 2013 R. Shnibha and A. Albarbar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vibration analysis is widely used for rotating machinery diagnostics; however measuring vibration of operational oil well pumps is not possible. The pump’s driver’s current signatures may provide condition-related information without the need for an access to the pump itself. This paper investigates the degree of relationship between the pump’s driver’s current signatures and its induced vibration. This relationship between the driver’s current signatures (DCS) and its vibration signatures (DVS) is studied by calculating magnitude-squared coherence and phase coherence parameters at a certain frequency band using continuous wavelet transform (CWT). The CWT coherence-based technique allows better analysis of temporal evolution of the frequency content of dynamic signalsandareasinthetime-frequencyplanewherethetwosignalsexhibitcommonpowerorconsistentphasebehaviourindicating a relationship between the signals. This novel approach is validated by experimental data acquired from 3 kW petroleum pump’s driver. Both vibration and current signatures were acquired under different speed and load conditions. The outcomes of this research suggest the use of DCS analysis as reliable and inexpensive condition monitoring tool, which could be implemented for oil pumps, real-time monitoring associated with condition-based maintenance (CBM) program. 1. Introduction Vibration monitoring is particularly suited to pumps due to the number of integrated rotating parts, which may show Pumps and their associated systems are essential in oil and gas additional movement when faults develop [7]. A more recent facilities for the efficient transportation of uids. fl Common development in pump condition monitoring is the applica- pumps found in these facilities include centrifugal, recip- tion of ultrasonic sensors [8]; introducing a new ultrasonic rocating, diaphragm, and rotary pumps [1]. The condition measurement based on acoustic emission analyses for high- monitoring of pumps and their associated systems is an estab- pressure process pumps. lished application of CBM and is an existing area of research The most obvious technique for obtaining a vibration [2]. Rohlfing [ 3] provides three examples in the oil and gas signal from pump driver’s “induction motor” is by direct industry wherepump’sCBM hasbeeneeff ctivelyimple- measurement using vibration transducers (usually acceler- mented. Azadeh et al. in [4]havedeveloped adiagnostic ometers) mounted on the driver. This requires a high-per- mechanism for pump failures in which pump operating formance vibration transducer capable of withstanding harsh problems fall into two categories: (1) hydraulic problems that environmental conditions and which can cost several hun- suggest the pump may fail to deliver liquid, deliver insuffi- dred dollars. When a large number of machines are con- cient capacity, develop insufficient pressure, or lose its prime cerned andmorethanone transducer is required perma- at starting and (2) mechanical problems that are characterised chine, thetotal cost canbehigh. Amajor disadvantage of by the consumption of excessive power or development of vibration monitoring is that it requires access to the machine. mechanical difficulties at the seal chambers or bearings; in For accurate measurements, sensors should be mounted either case vibration, noise, or breakage may occur. Fatigue is rigidly on the machine, which requires expertise and trained a common cause of pump failure [5, 6]. personnel. 2 Advances in Acoustics and Vibration An increasing number of pumps are installed with elec- phase of the supply. Figure 2 shows the RMS values of current trical motors as their prime driver. This development has and vibration for different motor loads; it can be seen that the introduced new possibilities for condition monitoring by the stator current and vibration signals are inversely correlated. use of driver’s electric signals such as current and voltage. A common approach for extracting information concern- In response an indirect method, called sensorless detec- ing frequency features of a periodic signal is to transform the tion and diagnosis intended for mechanical equipment driv- signal to the frequency domain using the discrete Fourier en by AC induction motors, has been developed over the past transform. 