sensors Article
Quantitative Index and Abnormal Alarm Strategy Using Sensor-Dependent Vibration Data for Blade Crack Identification in Centrifugal Booster Fans Jinglong Chen 1, *, Hailiang Sun 2 , Shuai Wang 1 and Zhengjia He 1 1 2
*
State Key Laboratory for Manufacturing and Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China;
[email protected] (S.W.);
[email protected] (Z.H.) Beijing Institute of Astronautical Systems Engineering, Beijing 100076, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-186-2938-5667
Academic Editor: Vittorio M. N. Passaro Received: 8 March 2016; Accepted: 27 April 2016; Published: 9 May 2016
Abstract: Centrifugal booster fans are important equipment used to recover blast furnace gas (BFG) for generating electricity, but blade crack faults (BCFs) in centrifugal booster fans can lead to unscheduled breakdowns and potentially serious accidents, so in this work quantitative fault identification and an abnormal alarm strategy based on acquired historical sensor-dependent vibration data is proposed for implementing condition-based maintenance for this type of equipment. Firstly, three group dependent sensors are installed to acquire running condition data. Then a discrete spectrum interpolation method and short time Fourier transform (STFT) are applied to preliminarily identify the running data in the sensor-dependent vibration data. As a result a quantitative identification and abnormal alarm strategy based on compound indexes including the largest Lyapunov exponent and relative energy ratio at the second harmonic frequency component is proposed. Then for validation the proposed blade crack quantitative identification and abnormality alarm strategy is applied to analyze acquired experimental data for centrifugal booster fans and it has successfully identified incipient blade crack faults. In addition, the related mathematical modelling work is also introduced to investigate the effects of mistuning and cracks on the vibration features of centrifugal impellers and to explore effective techniques for crack detection. Keywords: fault diagnosis; blade crack; vibration signal analysis; quantitative identification; centrifugal booster fan
1. Introduction Blast furnace gas (BFG) is a byproduct of iron-making. With the great expansion of the iron and steel industry, the production of BFG during iron-making has increased remarkably [1]. However, blast furnace gas is characterized in low calorific value, difficulty to burn and combustion instability as a power fuel [2], so how to deal effectively with blast furnace gas is a problem that puzzles iron and steel enterprises. In recent years, many enterprises have taken to use BFG to generate electricity in order to save energy and improve benefits. Centrifugal booster fans are important pieces of equipment which are used to pressurize BGF and make sure it can go into the furnace safely and combust stably. Faults occurring on the centrifugal booster fan may lead to accidents such as unstable combustion in the furnace, blow outs, downtime, and potentially huge economic losses. Blade crack faults (BCFs) are among the typical faults in centrifugal booster fans. Different cracks arise after long running due to the resonance, decreased anti-fatigue capability because of manufacture problems, installation issues or the work conditions [3]. This may result in blades breaking off and the unit being damaged. For a rotor system, the stiffness of the shaft would display cyclical behavior
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once blade cracks occur [4], and the dynamic response signal will show obvious non-stationary characteristics. Fault diagnosis for blade crack faults is a difficult problem that has drawn the attention of top scholars and outstanding engineers worldwide [5–8]. Lots of related studies have been reported in important journals [9–11]. These researches place emphasis on the following aspects: first, the method based on finite element tries to model the cracked rotor and analyze the fracture mechanisms and the effect on the structural dynamic properties [12–14]. This method can provide a theoretical basis for the blade damage identification method based on vibration signals. Second, the method combining the finite element modeling with the signal processing makes use of the finite element method to analyze the dynamic responses of the cracked rotor [15]. Then a signal processing method such as the wavelet transform is applied to analyze the dynamic response and lock the position and size of the crack damage. However, the practical engineering of these methods, especially for blade crack fault detection under operation conditions, should be studied further. On-line and off-line condition monitoring systems have been widely used for important rotating machinery [16–20]. Vibration signals can be acquired effectively in various industrial fields [21–24]. The key issue is how to extract the characteristics of the blade crack faults using appropriate signal analysis methods [25]. In this paper, operation condition information is monitored and collected in a timely way based on a condition monitoring and fault diagnosis system for centrifugal booster fans. Firstly, aiming at the problem of energy leak in FFT, amplitude and phase can be accurately estimated by an discrete spectrum interpolation method [26,27]. The short time Fourier transform (STFT) can effectively pick out the non-stationary components in vibration signals [28]. The discrete spectrum interpolation method and STFT are integrated in the testing system. They are used to extract the features of the blade cracks of generator centrifugal booster fans in a certain iron and steel group’s power plants. Finally, for the purpose of quantitative identification and abnormality alarming for blade crack faults, a quantitative identification and abnormality alarm strategy based on compound indexes including the largest Lyapunov exponent (LLE) [29,30] and relative energy ratio at the second harmonic frequency component is proposed in this paper. The proposed method is applied to an accident caused by blade crack faults using historical data. The results demonstrate that this method could quantitatively identify blade cracks in booster fans successfully. In addition, related work on mathematical modelling is also introduced to investigate the effects of mistuning and cracks on the vibration features of centrifugal impellers and to explore effective techniques for crack detection. The rest of the paper is organized as follows: in Section 2, the fault identification method for thermal generator sets is introduced. In Section 3, a case study via blade crack faults of centrifugal booster fans is presented. In Section 4, a quantitative identification and abnormality alarm strategy for blade crack faults is proposed. In Section 5, mathematical modelling for revealing vibration signal properties is introduced. In Section 6, some conclusions are provided. 2. Fault Identification Method for Thermal Generator Sets 2.1. Centrifugal Booster Fans in Thermal Generator Sets Blast furnace gas (BFG) is a byproduct of iron-making, whose production in iron-making increases from year to year due to the growth of the iron and steel industry. The blast furnace gas (BFG) is a recyclable energy gas and plays an important role in the energy consumption of iron and steel works. The power station in a steel-making plant would try to generate energy with the BFG. The Unit 4 studied in this work is the first 350 MW unit in the world which fully combusts the BFG. Its combustion ability can reach 1 million m3 /h. The structure of the Unit 4 is displayed in Figure 1 [31]. Unit 4 is equipped with three dual-speed BFG booster fans. These fans are of importance and used to pressurize the BFG and make sure it can go into the furnace safely and combust stably. There is a tower-type once-through boiler in this unit, along with 18 compound gas burners that are well-distributed in three tiers. They are arranged separately on the front wall and back wall of the boiler.
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Control damper Original BFG gas
Low speed booster fan
A High speed booster fan
Heater
Gas burner
B Venturi meter BFG
entrance C shutting door
Figure Figure1.1.The Thestructure structureofofthe theBFG BFGbooster boosterfans fansininUnit Unit4.4.
Thetechnological technologicalprocess processofofthe theBFG BFGrecovery recoveryunit unitisisasasfollows: follows: The The original original BFG The BFG gas gas from from the the the the Department Department of ofEnergy Energyand andEnvironment EnvironmentProtection Protection→ Venturi meter → BFG entrance shutting door → BFG booster fan entrance control damper Ñ Venturi meter Ñ BFG entrance shutting door Ñ BFG booster fan entrance control → threeÑBFG booster fans → BFG fan exit isolation damper →Ñ Heater damper three BFG booster fans Ñ booster BFG booster fan exit isolation damper Heaterand andits bypass → BFG gas burners. its bypass Ñ BFG gas burners. The speed of the booster fans is divided into two levels: 744 r/min and 993 r/min, whose The speed of the booster fans is divided into two levels: 744 r/min and 993 r/min, whose corresponding powers are 730 kW and 1650 kW, respectively. corresponding powers are 730 kW and 1650 kW, respectively. 2.2.Condition ConditionData DataAcquisition AcquisitionTesting TestingFramework Framework 2.2. Foriron-making, iron-making,lots lotsofofrotating rotating machineries work complex process going from For machineries work in in thethe complex process going from ironiron ore ore to to steel products. The equipment is in long-term use under complex conditions. This may lead steel products. The equipment is in long-term use under complex conditions. This may lead to variousto various typesand of fault cause huge economy losses. However, the key parts of machinery rotating machinery types of fault causeand huge economy losses. However, the key parts of rotating are not are not stationary and therefore not easy to change, so it is crucial to carry out effective condition stationary and therefore not easy to change, so it is crucial to carry out effective condition monitoring monitoring and fault diagnosis. and fault diagnosis. Forthis thispurpose, purpose,aatesting testingsystem systemfocusing focusingon onextracting extractingabnormal abnormalcondition conditioninformation informationfrom from For vibration signals is designed. This system works on the data acquisition level and network database vibration signals is designed. This system works on the data acquisition level and network database levelofofthe thetesting testingand anddiagnosis diagnosissystem systemininthe thesteel-making steel-makingplant. plant. Off-line Off-line vibration vibration tests testsare are level conductedfor foriron ironand andsteel steelsmelting smeltingmechanical mechanicalequipment equipmentusing usingportable portabledata dataacquisition acquisitiondevices devices conducted such as the CSI2320, Telesens8823, SONY-EX and so on. These devices can selectively implement such as the CSI2320, Telesens8823, SONY-EX and so on. These devices can selectively implement hardwareintegration integrationfor foracquired acquiredsignals signalstotosave savethe thedata. data. hardware Thefeature featureinformation informationcan canbe beextracted extractedfrom fromthe thevibrations vibrationsby bymeans meansofoftraditional traditionalspectrum spectrum The analysis, characteristic spectrum analysis and special feature extraction modules in the testing analysis, characteristic spectrum analysis and special feature extraction modules in the testing system. system. Thetrends change trendsfeatures of thesecan features can used judge the workingofcondition of the The change of these be used to be judge thetoworking condition the equipment equipment and the appearance of incipient faults. The software interface of the testing system and the appearance of incipient faults. The software interface of the testing system is implementedis implemented based on The Labview 7.0.system The testing system is programmed with mixtureand of Labview based on Labview 7.0. testing is programmed with a mixture of aLabview Visual and Visual Studio routines with consideration to execution efficiency. In addition, SQL Server Studio routines with consideration to execution efficiency. In addition, SQL Server is introduced as theis introduced as the extended to access the internal database. tipsfunctions about the extended interface to access interface the internal database. Moreover, simpleMoreover, tips aboutsimple the main main functions are available in a help module. The system also has other functions such as saving are available in a help module. The system also has other functions such as saving results, report results, report generation, etc.flow Thediagram whole work diagram of the quantitative identification generation, etc. The whole work of the flow quantitative identification research framework is research framework is displayed in Figure 2. displayed in Figure 2.
