Investigation on Planetary Bearing Weak Fault Diagnosis Based on a Fault Model and Improved Wavelet Ridge

energies

Article Investigation on Planetary Bearing Weak Fault Diagnosis Based on a Fault Model and Improved Wavelet Ridge

Hongkun Li *, Rui Yang ID , Chaoge Wang and Changbo He

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China; [email protected] (R.Y.); [email protected] (C.W.); [email protected] (C.H.) * Correspondence: [email protected]; Tel.: +86-0411-8470-6561

Received: 12 April 2018; Accepted: 9 May 2018; Published: 17 May 2018

Abstract: Rolling element bearings are of great importance in planetary gearboxes. Monitoring their operation state is the key to keep the whole machine running normally. It is not enough to just apply traditional fault diagnosis methods to detect the running condition of rotating machinery when weak faults occur. It is because of the complexity of the planetary gearbox structure. In addition, its running state is unstable due to the effects of the wind speed and external disturbances. In this paper, a signal model is established to simulate the vibration data collected by sensors in the event of a failure occurred in the planetary bearings, which is very useful for fault mechanism research. Furthermore, an improved wavelet scalogram method is proposed to identify weak impact features of planetary bearings. The proposed method is based on time-frequency distribution reassignment and synchronous averaging. The synchronous averaging is performed for reassignment of the wavelet scale spectrum to improve its time-frequency resolution. After that, wavelet ridge extraction is carried out to reveal the relationship between this time-frequency distribution and characteristic information, which is helpful to extract characteristic frequencies after the improved wavelet scalogram highlights the impact features of rolling element bearing weak fault detection. The effectiveness of the proposed method for weak fault recognition is validated by using simulation signals and test signals.

Keywords: planetary bearing; fault model; improved wavelet scalogram; initial impact feature; wavelet ridge

1. Introduction Condition monitoring, fault diagnosis and periodic maintenance are indispensable steps to maintain the normal operation of rotating machinery [1,2]. Fault diagnosis and pattern recognition are very important and difficult parts of this task. At present, signal processing methods are the most popular fault diagnosis techniques due to the fact vibration signals are easily collected by sensors, but feature extraction is an essential and complex question for fault diagnosis of rotating machinery based on vibration signal analysis due to the fact vibration signals are usually disturbed by the external circumstances in practical engineering applications. Meanwhile, early fault signal characteristics are very weak and may even be buried under strong background noise, so with traditional signal processing methods it is difficult to analyze this kind of signal. Many researchers have focused their attentions on weak impact feature extraction. Due to their compact structure, accurate transmission ratio, smooth transmission, and high efficiency, planetary gearboxes are widely used in automobile, aerospace, petrochemical and other machinery manufacturing industries. Rolling element bearings are the most extensively available components in planetary gearboxes [2]. The planetary gearbox differs from the traditional parallel

Energies 2018, 11, 1286; doi:10.3390/en11051286 www.mdpi.com/journal/energies Energies 2018, 11, 1286 2 of 23 gearbox in that it consists of one or more planet gears, a sun gear, and a planet carrier. Planet gears are distributed around the sun gear and driven by the planet carrier to rotate around the sun gear and also rotate around their own rotation axis. Due to the complexity of the planetary gear structure and operation environment, vibration signal components are no longer as simple as in a parallel gearbox. In the event of planetary gearbox failure, vibration mechanism and vibration signal processing methods are two main research areas. The study of mechanical equipment failure mechanisms is the basis for other research. This provides theoretical support for investigating the changes of running component parameters and vibration signal models before and after a failure. The research on the vibration mechanism of gearboxes is mainly focused on dynamic models and vibration signal response analysis. McFadden and Smith [3] indicated that asymmetrical sidebands are contained in the vibration spectrum of planetary gearboxes and explained that this phenomenon is affected by the number of gear ring teeth and the number of planet gears. Mosher [4] established the kinetic equations of planetary gearboxes and predicted that the frequency component with larger amplitude in the frequency spectrum is located at the meshing frequency and its multiplied frequency, and there is a sideband near it. Inalpolat and Kahraman [5] developed a mathematical model to study the sideband mechanism in the spectrum of planetary vibration signals, thereby classifying planetary gearboxes into five types. Subsequently, they added dynamic meshing stiffness and damping analysis to their original model to perfect the mathematical model of planetary gearboxes [6]. However, they considered that the planet gears transmit vibration to sensors only within 1/3 rotation cycle and the vibration signal intensity at other times was considered to be zero. Feng et al. [7] established a vibration signal model of a planetary gearbox in normal operation state and in local and distributed fault states. However, in the establishment of the vibration signal model, only the motion of a single planetary gear is taken into consideration. Neither the manufacturing error nor the meshing phase difference is analyzed in the meshing process. Liang et al. [8] simulated the vibration source signal by constructing kinetic equations. Then, an improved Hamming window function is used to show the influence of signal transmission path on the vibration signal model. Taking comprehensive consideration of each source signal vibration, the vibration signal simulation model of a planetary gearbox in a healthy state and with a cracked sun gear are built. Lei et al. [9] analyzed the vibration signal transmission path and meshing phase of planetary gearboxes according to the transmission mechanism and meshing principle and established the vibration signal model for normal operation and fault conditions. Ding et al. [10] adopted the ring gear modal experimental to verify the variation of vibration amplitude with the transmission path. To summarize, it can be seen that the research on planetary gearboxes based on vibration mechanisms is mostly carried out under the conditions of healthy state or gear failure. The literature on failure mechanism of planetary bearing is very scarce. Most researchers are accustomed to establishing kinetic equations to study vibration mechanisms. However, the kinetic equations are based on the assumption of multiple parameters. In the event of a fault, the structural properties will change and the value of parametric variation cannot be accurately measured. Therefore, it is only applicable to the analysis of healthy gearbox and the fault characteristics cannot be described accurately and completely [2]. In addition, it will take a long time from the start of the fault to when significant impact features are discovered, which may lead to huge economic losses and even casualties. It would make no sense to detect planetary bearing faults by using traditional fault diagnosis methods when the machine has already broken down. Therefore, weak impact feature identification in rolling element bearings is significant to reduce downtime and safety hazards during the running period. Nowadays, a large number of modern approaches have been proposed, such as kurtograms [11,12], improved kurtograms and spectral kurtosis [13–15], cyclostationary analysis [16,17], empirical mode decomposition [18], synchronous averaging technology [19,20], sparse models [21,22], stochastic resonance [23], wavelet spectra [24], etc. Energies 2018, 11, 1286 3 of 23

When the impact characteristics of a signal are relatively weak or submerged in fault-independent interference components, they require further analysis [13]. In some circumstances, time domain analysis and frequency domain analysis are available for simple fault signal processing. However, most bearing fault signals are both non-stationary and non-linear. As a result, time-frequency distribution will be a good choice [25]. Nowadays, the main time-frequency distribution methods, including Short Time Fourier Transform and Wigner-Ville distribution, are very useful for dynamic vibration signal analysis. However, they have a poor performance in extracting weak impact features in the case where the signal is disturbed by strong background noise. To solve this problem, another method named Hilbert spectrum is studied in signal processing [26,27], although it has a limitation in time-frequency resolution. The background noise can be ignored as an insignificant ingredient because it has nothing to do with fault diagnosis. Based on this property, several new time-frequency analysis methods are extensively used in weak fault feature extraction. For example, wavelet analysis has multi-resolution characteristics and can detect well the local characteristics of signals. It is very effective in analyzing non-stationary signals. Recently, lots of improved algorithms have been put forward, that are very effective for a more complicated signal analysis. He proposed a novel method which takes the time-frequency manifold as a template to do a correlation matching to enhance the periodic faults’ impulse components [28]. Time domain averaging can reduce the random disturbance and retain the periodic components. Time synchronous averaging carried out for wavelet coefficients in the range of all scales obtained by wavelet transform [29] can improve the time-frequency resolution. This is a feasible approach for noisy fault signal feature extraction. Besides that, time-frequency distribution based on wavelet scalograms can reveal signal feature information. In some cases, it fails to detect the fault symptoms and lacks concentration. The improved scalograms named reassigned wavelet scalograms have a better concentration in time-frequency plane [30]. In this paper, to extract fault features more accurately, the fault mechanism of planetary bearings is studied. A fault simulation model is established considering the influence of various factors. Then, a new method of improved wavelet scalograms is proposed. It combines time-frequency distribution reassignment with synchronous averaging. A wavelet ridge is also determined to improve the veracity of mechanical fault feature recognition. The organization of the rest paper is as follows: in Section2, the fault mechanism is introduced and the simulation signal model is established. The theory of the proposed method is introduced in Section3. Section4 presents the validity of the proposed method by simulated signal and weak rolling element bearing fault data. Finally, some major concluding remarks are given in Section5.

2. Fault Model

2.1. Meshing Frequency and Fault Characteristic Frequency During the operation of a planetary gearbox, the planet carrier and sun gear rotate in the same direction. The planet gear and the sun gear mesh with each other, whilst the planet gears orbit around their own rotation axis and the planet carrier, as shown in Figure1. It is assumed that the rotation direction of planet carrier is positive and its rotational frequency is fc, the planetary gear rotational (r) frequency is fp, the absolute rotation frequency of planet gear is fp . The gear ring is connected with the case and it rotates with respect to the planet carrier at the frequency fc. The meshing frequency of planetary gearbox is: fm = Nr fc = Ns( fs − fc) = Np( fp + fc) (1) wherein fm is the meshing frequency of planetary gearbox; Nr is the number of gear ring tooth; Ns is the number of sun gear tooth; Np is the number of planet gear tooth; fs is the rotational frequency of sun gear. Energies 2018, 11, 1286 4 of 23 Energies 2018, 11, x 4 of 23

2 1 4 5 6

FigureFigure 1. 1.PlanetaryPlanetary gearbox model model diagram. diagram.