20 years and is growing rapidly [9]. Sensorless detection and In Figure 3, the same signals as those shown in Figure 1 diagnosis technology monitors the state of devices and are presented in the frequency domain; it is assumed that the motors using the motor’s stator current rather than external original signals are stationary. sensors to detect vibration, noise, and so forth. Besides the The coherence is a function of the power spectral density original power frequency, the stator current contains a large (𝑃 and𝑃 )of𝑥 and𝑦 and the cross power spectral density 𝑥𝑥 amount of information relative to mechanical faults. (𝑃 ) of𝑥 and𝑦 is given by eTh relationship between the magnitudes of stator current 󵄨 󵄨 2 󵄨 󵄨 harmonics, the magnitudes of the vibration harmonics, and 󵄨 󵄨 𝑃 (𝑓) 󵄨 󵄨 󵄨 󵄨 (1) specific machine faults types has been closely studied [ 9, 10]. 𝐶 (𝑓)= . 𝑃 (𝑓)𝑃 (𝑓) 𝑥𝑥 In [9] the relationship between the vibration and current har- monic magnitudes for a source of known vibration frequency Coherence is a function of frequency with𝐶 (𝑓) ranging was investigated to determine the feasibility of setting a limit between 0 and 1 and indicates how well signal𝑥 corresponds or standard on the current harmonics due to these vibrations. to signal𝑦 at each frequency. The degree of synchronization It was concluded that for a given vibration frequency, the in stator current signal and vibration signal is commonly harmonic’s RMS vibration level and RMS current level are characterized by coherence phase and magnitude-squared monotonically related. In [10] a method for sensorless on-line coherence (DSC). vibration monitoring of an induction motor was proposed andinitially evaluatedinthe laboratory anditwas shown 2.1. Welch’s Method. An improved estimator of the PSD is the that vibration information can be gained in a sensorless fash- oneproposedbyWelch [8]. The method consists of dividing ion by utilizing the nearly linear relation between a par- the time series data into (possibly overlapping) segments, ticular vibration spectral component and its corresponding computing a modified periodogram of each segment and current harmonics. This forms the basis for using the current then averaging the PSD estimates. eTh result is Welch’s PSD harmonics as an indicator of motor vibrations. eTh constant estimate. of proportionality between the current and vibration har- The averaging of modified periodograms tends to monics is obtained by measuring a baseline value of vibration. decrease the variance of the estimate relative to a single peri- Baseline measurements were taken for the normal operation odogram estimate of the entire data record. Although overlap condition of the test machine, which included a small exter- between segments tends to introduce redundant information, nally induced vibration. Experimental results clearly showed this effect is diminished by the use of a nonrectangular win- that this method is applicable for a wide range of vibrations. dow, which reduces the importance or weight given to the end The cause and effect relationship between two signals or samples of segments (the samples that overlap). the commonality between them is generally estimated using However, as mentioned above, the combined use of short the coherence function. Reference [11] presents the results of data records and nonrectangular windows results in reduced a case study of motor bearing degradation caused by accel- resolution of the estimator. In summary, there is a trade-off erated electrical discharge machining under the seven aging between variance reduction and resolution. One can manip- cycles. To identify bearing damage using the motor current ulate the parameters in Welch’s method to obtain improved signal,thecoherencefunctionbetweenthemotorcurrentand estimates relative to the periodogram, especially when the vibration signature was computed. The largest values of the SNR is low. coherence amplitude, when driver’s current and vibration sig- The coherence function between the motor current and nals were correlated, were located at the dynamic eccentricity vibration signature was computed and plotted in Figure 4. and bearing defect. The largest coherence amplitude values, where motor current This paper aims at investigating the use of current signal and vibration signals are best correlated, are located at 50 and as reliable and inexpensive tool for pump’s CBM programs. It 100 Hz. is organised as follows. In Section 2 wavelet coherence is pre- sented. Some experimental data for validating this approach are presented and some results are given. Concluding remarks 2.2. Wavelet-Based Approach. eTh CWTallowsanalysisof appear in Section 3. the temporal evolution of the frequency content of a given signal or time series. eTh application of the CWT to two time series and the cross-examination of the two decompositions can reveal localized similarities in time and scale. Areas in 2. Coherence Analysis Technique the time-frequency plane where the two time series exhibit Figure 1 shows the time domain of current and vibration sig- common power or consistent phase behaviour indicate a nals for a healthy motor and motor with a 20 V drop in one relationship between the signals. 