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Acquired signals Acceleration sensor
Centrifugal booster fans
Interpolation method for discrete spectrum Discrete Fourier Transform
ˆ0 arctan( I k / Rk ) f 1
Aˆ 0 X w ( k ) / W (f 1 )
Mathematical modelling Short Time Fourier Transform
Abnormal status detection
STFTx ( , f )
x(t ) w* (t ) e j2πft dt
Mathematical Formulation Mp Cp Kp = b fnl (p)
Largest Lyapunov exponent
max
1 M L( t ) ln k tm t0 k 1 L(tk 1 )
Quantitative identification method 2
A2 f K 2 f j 2 Aif
i 1~ j,
Historical running dada Evaluation threshold
i 1
Frequency-based methods for detection of crack
Maintenance decision Status alarm
Overhaul downtime
Trend forecast
Figure Figure 2. 2. The The work work flow flow diagram diagram of of the the research research framework. framework.
2.3. 2.3. Fault Fault Feature Feature Extraction Extraction Method Method 2.3.1. 2.3.1. Discrete Discrete Spectrum Spectrum Interpolation Interpolation Method Method The concerning The rotating rotating frequency frequency and and its its harmonic harmonic components components can can reflect reflect the the fault fault features features concerning misalignment, misalignment, rub-impact, rub-impact, and and dynamic dynamic unbalance. unbalance. These These features features are are usually usually extracted extracted from from the the frequency spectrum. However, the FFT and spectrum analysis would cause energy leakage owing to frequency spectrum. However, the FFT and spectrum analysis would cause energy leakage owing time domain truncation and the interference of noise [26,27]. This may lead to great errors in the to time domain truncation and the interference of noise [26,27]. This may lead to great errors in the frequency, amplitude and and phase phase in inthe theFFT FFTand andspectrum spectrumanalysis. analysis.In Inorder ordertotoimprove improvethe theaccuracy, accuracy,a frequency, amplitude adiscrete discretespectrum spectruminterpolation interpolationmethod methodis is adopted. adopted. Let harmonic signal signal sequence sequence with with frequency frequency ff00,, amplitude amplitude A A00 and θ00.. Suppose Let x(t) x(t) be be aa harmonic and phase phase θ Suppose the and phase phase first firstcalculated calculatedby byaaDiscrete DiscreteFourier FourierTransform Transform (DFT) and then corrected the amplitude amplitude and (DFT) and then corrected by by the interpolation method are: the interpolation method are: ˆ A ∇ f 11q (1) ˆ 0 “ XXw(pkq{Wp (1) A k ) / W ( f ) 0 w θˆˆ0 “ arctanpI ∇ff11 (2) arctan( I kk {R / Rkkq)` π (2) 0 where Xw pkq means the kth line of the harmonic signal, i.e., the maximum value of the main ( kf)1 ) means where the kth of the harmonic signal, the maximum value ofwith the main lobe. lobe. X W(w ∇ expresses theline frequency spectrum for i.e., a rectangular window the value 1 1 )/ (π ∇f 1 ). R and I represent the real and imaginary parts of DFT,1 respectively. 1 W( ∇ f ) = sin(π ∇ f k k for a rectangular window with the value W(f ) = sin(f1)/ W(f ) expresses the frequency spectrum 1 When rotating frequency is input, the system would correct and phaseWhen of the rotating rotating (f ). Rk and Ik represent the real and imaginary parts the of amplitude DFT, respectively. frequency is and search the accurate phase ofand its phase harmonic components automatically. frequency input, thefor system would amplitude correct theand amplitude of the rotating frequency and search for the accurate amplitude and phase of its harmonic components automatically. 2.3.2. Short Time Fourier Transform 2.3.2.Blade Short crack Time fault Fourier Transform diagnosis is a problem that troubles scholars and engineers at home and abroad. Online and offline condition monitoring systems widely scholars used in rotating machinery. The and key Blade crack fault diagnosis is a problem thatare troubles and engineers at home problem is how to choose the method to process the signal from the industrial field and obtain the abroad. Online and offline condition monitoring systems are widely used in rotating machinery. The faultproblem characteristics. The short-time Fourierto transform (STFT) is one ofthe the industrial earliest and theand mostobtain basic key is how to choose the method process the signal from field methods used for time-frequency analysis [28]. The STFT is one of the most widely used algorithms in the fault characteristics. The short-time Fourier transform (STFT) is one of the earliest and the most signal processing and fault diagnosis based on a detailed Fourier transform centered at each time point. basic methods used for time-frequency analysis [28]. The STFT is one of the most widely used In STFT, theinsignal comparedand withfault window functions concentrated both the time and algorithms signalisprocessing diagnosis basedthat on are a detailed Fourierintransform centered at each time point. In STFT, the signal is compared with window functions that are concentrated in
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frequency Sensors domains. 2016, 16, 632 The STFT algorithm and the window function can be mathematically represented 5 of 20 both the time and frequency domains. The STFT algorithm and the window function can be as follows: r `8 f t dt both the time and frequency domains. the window function can be STFT pτ, f q “The´8STFT xptqalgorithm w˚ pt ´ τqand e´j2π mathematically represented asxfollows: ” ı˚ r `8 mathematically represented as follows: j 2π f t (3) “ ´8 xptq wpt ´ τqe dt * j2πft
STFTx (, f ) x(t ) w (t ) e dt * j2π j ft2π STFTx (, f ) “ xxptq, x ( t ) w ( t ) e d t f ty wpt ´ τqe (3) x(t ) w(t )e j 2πft dt j 2πft (3) w(t j)e where, w(t) is the window function which hasa user time duration; and x(t) is the waveform x(t )defined d t x(t ), w(t )e 2πft signal in the time domain. x(t ), w(t )e j 2πft where, w(t) is the window function which has a user defined time duration; and x(t) is the 3. Case Study via Crack Fault of awhich Centrifugal Booster Fantime duration; and x(t) is the where, w(t)signal is Blade the window function has a user defined waveform in the time domain. waveform signal in the time domain. Unit 4 in a power station composed of three imported fans. As shown in Figure 3 [24,31,32], 3. Case Study via Blade Crack Fault of a Centrifugal Booster Fan the rotor blades No. A fan are Fault welded the entrance control 3. Case Studyinvia Blade Crack of aon Centrifugal Booster Fan damper. The three fans performed Unit 4 in a power station composed of three imported fans. Asoverhauled shown in Figure 3 [24,31,32], the 2011, well since they were first used in production, and had never been before. On 20 July 4 inina No. power station composed of three imported fans. As shown in Figure [24,31,32],well the rotorUnit blades A fan are welded on the entrance control damper. The three fans 3performed the rotor broke apart during the process of switching fromdamper. low speed to high speed. Pieces of the rotor in No. fan are welded on the control Thebefore. three fans performed since blades they were firstAused in production, andentrance had never been overhauled On 20 July 2011,well the blades flew out of the volute. Figures from are shown in Figure 4 [24,31,32]. The the bearing since first usedthe in production, andthe hadscene never been overhauled 20 July 2011, rotor they brokewere apart during process of switching from low speed to highbefore. speed.On Pieces of the blades box in theout drive end andFigures the crushed. the main shaft isbearing seriously deformed. rotor broke apart during thecoupling process ofare switching fromBesides, low speed high speed. Pieces of the flew of the volute. from the scene are shown in Figure 4to[24,31,32]. The boxblades in the By analysis of the causes that produced the accident, we find that there were blade cracks in the flew the the volute. Figures the scene are shown in shaft Figure [24,31,32]. The bearing box in the driveout endofand coupling arefrom crushed. Besides, the main is 4seriously deformed. By analysis of drive end and coupling are crushed.we Besides, thethere mainwere shaftblade is seriously By analysis of booster thefans. causes thatthe produced the accident, find that cracks deformed. in the booster fans. the causes that produced the accident, we find that there were blade cracks in the booster fans.
