Combination of envelope spectra and generative topographic mapping to diagnose bearing fault and evaluate degradation of bearing

Abstract

Functioning of rotary machines is based on the rolling element of bearings, as the rolling element of bearings is the crucial component of rotary machines. The entire operation of rotating machines may terminate after the failure of rolling element of bearings. Hence, the failure of rotating machine is mainly dependent on rolling element of bearings, as it may break down an entire process. Thus, to avoid the sudden breakdown of rotary machines, it is essential to develop the diagnostics and prognostics methodology of rolling element of bearings. In the rolling element of bearings, the assessment of bearing degradation and fault diagnostics is essential to establish the methodology. Therefore, this article proposes a methodology to diagnose the fault of bearing and to assess bearing degradation. Envelope spectra and generative topographic mapping are used together in the methodology proposed. Selection of frequency band based on encompasses one or more resonance of spectrum of signal. Envelope spectrum is determined to pick up the peak of the characteristic fault frequencies of the bearing. Extraction of features from time and frequency domains is used in the generative topographic mapping. Health degradation index, an index termed as evolution of bearing degradation, is obtained from the classifier of generative topographic mapping. Experiments were conducted on the bearings to verify the proposed methodology. On interpreting the obtained results, it was found that the proficiency of proposed methodology is most appropriate for detecting the fault in bearing and tracking the degradation evolution in bearings.

Keywords

Bearing dynamics fault detection envelop spectrum analysis bearing degradation assessment generative topographic mapping

Introduction

Reduction of downtime is possible by implanting an appropriate methodology. It may be possible to avoid the breakdown and enhance the functioning of rotary machineries. The breakdown of rotary machinery leads to a decrease in the productivity of machine and an increase in the maintenance cost. Such a breakdown is due to the failure of rolling element bearing. Hence, the bearings form the key element of rotary machinery. According to the Electric Power Research Institute (EPRI) report, 40% of electric motor failures are due to the failure of bearings.¹ In order to ensure the incessant operation of machinery, the lifetime of the bearing needs to be monitored carefully. As per industry standards,² the fatal failure size of defect in bearing can be detected using diagnostic methods. Two important aspects of condition monitoring of bearings, namely bearing diagnostics and prognostics, are suggested.³ Prevention of bearing failure is possible by implementing a proper condition monitoring technique. The changes in the signal are explored with the signal processing techniques that identify an incipient and propagation of fault in bearing. The detection and classification of fault in bearings is suggested in the bearing diagnostics. The bearing degradation over the lifetime and remaining useful life are presented in the bearing prognostics. Hence, diagnostics and prognostics are the powerful techniques in the condition monitoring of bearing.

Diagnosis of bearing fault by the vibration methods is expressed in Jardine et al.⁴ and Heng et al.⁵ The changes in the vibration characteristics level, when defect develops in bearing, are represented with the root mean square (RMS) and kurtosis, as suggested by Yang and Chen⁶ and Aini et al.⁷ Fault frequency of bearing is recognized to transform the vibration signal in frequency domain. This method is effective to diagnose the bearing fault.^8–11 Envelope analysis is used to identify an impulse of demodulating high frequency cause by fault developed in gear mesh.^12–15

The assessment of performance degradation is observed using the health degradation indicator (HDI) of bearing, which will help warn of the unexpected increase in the severity of degradation. Feature extraction from bearing vibration signal is made easy with the signal processing techniques.¹⁶ The data obtained from the experiment can be used to model the linear relationship between variables using the traditional probabilistic method. But most of the industrial process variables needed to model are characteristically in a non-linear relationship. In the literature, it has been seen that the some of the probabilistic non-linear models are proposed for the non-linear processes. Bishop et al.¹⁷ proposed generative topographic mapping (GTM) for non-linear process monitoring. The variable observed in high-dimensional into low-dimensional features seeks the explanation using GTM.¹⁸ The data process monitoring and visualization areas are applied in the GTM.^19,20 Therefore, using GTM, it was possible to effectively extract the non-linear features from the data space. The maximum likelihood with expectation-maximization (EM) algorithm determined the model parameters.²¹

The transient forces are generated due to the impulsive action by the rolling element, which passes over local fault. Capturing of transient forces would be possible with proper diagnostic techniques. The impact at regular interval due to constant input speed of bearing is known as the characteristic fault frequency. Higher frequency develops in the system due to the resonance of excitation of impacts than due to any other machine element. This resonance is considered as the amplitude modulation of characteristic fault frequency. The envelope spectra can be applied to the resonance of higher energy frequencies with band-pass filter to detect the defect in the bearing. Bearing degradation shown with the HDI of bearing is needed to capture and avoid the unexpected increase in the severity of degradation. The extracted fault features express the health status of bearing. Sensitivity of feature changes in the fault is great concerned to determine the proper process of extraction feature. Initiation and propagation of fault reveals the degradation in bearing. So, it is more challenging to select based on the fault feature of bearing.