𝑥𝑦 𝑦𝑦 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑦𝑦 Advances in Acoustics and Vibration 3 1 0.3 0.2 0.5 0.1 −0.1 −0.2 −0.5 −0.3 −1 −0.4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time (s) Time (s) (a) Healthy motor with symmetrical supply voltages (b) Healthy motor with symmetrical supply voltages 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time (s) Time (s) (c) One-phase supply voltage with 20 volts drop (d) One-phase supply voltage with 20 volts drop Figure 1: Time waveforms of current and vibration signatures. Effect of load on RMS values for current and vibration 0.4 0.3 0.2 0.1 0 0 0 25% 50% 75% 100% Motor load (kW) Current Vibration Figure 2: Inu fl ence of load on the RMS current and vibration signals. Current (A) Current (A) RMS current Amplitude (g) Amplitude (g) RMS vibration 4 Advances in Acoustics and Vibration −50 −50 −100 −100 −150 −150 −200 −200 −250 −250 50 100 150 200 250 50 100 150 200 250 Frequency (Hz) Frequency (Hz) (a) Healthy motor with symmetrical supply voltages (current) (b) Healthy motor with symmetrical supply voltages (vibration) −50 −50 −100 −100 −150 −150 −200 −200 −250 −250 −300 50 100 150 200 250 50 100 150 200 250 Frequency (Hz) Frequency (Hz) (c) One-phase supply voltage with 20 volts drop (current) (d) One-phase supply voltage with 20 volts drop (vibration) Figure 3: Frequency spectra of current and vibration signatures. Coherence estimate via Welch The wavelet coherence of two time series 𝑥 and𝑦 is 0.7 𝑆[𝐶 (𝑎,𝑏 )𝐶 (𝑎,𝑏 )] 0.6 (2) 󵄨 󵄨 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 √ √ 󵄨 󵄨 𝑆( 󵄨𝐶 (𝑎,𝑏 )󵄨 ) 𝑆( 𝐶 (𝑎,𝑏 ) ) 󵄨 󵄨 𝑥 𝑦 󵄨 󵄨 0.5 󵄨 󵄨 0.4 where𝐶 (𝑎,)𝑏 and𝐶 (𝑎,)𝑏 denote the continuous wavelet 𝑥 𝑦 transforms of𝑥 and𝑦 at scale𝑎 and position𝑏 .Thesuper- 0.3 script∗ is the complex conjugate and𝑆 is a smoothing opera- tor in time and scale. 0.2 Figure 5 presents the CWT analysis of the current and vibration signals when a 3 kW driver running at healthy con- 0.1 dition. eTh top left gur fi e shown the CWT modulus of current signal and the top right figure shown the cwt modulus of vibration signal. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 eTh common periods of the current and vibration signals Frequency (kHz) at scales 32 and 16, respectively are clearly detected in the Figure 4: Coherence between DCS and DVS signals. moduli of the individual wavelet spectra and frequency at 50 and100Hz as showninFigure 5. For jointly stationary time series, the cross spectrum and The wavelet spectrum, defined for each signal, is charac- associated coherence function based on the Fourier trans- terized by the modulus and the phase of the CWT obtained form are used to detect common behaviour in the frequency usingthe complexvaluedwavelet.Theindividualwavelet domain. In the general nonstationary case, wavelet-based spectra are denoted as𝐶 (𝑎,)𝑏 and𝐶 (𝑎,)𝑏 .Thetwo decom- 𝑥 𝑦 counterparts can be defined to provide time-localized alter- positions are the same, up to a translation, since the CWT natives. is translation-invariant. To examine the relationship between Magnitude Power spectrum (dB) Power spectrum (dB) Power spectrum (dB) Power spectrum (dB) Advances in Acoustics and Vibration 5 Current signal Vibration signal 0.2 0.5 −0.5 −0.2 1000 2000 3000 4000 1000 2000 3000 4000 (a) (b) Continuous wavelet transform (CWT) Continuous wavelet transform (CWT) Modulus Modulus 50.0 50.0 44.6 44.6 39.1 39.1 33.7 33.7 28.2 28.2 22.8 22.8 17.3 17.3 11.9 11.9 6.4 6.4 1.0 1.0 1000 2000 3000 4000 1000 2000 3000 4000 (c) (d) Angle Angle 50.0 50.0 44.6 44.6 39.1 39.1 33.7 33.7 28.2 28.2 22.8 22.8 17.3 17.3 11.9 11.9 6.4 6.4 1.0 1.0 1000 2000 3000 4000 1000 2000 3000 4000 (e) (f) Figure 5: Continuous wavelet transform of DCS and DVS signals. the two signals in the time-scale plane, consider the wavelet where 𝑆 is a smoothing operator in time and scale. eTh cross spectrum𝐶 (𝑎,)𝑏 , which is defined as wavelet coherence can be interpreted as the square of the local correlation coefficient in the time-scale plane. eTh common period of the signals at scale 192 is clearly (3) 𝐶 (𝑎,𝑏 )=𝐶 (𝑎,𝑏 )𝐶 (𝑎,𝑏 ). 