Taking Takinga fixed a ﬁxedpoint point on onthe the housing housing as as reference referenc point,e point, the rotational the rotational frequency frequency of planetary of gear planetary = gear innerinner ring ring isis equal equal to theto planetarythe planetary carrier’s, thatcarrier’s, is fpi = thatfc. The is rotational ffpi c frequency. The rotational of planetary frequency gear of outer ring relative to the reference point is the absolute rotational frequency of planetary gear, that is planetary gear(r )outer ring relative to the reference point is the absolute rotational frequency of f = f = f Z /Z . po p c r p ==()r planetary gear,The rotationalthat is ff frequencypo p of fZZ planetc r/ gear p . bearing cage relative to this reference point is:

The rotational frequency of planet gear (bearin− d g) cage relatived to this reference point is: fpi 1 D cos α fpo(1+ D cos α) fcg = 2 + 2 f dd(1− d cos α) f Z (1− d cos α) = −+c D αα+ c r D , (2) ffpi(1cos2 ) po (12Zp cos ) =+f DD(Z +Z )+ f (Z −Z ) d cos α fcg = c r p c r p D 222Zp dd The characteristic frequency of planet−− gearαα bearing outer ring is: ffZccr(1cos ) (1 cos ) =+DD d d (2) fc Zr fc(1− cos α) fc Zr(1+ cos α) f = N( f − f )22 = N( − D Zp − D ) BPFO po cg Zp 2 2Zp f (Z +Z )+ f (Z −Z ) d cos α = N( fc Zr − c r p c r p D d ) , (3) Zp +−2Zp α fZcr(+) Z p fZ cr ( Z p ) cos f (Z −Z )(1− d cos α) = N(=c r p D ) D 2Zp 2Z p , The characteristic frequency of planet gear bearing inner ring is: The characteristic frequency of planet gear bearing outer ring is: f (1− d cos α) f Z (1+ d cos α) f = N( f − f ) = N( f − c D + c r D ) BPFI pi cg c 2 2Zp f (Z +Z )+ f (Z −Z )ddd c r p cf(1-r p D cosαα)fZ(1+ cosα) = N( fc − 2Zccr) , (4) fZcr p DD f =N(f -ff ( )=N(Z −Z )(1+ d cos -α) - ) BPFO= poN cgc p r D 2ZpZ2pp 2Z d wherein, N is the number of rolling elements,f(Z+Z)+f(Z-Z)d is the roller diameter,cosDαis the bearing pitch diameter, fZ cr p cr p α is the contact angle. f=N(cg is thecr rotational - frequency of bearingD cage. fBPFO ) and fBPFI represent the (3) characteristic frequencies of planetZ2Zpp gear bearing outer ring and planet gear bearing inner ring. d 2.2. Fault Simulation Model f(Z-Z)(1- cosα) cr p D During the operation=N( of planetary gearbox, the meshing ) points are shown as points 1 to 6 in 2Z Figure1. In order to analyze the fault mechanism,p the vibration signal model is established, in the case of sensors are installed on the housing. In the paper, only time-varying transmission paths are considered, Thethat characteristic is shown in Figurefrequency2. The of positions planet ofgear all planetary bearing gearsinner are ring constantly is: changing during the operation. The transmission path of vibration signal generated by each meshing points to sensor dd f(1- cosα)fZ(1+ cosα) ccrDD fBPFI=N(f pi -f cg )=N(f- c + ) 22Zp d f(Z+Z)+f(Z-Z)cr p cr p cosα D (4) =N(f-c ) 2Zp d fcp (Z - Z r )(1+ cosα) =N D 2Z p ,

Energies 2018, 11, x 5 of 23 wherein, N is the number of rolling elements, d is the roller diameter, D is the bearing pitch diameter, α is the contact angle. fcg is the rotational frequency of bearing cage. fBPFO and fBPFI represent the characteristic frequencies of planet gear bearing outer ring and planet gear bearing inner ring.

2.2. Fault Simulation Model During the operation of planetary gearbox, the meshing points are shown as points 1 to 6 in Figure 1. In order to analyze the fault mechanism, the vibration signal model is established in the case of sensors are installed on the housing. In the paper, only time-varying transmission paths are considered, that is shown in Figure 2. The positions of all planetary gears are constantly changing during the operation. The transmission path of vibration signal generated by each meshing points to sensor is time-varying. Vibration transmission is mainly through these two paths. The first Energies 2018, 11, 1286 5 of 23 time-varying path starts from planet gear bearing to planet gear, and then to gear ring. Finally, it passes through the case to sensors. The second path has one more transmission of sun gear to planet is time-varying. Vibration transmission is mainly through these two paths. The first time-varying gear thanpath the starts first from one, planet and gear its bearingtransmission to planet leng gear,th and does then tonot gear change ring. Finally, with itthe passes rotation through of planet carrier. Therefore,the case to sensors. the second The second path pathis an has equal-pr one moreoportional transmission attenuation of sun gear toprocess planet gearrelative than theto the first one. Assumingfirst one, that and itsthe transmission amplitude length modulation does not changeeffect withproduced the rotation by the of planetfirst path carrier. is Therefore,wi (t) , and then the second path is an equal-proportional attenuation process relative to the first one. Assuming that the the amplitude modulation effect produced by the second path can be expressed as κw(t)i , where amplitude modulation effect produced by the first path is wi(t), and then the amplitude modulation 0<κ <1effect. produced by the second path can be expressed as κwi(t), where 0 < κ < 1.

Figure 2.Figure Vibration 2. Vibration signal signal transmission transmission in inplanetary planetary gear gearbox.box. 1.1. From planet planet gear gear to gearto gear ring, ring, to housing,to then housing, to sensors; then to sensors; 2. From 2. sun From gear sun gearto planet to planet gear, gear, to to gear gear ring,ring, to to housing, housing, then then to sensors. to sensors.

2.1.1. Amplitude2.2.1. Amplitude Modulation Modulation Model Model As the planetary gear moves from the bottom to the top of gearbox housing, it comes closer to As thethe sensorplanetary and thegear impact moves characteristic from the strength bottom of to vibration the top signals of gearbox collected housing, by the sensor it comes show closer to the sensoran and increasing the impact trend. characteristic As the planetary strength gear moves of vibration from the top signals to the bottomcollected of gearboxby the sensor housing, show an increasingthe trend. planetary As gears the areplanetary farther away gear from moves the sensor, from and the the top vibration to the signals bottom collected of gearbox by the sensor housing, the planetaryare gears weaker. are The farther amplitude away modulation from the effect sensor, can be and expressed the vibration as the following signals equation: collected by the sensor

2 are weaker. The amplitude modulationβ(mod(w ceffectt+ψi,2π )can−π) be expressed as the following equation: wi(t) = e (α − (1 − α) cos(2π fct + ψi)), (5) 2 wt=e()β((mod wci t+ψ ,2π ))-π (α -1- (α )cos(2πft+ψ )) (5) wherein, wi(t) representsici the amplitude modulation effect of planetary gearbox, during operation. α and β indicate the intensity of modulation effect. ψi is the phase difference between the i − th planet wherein,gear wi (t)and represents the first planet the gear. amplitude The different modulation modulation effe effectsct of caused planetary by different gearbox values during of α and operation.β α are shown in Figure3. and β indicate the intensity of modulation effect. ψi is the phase difference between the ith- planet In Figure3a, in the case of α > 1, all of the amplitudes are greater than 1. If α < 0.5, the amplitudes gear andare the less first than planet 0. Neither gear. ofThe these different two conditions modulation satisfies effects the requirements. caused by different Therefore, values it can beof α and β are showndetermined in Figure that 3.α is in the range of [0.5, 1]. In Figure3b, when α is fixed and β > 0, the amplitudes are greater than 1. The larger the value of β, the greater the effect of amplification. When 0.5 < α < 1 and β < 0, all of the amplitude concentrated between [0, 1], that meets the requirements.

Energies 2018, 11, 1286 6 of 23 Energies 2018, 11, x 6 of 23

(a) (b)

(c)

Figure 3. The influence of different α and β values on exponential decay function; (a) The Figure 3. The inﬂuence of different α and β values on exponential decay function; (a) The exponential α exponentialdecay effect decay of different effect of values different of α;( valuesb) The of exponential ; (b) The decayexponential effect ofdecay different effect values of different of β;( valuesc) The β α β ofexponential ; (c) The decay exponential effect of decay different effect values of different of α and valuesβ. of and .

α > α 2.2.2.In Meshing Figure Force3a, in Modelthe case of 1, all of the amplitudes are greater than 1. If <0.5, the amplitudes are less than 0. Neither of these two conditions satisfies the requirements. Therefore, it α α β > can be determined that is in the range ofJ [0.5, 1]. In Figure 3b, when is fixed and 0 , the amplitudes are greater thanF ir1.( tThe) = largerA1 ∑thecos value(j2π fofmt +β φ, irthe(t) greater + θj) the effect of amplification. j=1 When 0.5<<α 1 and β < 0 , all of the amplitudeJ concentrated between, [0, 1], that meets the (6) = A1 ∑ cos(j2πZr fct + φir(t) + θj) requirements. j=1

2.2.2. Meshing Force Model J Fis(t) = A2 ∑ cos(j2π fmt + φis(t) + θj) j=1 J J , (7) = A (j Z f t + (t) + ) Ft=Air() 12 ∑ cos(cos j2π2ft+ mπ rφc ir ()t+φθis j ) θj j=1j=1 , (6) J φir(t) = 2πZr(i − 1)/3, (8) =A1rcirj cos( j2πZft+φ (t+ )θ ) j=1 φis(t) = 2πZs(i − 1)/3, (9)

J with Fir(t) and Fis(t) meaning the meshing forces generated by the i − th planetary gear engaged with Ft=A() cos( j2πft+φ ()t+θ ) the ring gear and the sun gear, respectively.is 2A and A mindicate is the j size of meshing force. fm is meshing j=1 1 2 , (7) frequency. J is the maximum harmonic order,J j = 1, 2 ··· J. θj is the phase of the j − th harmonic. φir(t) and φis(t) are the meshing phase=A differences2rcisjcos( j2π generatedZft+φ ( byt+ ) theθ )i − th planet gear engaged with the ring gear and the sun gear, respectively.j=1 This is affected by whether the number of ring gear teeth and the number of planet gears are integral multiples and whether the phase difference between the (8) adjacent planet gears is equal. The meshingφir() phaset=2π differenceZi-1/3 r ( ) can, be calculated by Equations (8) and (9).

2.2.3. Fault Step-Impact Model φis()t=2πZi-1/3 s ( ) , (9) In the case where a bearing outer ring is cracked or peeled off, its geometry will change. As shown with Ftir () and Ftis () meaning the meshing forces generated by the ith- planetary gear engaged in Figure4, when the rolling element is in contact with the fault position, the components of the with the ring gear and the sun gear, respectively. A1 and A2 indicate the size of meshing force.

fm is meshing frequency. J is the maximum harmonic order, j =1,2J . θ j is the phase of the

Energies 2018, 11, x 7 of 23

j -th harmonic. φir ()t and φis ()t are the meshing phase differences generated by the ith- planet gear engaged with the ring gear and the sun gear, respectively. This is affected by whether the number of ring gear teeth and the number of planet gears are integral multiples and whether the phase difference between the adjacent planet gears is equal. The meshing phase difference can be calculated by Equations (8) and (9).

2.2.3. Fault Step-Impact Model Energies 2018, 11, 1286 7 of 23 In the case where a bearing outer ring is cracked or peeled off, its geometry will change. As shown in Figure 4, when the rolling element is in contact with the fault position, the components of vibrationthe vibration signal signal will will change. change. Reference Reference [31 [31]] proposed proposed that that the the bearing bearing fault fault signal signal isis composedcomposed of two parts. A step signal will be generated when th thee rolling element enters the fault position and an impact signal will be generated when the rolling elementelement leaves the fault position. The fault size can be quantiﬁedquantified basedbased on on the the time time difference difference between between the the step step signal signal and theand impact the impact signal. signal. In this In paper, this paper,the research the research mainly focusesmainly onfocuses bearing on faultsbearing occurred faults occurred at an early at stage, an early that stage, is the that case is of the a small case faultof a smallsize. The fault rolling size. elementThe rolling does element not come does into not contact come with into the contact lowest with end the of thelowest bearing end outerof the ring bearing fault whenouter itring passes fault through when it the passes fault through area. It is the assumed fault area. that theIt is frequency assumed ofthat bearing the frequency cage is fcg of, the bearing outer cagering faultis fcg width, the outer is l0 andring Dfaulto is thewidth outer is ringl0 and diameter. Do is the Therefore, outer ring the diameter. time difference Therefore, between the time the differencerolling element between entering the rolling and leaving element the en faulttering area and is: leaving the fault area is:

ll00 ∆Δt0t=0= , (10) (10) 2πDfDocgo fcg

(a) (b)

Figure 4. 4. SchematicSchematic diagram diagram of ofrollin rollingg element element fault fault position. position. (a) Schematic (a) Schematic diagram diagram of the rolling of the elementrolling element entering entering the fault the location; fault location; (b) Schematic (b) Schematic diagram diagramof the rolling of the element rolling leaving element the leaving fault location.the fault location.