Figure 3. Photos of the rotor blades of No. A fan.
Figure 3. Photos of the rotor blades of No. A fan. Figure 3. Photos of the rotor blades of No. A fan.
Figure 4. Site photos of the broken blade. Figure 4. Site photos of the broken blade.
Figure 4. Site photos of the broken blade. In order to extract the vibration characteristics during crack growth and gain experience for In order to extractonthe characteristics and gain experience condition monitoring thevibration same type of unit, the during testing crack systemgrowth as described before is usedfor to In orderthe to extract the vibration characteristics during crack growth gain condition monitoring on the same4 type of unit, system as described before isaccident), used to for analyze historical data. From August 2010 the to 6testing July 2011 (nearly a yearand before theexperience analyze the historical data. From 4type August 2010 to July (nearly a year before condition monitoring the of unit, the6 testing system as described before is used to data acquisition of on the twosame bearings, which support the 2011 rotor, was carried out by the theaccident), industry data acquisition ofdata. the two which rotor, out bythe the industry analyze the historical Frombearings, 4atAugust 2010support tofrequency 6 Julythe 2011 yearthe before technological service company the sampling of (nearly 2560was Hzacarried with length ofaccident), 4096, ten data technological service company at the sampling frequency of 2560 Hz with the length of 4096, ten times, including low speed (744 r/min) and high speed (993 r/min). Three groups of sensors are used, acquisition of the two bearings, which support the rotor, was carried out by the industry technological times, including speed (744 r/min) and high speed r/min). Three groups of sensors are used, as shown in Figure service company at low the5.sampling frequency of 2560 Hz(993 with the length of 4096, ten times, including as shown in Figure 5.
low speed (744 r/min) and high speed (993 r/min). Three groups of sensors are used, as shown in Figure 5.
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Figure three groups groupsof ofsensors. sensors. Figure5.5.A Asketch sketch of the three
Two sensors mounted two bearings. The vibration data from horizontal, vertical Two sensors areare mounted onon thethe two bearings. The vibration data from thethe horizontal, vertical and and radial directions are obtained in each sensor. In addition, a sensor is mounted on the motor. The radial directions are obtained in each sensor. In addition, a sensor is mounted on the motor. The data data the direction vertical direction is there. obtained details about the running status dataare from thefrom vertical is obtained Thethere. detailsThe about the running status data acquisition acquisition are shown in Table 1, where D means low speed, Dh means high speed and GE means shown in Table 1, where D means low speed, Dh means high speed and GE means the amplitude of the the amplitude of the envelope in g. A means radial direction, H indicates horizontal direction and V envelope in g. A means radial direction, H indicates horizontal direction and V is the vertical direction. is the vertical direction. Table 1. The acquired running sensor-dependent vibration data. Table 1. The acquired running sensor-dependent vibration data.
No. No. 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
Date Date 10-08-04 10-08-04 10-10-09 10-10-09 10-12-09 10-12-09 11-01-13 11-01-13 11-03-08 11-03-08 11-04-07 11-04-07 11-04-12 11-04-12 11-05-11 11-05-11 11-06-09 11-06-09 11-07-06 11-07-06
2# 2# AA HH mm/s mm/s mm/s mm/s DD DD DD DD DD DD DD DD DD DD Dh Dh Dh Dh Dh Dh Dh Dh D D D D DD DD DD D D
V GE GE D D D D D D D D Dh Dh Dh Dh D D D D D
3# 3# AA HH mm/s mm/s mm/s mm/s DD DD DD DD DD DD DD DD DD DD Dh Dh Dh Dh Dh Dh DhD Dh D DD DD DD DD D D
VV GEGE DD DD DD DD DD DhDh Dh DhD DD DD D
The vibration data from the drive end in the horizontal direction is chosen for analysis. We can The vibration data from the drive end in the horizontal direction is chosen for analysis. We can see that there is no a distinctive trend that would allow identifying crack faults. The waterfall plot see that there is no a distinctive trend that would allow identifying crack faults. The waterfall plot based on FFT of No. 2H sensor data from 0 to 100 Hz is displayed in Figure 6 [23,24]. We can clearly based on FFT of No. 2H sensor data from 0 to 100 Hz is displayed in Figure 6 [23,24]. We can clearly observe the rotating frequency and its harmonic components. As time goes by, the amplitude of the observe the rotating frequency and its harmonic components. As time goes by, the amplitude of the rotating frequency increases first and then drops. However, the amplitude of the second harmonic rotating frequency increases first and then drops. However, the amplitude of the second harmonic drops first andand then increases. From thethe point of view of the dynamics, thethe cracks close andand open twice drops first then increases. From point of view of the dynamics, cracks close open in twice one cycle of the rotation. The amplitude of the vibration response signal changes twice due to in one cycle of the rotation. The amplitude of the vibration response signal changes twice duethe change of the rotor large from to small, thesmall, characteristics of the amplitude in the to the change of stiffness the rotorfrom stiffness largesoto so the characteristics of theincrease amplitude second harmonic could be used tocould indicate a blade crack. a blade crack. increase in the second harmonic be used to indicate In In order to obtain more accurate amplitude and phase data dataofofthe therotating rotatingfrequency frequency and order to obtain more accurate amplitude and phase and thethe second harmonic component, a discrete spectrum interpolation method is used to analyze the signals second harmonic component, a discrete spectrum interpolation method is used to analyze the of signals the No.of2the and No. 3 sensors in theinhorizontal direction. TheThe result after correction No. 2 and No. 3 sensors the horizontal direction. result after correctionisisshown shown in in Figure 7 [23,24]. As shown in Figure theamplitude amplitudeof of the the rotating rotating frequency and Figure 7 [23,24]. As shown in Figure 7, 7, the frequencyfirst firstincreased increased and then decreased. The amplitude of the second harmonic component increases constantly. A breathing then decreased. The amplitude of the second harmonic component increases constantly. A breathing crack consideredtotoappear appearfrom fromJanuary January to to March March in rotor crack is is considered in 2011 2011 according accordingtotothe theresults. results.The The rotor stiffness changes twice in every rotating cycle. This leads to the increase of the amplitude of the second harmonic component.
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stiffness changes the Sensors 2016, 16, 632 twice in every rotating cycle. This leads to the increase of the amplitude of 20 stiffness changes twice in every rotating cycle. This leads to the increase of the amplitude of7 ofthe second Sensorsharmonic 2016, 16, 632component. 7 of 20 second harmonic component. stiffness changes twice in every rotating cycle. This leads to the increase of the amplitude of the second harmonic component.
Figure Figure 6. 6. The The FFT FFT waterfall waterfall from from 00 to to 100 100 Hz Hz of of the the No. No. 2H 2H sensor. sensor. Figure 6. The FFT waterfall from 0 to 100 Hz of the No. 2H sensor. Figure 6. The FFT waterfall from 0 to 100 Hz of the No. 2H sensor.
Figure 7. The trends of the amplitude value by the interpolation method for the discrete spectrum of Figure 7. The trends Figure trends of of the the amplitude amplitude value value by by the the interpolation interpolation method method for for the the discrete discrete spectrum spectrum of of the No. 7. 2HThe sensor. 7. sensor. The trends of the amplitude value by the interpolation method for the discrete spectrum of theFigure No. 2H 2H the No. 2H sensor.
For the purpose of extracting the non-stationary characteristics of the cracked rotor, STFT is For the purpose of extracting the non-stationary characteristics of the cracked rotor, STFT is employed to process the signal of the and cracked rotor. The result displayed in STFT Figure 8. ForFor the purpose of extracting thenormal non-stationary characteristics theiscracked cracked rotor, STFT is the purpose extracting non-stationary characteristics ofofthe rotor, is 8. employed to process theofsignal of thethe normal and cracked rotor. The result is displayed in Figure From the figure, we can find a component that is obviously equal to the rotation frequency of the fan, employed to process the signal of the normal and cracked rotor. The result is displayed in Figure 8. employed to process signal of the normal cracked rotor. result is displayed in Figure From the figure, we canthe find a component that and is obviously equalThe to the rotation frequency of the 8. fan, which is 12.5 Hz. When cracks appear, there is a discontinuous line at the frequency of the second From thethe figure, we can that obviouslyequal equal tothe the rotationfrequency frequency of the From figure, we canfind findaacomponent component thatisis isaobviously thesecond fan,fan, which is 12.5 Hz. When cracks appear, there discontinuous to line atrotation the frequency ofofthe harmonic, which fluctuates at the rotating frequency and its second harmonic, as shown in Figure which is 12.5 Hz. When cracks appear, there is a discontinuous line at the frequency of the second which is 12.5 Hz. When cracks appear, there is a discontinuous line at the frequency of the second 9. harmonic, which fluctuates at the rotating frequency and its second harmonic, as shown in Figure 9. harmonic, which fluctuates its second secondharmonic, harmonic,asasshown shownininFigure Figure harmonic, which fluctuatesatatthe therotating rotatingfrequency frequency and and its 9. 9.
Figure 8. The STFT result acquired from a normal blade. Figure8.8.The TheSTFT STFTresult result acquired acquired from Figure from aanormal normalblade. blade. The STFT result
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Figure9.9.The TheSTFT STFTresult result acquired acquired from Figure from aa cracked crackedblade. blade. Figure 9. The STFT result acquired from a cracked blade.