𝑥 𝑦 detected using Freq = scal2frq(192,“mother wavelet”, 1/sample frequency); note that this corresponds to a frequency of The magnitude of the wavelet cross spectrum can be inter- 50 Hz. preted as the absolute value of the local covariance between eTh arrows in the gfi ure represent the relative phase the two time series in the time-scale plane, as shown in between the two signals as a function of scale and posi- Figure 6. In this example, this nonnormalized quantity high- lights the fact that both signals have a significant contribution tion. eTh relative phase information is obtained from around scales 32, 16 at all positions. the smoothed estimate of the wavelet cross spectrum, Figure 7 displays wavelet coherence and the empirical wa- 𝑆(𝐶 (𝑎,))𝑏 .Theplotofthe relative phases is superimposed velet coherence for𝑥 and𝑦 is defined as the following ratio: on the wavelet coherence. eTh relative phase information produces a local measure of the delay between the two time series.Notethatfor scales around 16,28, and32, thedirection 𝑆(𝐶 (𝑎,𝑏 )) (4) of thearrowscapturesthe relative phasedieff rence between 2 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 √ √ 𝑆( 󵄨𝐶 (𝑎,𝑏 )󵄨 ) 𝑆( 󵄨𝐶 (𝑎,𝑏 )󵄨 ) 𝑥 𝑥 the two signals. 󵄨 󵄨 󵄨 󵄨 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑥𝑦 6 Advances in Acoustics and Vibration Current and vibration signals at healthy condition 3. Conclusions 0.5 This study utilized Welch’s method and continuous wavelet techniques for spectral estimation to investigate the coher- −0.5 ence between the driver’s current DCS and the driver’s vibra- 500 1000 1500 2000 2500 3000 3500 4000 tion signatures. eTh coherence between DCS and DVS signals Samples was investigated at a particular frequency and in dieff rent (a) frequency bands. Both signals are completely coherent if the Wavelet cross spectrum (WCS) magnitude-squared coherence is equal to 1; if MSC is equal Modulus to zero, then both signals are independent of each other. eTh results show that both signals are coherent at the frequencies 313 at which the magnitude-squared coherence (DSC) is greater than 0.5 and both signals are incoherent (less coherent) if DSC is less than 0.5. Wavelet coherence analysis greatly facil- itates the detection of the quasiperiodic component indicative 500 1000 1500 2000 2500 3000 3500 4000 of a system anomaly. Wavelet cross spectrum and wavelet Samples coherence are useful to reveal localized similarities between (b) DCS and DVS signals in the time-scale plane and to interpret Angle theresults.Theresults of this work show thepossibility to 417 estimate the DVS signal information from the DCS signal. References 500 1000 1500 2000 2500 3000 3500 4000 [1] B. K. N. Rao, Handbook of Condition Monitoring, Elsevier, Ox- ford, UK, 1996. Samples [2] P. A. Higgs, R. Parkin, M. Jackson et al., “A survey on condition (c) monitoring systems in industry,” in Proceedings of the 7th Bien- nial ASME Conference Engineering Systems Design and Analysis, Figure 6: Wavelet cross spectrum of DCS and DVS signals. Manchester, UK, 2004. [3] G. Rohlfing, “Condition monitoring of multiphase pumps,” World Pumps,vol.2010, no.4,pp. 34–39, 2010. [4] A. Azadeh, V. Ebrahimipour, and P. Bavar, “A fuzzy inference Current and vibration signals at healthy condition system for pump failure diagnosis to improve maintenance pro- 0.5 cess: the case of a petrochemical industry,” Expert Systems with Applications,vol.37, no.1,pp. 627–639, 2010. −0.5 500 1000 1500 2000 2500 3000 3500 4000 [5] R. Ocampo, “Fatigue failures in pumps: part 1,” World Pumps, vol. 2008,no. 500, pp.42–45,2008. Samples [6] R. Ocampo and B. Ruiz, “Fatigue failures in pumps: part 2,” (a) World Pumps,vol.2008, no.502,pp. 18–21, 2008. Wavelet coherence of current and vibration signals [7] C. Hansford, “Condition monitoring: combating down time with vibration sensors,” World Pumps,no. 428, pp.50–53,2002 [8] A. Pu¨ttmer, “New applications for ultrasonic sensors in process industries,” Ultrasonics, vol. 44, supplement 1, pp. e1379–e1383, [9] X.Shi,J.Shao, J. Si,and B. Li,“Experiment andsimulationof rotor’s torsional vibration based on sensorless detection tech- nology,” in Proceedings of the IEEE International Conference on Automation and Logistics, pp. 2673–22678, 2008. [10] C. M. Riley, K. Lin Brian, and T. G. Habetler, “A method for sensorless on-line vibration monitoring of induction machines,” 500 1000 1500 2000 2500 3000 3500 4000 IEEE Transactions on Industry Applications,vol.34, no.6,pp. Samples 1240–1245, 1998. [11] S. Seker, E. Ayaz, and E. Turk ¨ can, “Elman’s recurrent neural 0.2 0.4 0.6 0.8 network applications to condition monitoring in nuclear power (b) plant and rotating machinery,” Engineering Applications of Arti- ficial Intelligence ,vol.16, no.7-8,pp. 647–656, 2003. Figure 7: Wavelet coherence of DCS and DVS signals. 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Petroleum Pumps’ Current and Vibration Signatures Analysis Using Wavelet Coherence Technique