The step response model can be expressed as: The step response model can be expressed as: ecos2-c() t-kT((πf t-kT )) +e -c () t-kT t-kT+Δt>0 (  n0 xt=step () −c(t−kT) −c(t−kT) , (11) e cos(2π fn(t − kT)) + e t − kT + ∆t0 > 0 xstep(t) =  0 t-kT+Δt<00 , (11) 0 t − kT + ∆t0 < 0 The impact signal model can be expressed as:

The impact signal model can be expressed-c() t-kT as: ecos2(2.5())πfn / t-kT t-kT>0 xt=imp ()(  , (12) e−c(t−kT) cos0(2π f /2.5(t − kT)) t-kT<0t − kT > 0 x (t) = n , (12) imp 0 t − kT < 0 The bearing failure vibration signal model: The bearing failure vibration signal model: xt() =kx1step () t +kx 2imp () t , (13) ( ) = ( ) + ( ) wherein, c is the attenuation coefficient,x t k 1xkstep1 andt kk22 xareimp tthe, contributions of the above two (13) vibration signal models. The step signal, impact signal, and bearing fault simulation model for a wherein, c is the attenuation coefﬁcient, k and k are the contributions of the above two vibration rotating cycle are shown in Figure 5a–c, respectively.1 2 signal models. The step signal, impact signal, and bearing fault simulation model for a rotating cycle are shown in Figure5a–c, respectively.

Energies 2018, 11, 1286 8 of 23

Energies 2018, 11, x 8 of 23

(a) (b)

(c)

Figure 5. Single-cycle bearing fault signal model. (a) The time domain waveform of step signal; (b) Figure 5. Single-cycle bearing fault signal model. (a) The time domain waveform of step signal; (b) The The time domain waveform of impact signal; (c(c) ) The time domain simulation waveform for time domain waveform of impact signal; (c) The time domain simulation waveform for quantitative quantitativeFigure 5. Single-cycle analysis of bearing bearing fault faults. signal model. (a) The time domain waveform of step signal; (b) analysis of bearing faults. The time domain waveform of impact signal; (c) The time domain simulation waveform for Thequantitative simulated analysis signal of is bearing determined faults. as c =100, T =0.1, fn = 500 . The multi-cycle time domain waveformThe simulated of bearing signal attenuation is determined impact as signalc = 100is shown, T = 0.1in Figure, fn = 6a.500 In. The Figure multi-cycle 6b, the frequency time domain spectrum analysis is continued, and the carrier frequencies of 500 Hz and 200 Hz are the center waveformThe of simulated bearing attenuationsignal is determined impact as signal c =100 is,shown T =0.1 in, f Figuren = 5006. a.The In multi-cycl Figure6eb, time the domain frequency spectrumfrequencieswaveform analysis ofwith bearing is the continued, attenuationimpact characteristic and impact the carriersignal frequency is frequencies shown 10 in HzFigure ofand 500 6a. its HzIn multiplication Figure and 200 6b, Hzthe frequencyfrequency are the center component is the side frequency spread over the entire frequency axis. In the envelope spectrum, frequenciesspectrum with analysis the impact is continued, characteristic and the frequency carrier frequencies 10 Hz and of its 500 multiplication Hz and 200 Hz frequency are the componentcenter only the impact characteristic frequency is demodulated. is thefrequencies side frequency with spreadthe impact over thecharacteristic entire frequency frequency axis. 10 In Hz the and envelope its multiplication spectrum, onlyfrequency the impact component is the side frequency spread over the entire frequency axis. In the envelope spectrum, characteristic frequency is demodulated. only the impact characteristic frequency is demodulated.

(a)

(b)

(c)

Figure 6. The analytic results of bearing multi-cycle fault signal; (a) The time domain waveform; (b) The frequency domain waveform; (c) The envelope(c )spectrum. Figure 6. The analytic results of bearing multi-cycle fault signal; (a) The time domain waveform; (b) Figure 6. The analytic results of bearing multi-cycle fault signal; (a) The time domain waveform; The frequency domain waveform; (c) The envelope spectrum. (b) The frequency domain waveform; (c) The envelope spectrum.

Energies 2018, 11, x 9 of 23 Energies 2018, 11, 1286 9 of 23 2.2.4. Bearing Load Zone Pass Effect 2.2.4.During Bearing the Load operation Zone Pass of planetary Effect gearbox, bearing capacity can be considered as the radial (s) force Duringalong the the radial operation direction. of planetary In the case gearbox, of the bearing rotation capacity frequency can of be bearing considered outer as ring the radialis fo force, the (s) (s) anglealong deviating the radial from direction. the vertical In the case centerline of the rotationis β =2π frequencyfto . The offunction bearing expression outer ring of is fitso ,load the angleover (s) timedeviating is: from the vertical centerline is β = 2π fo t. The function expression of its load over time is: 1 (  1 (s)(s) (s) (s) QQ1-cos2max{1{()}− cos(2ππft, of t)}, 2π 2ft<π of tβ< max|β | Q(t) =Qt() =max 22ε o o , max ,(14) (14) 0, other  0, other wherein, Q is the maximum load density; ε is the load distribution factor; β is the maximum wherein, Qmaxmax is the maximum load density; ε is the load distribution maxfactor; βmax is the maximumangle of load angle distribution of load distribution area. In the area. process In the of bearingprocess rotation,of bearing there rotation, is not there always is loadnot always distribution load distributionin one period. in one The period. load will The be load zero will when be thezero rotation when the angle rotation exceeds angle the exceeds maximum the load maximum distribution load distributionarea. The maximum area. The maximum angle of load angle distribution of load distri areabution can bearea controlled can be cont byrolled changing by changing the size oftheε . sizeIn addition, of ε . In addition, the magnitude the magnitude of modulated of modulated vibration signalvibration by loadsignal distribution by load distribution is changed is by changed resizing bythe resizing maximum the loadmaxi density.mum load density. (s) Assuming t = (0 : n)/6000 (n is the sampling length), fo (s) = 100, Qmax = 1. In the case of Assuming t=(0:n)/6000 (n is the sampling length), fo = 100 , Qmax =1. In the case of ε = 0.6/0.5/0.4/0.3, the signal model is shown in Figure7. As can be seen from ﬁgure, ε < 0.5 meets ε = 0.6 / 0.5 / 0.4 / 0.3 , the signal model is shown in Figure 7. As can be seen from figure, ε < 0.5 the requirements. meets the requirements.

Figure 7. The effect of load distribution factor on vibration waveform. Figure 7. The effect of load distribution factor on vibration waveform. 2.2.5. Outer Ring Fault Vibration Signal Model 2.2.5. Outer Ring Fault Vibration Signal Model The vibration signal model of planet gear bearing outer ring fault is established based on the aboveThe analysis. vibration In the signal process model of of modeling, planet gear bearing bearing load outer zone ring passing fault is effect, established step-impact based signal on the model,above analysis.and the effect In the of process transmission of modeling, paths bearing are all loadtaken zone into passing account. effect, Considering step-impact only signal one planet model, gearand bearing the effect outer of transmission ring failed, the paths vibration are all signal taken model into account. is shown Considering as follows: only one planet gear bearing outer ring failed, the vibration signal model is shown as follows:

xt=wtQtkxoi() () ()[ 1step2imp () t+kx ()] t, (15) xo(t) = wi(t)Q(t)[k1xstep(t) + k2ximp(t)], (15) 2 wherein, wt=e()β(mod(wci t+ψπ,2 )-π) (αα -(1- )cos(2πft+))ψ ici2 β(mod(wct+ψ ,2π)−π) wherein, wi(t) = e i (α − (1 − α) cos(2π fct + ψi))  1 Q{1-cos(2πf(s) t)} ( maxn o (s) o Q(t) =  Q 12ε− 1 cos(2π f t)  max 2ε o Q(t) =  0 q(t) < 0 0 q(t) < 0 (s) f =4, β =-3, α =0.6, θφψ、、 =0, ε =0.2, Q =0.2, f = 500 , f =6. The repeat period c max n (s) o f = β = − α = θ φ ψ = ε = Q = f = f = ofc impact4, simulation3, 0.6signal, , is, 100 0ms., The0.2 ,timemax domain0.2, andn 500frequenc, o y domain6. The repeat waveforms period of of simulatedimpact simulation signal are signal shown is 100in Figure ms. The 8. time domain and frequency domain waveforms of simulated signal are shown in Figure8.

Energies 2018, 11, 1286 10 of 23 Energies 2018, 11, x 10 of 23

0.01

Amplitude(g) -0.01

0 0.5 1 1.5 2 Time (s)

(a) x 10-3 5 ()s − focf fob 4 fc

3 +−()s fobff o c 2 f f ()s c f + f 2 o ob c

Amplitude(g) 1

0 0 5 10 15 20 25 30 35 40 45 50 Frequency(Hz)

(b)

Figure 8. The analytic results of planet gear bearing outer ring fault simulation signal. (a) Time Figure 8. The analytic results of planet gear bearing outer ring fault simulation signal. (a) Time domain domain waveform of simulation signal; (b) The envelope spectrum of simulation signal. waveform of simulation signal; (b) The envelope spectrum of simulation signal.

By analyzing the time domain waveform and frequency spectrum of the planet gear bearing outer Byfault analyzing simulation the signal, time domain it can be waveform seen that anda series frequency of impact spectrum signals ofare the generated planet gear in the bearing time domainouter fault signal simulation when the signal, bearing it canfails. be The seen load that of a the series bearing of impact is changed signals due are generatedto the rotation in the of timethe planetdomain gear signal bearing when outer the ring, bearing thereby fails. causing The load an in oftermittent the bearing distribution is changed of the due impact to the signal. rotation The of naturalthe planet frequency gear bearing of the outer bearing ring, therebyand its causingside frequency an intermittent components distribution appear of thein the impact envelope signal. spectrum.The natural In frequencyaddition, there of the are bearing side frequency and its sidecomp frequencyonents that components are the characteristic appear in thefrequency envelope of bearing,spectrum. carrier In addition, frequency there and are bearing side frequency rotation frequency. components In thatsummary, are the wh characteristicen the outer frequency ring of of bearing, carrier frequency and bearing rotation frequency. In summary,±± when ±()s the outer ring of planetary bearing fails, the frequency peak mainly appears in flfmfnfnobc o ((ls), m , n are planetary bearing fails, the frequency peak mainly appears in f ± l f ± m f ± n f (l, m, n are positive integers). n ob c o positive integers).