The photo of the cracked rotor on the A fan is displayed in Figure 10. In this section, two The photo of the cracked rotor on the A fan is displayed in Figure 10. In this section, two methods The photo of thetocracked on the A fault. fan isThe displayed in characteristics Figure 10. In of this two methods are applied analyze rotor the blade crack vibration thesection, crack blade are applied to analyze the blade crack fault. The vibration characteristics of the crack blade are methods are applied to help analyze the bladethe crack fault. The vibrationperformance characteristics the crack blade are extracted. This can to improve qualitative diagnosis forofblade cracks, but extracted. This This can can helphelp to improve thethe qualitative diagnosis performance forblade bladecracks, cracks, but are extracted. to improve qualitative diagnosis performance for but the detection of crack damage based on the vibration signals is still less studied, and especially thethe detection based the signals stillless lessplans studied, andespecially especially detectionofidentification ofcrack crackdamage damage based on the vibration vibration signals isis still studied, and aquantitative method foron scheduling reasonable maintenance is lacking. aquantitative identification method for scheduling reasonable maintenance plans is lacking. aquantitative identification method for scheduling reasonable maintenance plans is lacking.
Figure 10. The photo of the cracked blade in the A fan. Figure 10. The photo of the cracked blade in the A fan. Figure 10. The photo of the cracked blade in the A fan.
4. Quantitative Identification and Abnormality Alarm Strategy for Blade Crack Faults 4. Quantitative Identification and Abnormality Alarm Strategy for Blade Crack Faults 4. 4.1. Quantitative Identification and Abnormality The Proposed Quantitative Identification Method Alarm Strategy for Blade Crack Faults 4.1. The Proposed Quantitative Identification Method Sun Quantitative and Chen have attemptedMethod to propose a quantitative identification index for blade 4.1. TheBoth Proposed Identification Both Sun and Chen have attempted to propose a quantitative identification for blade crack identification and have obtained certain achievements [24,31], but a more index comprehensive Both Sun and Chen have attempted to propose a quantitative identification index for blade crack identification and have certainidentification achievementsindex [24,31], but be a more comprehensive abnormality alarm strategy viaobtained a quantitative should proposed to indicate crack identification and have obtained certain achievements [24,31], but a more comprehensive abnormality alarm strategy via a quantitative identification index maintenance, should be proposed to indicate the unbalancedness and implement necessary condition-based so a quantitative abnormality alarm strategy via quantitative identification shouldindex be proposed toindex indicate the unbalancedness and implement necessary condition-based maintenance, so quantitative identification method based onacompound indexes includingindex a traditional anda new is theidentification unbalancedness and implement necessary condition-based maintenance, so new a quantitative method based on compound indexes including a traditional index and index is developed in this section. identification based on compound indexes including a traditional index and new index developed inmethod this section. is developed in Lyapunov this section. 4.1.1. Largest Exponent Algorithm 4.1.1. Largest Lyapunov Exponent Algorithm Lyapunov exponents, whichAlgorithm measure the exponential rates of divergence or convergence of 4.1.1. Largest Lyapunov Exponent Lyapunov exponents, the exponential rates of divergence or processes. convergence of nearby trajectories in state which space, measure are generally calculated to characterize chaotic If the Lyapunov exponents, which theexponents exponential rates of divergence orthe convergence ofIfnearby nearby trajectories state space, are generally calculated to characterize chaotic processes. the largest value in the in spectrum ofmeasure Lyapunov is positive, it means that system is chaotic. trajectories in state space, aretogenerally calculated to characterize chaotic processes. If value largest value in the spectrum of Lyapunov exponents is positive, it means that the system is chaotic. The largest value equal zero indicates periodic or quasi-periodic dynamics. If the all largest Lyapunov largest value equal to zero indicates periodic itormeans quasi-periodic dynamics. If all Lyapunov in The the spectrum of Lyapunov exponents is positive, that the system is chaotic. The largest
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value equalare to negative zero indicates periodic or quasi-periodic If all Lyapunov are exponents then the stable critical point is an dynamics. attractor [29,30]. Among all exponents the Lyapunov negative then the stable critical point is an attractor [29,30]. Among all the Lyapunov exponents, exponents, the Largest Lyapunov Exponent (LLE) has aroused considerable interest for its the Largest Lyapunov Exponent (LLE) aroused considerable for its significant significant practical applications. Thehas LLE has been appliedinterest to many fields for its practical notable applications. has been applied to many as fields for its notable In thisof paper, the capabilities. InThe thisLLE paper, the LLE is calculated an indicator of thecapabilities. chaotic behavior the load LLE is calculated as an indicator of the chaotic behavior of the load demand by using Wolf’s algorithm, demand by using Wolf’s algorithm, which is given as [29,30]: which is given as [29,30]: 1 MM L '(1tk ) ÿln L pt q max 1 (4) λmax “ tm t0 k 1 lnL(tk 1 k) (4) t m ´ t0 Lptk´1 q k “1 where L’(tk) and L’(tk–1) mean the Euclidean distances computed between the nearest neighboring where L’(tk ) and L’(tk–1 ) mean the Euclidean distances computed between the nearest neighboring points on the different trajectories of the attractor at the tk and tk–1 time steps, respectively [29]. m points on the different trajectories of the attractor at the tk and tk–1 time steps, respectively [29]. indicates the number of replacement steps or iteration number. Details on the calculation parameter m indicates the number of replacement steps or iteration number. Details on the calculation parameter selection are given in [29,30]. The negative value of LLE indicates normal condition and a positive selection are given in [29,30]. The negative value of LLE indicates normal condition and a positive value of LLE indicates non-linear conditions, then the value of LLE can be used to initially identify value of LLE indicates non-linear conditions, then the value of LLE can be used to initially identify the mechanical system state. According to the principle of the LLE algorithm, the LLE value of the mechanical system state. According to the principle of the LLE algorithm, the LLE value of condition data from the No. 2H sensor is computed and displayed in Figure 11. We can observe that condition data from the No. 2H sensor is computed and displayed in Figure 11. We can observe the non-linear condition appears after 9 December 2010. In addition, we still need a quantitative that the non-linear condition appears after 9 December 2010. In addition, we still need a quantitative identification index to confirm the degree of crack fault. identification index to confirm the degree of crack fault.
Figure 11. The LLE value of the A fan No. 2H sensor. Figure 11. The LLE value of the A fan No. 2H sensor.
4.1.2. Relative Energy Energy Ratio 4.1.2. Relative Ratio at at Second Second Harmonic Harmonic Frequency Frequency Component Component According According to to the the previous previous section, section, we we know know that that when when aa breathing breathing crack crack appears, appears, the the rotor rotor stiffness changes twice from large to small in one rotation cycle [33,34]. The amplitude of the stiffness changes twice from large to small in one rotation cycle [33,34]. The amplitude of the second second harmonic harmonic increases increases obviously obviously and and the the phase phase also also changed changed significantly significantly [35,36]. [35,36]. Hence, Hence, aa quantitative quantitative identification method for blade crack fault and a new index are proposed to describe the identification method for blade crack fault and a new index are proposed to describe the degree degree of of damage accurately when cracks grow. The new index is called relative energy ratio at second harmonic damage accurately when cracks grow. The new index is called relative energy ratio at second frequency componentcomponent and expressed as K2f : harmonic frequency and expressed as K2f: f K2Kf 2 “
” ı22 AA2 f 2f j
2 j ” ř Aif ı2 Ai f i 1
11 ~„j j ii “
(5) (5)
i “1
where Af means the amplitude of the rotating frequency and A2f means the amplitude of the second harmonic. Moreover, we can find that the energy of the first six order harmonic accounts for more
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16, 632 10 of 20 thanSensors 98%2016, of the total energy, so based on the amplitude of the frequency spectrum, the parameter j is selected as j = 6. The value of Kthe 2f can be used thefrequency degree of the where a andofb the aresecond thresholds where Af means amplitude of to thejudge rotating and A2fdamage, means the amplitude thatharmonic. need to beMoreover, decided:we can find that the energy of the first six order harmonic accounts for more
(1) (2) (3)
than 98% of the total energy, so based on the amplitude of the frequency spectrum, the parameter j is 0 ≤ K2f < a : Normal condition, the rotor works well; selected as j = 6. a≤K 2f ≤ b: Incipient fault, a crack appears, but is small; The value of K2f can be used to judge the degree of the damage, where a and b are thresholds that K 2f > b: Serious fault, many cracks appear. need to be decided:
During period of August 2010 to December 2010, the rotor worked normally and we can (1) 0 ď Kthe 2f < a : Normal condition, the rotor works well; think of this fan remained at the normal level of 0.9 during this time. The values of K2f (2)thea reliability ď K2f ď b: Incipient fault, a crack appears, but is small; calculated the bearing the horizontal direction are shown in Figure 12. According to the (3) K2ffrom > b: Serious fault, data manyincracks appear. LLE value in Figure 11, a non-linear condition appeared after 9 December 2010. Moreover, During the period of August 2010 to December 2010, the rotor worked normally and we can according to the value of K2f in December 2010 to March 2011 as shown in the Figure 12, the think the reliability of this fan remained at the normal level of 0.9 during this time. The values of K2f threshold value a bearing is preliminarily as 0.02.areOn 20 July 2011,12.the rotor broke calculated fromofthe data in thedetermined horizontal direction shown in Figure According to theapart during process of 11, switching from low speed to high speed (from 744 to 993according r/min) and LLEthe value in Figure a non-linear condition appeared after 9 December 2010.r/min Moreover, to we can the think that the reliability of this fan has decreased to 0 at this time. Based on the lowest reliability value of K2f in December 2010 to March 2011 as shown in the Figure 12, the threshold value of a is requirement thedetermined steel-making plant forthe safe running and during the value of K2f inofbearing 2 from preliminarily as 0.02. On requires 20 July 2011, rotor broke apart the process switching speed2011, to high 744 r/min to 993 and we cansake thinkand thatthe the reliability reliability ofof this Junefrom 2011low to July thespeed value(from of b is determined asr/min) 0.25 for safety’s has decreased 0 at on themapping lowest reliability requirement theso steel-making fan this has fan decreased belowto0.5 atthis thistime. timeBased by linear relationship analysis, the parameters plant requires for safe running and the value of K in bearing 2 from June 2011 to July 2011, the value 2f are determined completely. of b is determined as 0.25 for safety’s sake and the reliability of this fan has decreased below 0.5 at this a the 0.02 time by linear mapping relationship analysis, so parameters are determined completely.