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Copyright © 2013 Rmdan Shnibha and Alhussein Albarbar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 659650, 6 pages http://dx.doi.org/10.1155/2013/659650 Research Article Petroleum Pumps’ Current and Vibration Signatures Analysis Using Wavelet Coherence Technique Rmdan Shnibha and Alhussein Albarbar Advanced Industrial Diagnostic Centre, Digital Signal Processing Research Group, School of Engineering, Manchester Metropolitan University, Manchester M1 5GD, UK Correspondence should be addressed to Alhussein Albarbar; a.albarbar@mmu.ac.uk Received 28 February 2013; Revised 7 May 2013; Accepted 22 May 2013 Academic Editor: K. M. Liew Copyright © 2013 R. Shnibha and A. Albarbar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vibration analysis is widely used for rotating machinery diagnostics; however measuring vibration of operational oil well pumps is not possible. The pump’s driver’s current signatures may provide condition-related information without the need for an access to the pump itself. This paper investigates the degree of relationship between the pump’s driver’s current signatures and its induced vibration. This relationship between the driver’s current signatures (DCS) and its vibration signatures (DVS) is studied by calculating magnitude-squared coherence and phase coherence parameters at a certain frequency band using continuous wavelet transform (CWT). The CWT coherence-based technique allows better analysis of temporal evolution of the frequency content of dynamic signalsandareasinthetime-frequencyplanewherethetwosignalsexhibitcommonpowerorconsistentphasebehaviourindicating a relationship between the signals. This novel approach is validated by experimental data acquired from 3 kW petroleum pump’s driver. Both vibration and current signatures were acquired under different speed and load conditions. The outcomes of this research suggest the use of DCS analysis as reliable and inexpensive condition monitoring tool, which could be implemented for oil pumps, real-time monitoring associated with condition-based maintenance (CBM) program. 1. Introduction Vibration monitoring is particularly suited to pumps due to the number of integrated rotating parts, which may show Pumps and their associated systems are essential in oil and gas additional movement when faults develop [7]. A more recent facilities for the efficient transportation of uids. fl Common development in pump condition monitoring is the applica- pumps found in these facilities include centrifugal, recip- tion of ultrasonic sensors [8]; introducing a new ultrasonic rocating, diaphragm, and rotary pumps [1]. The condition measurement based on acoustic emission analyses for high- monitoring of pumps and their associated systems is an estab- pressure process pumps. lished application of CBM and is an existing area of research The most obvious technique for obtaining a vibration [2]. Rohlfing [ 3] provides three examples in the oil and gas signal from pump driver’s “induction motor” is by direct industry wherepump’sCBM hasbeeneeff ctivelyimple- measurement using vibration transducers (usually acceler- mented. Azadeh et al. in [4]havedeveloped adiagnostic ometers) mounted on the driver. This requires a high-per- mechanism for pump failures in which pump operating formance vibration transducer capable of withstanding harsh problems fall into two categories: (1) hydraulic problems that environmental conditions and which can cost several hun- suggest the pump may fail to deliver liquid, deliver insuffi- dred dollars. When a large number of machines are con- cient capacity, develop insufficient pressure, or lose its prime cerned andmorethanone transducer is required perma- at starting and (2) mechanical problems that are characterised chine, thetotal cost canbehigh. Amajor disadvantage of by the consumption of excessive power or development of vibration monitoring is that it requires access to the machine. mechanical difficulties at the seal chambers or bearings; in For accurate measurements, sensors should be mounted either case vibration, noise, or breakage may occur. Fatigue is rigidly on the machine, which requires expertise and trained a common cause of pump failure [5, 6]. personnel. 2 Advances in Acoustics and Vibration An increasing number of pumps are installed with elec- phase of the supply. Figure 2 shows the RMS values of current trical motors as their prime driver. This development has and vibration for different motor loads; it can be seen that the introduced new possibilities for condition monitoring by the stator current and vibration signals are inversely correlated. use of driver’s electric signals such as current and voltage. A common approach for extracting information concern- In response an indirect method, called sensorless detec- ing frequency features of a periodic signal is to transform the tion and diagnosis intended for mechanical equipment driv- signal to the frequency domain using the discrete Fourier en by AC induction motors, has been developed over the past transform. 