3.3. Theory Theory of of Proposed Proposed Method Method

3.1.3.1. Improved Improved Wavelet Wavelet Scalogram Scalogram AsAs a a method method of of time-frequency time-frequency analysis, analysis, wavelet wavelet transform transform is is adaptive. adaptive. It It can can eliminate eliminate noise noise andand identify identify weak weak fault fault information. information. The The different different scale scale value value corresponds corresponds to to different different frequency frequency bandband [32]. [32]. Select Select the the appropriate appropriate scale scale value value according according to to the the frequency frequency characteristics characteristics of of the the signal signal beingbeing analyzed. SupposeSuppose ψψ(t(t)) is ais wavelet a wavelet basis basis function, function, then itthen must it satisfymust ﬁnitesatisfy energy finite theorem,energy theorem,i.e., ψ(t) ∈i.e.,L 2(ψR(t)) and∈ L2 (R) its Fourier and itstransform Fourier transformψˆ(ω) should ψˆ (ω respect) should the respect following the following rule: rule: Z 2 2 C C== ψˆ(ω)) //ωω|ωd|dω<∞< ∞,(16) (16) ψ ψ  , whereinwherein ψψ((t)t) is is a a wavelet wavelet basic basic function, function, alsoalso calledcalled mothermother wavelet.wavelet. In general, the the signal signal analysis analysis resultsresults depend depend on on the the ψ((t)t).. ReferenceReference [[33]33] discussed the effect effect of of different different mother mother wavelets wavelets on on the the analysisanalysis result result of of bearing bearing impact impact failure failure signal. signal. It It is is verified veriﬁed that that the the Morlet Morlet wavelet wavelet can can better better match match thethe characteristic characteristic of of bearing bearing impact impact faults. faults. Therefore, Therefore, with with a a Morlet Morlet wavelet wavelet as as mother mother wavelet wavelet to to performperform wavelet wavelet transform transform in in the the following following section, section, a a series series of of wavelet wavelet functions functions can can be be obtained obtained by by translatingtranslating and stretchingstretching the the mother mother wavelet waveletψ( t)ψ.(t) Taking. Takinga as itsa scaleas its factor scale and factorb as and the translationb as the translation factor, the deformation of a basic wavelet function ψa,b (t) can be obtained by changing the values of a and b according to the following Equation (17):

Energies 2018, 11, 1286 11 of 23

factor, the deformation of a basic wavelet function ψa,b(t) can be obtained by changing the values of a and b according to the following Equation (17):

t − b ψ (t) = |a|−1/2ψ , (17) a,b a

ψa,b(t) is not unique, and its shape and position vary in relation to the values of a and b. The continuous wavelet transform was carried out on x(t) ∈ L2(R), that can be expressed in the general form of Equation (18): Z t − b W (a, b; ψ) = a−1/2 x(t)ψ∗ dt, (18) x a wherein, ψ(t) is an analytical wavelet, ψ∗(t) is the complex conjugate of ψ(t). This definition is analogous to the Fourier transform. What makes them different is the basis function: a Fourier transform is performed on cosine functions and wavelet transform depends on the wavelet function. The wavelet transform is reversible and the original signal can be restored by using the wavelet transform coefficients: 1 −2 x(t) = x a Wx(a, b; ψ)ψa,b(t)dadb, (19) Cψ wherein SG(a, b; ψ) can be defined as the modulus of continuous wavelet transform coefficients, which is given by Equation (20).

2 SG(a, b; ψ) = |Wx(a, b; ψ)| , (20)

As mentioned earlier, a wavelet scalogram lacks concentricity just like other traditional time-frequency analysis method. Taking a simple chirp signal as an example, its time domain waveform is shown in Figure9a, its time-frequency spectra obtained by a wavelet scalogram is shown in Figure9b. It can be seen that the time-frequency ridge spreads over a wide area. Applying the reassigned wavelet scalogram to calculate the time-frequency ridge and the result is shown in Figure9c. The picture displays that the time-frequency feature concentrated in a ﬁne straight line. Given the problem that the time-frequency resolution of wavelet scalogram is low, a reassigned method was put forward. The theoretical deﬁnition of improved wavelet scalogram is expressed as:

ˆ 2 ˆ RSG(aˆ, b; ψ) = x (aˆ/a) SG(a, b; ψ) × δ(b − b0(a, b))δ(aˆ − a0(a, b))dadb, (21)

0 0 0 wherein, b (a, b), ψ (t), ψˆ(t) and ω0/a (a, b) can be calculated by the following equations: ( ) W (a, b; ψ0)W∗(a, b; ψ) 0( ) = − x x b a, b b Re a 2 , (22) |Wx(a, b; ψ)| ( ) ω ω W (a, b; ψˆ)W∗(a, b; ψ) 0 = 0 + x x 0 Im 2 , (23) a (a, b) a 2πa|Wx(a, b; ψ)| dψ ψ0(t) = tψ(t), ψˆ(t) = (t), (24) dt wherein, ω0 is the central angular frequency. SG(ai, bj; ψ) represents the average of the energy density of a region which takes (ω0/ai, ait + bj) as geometric center. Due to the fact the energy of this region is not geometrically symmetric and it is unreasonable to concentrate the average energy at the geometric center, at this time, some people advise placing the average energy at the local energy center (ω0/aˆi, aˆit + bˆj). Then, a reassigned wavelet scalogram is put forward based on this principle. Energies 2018, 11, 1286 12 of 23 Energies 2018, 11, x 12 of 23

1 0 -1

Amplitude 0100200 300 400 500 600 700 800 900 1000 Time(s)

(a) 0.5

0.4

0.3

0.2

Frequency(Hz) 0.1

0 0 200 400 600 800 1000 Time(s)

(b) 0.5

0.4

0.3

0.2

Frequency(Hz) 0.1

0 0 200 400 600 800 1000 Time(s)

(c)

Figure 9. Simulation signal analysis. (a) Time waveform for a line modulation signal; (b) Wavelet Figure 9. Simulation signal analysis. (a) Time waveform for a line modulation signal; (b) Wavelet scalogram for the line modulation signal; (c) Reassigned wavelet scalogram. scalogram for the line modulation signal; (c) Reassigned wavelet scalogram. For a detailed description about reassigned wavelet scalograms readers can consult [30]. In FigureFor 9c a is detailed easier to descriptionidentify how about the frequency reassigned changes wavelet with scalograms time compared readers with can Figure consult 9b. [The30]. Inreassigned Figure9c wavelet is easier scalogram to identify is howa spectrogram the frequency that can changes improve with readability time compared after the with average Figure value9b. Theis concentrated reassigned waveletat the local scalogram energy is center. a spectrogram The mechanical that can improvevibration readability signal generally after the exhibits average a valueperiodic is concentrated characteristic, at and the localits cycle energy is variable center. with The mechanicalthe change of vibration device operating signal generally state. Moreover, exhibits a periodicthe operation characteristic, of equipment and its is cycle vulnerable is variable to withexternal the changeinterferences. of device Therefore, operating vibration state. Moreover, signals thecollected operation from of rotating equipment machinery is vulnerable are often to external mixed interferences. with external Therefore, interference vibration that is signals independent collected of fromfault rotatinginformation. machinery The fault are feature often mixedis weak with when external the external interference interference that is independentis strong [16,17]. of fault The information.traditional time The domain fault feature and frequency is weak when domain the external analysis interference method is unable is strong to [ 16diagnose,17]. The weak traditional faults. timeThen, domain a series and of new frequency methods domain are exploited analysis method to impr isove unable the accuracy to diagnose of the weak fault faults. diagnosis, Then, asuch series as ofthe new synchronous methods are averaging exploited technique to improve [19,34]. the accuracy of the fault diagnosis, such as the synchronous averagingLet us technique suppose [19x(n,34) ]. is a periodic vibration signal that was collected by a transducer mounted Let us suppose x(n) is a periodic vibration signal that was collected by a transducer mounted on a on a rotary machine, and its sampling point is N , where n = 1,2, N . If the vibration signal x is rotary machine, and its sampling point is N, where n = 1, 2, ··· N. If the vibration signal x is formed by formed by M stationary components with the period is P, then N=P× M. We choose the Morlet M stationary components with the period is P, then N = P × M. We choose the Morlet wavelet as the wavelet as the basic wavelet to perform wavelet transform on x(n) . After that, we can implement basic wavelet to perform wavelet transform on x(n). After that, we can implement the reassignment processthe reassignment to obtain the process reassigned to obtain wavelet the scalogram reassignedRTFR wavelet( f , p) scalogramaccording toRTFR(f,p) Equation (21),according where tof representsEquation (21), the frequencywhere f and representsp represents the frequency the time. and If thep signalrepresents contains the time.F frequencies, If the signal then contains the size ofF RTFRfrequencies,( f , p) is thenF × theP. size of RTFR(f,p) is FP× . Based on the reassigned wavelet scalogram, we can use time synchronous averaging for further analysis. Each row corresponds to a frequency value, which is made up of repeating M components with a period P. An average for each cycle and time synchronous RTFR(f,p) can be obtained.

Energies 2018, 11, 1286 13 of 23

Based on the reassigned wavelet scalogram, we can use time synchronous averaging for further analysis. Each row corresponds to a frequency value, which is made up of repeating M components with a period P. An average for each cycle and time synchronous RTFR( f , p) can be obtained. Energies 2018, 11, x 13 of 23 1 M−1 RTFR( f , p) = RTFR( f , p + mP), (25) M1 ∑M-1 RTFR(f,p) =m=0 RTFR(f,p + mP) , (25) M m=0 The process diagram of the improved wavelet scalogram is displayed in Figure 10. The process diagram of the improved wavelet scalogram is displayed in Figure 10.

Figure 10. Multi-scales synchronous averaging of reassigned wavelet scalogram. Figure 10. Multi-scales synchronous averaging of reassigned wavelet scalogram. The value of P directly affects the synchronous averaging results. The characteristic frequency Theis related value toof theP runningdirectly speed affects of the rolling synchronous element averagingbearings. However, results. in The some characteristic cases, they are frequency not is relatedinteger to multiple the running relationships, speed of so the a one-step rolling elementrounding bearings. process is However,usually added. in some cases, they are not integer multiple relationships, so a one-step rounding process is usually added. T-TK1× NUM = round fS , (26) K-1 T − T NUM = round K 1 × f , (26) We can intercept the K impact series to calculateK − 1 the avSerage points per cycle. In normal circumstances, we can select a proper signal length according to the signal characteristics for analysis Wein order can interceptto improve the theK computationalimpact series efficiency, to calculate whilst the not average affecting points the analysis per cycle. results. In The normal circumstances,computational we efficiency can select of Equation a proper (26) signal will be length low in the according case where to theK is signal large. In characteristics addition, the for analysiscalculation in order accuracy to improve will decrease the computational in the case where efficiency, K is small. whilst Therefore, not affecting when defining the analysis the value results. The computationalof K , the calculation efficiency accuracy ofEquation and computational (26) will complexity be low in should the case both where be takenK is into large. account. In addition, the calculation accuracy will decrease in the case where K is small. Therefore, when defining the value 3.2. Wavelet Ridge of K, the calculation accuracy and computational complexity should both be taken into account. The wavelet ridge [35] is a sequence points extracted from the maximum of modulus values of 3.2. Waveletwavelet Ridgecoefficients of time-frequency spectra at each moment. By analyzing the wavelet ridge, we can explore the composition of signal components, like signal envelope, instantaneous amplitude, Theand waveletinstantaneous ridge phase. [35] is Suppose a sequence x(t) points is a extractedsimple cosine from signal. the maximum It can be of expressed modulus as values of waveletx(t) = A(t)cos coefficients(φ(t)) and of its time-frequency analytic signal can spectra be calculated at each by the moment. following Byexpression. analyzing the wavelet ridge, we can explore the composition of signal components, like signal envelope, instantaneous jφx (t) (27) amplitude, and instantaneous phase.x(t) Suppose = x(t) + jxxHx(t (t)) is = a A simple (t)e cosine, signal. It can be expressed as x(t) = A(t) cos(φ(t)) and its analytic signal can be calculated by the following expression. wherein, xH (t) is the Hilbert transform of x(t) . A(t) > 0 is the instantaneous amplitude and φ(t) is

the instantaneous phase angle. The complex form of the mother wavelejφx(t)t is denoted as Equation (28). xe(t) = x(t) + jxH(t) = Ax(t)e , (27)  jφψ (t) (28) ψ(t) = Aψ (t)e , wherein, xH(t) is the Hilbert transform of x(t). A(t) > 0 is the instantaneous amplitude and φ(t) is the instantaneousSelect ψ(t) phase as the angle. wavelet The basis complex for wavelet form transform of the mother to analyze wavelet signal is denotedx(t) . as Equation (28).