b 0.25
a “ 0.02
(6)
(6)
testing system and works well in the condition The new index K2f is introduced into the b “ 0.25 monitoring of the centrifugal booster fan. The results in Figure 12 could be employed to analyze the The new index K2f is introduced into the testing system and works well in the condition condition of No. A fan. monitoring of the centrifugal booster fan. The results in Figure 12 could be employed to analyze the Based on the proposed quantitative identification method, the abnormality alarm strategy can condition of No. A fan. be obtained, and then the condition-based maintenance actions can be arranged reasonably to Based on the proposed quantitative identification method, the abnormality alarm strategy can be ensure safe and andthen reliable operation. Tomaintenance sum up, the proposed quantitative identification obtained, the condition-based actions can be arranged reasonably to ensure and abnormality alarm operation. strategy procedure using sensor-dependent vibration data for blade crack safe and reliable To sum up,ofthe proposed quantitative identification and abnormality identification in centrifugal booster fans can be summarized by the flow chart displayed in alarm strategy procedure of using sensor-dependent vibration data for blade crack identification in Figure 13. centrifugal booster fans can be summarized by the flow chart displayed in Figure 13. 0.35
0.3
Amp
0.25
0.2
0.15
0.1
0.05
0 10-10-09
10-12-09
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Date Figure The resultsby bythe thequantitative quantitative identification index on on A fan No.No. 2H sensor. Figure 12.12. The results identification index A fan 2H sensor.
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Acquired sensor data
Largest Lyapunov exponent No
Relative energy ratio
K 2 f [a, b]?
LLE 0? Yes
No
Yes
Early fault warning
Compute
LLE & K 2 f No
LLE 0 & K 2 f b ? Yes Serious fault and maintenance outage Figure 13. The flow chart of the proposed quantitative identification and abnormality alarm strategy. Figure 13. The flow chart of the proposed quantitative identification and abnormality alarm strategy.
Meanwhile, the process of the proposed method and strategy for the mentioned engineering
Meanwhile, the process of the proposed method and strategy for the mentioned engineering tasks tasks in the power station can be summarized as follows: in the power station can be summarized as follows:
(1) Collect the sensor-dependent running condition vibration data; Pre-process this vibration data using the discretevibration spectrum data; interpolation method and STFT; (1)(2)Collect the sensor-dependent running condition (3) Compute the LLE and the relative energy ratio K2f. (2) Pre-process this vibration data using the discrete spectrum interpolation method and STFT; (4) Confirm the degree of blade crack fault(s) of the centrifugal booster fan; (3) Compute the LLE and the relative energy ratio K2f . (5) Conduct the corresponding maintenance management activities based on the fault degree.
(4) Confirm the degree of blade crack fault(s) of the centrifugal booster fan; (5)4.2.Conduct corresponding maintenance management activities based on the fault degree. Test and the Validation
also examined the performance of statistical features reported in the literature as 4.2. TestWe andhave Validation
comparisons to validate the performance of the proposed method. Some of the feature parameters Webeen havedemonstrated also examined performance of statistical features reported the literature have to bethe ineffective in previous publications, but in differentinpapers, different as comparisons to validate the performance proposed method. Some the feature parameters feature parameters are applied according of to the experience accumulated byof different researchers. In have been demonstrated to be ineffective in previous publications, but in different papers, different the different applications, different feature parameters give different diagnosis performance. Thus, manyparameters feature parameters are calculated in this study. In total, 21 feature values obtained, shown feature are applied according to the experience accumulated by are different researchers. in Table 2. These features are adopted to indicate the faulty from the acquired vibration In the different applications, different feature parameters givecondition different diagnosis performance. Thus, signals. Theparameters results are displayed in Figures 14–19. According to feature the analyzed figures, some results many feature are calculated in this study. In total, 21 values are obtained, shown can be2.obtained. First, obvious trendstocannot be found the majority the mentioned 21 vibration feature in Table These features are adopted indicate the faulty conditionoffrom the acquired parameters, exceptare for the feature values F14–19. 4, F11 and F12, so they of no use for blade signals. The results displayed in Figures According to theare analyzed figures, somecrack results identification in centrifugal booster fans. Next, the feature values of F 4, F11 and F12 show relatively can be obtained. First, obvious trends cannot be found the majority of the mentioned 21 feature clear trendsexcept compared to the remaining feature but the proposed strategy using compound parameters, for the feature values F4 , Fvalues, 11 and F12 , so they are of no use for blade crack feature parameters can indicate the running condition of centrifugal booster fans by hierarchical identification in centrifugal booster fans. Next, the feature values of F4 , F11 and F12 show relatively descriptions. The contrastive results demonstrate the effectiveness of the proposed strategy for the clear trends compared to the remaining feature values, but the proposed strategy using compound engineering task at hand.
feature parameters can indicate the running condition of centrifugal booster fans by hierarchical descriptions. The contrastive results demonstrate the effectiveness of the proposed strategy for the engineering task at hand.
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Table 2. The contrastive feature parameters.
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Parameter N
N
n 1
n 1
x ( n ) parameters. Table 2. The contrastive feature x (n) F1 x _ m
N
Category
( x (n) F ) N 1
n“1
n“1
x_r “
N
N
N
N
N
x(n)
N ř n=1 ( x ( n )) 2 ( x(nq)2 x _ m)3 x_av = pxpnq´F n 1 x _ rm s F3 x _ n“1 skew n 1 1 3 N N ( N 1) x _x_peak std x_std “ “ max |xpnq| x_pp “ maxpxpnqq ´ minpxpnqq N´1
d
F2 “
Time-Domain index
)2
max( x ( n )) min( x ( n )) x _ peak max x(n) x_ppN= ? 2 N ř ˆ ř |xpnq| ˙ xpnq
n 1
F1 “ x_m “
Time-Dom ain index
N
Parameter
2
1
F2 x _ std
x_r (
N
N
d
nN) pxpnq´x_mq x _ m) 3 ( x (ř 4
N ř
N ř
2
|xpnq| x _ peakpxpnqq x _ peak F5 x _ crest F6 x _ yu n“1 _ kur n 1 n“1 n“1 x_av “ x _ r N FF34 “x x_skew x_rms “ x _ rms 3 (“ N 1)pN´1qx_std x _ std 4 N
N ř x _ peak x4_ rms F8 x _ impx_peak F7 xpxpnq´x_mq _ shape x_peak n“1 x _ av F4 “ x_kur “ pN´1qx_std4 x _Fav “ x_crest “ 5 x_rms F6 “ x_yu “ x_r K
K
F9 K ř
F9 “
s(k ) k 1
K spkq
k“1
2
F10
F11
K 1
pspkq´F9 q2
k“1
k 1
F11 “
K´1
K
4
9
k 1
K ř
K
3
9
F10 “
K
K
x_peak F ) “x_rms ( s( k ) F8F “ ) x_imp F7( s(“k ) x_shape s ( k ) x_av F ) (“ x_av
f s(k )
9
F12 K Kř( pspkq´F F10 )3 9 q3 k“1 F12 ? 3 Kp F10 q
k 1
k
K2 ř F13 4k 1K KF 10 pspkq´F q
F13s(k“)
9
“
k“1 2 KF10
k 1
K ř
f k spkq
k“1 K ř
spkq FrequencyK F14 k“1 g g ( f F ) s( k ) 2 d f s ( k ) F f f k 18 F ř s ( k )ř K Ks ( k ) K K Domain ř fř ff k 1 2 4 Frequency-Domain p f k ´F13Kq2 spkq F f f k2 spkq F13 F f k“1 f k spkq f k“1F17f k spkq K K k“1 k“1 f f 4 F18 “ F15 “ e F16“f e s ( k )K index index F14 “ s (Kk ) Fs17(k )“df k s(k ) K K
2
k
14
13
K
k 1
2 k
k 1 K
15
K
ř
3
k
k 1
F19 “
k“1
13
k
13
KF143 3
KF14
F
2 k
k 1
spkq
k“1
F ) s3( k ) ř(Kf pf ´F q spkq
4 k
k 1 K
16
k 1
F19
K
ř
k“1
K
f k2 spkqk 1
4
K( f F 1 ) s ( k ) ř 3 k p f k ´F13 q4 spkq k 1
20 F20 “
k“1
KF1444
F21
k 1
K
spkq
K ř
k“1
k“1
(k ) (řKf k F13 )1/2 s1{2 p f k ´F13 q
k 1
F21 “
KF14
K ř
f k4 spkq
F14 F13
spkq
k“1
K KF? 14 F14
x(n) timedomain domain signal, n 2,= . 1,2,…,N; the sample point; is the of spectrum k= x(n)isis the time signal, n = 1, . . , N; N isN theissample point; s(k) is thes(k) spectrum x(n), k = 1,of2,x(n), . . . , K; K is the number of spectrum lines; f is the frequency of k-th spectrum line. K 1,2,…,K; K is the number of spectrum lines; fK is the frequency of k-th spectrum line. 0.35
3
0.3
2.8
0.25 2.6
0.2 0.15
2.4
0.1
2.2
0.05
2
0 1.8
-0.05 -0.1
1.6
1
2
3
4
5
6
1
(F1)
2
3
4
(F2) Figure 14. Cont.