20 years and is growing rapidly [9]. Sensorless detection and In Figure 3, the same signals as those shown in Figure 1 diagnosis technology monitors the state of devices and are presented in the frequency domain; it is assumed that the motors using the motor’s stator current rather than external original signals are stationary. sensors to detect vibration, noise, and so forth. Besides the The coherence is a function of the power spectral density original power frequency, the stator current contains a large (𝑃 and𝑃 )of𝑥 and𝑦 and the cross power spectral density 𝑥𝑥 amount of information relative to mechanical faults. (𝑃 ) of𝑥 and𝑦 is given by eTh relationship between the magnitudes of stator current 󵄨 󵄨 2 󵄨 󵄨 harmonics, the magnitudes of the vibration harmonics, and 󵄨 󵄨 𝑃 (𝑓) 󵄨 󵄨 󵄨 󵄨 (1) specific machine faults types has been closely studied [ 9, 10]. 𝐶 (𝑓)= . 𝑃 (𝑓)𝑃 (𝑓) 𝑥𝑥 In [9] the relationship between the vibration and current har- monic magnitudes for a source of known vibration frequency Coherence is a function of frequency with𝐶 (𝑓) ranging was investigated to determine the feasibility of setting a limit between 0 and 1 and indicates how well signal𝑥 corresponds or standard on the current harmonics due to these vibrations. to signal𝑦 at each frequency. The degree of synchronization It was concluded that for a given vibration frequency, the in stator current signal and vibration signal is commonly harmonic’s RMS vibration level and RMS current level are characterized by coherence phase and magnitude-squared monotonically related. In [10] a method for sensorless on-line coherence (DSC). vibration monitoring of an induction motor was proposed andinitially evaluatedinthe laboratory anditwas shown 2.1. Welch’s Method. An improved estimator of the PSD is the that vibration information can be gained in a sensorless fash- oneproposedbyWelch [8]. The method consists of dividing ion by utilizing the nearly linear relation between a par- the time series data into (possibly overlapping) segments, ticular vibration spectral component and its corresponding computing a modified periodogram of each segment and current harmonics. This forms the basis for using the current then averaging the PSD estimates. eTh result is Welch’s PSD harmonics as an indicator of motor vibrations. eTh constant estimate. of proportionality between the current and vibration har- The averaging of modified periodograms tends to monics is obtained by measuring a baseline value of vibration. decrease the variance of the estimate relative to a single peri- Baseline measurements were taken for the normal operation odogram estimate of the entire data record. Although overlap condition of the test machine, which included a small exter- between segments tends to introduce redundant information, nally induced vibration. Experimental results clearly showed this effect is diminished by the use of a nonrectangular win- that this method is applicable for a wide range of vibrations. dow, which reduces the importance or weight given to the end The cause and effect relationship between two signals or samples of segments (the samples that overlap). the commonality between them is generally estimated using However, as mentioned above, the combined use of short the coherence function. Reference [11] presents the results of data records and nonrectangular windows results in reduced a case study of motor bearing degradation caused by accel- resolution of the estimator. In summary, there is a trade-off erated electrical discharge machining under the seven aging between variance reduction and resolution. One can manip- cycles. To identify bearing damage using the motor current ulate the parameters in Welch’s method to obtain improved signal,thecoherencefunctionbetweenthemotorcurrentand estimates relative to the periodogram, especially when the vibration signature was computed. The largest values of the SNR is low. coherence amplitude, when driver’s current and vibration sig- The coherence function between the motor current and nals were correlated, were located at the dynamic eccentricity vibration signature was computed and plotted in Figure 4. and bearing defect. The largest coherence amplitude values, where motor current This paper aims at investigating the use of current signal and vibration signals are best correlated, are located at 50 and as reliable and inexpensive tool for pump’s CBM programs. It 100 Hz. is organised as follows. In Section 2 wavelet coherence is pre- sented. Some experimental data for validating this approach are presented and some results are given. Concluding remarks 2.2. Wavelet-Based Approach. eTh CWTallowsanalysisof appear in Section 3. the temporal evolution of the frequency content of a given signal or time series. eTh application of the CWT to two time series and the cross-examination of the two decompositions can reveal localized similarities in time and scale. Areas in 2. Coherence Analysis Technique the time-frequency plane where the two time series exhibit Figure 1 shows the time domain of current and vibration sig- common power or consistent phase behaviour indicate a nals for a healthy motor and motor with a 20 V drop in one relationship between the signals. 