jφψ(t) ψe(t) = Aψ(t)e , (28)

Energies 2018, 11, 1286 14 of 23

Select ψe(t) as the wavelet basis for wavelet transform to analyze signal xe(t). 1 Z t − b 1 Z ∗ jφa,b(t) WTx(a, b; ψ) = √ x(t)ψe dt = √ Aa b(t)e dt, (29) 2 a e a 2 a , R R

( ∗ t−b Aa,b(t) = Ax(t)Aψ ( a ) t−b , (30) φa,b(t) = φx(t) − φψ( a ) ∗ wherein, ψe (t) is the conjugate of ψe(t). The Morlet wavelet is selected as the basis function to match the vibration signal. That is determined by its downhill feature in a short time. This is very similarity to the bearing fault impulse signal. The expression of a Morlet complex wavelet is:

1 t2 ψ(t) = p exp(− )exp(i2π fct), (31) π fb fb wherein, fb is the frequency band range and fc is wavelet center frequency. There is a one-to-one correspondence between wavelet transform scale and frequency. Suppose sampling frequency is fs and the corresponding frequency of scale a is:

f f f = s c , (32) a After obtaining the wavelet ridge line, the wavelet coefficients corresponding to wavelet ridge line position can be deduced by inverse transform formula [28,36]. p a (b) _∗ WT (a (b), b) = r × A (b) × exp(iφ (b)) × (φ (ω ) + ξ(a, b)), (33) x r 2 x x ψ c _ ξ(a, b) is the modification item, which is usually small and often can be ignored. φ ψ(ω) is the Fourier _ transform of φψ(t), φ(ω) will reach maximum at the point of center frequency ωc. With a slight modification for Equation (33), the amplitude of continuous wavelet transform is approximately equal to: p a (b) _∗ |WT (a (b), b)| ≈ r × A (b) × φ (ω ), (34) x r 2 x ψ c In wavelet function expression, the parameter b represents time shift. Then, b can be taken as t. Accordingly, WTx(ar(b), b) can be represented as WTx(ar(t), t). In this circumstance, the instantaneous amplitudes of signal envelope are estimated by:

2|WTx(ar(t), t)| Ax(t) ≈ p , (35) ar(t)

In the same way, Equation (20) can be transformed into Equation (36).

a (t) SG (a (t), t) ≈ r (A (t))2, (36) x r 4 x

Afterwards, instantaneous amplitudes of signals x(t) can be approximated by Equation (37). p 2|WTx(ar(t), t)| 2 SGx(ar(t), t) Ax(t) ≈ p = p , (37) ar(t) ar(t)

It can be seen from Equation (37) that the wavelet ridge extraction process is actually a demodulation process. In the same way, the instantaneous amplitudes of signal can be estimated on Energies 2018, 11, 1286 15 of 23 the improvedEnergies 2018 wavelet, 11, x scalogram according to Equation (37). There are two different ways15 toof extract23 wavelet ridge. The one way is based on wavelet coefficient phase values and the other one is based on It can be seen from Equation (37) that the wavelet ridge extraction process is actually a modulusdemodulation values. By process. comparing In the bothsame analysisway, the results,instantaneous it can amplitudes be concluded of signal that thecan methodbe estimated on the on basis of waveletthe improved coefficient wavelet modulus scalogram is more according accurate. to Equation Because (37). the There local are maximum two different value ways of to theextract wavelet scalewavelet spectral ridge. coefficients The one andway is the based continuous on wavelet wavelet coefficient transform phase values coefficients and the isother the one same is based point that is theon wavelet modulus ridge values. point By comparing(ar(b), b) [ both28]. Inanalysis this paper, results, the it can wavelet be concluded ridge isthat extracted the method by calculatingon the the waveletbasis of coefficientwavelet coefficient modulus modulus maximum is more and a accurate. new time Because series the was local reconstructed. maximum value Reference of the [37] proposedwavelet a method scale spectral to extract coefficients the wavelet and the ridge contin whichuous minimizes wavelet transform the cost function.coefficients This is the method same only point that is the wavelet ridge point ()a(b),b [28]. In this paper, the wavelet ridge is extracted by needs the model information to extract ther ridge line. There are n − 1 cost functions along the wavelet ridgecalculating line described the wavelet in the followingcoefficient equation.modulus maximum and a new time series was reconstructed. Reference [37] proposed a method to extract the wavelet ridge which minimizes the cost function. This method only needs the model information to extract the ridge line. There are n1- cost functions CFk = −|W(a(k), k)|2+|a(k) − ar(k − 1)|2, k = 2, 3, ··· n, (38) along the wavelet ridge line described in the following equation.

The ridge point can be determined according22 to the Equation (38). If the (k − 1)th ridge point CFkr = -|W(a(k),k)| +|a(k) - a (k - 1)| ,k = 2,3, n , (38) (ar(k − 1), k − 1) was derived, then the k − th ridge point {(ar(k), k)} can be determined by minimizing The ridge point can be determined according to the Equation (38). If the (1)k − th ridge point the CFk. During run time, each choice is evaluated in order. When all of the items (from k = 2 to k = n) ((a k-1 ) ,k-1 ) was derived, then the k -th ridge point {()a(k),k} can be determined by minimizing are completelyr calculated, the instantaneous amplitudesr of the signal can be estimated according to

Equationthe CF (37).k . During Due to run the time, bearing each faultchoice signal is evaluate presentsd in periodicorder. When impact all of characteristics, the items (from whichk=2 to can be identifiedk=n) from are completely the instantaneous calculated, amplitude. the instantaneous In general, amplitudes a small of scale the bandsignal iscan chosen be estimated for analysis, therebyaccording improving to Equation the accuracy (37). Due of to wavelet the bearing ridge fault extraction signal presents results. periodic impact characteristics, Itwhich is indicated can be identified that 3 dB from bandwidth the instantaneous is the most ampl suitableitude. In choice general, according a small scale to the band characteristic is chosen of for analysis, thereby improving the accuracy of wavelet ridge extraction results. signal being analyzed. The region where the wavelet coefficients modulus is large usually corresponds It is indicated that 3 dB bandwidth is the most suitable choice according to the characteristic of to the range of the wavelet ridge. Besides, selecting a suitably small scale to analyze can reduce the signal being analyzed. The region where the wavelet coefficients modulus is large usually calculationcorresponds time. to the range of the wavelet ridge. Besides, selecting a suitably small scale to analyze can reduce the calculation time. 3.3. Incipient Feature Extraction Method The3.3. Incipient process Feature flow diagram Extraction of Method rolling element bearing fault diagnosis based on improving wavelet scalogramThe and process wavelet flow ridge diagram is shown of inrolling Figure element 11. This bearing method fault is readilydiagnosis available based foron rollingimproving element bearingwavelet fault scalogram feature extraction. and wavelet To ridge begin is with, shown the in improvedFigure 11. waveletThis method scalogram is readily plays available an important for role inrolling extracting element and bearing enhancing fault feature the impact extraction. characteristics. To begin with, In addition the improved to that, wavelet the wavelet scalogram ridge can identifyplays fault an important frequency role more in extracting clearly. This and methodenhancing does the impact not require characteristics. any prior In knowledge addition to ofthat, rolling elementthe wavelet bearing ridge fault can information, identify fault and frequency can be appliedmore clearly. to determine This method the do weakes not fault require characteristics any prior for knowledge of rolling element bearing fault information, and can be applied to determine the weak rolling element bearings. fault characteristics for rolling element bearings.

Data acquisition for vibration and speed signals

Continuous wavelet for Rolling element decomposition and bearing type ressigned wavelet scalogram

Multi-cycle sychronous Rolling bearing averaging for reassigned operational condition wavelet scalogram

Wavelet ridge determination Rolling bearing based on improved wavelet characteristic frequency scalogram

Early fault classification for rolling bearing based on characteristic frequency analysis

Figure 11. Flowchart for rolling element bearing fault diagnosis. Figure 11. Flowchart for rolling element bearing fault diagnosis.

Energies 2018, 11, 1286 16 of 23

Energies 2018, 11, x 16 of 23 4. Algorithm Veriﬁcation 4. Algorithm Verification 4.1. SimulationEnergies 2018 Analysis, 11, x 16 of 23 4.1. Simulation Analysis In4. Algorithm order to prove Verification the validity of the wavelet ridge line method based on improved wavelet scalogram,Inthe order simulation to prove signalthe validity that is of established the wavelet by ridge researching line method the characteristicsbased on improved of rolling wavelet element bearingscalogram,4.1.taken Simulation as the an Analysissimulation analytical signal object. that We is add establis Gaussianhed by white researching noise withthe characteristics a signal-to-noise of rolling ratio size element bearing taken as an analytical object. We add Gaussian white noise with a signal-to-noise of −15 dBIn into order the to planet prove gearthe validity bearing of outerthe wavelet ring fault ridge signal line method simulation based model on improved which iswavelet shown in ratio size of −15 dB into the planet gear bearing outer ring fault signal simulation model which is Figurescalogram,8a. The time-domain the simulation impact signal characteristicthat is establis signalhed by isresearching almost invisible the characteristics in Figure 12 of, whichrolling can shown in Figure 8a. The time-domain impact characteristic signal is almost invisible in Figure 12, representelement as abearing weak impacttaken as signal. an analytical object. We add Gaussian white noise with a signal-to-noise which can represent as a weak impact signal. ratio size of −15 dB into the planet gear bearing outer ring fault signal simulation model which is shown in Figure 8a. The time-domain0.04 impact characteristic signal is almost invisible in Figure 12, which can represent as a weak0.02 impact signal. 0 0.04 -0.02 Amplitude 0.02 -0.04 0 0.25 0.5 0.75 1 1.25 1.5 0 Time (s) -0.02 Amplitude (a) -0.04 -4 x0 10 0.25 0.5 0.75 1 1.25 1.5 3 Time (s)