5
6
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13 of 20 13 of 20 13 of 20 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 -0.1 -0.15 -0.15 -0.2 -0.21 1
2 2
3 3
4 4
5 5
6 6
(F3) (F3) Figure14. 14. The analyzed analyzed resultsof of A fanNo. No. 2H sensor sensor byFF1,, FF2 and 3. Figure and F Figure 14. The The analyzedresults results ofAAfan fan No.2H 2H sensorby by F11, F2 and FF33.. 2.5 2.5
2.5 2.5
2.4 2.4
2.4 2.4
2.3 2.3
2.3 2.3
2.2 2.2 2.1 2.1
2.2 2.2
2 2
2.1 2.1
1.9 1.9
2 2
1.8 1.8
1.9 1.9
1.7 1.7 1.6 1.61 1
2 2
3 3
4 4
5 5
6 6
1.8 1.81 1
2 2
3 3
(F4) (F4)
4 4
(F5) (F5)
3.6 3.6 3.4 3.4 3.2 3.2 3 3 2.8 2.8 2.6 2.6 2.4 2.4 2.2 2.21 1
2 2
3 3
4 4
5 5
6 6
(F6) (F6) Figure 15. The analyzed results of A fan No. 2H sensor by F4, F5 and F6. Figure15. 15. The Theanalyzed analyzedresults resultsof ofAAfan fanNo. No.2H 2Hsensor sensorby byFF4,, FF5 and F6. Figure 4 5 and F 6 .
5 5
6 6
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1.24 1.24
2.9 2.9 2.8 2.8 2.7 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2.1 2.1 2 1 2 1
1.22 1.22 1.2 1.2 1.18 1.18 1.16 1.16 1.14 1.14 1.12 1.12 1.1 1 1.1 1
2 2
3
4
3
5
4
(F7) (F7)
6
5
6
2
3
2
4
5 5
6 6
(F8) (F8)
-3
x 10 -3 14 x 10 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 1 5 1
4
3
2 2
3
4
3
5
4
6
5
6
(F9) (F9) Figure 16. The analyzed results of A fan No. 2H sensor by F7, F8 and F9. Figure A fan fan No. No.2H 2Hsensor sensorby byF7F, 7F,8Fand 8 and Figure16. 16.The Theanalyzed analyzed results results of of A F9. F 9 . -3
x 10 -3 7.5 x 10 7.5 7 7 6.5 6.5 6 6 5.5 5.5 5 5 4.5 4.5 4 4 3.5 3.5 3 3 2.5 1 2.5 1
2 2
3
4
3
4
5 5
6 6
44 44 42 42 40 40 38 38 36 36 34 34 32 32 30 30 28 28 26 26 24 1 24 1
2
3
2
4
3
(F10) (F10) 2000
4
5 5
(F11) (F11)
2000 1800 1800 1600 1600 1400 1400 1200 1200 1000 1000 800 800 600 1 600 1
2 2
3 3
4 4
5 5
6 6
(F12) (F12) Figure 17. The analyzed results of A fan No. 2H sensor by F10, F11 and F12. Figure 17.The Theanalyzed analyzedresults results of of A A fan fan No. No. 2H , F, 11F and F12. Figure 17. 2H sensor sensorby byFF1010 11 and F 12 .
6 6
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225 220
29
215
28
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27
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30 29 28 27 26
25
195
25
24
24
190 23
23
185 22
180 1175
1
2
2
3
4
3
4
5
5
6
21
6
22 1 21
1
2
2
3
(F13()F13) 365 360 355 350 345 340 335 330 325 320 315
3
4
4
5
5
6
6
(F14()F14)
365 360 355 350 345 340 335 330 325 320 1315
1
2
2
3
3
4
4
5
5
6
6
(F15()F15) Figure18. 18. Theanalyzed analyzedresults resultsof of A fanNo. No.2H 2Hsensor sensorby byFF13,, FF14 and FF15. . Figure Figure The 18. The analyzed resultsAoffan A fan No. 2H sensor by , Fand 14 and 13 F1314 15F15.
(F16()F16)
(F17()F17)
(F18()F18)
(F19()F19)
(F20()F20)
(F21()F21)
Figure 19. The analyzed results of A fan No. 2H sensor by F16–F21. Figure analyzed results A fan sensor F16–F21. Figure 19. 19. TheThe analyzed results of Aoffan No.No. 2H 2H sensor by Fby 16 –F 21 .
Moreover, thethe proposed blade crack quantitative identification method is applied to to identifying Moreover, proposed blade crack quantitative identification method is applied identifying Moreover, the proposed blade crack quantitative identification method is applied toprinciple identifying thethe running condition of No. A fan using A fan No. 3H sensor data. According to the of of running condition of No. A fan using A fan No. 3H sensor data. According to the principle the running condition of No. A fan using A fan No. 3H sensor data. According to the principle of the Largest Lyapunov Exponent algorithm, thethe LLE value of of thethe condition data from thethe No. 3H3H the Largest Lyapunov Exponent algorithm, LLE value condition data from No. LargestisLyapunov Exponent algorithm, the LLE value of the condition data from the No. 3H sensor is sensor computed and displayed in Figure 20. We can observe that the non-linear condition also sensor is computed and displayed in Figure 20. We can observe that the non-linear condition also computed and 9displayed in Figure 20. addition, We can observe that the condition also appears after appears after December 2010. In the index K 2f non-linear is also used to identify the running appears after 9 December 2010. In addition, the index K2f is also used to identify the running 9 December Infan addition, the index K3H also used toand identify the running conditionin ofFigure No. A fan condition of2010. No. A using A fan No. data thethe results areare displayed 21.21. 2f issensor condition of No. A fan using A fan No. 3H sensor data and results displayed in Figure From the result, we also can obtain a clear trend to indicate the blade cracks in centrifugal booster From the result, we also can obtain a clear trend to indicate the blade cracks in centrifugal booster
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using2016, A fan Sensors 16,No. 632 3H sensor data and the results are displayed in Figure 21. From the result, we also can 16 of 20
fansobtain and the value is above the abnormal warning stage after 8 March The result indicates a clear trend to indicate the blade cracks in centrifugal booster fans2011. and the value is above the that the proposed index and alarm strategy feasible. abnormal warning after March is 2011. The result indicates that the2011. proposed index indicates and alarmthat fans and the value isstage above the8 abnormal warning stage after 8 March The result strategy is feasible. the proposed index and alarm strategy is feasible. 0.01 0.008 0.01 0.006 0.008 0.004 0.006
Amp Amp
0.004 0.002 0.002 0
-0.002 0 -0.002 -0.004 -0.004 -0.006 -0.006 -0.008 -0.008 -0.01 10-10-09 -0.01 10-10-09
10-12-09 10-12-09
11-01-13
11-03-08
Date11-03-08 Date
11-01-13
11-06-09 11-06-09
11-07-06 11-07-06
Figure 20. The LLE value of A fan No. 3H sensor. Figure 20. The TheLLE LLEvalue valueofofAAfan fanNo. No.3H 3H sensor. Figure 20. sensor. 0.35 0.35 0.3 0.3 0.25 0.25
Amp Amp
0.2 0.2
0.15 0.15 0.10.1 0.05 0.05 0 0 10-10-09 10-10-09
10-12-09 10-12-09
11-01-13 11-01-13
11-03-08 11-03-08
11-06-09 11-06-09
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Date Date Figure The result thequantitative quantitativeidentification identification index AAfan Figure 21. The resultby bythe the quantitative index of of A No.No. 3H3H sensor. Figure 21.21. The result identification index offan fan No. 3Hsensor. sensor.