𝑥𝑦 𝑦𝑦 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑦𝑦 Advances in Acoustics and Vibration 3 1 0.3 0.2 0.5 0.1 −0.1 −0.2 −0.5 −0.3 −1 −0.4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time (s) Time (s) (a) Healthy motor with symmetrical supply voltages (b) Healthy motor with symmetrical supply voltages 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time (s) Time (s) (c) One-phase supply voltage with 20 volts drop (d) One-phase supply voltage with 20 volts drop Figure 1: Time waveforms of current and vibration signatures. Effect of load on RMS values for current and vibration 0.4 0.3 0.2 0.1 0 0 0 25% 50% 75% 100% Motor load (kW) Current Vibration Figure 2: Inu fl ence of load on the RMS current and vibration signals. Current (A) Current (A) RMS current Amplitude (g) Amplitude (g) RMS vibration 4 Advances in Acoustics and Vibration −50 −50 −100 −100 −150 −150 −200 −200 −250 −250 50 100 150 200 250 50 100 150 200 250 Frequency (Hz) Frequency (Hz) (a) Healthy motor with symmetrical supply voltages (current) (b) Healthy motor with symmetrical supply voltages (vibration) −50 −50 −100 −100 −150 −150 −200 −200 −250 −250 −300 50 100 150 200 250 50 100 150 200 250 Frequency (Hz) Frequency (Hz) (c) One-phase supply voltage with 20 volts drop (current) (d) One-phase supply voltage with 20 volts drop (vibration) Figure 3: Frequency spectra of current and vibration signatures. Coherence estimate via Welch The wavelet coherence of two time series 𝑥 and𝑦 is 0.7 𝑆[𝐶 (𝑎,𝑏 )𝐶 (𝑎,𝑏 )] 0.6 (2) 󵄨 󵄨 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 √ √ 󵄨 󵄨 𝑆( 󵄨𝐶 (𝑎,𝑏 )󵄨 ) 𝑆( 𝐶 (𝑎,𝑏 ) ) 󵄨 󵄨 𝑥 𝑦 󵄨 󵄨 0.5 󵄨 󵄨 0.4 where𝐶 (𝑎,)𝑏 and𝐶 (𝑎,)𝑏 denote the continuous wavelet 𝑥 𝑦 transforms of𝑥 and𝑦 at scale𝑎 and position𝑏 .Thesuper- 0.3 script∗ is the complex conjugate and𝑆 is a smoothing opera- tor in time and scale. 0.2 Figure 5 presents the CWT analysis of the current and vibration signals when a 3 kW driver running at healthy con- 0.1 dition. eTh top left gur fi e shown the CWT modulus of current signal and the top right figure shown the cwt modulus of vibration signal. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 eTh common periods of the current and vibration signals Frequency (kHz) at scales 32 and 16, respectively are clearly detected in the Figure 4: Coherence between DCS and DVS signals. moduli of the individual wavelet spectra and frequency at 50 and100Hz as showninFigure 5. For jointly stationary time series, the cross spectrum and The wavelet spectrum, defined for each signal, is charac- associated coherence function based on the Fourier trans- terized by the modulus and the phase of the CWT obtained form are used to detect common behaviour in the frequency usingthe complexvaluedwavelet.Theindividualwavelet domain. In the general nonstationary case, wavelet-based spectra are denoted as𝐶 (𝑎,)𝑏 and𝐶 (𝑎,)𝑏 .Thetwo decom- 𝑥 𝑦 counterparts can be defined to provide time-localized alter- positions are the same, up to a translation, since the CWT natives. is translation-invariant. To examine the relationship between Magnitude Power spectrum (dB) Power spectrum (dB) Power spectrum (dB) Power spectrum (dB) Advances in Acoustics and Vibration 5 Current signal Vibration signal 0.2 0.5 −0.5 −0.2 1000 2000 3000 4000 1000 2000 3000 4000 (a) (b) Continuous wavelet transform (CWT) Continuous wavelet transform (CWT) Modulus Modulus 50.0 50.0 44.6 44.6 39.1 39.1 33.7 33.7 28.2 28.2 22.8 22.8 17.3 17.3 11.9 11.9 6.4 6.4 1.0 1.0 1000 2000 3000 4000 1000 2000 3000 4000 (c) (d) Angle Angle 50.0 50.0 44.6 44.6 39.1 39.1 33.7 33.7 28.2 28.2 22.8 22.8 17.3 17.3 11.9 11.9 6.4 6.4 1.0 1.0 1000 2000 3000 4000 1000 2000 3000 4000 (e) (f) Figure 5: Continuous wavelet transform of DCS and DVS signals. the two signals in the time-scale plane, consider the wavelet where 𝑆 is a smoothing operator in time and scale. eTh cross spectrum𝐶 (𝑎,)𝑏 , which is defined as wavelet coherence can be interpreted as the square of the local correlation coefficient in the time-scale plane. eTh common period of the signals at scale 192 is clearly (3) 𝐶 (𝑎,𝑏 )=𝐶 (𝑎,𝑏 )𝐶 (𝑎,𝑏 ). 