2 (a) x 10-4 3 1 Amplitude 2 0 0 50 100 150 200 250 300 1 Frequency(Hz) Amplitude (b) 0 0 50 100 150 200 250 300 Figure 12. The analytic results of simulation signalFrequency(Hz) with Gaussian white noise. (a) The time domain Figure 12. The analytic results of simulation signal with Gaussian white noise. (a) The time domain waveform for noisy simulation signal; (b) The envelope(b) spectrum for noisy simulation signal. waveform for noisy simulation signal; (b) The envelope spectrum for noisy simulation signal. Subsequently,Figure 12. The analyticthe envelope results spectrumof simulation is applied signal with on Gaussianthe noise-added white noise. simulation (a) The timesignal. domain It can be Subsequently,seen waveformthat the envelope for the noisy envelope simulationanalysis spectrum issignal; infeasible (b) isThe appliedfor envelope weak on spectrumimpact the noise-added featurefor noisy extraction. simulation simulation signal.Then, a signal. wavelet It can transform is further performed on it. Its reassigned wavelet scalogram was calculated as shown in be seen thatSubsequently, the envelope the analysisenvelope isspectrum infeasible is applied for weak on the impact noise-added feature simulation extraction. signal. Then, It can a wavelet be Figure 13. Many spectral lines are scattereded in the wavelet scalogram spectrum. It is difficult to transformseen that is further the envelope performed analysis on it.is infeasible Its reassigned for weak wavelet impact scalogram feature extraction. was calculated Then, a aswavelet shown in recognize the impact information by adopting the reassigned wavelet scalogram method under the Figuretransform 13. Many is further spectral performed lines are on scattereded it. Its reassign ined the wavelet wavelet scalogram scalogram was spectrum. calculated as It shown is difﬁcult in to situation of strong noise. There are many interfering components in the spectrum because of outside recognizeFigure the 13. impact Many spectral information lines are by adoptingscattereded the in reassignedthe wavelet scalogram wavelet scalogram spectrum. It method is difficult under to the noise. In order to eliminate more irrelevant interference information, the scalogram is further recognize the impact information by adopting the reassigned wavelet scalogram method under the situationprocessed of strong by synchronou noise. Theres averaging are many where interfering the synchronous components cycle is in 400 the ms spectrum. The improved because wavelet of outside situation of strong noise. There are many interfering components in the spectrum because of outside noise.scalogram In order toof eliminatethe noisy simulation more irrelevant signal interferenceis shown in Figure information, 14. The theimpact scalogram characteristics is further from processed the noise. In order to eliminate more irrelevant interference information, the scalogram is further by synchronoustime-frequency averaging spectrum where can be the easi synchronously distinguished. cycle All is of 400 the ms. impact The components improved waveletin the simulation scalogram of processed by synchronous averaging where the synchronous cycle is 400 ms. The improved wavelet the noisysignal simulation are concentrated signal in is the shown frequency in Figure axis 14of .500 The Hz. impact The time characteristics interval between from the two time-frequency impact scalogram of the noisy simulation signal is shown in Figure 14. The impact characteristics from the spectrumpositions can is be 100 easily ms. That distinguished. is to say that All the of impact the impact period componentsis 100 ms and in the the impact simulation characteristic signal are time-frequency spectrum can be easily distinguished. All of the impact components in the simulation frequency is 10 Hz. This is consistent with the theoretical value of the simulation signal. concentratedsignal are in concentrated the frequency in the axis frequency of 500 Hz.axis Theof 500 time Hz. interval The time between interval thebetween two impactthe two positionsimpact is 100 ms.positions That is is to 100 say ms. that That the is impact to say period that the is 100impact ms period and the is impact100 mscharacteristic and the impact frequency characteristic is 10 Hz. 55 This isfrequency consistent is 10 with Hz. This the theoreticalis consistent valuewith the of theoretical the simulation value of signal. the simulation signal. 44

3355

2244 Frequency(kHz) Frequency(Hz) 1133

00 220 5050 100100 150150 200200 250250 300300 350350 400400

Frequency(kHz) TimeTime(ms) (ms) Frequency(Hz) 11

Figure 13. Time frequency 0reassignment0 distribution for the simulation signal. 0 5050 100100 150150 200200 250250 300300 350350 400400 Time (ms) Time(ms)

Figure 13. Time frequency reassignment distribution for the simulation signal. Figure 13. Time frequency reassignment distribution for the simulation signal.

Energies 2018, 11, 1286 17 of 23 Energies 2018, 11, x 17 of 23

55 Energies 2018, 11, x 17 of 23 Energies 2018, 11, x 44 17 of 23

33 55 55 22 44 44 Frequency(kHz) 100 ms

Frequency(Hz) 33 1 1 33 0 22 0 Frequency(kHz) 22 0 5050 100100100 ms 150150 200200 250250 300300 350350 400400 Frequency(Hz) Frequency(kHz) 100 ms 11 Time (ms) Frequency(Hz) Time(ms) 11

00 0 5050 100100 150150 200200 250250 300300 350350 400400 Figure 14. The scalogram00 under synchronous averaged. Figure 14. The scalogram0 5050 100100 150 under150TimeTime(ms)200 200(ms) synchronous250250 300300 350350 400400 averaged. Time (ms) Time(ms) Although it can be seen Figurethat the 14. wavelet The scalogram scale underspectrum synchronous has a periodic averaged. distribution characteristic in Although it can be seenFigure that the 14. The wavelet scalogram scale under spectrum synchronous has a averaged. periodic distribution characteristic Figure 14, the impact information is distributed in an area of the block rather than a point or a line. in Figure Although14, the impact it can be information seen that the wavelet is distributed scale spectrum in an has area a periodic of the block distribution rather characteristic than a point in or a The fault Althoughperiod cannot it can be be seen extracted that the waveletfrom Figure scale spectrum 14 correctly has a periodic using distributionthe synchronous characteristic averaging in Figure 14, the impact information is distributed in an area of the block rather than a point or a line. line. TheFigure fault 14, periodthe impact cannot information be extracted is distributed from Figure in an area14 correctly of the block using rather the than synchronous a point or averaginga line. scalogram.The fault There period is a cannotlack of be intuition extracted and from veracity. Figure 14In thiscorrectly paper, using further the synchronous analysis is performedaveraging by scalogram.The fault There period is a cannot lack of be intuition extracted and from veracity. Figure 14 In correctly this paper, using further the synchronous analysis is performedaveraging by using scalogram.the wavelet There ridge is aextraction lack of intuition method. and The veracity. analysis In this scale paper, area further is limited analysis to between is performed 6 and by 29 in usingscalogram. the wavelet There ridge is a extractionlack of intuition method. and veracity. The analysis In this scale paper, area further is limited analysis to is between performed 6 andby 29 termsusing of the the characteristic wavelet ridge frequency extraction rangemethod. obtained The analysis by the scale computational area is limited equation to between of 6wavelet and 29 inridge. in termsusing of the the wavelet characteristic ridge extraction frequency method. range The obtained analysis scale by the area computational is limited to between equation 6 and of 29 wavelet in The waveletterms of theridge characteristic extracted frequency from Figure range obtained14 is shown by the computationalin Figure 15. equation Then, ofthe wavelet instantaneous ridge. ridge.terms The of wavelet the characteristic ridge extracted frequency from range Figure obtained 14 is by shown the computational in Figure 15 equation. Then, of the wavelet instantaneous ridge. amplitudeThe wavelet and the ridge corresponding extracted from frequency Figure can14 is be shown calculated in Figure according 15. Then, to Equation the instantaneous (37) and the The wavelet ridge extracted from Figure 14 is shown in Figure 15. Then, the instantaneous amplitudeamplitude and and the the corresponding corresponding frequency frequency cancan be calculated according according to toEquation Equation (37) (37)and andthe the Fourieramplitude transform, and asthe shown corresponding in Figure frequency 16a,b. canThe be impact calculated characteristic according to frequency Equation (37)10 Hzand andthe its FourierFourier transform, transform, as shownas shown in in Figure Figure 16 16a,b.a,b. The impact impact characteristic characteristic frequency frequency 10 Hz 10 and Hz andits its multipliedFourier frequency transform, components as shown in are Figure clearly 16a,b. displayed The impact in Figure characteristic 16b. frequency 10 Hz and its multipliedmultiplied frequency frequency components components are are clearly clearly displayed displayed in Figure Figure 16b. 16b. multiplied frequency components are clearly displayed in Figure 16b. 1.41.4 1.41.4 1.21.21.41.4 1.21.2 111.21.2 11 1 0.8 1 0.80.80.8 0.80.8 0.60.60.60.6 0.60.6 Frequency(kHz) 0.40.40.4Frequency(kHz) 0.4 Frequency(kHz) Frequency(Hz) Frequency(Hz) 0.40.4

0.20.2Frequency(Hz) 0.20.2 0.20.2 00 00 0 100 200 300 400 0 00 100100 200 300300 404000 0 0 100100 Time200200 (ms) 300300 404000 0 100TimeTime(s)Time(ms)200 (ms) 300 400 Time(s)Time(ms)Time(s)Time(ms)Time (ms)

Figure 15. Wavelet ridge for improved wavelet scalogram. FigureFigureFigure 15. 15. 15.WaveletWavelet Wavelet ridge ridge ridge for for for improvedimproved improved waveletwavelet scalogram. scalogram. scalogram.

(a) 0.02 -3 (a) x 10 (a) 0.02 0.02 b) 1.5 -3 100 ms 100 ms 10x Hz10-3 b) 1.5x 10 50 Hz 100 ms 100 ms b) 1.5 1020 Hz Hz 0. 015 40 Hz50 Hz 100 ms 100 ms 10 Hz20 Hz 60 Hz 30 Hz40 50Hz Hz 0. 015 20 Hz 60 Hz 30 40Hz Hz 70 Hz 0. 015 1 60 Hz 1 30 Hz 7080 Hz Hz 0.01 1 70 Hz80 Hz 0.01 80 Hz

0.01 Amplitude(g) 0.5 Amplitude(g) Amplitude(g) 0. 005 Amplitude(g) 0.5 Amplitude(g) Amplitude(g) Amplitude(g) Amplitude(g) 0. 005 0.5 Amplitude(g) Amplitude(g) 0. 005 0 0 00 50 10 0 150 20 0 250 00 100 200 30 0 400 0 50 10 0 150 20 0 250 0 100 Time200 (ms) 30 0 400 Frequency (Hz) 0 Frequency (Hz) 0 Time (ms) 0 50 10 0 150 20 0 250 0 100 200(a) 30 0 400 (b) (a) (Frequencyb) (Hz) Time (ms) Figure 16. The analytic results of simulation signal’s instantaneous amplitude. (a) Time-domain Figure 16. The analytic(a ) results of simulation signal’s instantaneous(b amplitude.) (a) Time-domain waveform of simulation signal’s instantaneous amplitude; (b) The frequency domain waveform of Figurewaveform 16. The of analytic simulation results signal’s of simulationinstantaneous signal’s amplitude; instantaneous (b) The frequency amplitude. domain ( awaveform) Time-domain of Figuresimulation 16. The signal’sanalytic instantaneous results of simulationamplitude. signal’s instantaneous amplitude. (a) Time-domain waveformsimulation of simulation signal’s instantaneous signal’s instantaneous amplitude. amplitude; (b) The frequency domain waveform of waveformsimulation of signal’s simulation instantaneous signal’s instantaneous amplitude. amplitude; (b) The frequency domain waveform of When the signal is disturbed by strong background noise, fault information cannot be separated simulationWhen signal’s the signal instantaneous is disturbed amplitude. by strong background noise, fault information cannot be separated from the time domain waveform directly. The weak impact characteristics of the noisy simulation from the time domain waveform directly. The weak impact characteristics of the noisy simulation Whensignal were the signal clearly is and disturbed exactly byextracted strong by background adopting the noise, improved fault informationwavelet scalogram cannot algorithm be separated Whensignal the were signal clearly is disturbedand exactly by extracted strong backgrouby adoptingnd noise,the improved fault information wavelet scalogram cannot algorithmbe separated fromwhich the time is domainproposed waveform in this paper. directly. The Thereassigned weak impactwavelet characteristics scalogram was of used the noisyto enhance simulation impact signal from thewhich time is domainproposed waveform in this paper. directly. The reassigned The weak wavelet impact scalogram characteristics was used of tothe enhance noisy simulationimpact werefeatures clearly andfirstly. exactly Then, extracted the outside by adoptinginterference the components improved waveletwere decreased scalogram after algorithm synchronous which is signal featureswere clearly firstly. and Then, exactly the outside extracted interference by adopting components the improved were decreased wavelet scalogramafter synchronous algorithm averaging is performed on the reassigned wavelet scalogram. Finally, the visibility and the accuracy proposedaveraging in this is paper.performed The on reassigned the reassigned wavelet wavelet scalogram scalogram. was Finally, used to the enhance visibility impact and the features accuracy ﬁrstly. which is proposed in this paper. The reassigned wavelet scalogram was used to enhance impact features firstly. Then, the outside interference components were decreased after synchronous averaging is performed on the reassigned wavelet scalogram. Finally, the visibility and the accuracy