The proposed blade crack quantitativeidentification identificationmethod method also applied totoidentifying The proposed blade crack method is also applied identifying the The proposed blade crackquantitative quantitative identification is is also applied to identifying the the running condition of the remaining centrifugal booster fans. As mentioned previously, there are running conditionofofthe the remaining booster fans. fans. As mentioned previously, there are three running condition remainingcentrifugal centrifugal booster As mentioned previously, there are three fans in the unit 4. While No. A fan was being repaired, the quantitative identification index is is fans in the unit 4. While No. A fan was being repaired, the quantitative identification index is three fans in the unit 4. While No. A fan was being repaired, the quantitative identificationalso index also used toexamine examine the No.No. and No.These fans. These three fans have structures used to to examine the No. BNo. and C fans. threeThese fans have similar and functions, soand also used the BBand No. CC fans. three fansstructures have similar similar structures and functions, sobased the results based on the proposed method couldtobeanalyze employed to analyze the condition the results on the proposed method could be employed the condition of the No. B functions, so the results based on the proposed method could be employed to analyze the condition ofand theNo. No.CBfan. and After No. Ccalculation fan. After and calculation and comparison, the>values 0 andshow K2f = 0.024 show comparison, the values LLE 0 and LLE K2f =>0.024 that the of the No. B and No. C fan. After calculation and comparison, the values LLE > 0 and K2f = 0.024 show No. C fan is in an incipient fault state. It should be focused on and chosen for monitoring. In addition, that the No. C fan is in an incipient fault state. It should be focused on and chosen for monitoring. In that the No. C fan is in an incipient fault state. It should be focused on and chosen for monitoring. In the fault the status of the No.ofBthe fan No. (K2f B= fan 0.33) serious. It serious. needs to It beneeds repaired to repaired avoid it to addition, fault status (Kis2f much = 0.33)more is much more to be addition, theapart faultlike status of the During No. B fan (K 2f = 0.33) is much more serious. It needs to be repaired to breaking No. A fan. the checks, cracks were found in the No. B fan and No. C avoid it breaking apart like No. A fan. During the checks, cracks were found in the No. Bfan fanasand avoid it breaking apart like No. A fan. During the checks, cracks were found in the No. B fan and expected. Figures 22 and 23 [24]22 show Fromthe theresults. Figure 22a, wethe canFigure see that thewe crack No. C fan as expected. Figures andthe 23results. [24] show From 22a, cangrows see that No.the C fan as expected. Figures 22 and 23 [24] show the results. From the Figure 22a, we can see that from the fan entrance along thepropagates radial direction the damper. Figure 22b the crack grows fromand the propagates fan entrance and alongof the radial direction of shows the damper. theFigure crack 22b grows from the fan and entrance and propagates the radial direction damper. welding position ofthe the blade the damper. Stress concentration in this position led to of thethe growth shows welding position of the blade andalong the damper. Stress concentration in this Figure 22b shows the welding position of the blade and the damper. Stress concentration in at this position led to the growth and propagation of the crack. As shown in Figure 23, a crack appears position led to the growth anda propagation the in crack. shown in Figure 23, these a crack appears the welding position with length of 3–4ofcm No. As C fan. After detecting cracks, the at themaintenance welding position with a length of 3–4 cm in No. C fan. After detecting these cracks, the for No. B and No. C fans was carried out in March 2012 to prevent further accidents. maintenance forresults No. Bshow and No. C fans was carried out in March to prevent furtherdetection accidents. These above the effectiveness and robustness of the2012 proposed quantitative These above results show the effectiveness and robustness of the proposed quantitative detection
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Sensors 2016, 16, 632 and propagation of the crack. As shown in Figure 23, a crack appears at the welding position with17 a of 20
length of 3–4 Sensors 2016, 16, cm 632 in No. C fan. After detecting these cracks, the maintenance for No. B and No. C 17fans of 20 Sensors 2016, 16, 632 17 of 20 was carried out in show March the 2012effectiveness to prevent further These results show the effectiveness These above results andaccidents. robustness of above the proposed quantitative detection These above show the effectiveness and robustness theblade proposed quantitative detection and robustness of thefaults. proposed quantitative detection methodoffor crack faults. method for bladeresults crack These above results show the effectiveness and robustness of the proposed quantitative detection method for blade crack faults. method for blade crack faults.
(a)(a)
(b) (b) (b)
(a)
Figure 22. Photo of the crack on B fan. Figure 22. Photo of the crack on B fan.
Figure ofthe thecrack crackonon fan. Figure22. 22. Photo Photo of BB fan.
Figure 23. Photo of C fan after repair welding. Figure23. 23.Photo Photoof ofCCfan fanafter afterrepair repairwelding. welding. Figure
23. PhotoVibration of C fan after repair welding. 5. Mathematical ModellingFigure for Revealing Signal Properties 5. Mathematical Modelling for Revealing Vibration Signal Properties 5. Mathematical Modelling for Revealing Vibration Signal Properties An impeller consists for of a cover component, a disk component and several blades. The finite 5. Mathematical Modelling Vibration Signal Properties An impeller impeller consists of of aRevealing cover component, component, a disk disk component and several several blades. blades. The The finite finite An component and element model ofconsists an impelleraiscover depicted in Figurea 24. element model of an impeller is depicted in Figure 24. element model consists of an impeller depicted in Figure 24. An impeller of a is cover component, a disk component and several blades. The finite
element model of an impeller is depicted in Figure 24.
crack crack
crack
Cover Cover
Cover Blade Blade Disk Blade Disk
Figure 24. Finite element model of an impeller and a sector model. Figure Figure24. 24.Finite Finiteelement elementmodel modelof ofan animpeller impellerand andaasector sectormodel. model. Disk According to many experienced engineers, cracks initiate mostly at the weld toe on the cover
to many experienced engineers, cracks initiate weld located toe on the covera sidesAccording of the blade, as shown in Figure 24. In this paper, the mostly effects at ofthe a crack at such Figure 24. Finite element model of an impeller and a sector model. sides of the blade, as shown in Figure In thisare paper, the effects of cracks a crack at suchcan a position on the vibration response of the24. impeller of interest, and the at located other position position on the vibration response of the impeller are of interest, and the cracks at other position can be modeled without much modification. For details of the mathematical modelling and According many experienced engineers, cracks initiate mostly at the weld toe on the be modeled towithout much modification. details of the mathematical modelling andcover mathematical formulation readers may refer toFor [34,37]. formulation readers may refer to [34,37]. sidesmathematical of the blade, as shown in Figure 24. In this paper, the effects of a crack located at such a
position on the vibration response of the impeller are of interest, and the cracks at other position can
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According to many experienced engineers, cracks initiate mostly at the weld toe on the cover sides of the blade, as shown in Figure 24. In this paper, the effects of a crack located at such a position on the vibration response of the impeller are of interest, and the cracks at other position can be modeled without much modification. For details of the mathematical modelling and mathematical formulation readers may refer to [34,37]. After component synthesis, the equation of motion of a complete impeller is represented by: ..
.
Mp ` Cp ` Kp “ b ` fnl ppq
(7)
where M, C and K are the mass, damping and stiffness matrices of the ROM; p is the vector of the DOFs retained; b denotes the external excitation acting on the impeller; and fnl (p) is the nonlinear forces caused by intermittent contact of the crack surfaces. More details of this modelling process can be found in [34]. One of the most important issues for crack detection and identification is to search for sensitive indicators. An effective indicator should possess several features, such as robustness, monotonicity and industrial testability. The resonant frequencies discussed in the previous sections are potential indicators for the quantitative detection of crack faults. Two other kinds of frequency based indicators are studied in [34]. In sum, the frequency-based indicators for crack identification of centrifugal impellers were studied and discussed. However, an effective and reliable tool with a sensitive indicator for crack identification of impellers in operation still faces a lot of challenges at present. 6. Conclusions In this paper, a vibration analysis method for the purpose of detection and quantitative identification of blade crack faults based on the amplitude of the rotating frequency is proposed. Aiming at the problem of energy leakage in FFT, a discrete spectrum interpolation method is proposed to extract the amplitude and phase accurately first. Then a quantitative identification and abnormality alarm strategy based on compound indexes including the Largest Lyapunov Exponent and relative energy ratio of the second harmonic frequency component is proposed. The results show that the proposed method is feasible. In the future, more effective signal processing methods should be studied and used to extract the characteristics of blade crack faults. Moreover, dynamic modeling and analysis of cracked rotor blades is necessary and urgent in future work. More reasonable effective indexes could be constructed to indicate the crack initiation and propagation from the point of view of dynamic analysis. Furthermore, although the proposed method shows good performance, more reasonable parameter selection for the terms a and b in K2f should be studied based on plenty of running condition sensor-dependent vibration data in the future, and urgent demands, including quantitative diagnosis and fault location techniques, still remain to be established for scheduling reasonable maintenance plans. Acknowledgments: This research was supported financially by the Project of National Natural Science Foundation of China for Innovation Research Group (Grant No. 51421004), National Natural Science Foundation of China (Grant No. 51405379, 51505036), China Postdoctoral Science Foundation (Grant No. 2014M562396, Grant No. 2015T81017), Fundamental Research Funds for the Central Universities of China (Grant No. XJJ2015106) and Shaanxi Industrial Science and Technology Project (Grant No. 2015GY121). The authors would like to sincerely thank all the anonymous reviewers for their valuable comments that greatly helped to improve the manuscript. Special thanks are expressed to Binqiang Chen, Zitong Zhou and Zipeng Li for helping with describing and analyzing the experimental data. Author Contributions: H.S. and J.C. conceived and designed the experiments; H.S. performed the experiments; J.C., H.S. and S.W. analyzed the data; H.S. contributed materials; J.C. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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References 1. 2. 3. 4. 5.