𝑥 𝑦 detected using Freq = scal2frq(192,“mother wavelet”, 1/sample frequency); note that this corresponds to a frequency of The magnitude of the wavelet cross spectrum can be inter- 50 Hz. preted as the absolute value of the local covariance between eTh arrows in the gfi ure represent the relative phase the two time series in the time-scale plane, as shown in between the two signals as a function of scale and posi- Figure 6. In this example, this nonnormalized quantity high- lights the fact that both signals have a significant contribution tion. eTh relative phase information is obtained from around scales 32, 16 at all positions. the smoothed estimate of the wavelet cross spectrum, Figure 7 displays wavelet coherence and the empirical wa- 𝑆(𝐶 (𝑎,))𝑏 .Theplotofthe relative phases is superimposed velet coherence for𝑥 and𝑦 is defined as the following ratio: on the wavelet coherence. eTh relative phase information produces a local measure of the delay between the two time series.Notethatfor scales around 16,28, and32, thedirection 𝑆(𝐶 (𝑎,𝑏 )) (4) of thearrowscapturesthe relative phasedieff rence between 2 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 √ √ 𝑆( 󵄨𝐶 (𝑎,𝑏 )󵄨 ) 𝑆( 󵄨𝐶 (𝑎,𝑏 )󵄨 ) 𝑥 𝑥 the two signals. 󵄨 󵄨 󵄨 󵄨 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑥𝑦 6 Advances in Acoustics and Vibration Current and vibration signals at healthy condition 3. Conclusions 0.5 This study utilized Welch’s method and continuous wavelet techniques for spectral estimation to investigate the coher- −0.5 ence between the driver’s current DCS and the driver’s vibra- 500 1000 1500 2000 2500 3000 3500 4000 tion signatures. eTh coherence between DCS and DVS signals Samples was investigated at a particular frequency and in dieff rent (a) frequency bands. Both signals are completely coherent if the Wavelet cross spectrum (WCS) magnitude-squared coherence is equal to 1; if MSC is equal Modulus to zero, then both signals are independent of each other. eTh results show that both signals are coherent at the frequencies 313 at which the magnitude-squared coherence (DSC) is greater than 0.5 and both signals are incoherent (less coherent) if DSC is less than 0.5. Wavelet coherence analysis greatly facil- itates the detection of the quasiperiodic component indicative 500 1000 1500 2000 2500 3000 3500 4000 of a system anomaly. Wavelet cross spectrum and wavelet Samples coherence are useful to reveal localized similarities between (b) DCS and DVS signals in the time-scale plane and to interpret Angle theresults.Theresults of this work show thepossibility to 417 estimate the DVS signal information from the DCS signal. References 500 1000 1500 2000 2500 3000 3500 4000 [1] B. K. N. Rao, Handbook of Condition Monitoring, Elsevier, Ox- ford, UK, 1996. Samples [2] P. A. Higgs, R. Parkin, M. Jackson et al., “A survey on condition (c) monitoring systems in industry,” in Proceedings of the 7th Bien- nial ASME Conference Engineering Systems Design and Analysis, Figure 6: Wavelet cross spectrum of DCS and DVS signals. Manchester, UK, 2004. [3] G. Rohlfing, “Condition monitoring of multiphase pumps,” World Pumps,vol.2010, no.4,pp. 34–39, 2010. [4] A. Azadeh, V. Ebrahimipour, and P. Bavar, “A fuzzy inference Current and vibration signals at healthy condition system for pump failure diagnosis to improve maintenance pro- 0.5 cess: the case of a petrochemical industry,” Expert Systems with Applications,vol.37, no.1,pp. 627–639, 2010. −0.5 500 1000 1500 2000 2500 3000 3500 4000 [5] R. Ocampo, “Fatigue failures in pumps: part 1,” World Pumps, vol. 2008,no. 500, pp.42–45,2008. Samples [6] R. Ocampo and B. Ruiz, “Fatigue failures in pumps: part 2,” (a) World Pumps,vol.2008, no.502,pp. 18–21, 2008. Wavelet coherence of current and vibration signals [7] C. Hansford, “Condition monitoring: combating down time with vibration sensors,” World Pumps,no. 428, pp.50–53,2002 [8] A. Pu¨ttmer, “New applications for ultrasonic sensors in process industries,” Ultrasonics, vol. 44, supplement 1, pp. e1379–e1383, [9] X.Shi,J.Shao, J. Si,and B. Li,“Experiment andsimulationof rotor’s torsional vibration based on sensorless detection tech- nology,” in Proceedings of the IEEE International Conference on Automation and Logistics, pp. 2673–22678, 2008. [10] C. M. Riley, K. Lin Brian, and T. G. Habetler, “A method for sensorless on-line vibration monitoring of induction machines,” 500 1000 1500 2000 2500 3000 3500 4000 IEEE Transactions on Industry Applications,vol.34, no.6,pp. Samples 1240–1245, 1998. [11] S. Seker, E. Ayaz, and E. Turk ¨ can, “Elman’s recurrent neural 0.2 0.4 0.6 0.8 network applications to condition monitoring in nuclear power (b) plant and rotating machinery,” Engineering Applications of Arti- ficial Intelligence ,vol.16, no.7-8,pp. 647–656, 2003. Figure 7: Wavelet coherence of DCS and DVS signals. 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