Energies 2018, 11, 1286 18 of 23

Then, the outside interference components were decreased after synchronous averaging is performed on the reassigned wavelet scalogram. Finally, the visibility and the accuracy of time-frequency spectrum were improved by extracting the wavelet ridge. The wavelet coefficients characterize the similarity between the signal being analyzed and the basic wavelet. The larger the wavelet coefficients, Energies 2018, 11, x 18 of 23 the stronger the impact characteristic of the signal and the more obvious the fault feature. The modulus Energies 2018, 11, x 18 of 23 of waveletof time-frequency coefficients of spectrum every frequencywere improved points by extr at eachacting moment the wavelet can ridge. be highlighted The wavelet coefficients by extracting the waveletcharacterize ridgeof time-frequency onimproved the similarity spectrum wavelet between were improved scalogram. the signal by beinextr Byactingg furtheranalyzed the wavelet analysis and the ridge. basic of The the wavelet. wavelet wavelet Thecoefficients ridge, larger the it is easily to extractwaveletcharacterize the time coefficients, domainthe similarity the and stronger between frequency the the impact signal domain characbeing informationanalyzedteristic of and the the ofsignal basic fault and wavelet. impulses. the more The larger obvious By analyzing the the the wavelet coefficients, the stronger the impact characteristic of the signal and the more obvious the simulationfault signals,feature. The it can modulus be concluded of waveletthat coefficients the proposed of every frequenc methody ispoints suitable at each for moment detecting can be the weak highlightedfault feature. by Theextracting modulus the of wavelet wavelet ridgecoefficients on impr of everyoved frequencwavelet yscalogram. points at each By furthermoment analysis can be of impact features. thehighlighted wavelet ridge, by extracting it is easily the to wavelet extract ridge the time on impr domainoved andwavelet frequency scalogram. domain By further information analysis of of fault impulses.the wavelet By ridge,analyzing it is easilythe simulation to extract the signals, time domain it can andbe concluded frequency domainthat the information proposed ofmethod fault is 4.2. Experimental Study suitableimpulses. for detectingBy analyzing the weakthe simulation impact features. signals, it can be concluded that the proposed method is suitable for detecting the weak impact features. In order to further demonstrate the validity of the method presented in this research, signals 4.2. Experimental Study were collected4.2. Experimental from a planet Study gear rolling element bearing experiment rig for analysis. The schematic In order to further demonstrate the validity of the method presented in this research, signals diagram of theIn order bearing to further test-to-failure demonstrate experiment the validity of system the method is shown presented in in Figure this research, 17. On signals the laboratory were collected from a planet gear rolling element bearing experiment rig for analysis. The schematic bench, a frequencywere collected transformer from a planet drivesgear rolling one element motor bearing connected experiment to the rig for bearing analysis. housing, The schematic the planetary diagram of the bearing test-to-failure experiment system is shown in Figure 17. On the laboratory gearbox anddiagram the magneticof the bearing powder test-to-fail brakeure experiment through thesystem coupling. is shown Thein Figure planet 17. gearOn the bearing laboratory outer ring is bench,bench, a afrequency frequency transformer transformer drivesdrives one motormotor connectedconnected to to the the bearing bearing housing, housing, the the planetary planetary machinedgearboxgearbox with and a and 1 the mm the magnetic magnetic wide fault powder powder and brakebrake the bearingthrough the the surface coupling. coupling. structure The The planet planet after gear gear machiningbearing bearing outer outer failurering ring is is shown in Figuremachined 18machined. Planetary with with a a1 gearbox1mm mm wide wide parametersfault fault andand the arebearing shown surface surface in structure Tablestructure1. after After after machining machining calculation, failure failure theis shown is parameters shown of each componentin inFigure Figure 18. during18. Planetary Planetary operation gearbox gearbox parametersareparameters shown areare in shownshown Table in2 in. Table Table 1. 1. After After calculation, calculation, the the parameters parameters of of eacheach component component during during operation operation areare shown inin TableTable 2. 2.

(a)(a)

(a) (a) (b) (b) Electric Electricmotor motor

Coupling Coupling Coupling Coupling Coupling Pl anetary CouplingMagnetic Frequency Plummer block gearbox Powder Brake transformer housing Pl anetary Magnetic Frequency Plummer block gearbox Powder Brake transformer housing(b)

Figure 17. A bearing test-to-failure experiment(b system.) (a) System structure; (b) Test system. Figure 17. A bearing test-to-failure experiment system. (a) System structure; (b) Test system. Figure 17. A bearing test-to-failure experiment system. (a) System structure; (b) Test system.

Figure 18. Planetary bearing outer ring fault surface map.

Figure 18. Planetary bearing outer ring fault surface map. Figure 18. Planetary bearing outer ring fault surface map.

Energies 2018, 11, x 19 of 23

Table 1. Parameters for planetary gearbox and planetary bearing.

Energies 2018, 11, 1286 Configuration Parameter Parameter Values 19 of 23 The number of sun gear tooth 17 The number of planet gear tooth 35 TableThe 1.numberParameters of annular for planetary gear tooth gearbox and planetary 88 bearing. The number of planet gear 3 Conﬁguration Parameter Parameter Values Transmission ratio 6.176 TypeThe of number planetary of sun bearing gear tooth 17 204 NJ The number of planet gear tooth 35 The numberThe number of planetary of annular bearing gear tooth roller 88 11 The bearingThe number pitch of planetdiameter gear 343 mm Transmission ratio 6.176 The planetaryType rolling of planetary element bearing diameter 204 7.493 NJ mm The planetaryThe number bearing of planetary contact bearing angle roller 11 0° The bearing pitch diameter 34 mm The planetary rolling element diameter 7.493 mm TableThe planetary2. Parameters bearing for contact failure angle experiment 0system.◦ Characteristic Frequency Frequency Values Table 2. Parameters for failure experiment system. Sampling Frequency 12,800 Hz The rotation speed 1800 rpm Characteristic Frequency Frequency Values The rotation frequency of sun gear 30 Hz Sampling Frequency 12,800 Hz The rotation frequencyThe rotation of speed planet carrier 1800 4.86 rpm Hz The relativeThe rotation rotation frequency frequency of of sun planet gear gear 30 12.21 Hz Hz The absoluteThe rotation frequency frequency of planet of planet carrier gear 4.86 7.36 Hz Hz The relative rotation frequency of planet gear 12.21 Hz Bearing Theouter absolute ring rotationfault characteristic frequency of planet frequency gear 7.36 31.53 Hz Hz BearingBearing inner outerring ringfault fault characteristic characteristic frequency 31.53 49.38 Hz Hz Bearing inner ring fault characteristic frequency 49.38 Hz In the event of a failure in the outer ring of planet gear, the fault characteristics of the original accelerationIn thesignal event measured of a failure by in sensors the outer are ring weak of planet because gear, of the the fault long characteristics and complex of the transmission original acceleration signal measured by sensors are weak because of the long and complex transmission path between the vibration source and measuring points. There is no impact characteristic in its time path between the vibration source and measuring points. There is no impact characteristic in its domain waveform as shown in Figure 19a. Only the rotation frequency component of the shaft and time domain waveform as shown in Figure 19a. Only the rotation frequency component of the shaft its differentialand its differential frequency frequency components components with with the theplanetary planetary carrier carrier exist exist inin FigureFigure 19 b.19b. The The fault fault characteristiccharacteristic frequency frequency cannot cannot be be determined determined from from the frequencyfrequency domain domain waveform. waveform.

(a) 20 10

-10 Amplitude(g) -20 0 1 2 3 4 5 Time (s)

(a)

(b) 0.5 f1=fs-fc f f f =f 0.4 4 5 2 r f3 f3=2fr f f f4=3fr 0.3 2 7 f5=4fr

f6=5fr 0.2 f7=6fr f1 f6 f8 Amplitude(g) f8=7fr-4f c f 9 f =7f 0.1 9 r

0 0 100 200 300 400 500 Frequency (Hz)

(b)

Figure 19. Cont.

Energies 2018, 11, 1286 20 of 23 Energies 2018, 11, x 20 of 23

Amplitude(g) 0 0.25 -0. 125 0 0.2 0.4 0.6 0.8 1 0 Time (s) -0. 125 0 0.2 0.4 0.6 0.8 1 (cTime) (s)

Figure 19. The analytic results of planetary bearing(c) outer ring fault. (a) The time domain waveform Figure 19. The analytic results of planetary bearing outer ring fault. (a) The time domain waveform of of planetaryplanetaryFigure 19.bearing bearing The analytic outer outer ring results ring fault; fault; of planetar ( (bb)) The They envelope bearing outer spectrum spectrum ring offault. of planetary planetary (a) The bearing time bearing domain outer outer ringwaveform ring fault; fault; (c) The(ofc) keyplanetary The phase key phase bearing signal signal outerof sun of sunring gear gearfault; input input (b) Theshaft. shaft. envelope spectrum of planetary bearing outer ring fault; (c) The key phase signal of sun gear input shaft.

It canIt be can concluded be concluded that that the the traditional traditional spectral spectral analysis analysis methodmethod is is inapplicable inapplicable for for planetary planetary It can be concluded that the traditional spectral analysis method is inapplicable for planetary bearingbearing fault faultdiagnosis. diagnosis. According According to the to key the phase key phase signal signal shown shown in Figure in Figure 19c, the19c, sun the gear sun rotation gear bearing fault diagnosis. According to the key phase signal shown in Figure 19c, the sun gear rotation frequencyrotation of frequencythe planetary of the gearbox planetary is gearbox approximately is approximately 29.7 Hz. 29.7 Defining Hz. Deﬁning a value a value of 127 of 127ms msas the frequency of the planetary gearbox is approximately 29.7 Hz. Defining a value of 127 ms as the as the synchronization cycle, afterward, a continuous wavelet transform was carried out on the synchronizationsynchronization cycle, cycle, afterward, afterward, a continuousa continuous waveletwavelet transform waswas carried carried out out on onthe the planet planet planet bearing outer ring vibration signals. Twenty synchronization period signals were selected bearingbearing outer outer ring ringvibration vibration signals. signals. Twenty Twenty synchron synchronizationization period signals signals were were selected selected for foranalysis. analysis. for analysis. The wavelet scalogram and improved wavelet scalogram results are displayed in The waveletThe wavelet scalogram scalogram and and improved improved wavelet wavelet scalscalogramogram results areare displayed displayed in inFigure Figure 20a,b, 20a,b, Figure 20a,b, respectively. respectively.respectively.