6. 7. 8.
9. 10. 11.
12. 13. 14. 15. 16. 17. 18. 19. 20.
21. 22.
23.
Zhao, J.; Wang, W.; Liu, Y.; Pedrycz, W. A two-stage online prediction method for a blast furnace gas system and its application. IEEE Trans. Control Syst. Technol. 2011, 19, 507–520. [CrossRef] Steer, J.M.; Marsh, R.; Greenslade, M.; Robinson, A. Opportunities to improve the utilisation of granulated coals for blast furnace injection. Fuel 2015, 151, 40–49. [CrossRef] Wan, Z.G.; Cao, H.R.; Zi, Y.Y.; He, W.P.; He, Z.J. An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack. Eng. Fail. Anal. 2014, 42, 157–177. [CrossRef] Luo, Y.G.; Zhang, S.H.; Wu, B.; Hu, H.Y. Stability analysis of nonlinear stiffness rotor-bearing system with pedestal looseness fault. Appl. Mech. Mater. 2013, 483, 285–288. [CrossRef] Corbetta, M.; Sbarufatti, C.; Manes, A.; Giglio, M. Real-time prognosis of crack growth evolution using sequential Monte Carlo methods and statistical model parameters. IEEE Trans. Reliabil. 2015, 64, 736–753. [CrossRef] Sun, C.; Zhang, Z.S.; Guo, T. A novel manifold–manifold distance index applied to looseness state assessment of viscoelastic sandwich structures. Smart Mater. Struct. 2014, 23, 065019. [CrossRef] Liu, C.; Jiang, D.X.; Chu, F.L. Influence of alternating loads on nonlinear vibration characteristics of cracked blade in rotor system. J. Sound Vib. 2015, 353, 205–219. [CrossRef] Wang, Y.F.; Liang, M.; Xiang, J.W. Damage detection method for wind turbine blades based on dynamics analysis and mode shape difference curvature information. Mech. Syst. Signal Process. 2014, 48, 351–367. [CrossRef] Yang, B.; Sun, D.B. Testing, inspecting and monitoring technologies for wind turbine blades: A survey. Renew. Sustain. Energy Rev. 2013, 22, 515–526. [CrossRef] Kubiak, J.; Urquiza, G.; Rodriguez, J.A.; González, G.; Rosales, I.; Castillo, G.; Nebradt, J. Failure analysis of the 150 MW gas turbine blades. Eng. Fail. Anal. 2009, 16, 1794–1804. [CrossRef] Li, F.; Wang, J.X.; Chyu, M.K.; Tang, B.P. Weak fault diagnosis of rotating machinery based on feature reduction with Supervised Orthogonal Local Fisher Discriminant Analysis. Neurocomputing 2015, 168, 505–519. [CrossRef] Zhang, X.L.; Wang, B.J.; Chen, X.F. Operational safety assessment of turbo generators with wavelet rényi entropy from sensor-dependent vibration signals. Sensors 2015, 15, 8898–8918. [CrossRef] [PubMed] Witek, L. Crack propagation analysis of mechanically damaged compressor blades subjected to high cycle fatigue. Eng. Fail. Anal. 2011, 18, 1223–1232. [CrossRef] Infante, V.; Silva, J.M. Case studies of computational simulations of fatigue crack propagation using finite elements analysis tools. Eng. Fail. Anal. 2011, 18, 616–624. [CrossRef] Lin, L.; Chu, F.L. HHT-based AE characteristics of natural fatigue cracks in rotating shafts. Mech. Syst. Signal Process. 2012, 26, 181–189. [CrossRef] Fang, D.; Zivanovic, R. Condition Monitoring of an Induction Motor Stator Windings Via Global Optimization Based on the Hyperbolic Cross Points. IEEE Trans. Ind. Electron. 2015, 62, 1826–1834. Qiao, W.; Lu, D. A survey on wind turbine condition monitoring and fault diagnosis. IEEE Trans. Ind. Electron. 2015, 99, 6536–6545. [CrossRef] Liu, K.; Zhu, Z.Q. Position-offset-based parameter estimation using the adaline NN for condition monitoring of permanent-magnet synchronous machines. IEEE Trans. Ind. Electron. 2015, 62, 2372–2383. [CrossRef] Seungdeog, C.; Pazouki, E.; Baek, J.; Bahrami, H.R. Iterative condition monitoring and fault diagnosis scheme of electric motor for harsh industrial application. IEEE Trans. Ind. Electron. 2015, 62, 1760–1769. Chen, J.L.; Zhang, C.L.; Zhang, Y.Y.Z.X.Y.; He, S.L.; Yang, Z. Planetary gearbox condition monitoring of ship-based satellite communication antennas using ensemble multiwavelet analysis method. Mech. Syst. Signal Process. 2015, 54–55, 277–292. [CrossRef] Gao, Z.W.; Ding, S.X.; Cecati, C. Real-time fault diagnosis and fault-tolerant control. IEEE Trans. Ind. Electron. 2015, 62, 3752–3756. [CrossRef] Gao, Z.W.; Cecati, C.; Ding, S.X. A survey of fault diagnosis and fault-tolerant techniques—Part II: Fault diagnosis with knowledge-based and hybrid/active approaches. IEEE Trans. Ind. Electron. 2015, 62, 3768–3774. [CrossRef] Sun, H.L.; Zi, Y.Y.; He, Z.J.; Chen, B.Q.; Liu, H. Feature extraction and its application for nonstationary features of the blade crack in centrifugal booster fans. J. Vib. Shock 2012, 31.
Sensors 2016, 16, 632
24. 25.
26. 27. 28.
29.
30. 31. 32.
33. 34. 35. 36. 37.
20 of 20
Sun, H.L. Theory and Applications of Adaptive Multiwavelets in Quantitative Fault Diagnosis for Key Mechanical Equipments. Ph.D. Dissertation, Xi’an Jiaotong University, Xi’an, China, 2013. Kim, Y.H.; Youn, Y.W.; Hwang, D.H.; Sun, J.H.; Kang, D.S. High-Resolution Parameter Estimation Method to Identify Broken Rotor Bar Faults in Induction Motors. IEEE Trans. Ind. Electron 2013, 60, 4103–4116. [CrossRef] Amar, M.; Gondal, I.; Wilson, C. Vibration spectrum imaging: A novel bearing fault classification Approach. IEEE Trans. Ind. Electron 2015, 62, 494–502. [CrossRef] Lin, H.B.; Ding, K. Energy based signal parameter estimation method and a comparative study of different frequency estimators. Mech. Syst. Signal Process. 2011, 25, 452–462. Sharma, G.K.; Kumar, A.; Rao, C.B.; Jayakumar, T.; Raj, B. Short time fourier transform analysis for understanding frequency dependent attenuation in austenitic stainless steel. NDT E Int. 2013, 53, 1–7. [CrossRef] Wang, J.Z.; Jia, R.L.; Zhao, W.G.; Wu, J.; Dong, Y. Application of the largest Lyapunov exponent and non-linear fractal extrapolation algorithm to short-term load forecasting. Chaos Solitons Fractals 2012, 45, 1277–1287. [CrossRef] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from A time series. Phys. D Nonlinear Phenom. 1985, 16, 285–317. [CrossRef] Chen, B.Q. Overcomplete Frame Expansion Based on Wavelet Tight Frame and Its Applications to Machinery Fault Detection and Diagnosis. Ph.D. Dissertation, Xi’an Jiaotong University, Xi’an, China, 2013. Chen, B.Q.; Zhang, Z.S.; Zi, Y.Y.; He, Z.J.; Sun, C. Detecting of transient vibration signatures using an improved fast spatial-spectral ensemble kurtosis kurtogram and its applications to mechanical signature analysis of short duration data from rotating machinery. Mech. Syst. Signal Process. 2013, 40, 1–37. [CrossRef] Chen, J.L.; Zi, Y.Y.; He, Z.J.; Wang, X.D. Adaptive redundant multiwavelet denoising with improved neighboring coefficients for gearbox fault detection. Mech. Syst. Signal Process. 2013, 38, 549–568. [CrossRef] Wang, S.; Zi, Y.Y.; Li, B.; Zhang, C.L.; He, Z.J. Reduced-order modeling for mistuned centrifugal impellers with crack damages. J. Sound Vib. 2014, 333, 6979–6995. [CrossRef] Kuang, J.; Huang, B. The effect of blade crack on mode localization in rotating bladed disks. J. Sound Vib. 1999, 227, 85–103. [CrossRef] Fang, X.; Tang, J.; Jordan, E.; Murphy, K. Crack induced vibration localization in simplified bladed-disk structures. J. Sound Vib. 2006, 291, 395–418. [CrossRef] Wang, S.; Zi, Y.Y.; Wan, Z.G.; Li, B.; He, Z.J. Effects of multiple cracks on the forced response of centrifugal impellers. Mech. Syst. Signal Process. 2015, 60–61, 326–3438. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