（a）（a） 66 66 （b）（b）66 5 55 5 5 55 4 4 4 4 4 4 33 3 3 3 3

Frequency(kHz) 2 2 2 Frequency(kHz) Frequency(kHz) 2 2 2 Frequency(Hz)

1 Frequency(kHz) 1 Frequency(Hz) 1 Frequency(Hz) 1

1 Frequency(Hz) 0 1 0 20 40 60 80 100 120 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 Time (ms) 0 20 40 60 80 100 120 0 20 40 60Time(ms)80 100 120 0 Time (ms) 0 20 40 60 80 100 120 0 2020 4040Time(ms) 60 8080 100100 120120 Time(ms)Time (ms) (a) Time(ms)(Timeb) (ms) (a) (b) Figure 20. Improved wavelet scalogram analysis. (a) Reassigned scalogram; (b) Scalogram under Figure 20. Improved wavelet scalogram analysis. (a) Reassigned scalogram; (b) Scalogram under Figuresynchronoussynchronous 20. Improved averaged.averaged. wavelet scalogram analysis. (a) Reassigned scalogram; (b) Scalogram under synchronous averaged. The wavelet ridges extracted from the improved wavelet scalogram are shown in Figure 21. The instantaneousThe wavelet amplitude ridgesextracted and the corresponding from the improved frequency wavelet were scalogram calculated are according shown in to Figure Equation 21. The wavelet instantaneous ridges amplitudeextracted from and thethe correspondingimproved wavelet frequency scalogram were are calculated shown in according Figure 21. to The instantaneous(37) and the amplitude Fourier transform and the formula,corresponding as shown frequency in Figure were 22. It calculatedcan be observed according that the to planetEquation Equationbearing outer (37) and ring the fault Fourier characteristic transform formula,frequency as shownis 31.53 in Hz Figure and 22 its. It multiplied can be observed frequency that (37) and the Fourier transform formula, as shown in Figure 22. It can be observed that the planet thecomponents planet bearing are clearly outer displayed ring fault characteristicin the partial frequencyenlarged view is 31.53 in the Hz andupper its right multiplied corner frequency of Figure bearing outer ring fault characteristic frequency is 31.53 Hz and its multiplied frequency components22b. For planet are gear clearly bearing displayed fault insignals the partial with enlargedweak impact view characteristics, in the upper right the fault corner features of Figure cannot 22b. componentsForbe directly planet are gear extractedclearly bearing displayed by fault envelope signals in spectrumthe with partial weak analysis. enla impactrged Fortunately, characteristics, view in thethe theupperinterference fault right features informationcorner cannot of Figure be is 22b. Fordirectlyweakened planet extracted gearby using bearing by envelope the improvedfault spectrum signals wavelet analysis. with scalogramweak Fortunately, impact analysis. thecharacteristics, interference This method information the is fault very features iseffective weakened cannotfor be directlybyenhancing using extracted the the improved impact by characteristics.envelope wavelet scalogram spectrum Subsequently, analysis. analysis. wavelet This Fortunately, method ridge is extraction very theeffective interference is also for considered enhancing information as the an is weakenedimpactimproved by characteristics. usingenvelope the extraction improved Subsequently, method wavelet waveletand thescalogram ridgefaint fault extraction analysis. information is alsoThis under consideredmethod external is as very interference an improvedeffective is for envelope extraction method and the faint fault information under external interference is extracted enhancingextracted the byimpact a series characteristics. of transforms. Subsequently,The above desc riptionwavelet indicates ridge extraction that the proposed is also considered method has asa an by a series of transforms. The above description indicates that the proposed method has a better improvedbetter envelope performance extraction in planet method gear bearing and the weak faint fault fault diagnosis. information Based under on the external above interferencesimulation is performance in planet gear bearing weak fault diagnosis. Based on the above simulation signal and extractedsignal by and a series vibration of transforms. signal analysis, The it above can be desc concludedription thatindicates the proposed that the method proposed is effective method for has a vibrationrolling element signal analysis,bearing analysis. it can be concluded that the proposed method is effective for rolling element betterbearing performance analysis. in planet gear bearing weak fault diagnosis. Based on the above simulation signal and vibration signal analysis, it can be concluded that the proposed method is effective for rolling element bearing analysis.

Energies 2018, 11, 1286 21 of 23

Energies 2018, 11, x 21 of 23 Energies 2018, 11, x 21 of 23 4.54.5 4.54.5

44 44

3.53.5 Frequency(kHz) 3.53.5 Frequency(kHz) Frequency(Hz)

Frequency(Hz) 33 0 20 40 60 80 100 120 330 20 40 60 80 100 120 0 2020 4040 Time(ms)Time 6060 (ms) 80 100 120120 Time(ms)Time (ms)

Figure 21. Wavelet ridge of bearing signals for improved wavelet scalogram. Figure 21. WaveletWavelet ridge of bearing signal signalss for improved wavelet scalogram.

(a)4.23.5 (a)4.23.5 3.93 3.93 3.62.5 3.62.5 3.32 2 Amplitude(g) 3.3

Amplitude(g) 1.53

Amplitude(g) 1.53 Amplitude(g) 2.71 0 20 40 60 80 100 120 2.710 20 40 60 80 100 120 0 2020 4040 Time(ms)Time6060 (ms) 8080 100100 120120 Time(ms)Time (ms) (a) (b)0.70.7 (a) (b)0.70.7 (b) 1 0.6 f =31.53=f 0.6 (b) 1 1 ob 0.8 f =63.05=2f 0.60.6 f f2 =31.53=f ob f1 3 f 1=94.58=3fob 0.50.5 0.8 f2 f3 =63.05=2fob f f4 2 ob 0.6 f1 3 f4=126.1=4fob f5 f =94.58=3f 0.50.5 f2 3 ob f4 f5=189.2=6fob 0.6 f4=126.1=4fob 0.4 f5f 0.4 0.4 6 f6=220.7=7fob f5=189.2=6fob 0.4 f 0.4 Amplitude(g) 0.4 6 f6=220.7=7fob 0.30.3 0.2 Amplitude(g) Amplitude(g) 0.30.3 0.2 Amplitude(g) 0.20.2 0

Amplitude(g) 0 100 200 300 400 500 600 Amplitude(g) 0.20.2 0 Frequency (Hz)

Amplitude(g) 0 100 200 300 400 500 600 0.10.1 Frequency (Hz) 0.10.1 00 0 10001000 20002000 30003000 40004000 50005000 60006000 00 0 10001000 20002000Frequency(Hz)Frequency30003000 (Hz) 40004000 50005000 60006000 Frequency(Hz)Frequency (Hz) (b) (b) Figure 22. Time domain waveforms and spectrum of instantaneous amplitude. (a) Waveform of Figure 22. Time domain waveforms and spectrum of instantaneous amplitude. (a) Waveform of instantaneousFigure 22. Time amplitude; domain ( waveformsb) Spectrum and for instantaneous spectrum of instantaneous amplitude. amplitude. (a) Waveform of instantaneous amplitude; (b) Spectrum for instantaneous amplitude. instantaneous amplitude; (b) Spectrum for instantaneous amplitude. 5. Conclusions 5. Conclusions A vibration signal model is established by studying the vibration mechanism of planetary A vibration signal model is established by studying the vibration mechanism of planetary gearboxes,A vibration which is signal the theoretical model is establishedbasis of diagno bysing studying the running the vibration state of planetary mechanism bearings of planetary based gearboxes, which is the theoretical basis of diagnosing the running state of planetary bearings based ongearboxes, the vibration which signal is the analysis theoretical method. basis ofIn diagnosingaddition, bearing the running fault signal state ofweak planetary impact bearings feature on the vibration signal analysis method. In addition, bearing fault signal weak impact feature determinationbased on the is vibration very difficult signal be analysiscause of method.the excessive In addition,interference. bearing This pape faultr applied signal weaksynchronous impact determination is very difficult because of the excessive interference. This paper applied synchronous averagingfeature determination to reduce the isdisturbed very difficult components because of a of reassigned the excessive wavelet interference. scalogram which This paperis obtained applied by averaging to reduce the disturbed components of a reassigned wavelet scalogram which is obtained by multi-scalesynchronous analysis averaging of a continuous to reduce the wavelet disturbed transf componentsormation, thus of aimproving reassigned the wavelet wavelet scalogram time-frequency which multi-scale analysis of a continuous wavelet transformation, thus improving the wavelet time-frequency resolution.is obtained As by multi-scalewell, a wavelet analysis ridg ofe ais continuous extracted waveletto improve transformation, the visibility thus and improving the accuracy the waveletof the resolution. As well, a wavelet ridge is extracted to improve the visibility and the accuracy of the time-frequencytime-frequency resolution.spectrum. The As well,results a wavelet indicated ridge that is extractedthe newly to improveproposed the method visibility has and good the time-frequency spectrum. The results indicated that the newly proposed method has good performanceaccuracy of the in time-frequencydetecting of planet spectrum. gear Thebearing results weak indicated fault signals that the that newly are proposed usually methodmasked hasby performance in detecting of planet gear bearing weak fault signals that are usually masked by externalgood performance disturbances. in detecting This method of planet can contribute gear bearing to weakweak faultimpact signals characteristic that are usually determination masked byin external disturbances. This method can contribute to weak impact characteristic determination in bearingexternal fault disturbances. analysis. This method can contribute to weak impact characteristic determination in bearing fault analysis. Author Contributions: R.Y. built the simulation model, analyzed the data and wrote the paper; H.L. conceived Author Contributions: R.Y. built the simulation model, analyzed the data and wrote the paper; H.L. conceived theAuthor signal Contributions: processing methodR.Y. built and the guid simulationed the writing model, of analyzedpaper; C.W. the dataand C. andH. wrotedesigned the paper;the test H.L. stand conceived and the the signal signal processing processing method method and and guid guideded the the writing writing of ofpaper; paper; C.W. C.W. and and C.H. C.H. designed designed the thetest teststand stand and andthe experiments. experiments.the experiments. Acknowledgments: This research was funded by Chinese National Science Foundation (No. 51575075) and Acknowledgments: This research waswas fundedfunded byby ChineseChinese NationalNational ScienceScience FoundationFoundation (No.(No. 51575075) and ChineseChinese National National Science Science Foundation Foundation (No. (No. 51175057), 51175057), and and Collaborative Innovation Innovation Center Center of of Major Machine Chinese National Science Foundation (No. 51175057), and Collaborative Innovation Center of Major Machine ManufacturingManufacturing in Liaoning for this research is gratefully acknowledged. Manufacturing in Liaoning for this research is gratefully acknowledged. ConflictsConflicts of Interest: The authors declare no conflictconflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

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