As a novel approach, the two aspects are presented together in this article. This article addresses both fault diagnosis of bearing and performance degradation assessment of bearings. The novelty of this article is in the use of envelope spectra, which determine the characteristic fault frequency of the bearings and bearing degradation using GTM. The selection of frequency band from the spectrum of signal containing one or more resonance is based on the presence of energy in the band. The highest resonance energy band would be the best frequency band for the envelope spectra. Tracking of bearing damage evolution using the exponential value of health degradation indicator (HDIE) is mapped with GTM. Sensitivity and effectiveness of features in time and frequency domains are observed using this methodology. The selection of fault features should be such that it warns of an incipient fault and manages to show the severity of the increase in degradation.

Technical background

Theory of envelope spectra

The selection of frequency band from the spectrum of signal for analyzing envelope spectra is a key part for detecting the fault in bearing. The response effectively includes the fault induced in the bearing. The propagation of fault is shown in the frequency band. In their literature, Sun and Tang²² and Urbanek et al.²³ proposed an improvement of sensitivity of envelope spectra. The modulating signal may contain several parts as it is difficult to diagnose the fault. The non-linear distortion of signals is contained in the modulated signal as the fault-induced strength is observed to be very weak. The defective bearings generate a distinctive impulse every time the fault encounters into a mating surface. The demodulation can be carried out for separating the modulating signal from the modulated signal.

In continuous operation, the system response can be presented due to the train of impact as

x (t) = \sum_{- n}^{+ n} δ (t - k T_{d}) A (t) * e^{- \frac{t}{T_{e}}} \sin (ω_{r} t + θ_{0})

(1)

where $T_{d} = 1 / f_{d}$ , $f_{d}$ is the bearing fault frequency, A(t) is the bearing load distribution effect, $δ (t)$ is the Dirac delta function, $θ_{0}$ is the initial phase angle, and $T_{e}$ is the trigger time of the impulse..

When impulsive train signal of fault is induced in bearing

a_{m} (t) = \sum_{- n}^{+ n} δ (t - k T_{d}) A (t) * e^{- \frac{t}{T_{e}}}

(2)

The resonance frequency ( $ω_{r}$ ) is shown in carrier of signal ( $c (t)$ )

c (t) = a_{m} (t) \sin (ω_{r} t + θ_{o})

(3)

The frequency of amplitude modulated signal is the fault frequency. Demodulation is done to separate the modulating signal. Indication of vibration signal for the resonance of bearing can be identified by the envelope spectra. The resonance of bearing is contained in the frequency band. The modulating signal approximates the envelope signal using equations (2) and (3). The envelope signal spectrum lies in range of ± $ω_{r}$ . The vibration modes with spectral magnitude energy are indicated in Figure 1(a). This energy can be used for the bandwidth. The characteristic fault frequency is associated with the first term of equation (2) as represented in the envelope signal. The details of the methodology are shown in Figure 1(b).

Figure 1.

(a) Frequency band for envelope spectra. (b) Procedure of envelope spectra.

Characteristic fault frequencies

The outer race of bearing is considered as stationary to compute the characteristic fault frequencies of defect at cage, ball, outer, and inner race. The characteristic fault frequency is represented as given in equations (4)–(7)

F_{O R F} = \frac{N_{b}}{2} F_{s} (1 - \frac{D_{b} \cos (θ)}{D_{p}})

(4)

F_{I R F} = \frac{N_{b}}{2} F_{s} (1 + \frac{D_{b} \cos (θ)}{D_{p}})

(5)

F_{B F} = \frac{D_{p}}{D_{b}} F_{s} (1 - \frac{D_{b}^{2} \cos^{2} (θ)}{D_{p}^{2}})

(6)

F_{C F} = \frac{1}{2} F_{s} (1 - \frac{D_{b} \cos (θ)}{D_{p}})

(7)

where F_ORF is the outer race fault, F_IRF is inner race fault, F_BF is the ball fault frequency, and F_CF is the cage fault frequency; D_p is pitch circle diameter; F_s is speed of shaft; D_b is ball diameter; θ is ball contact angle; and N_b is number of balls.

GTM

The non-linear parametric function is mapped using the GTM. Every point in latent space x is mapped in the data space using the function y(x, W). The points x in latent space map corresponding points y(x, W) in data space. W is a matrix parameter that represents weights and biases. The main aim of GTM is the probability distribution of Data-dimensional space (D-space) in terms of Latent-dimensional space (L-space). The probability distribution p(t) representation of D-space ( $t = (t_{1}, t_{2}, t_{3}, \dots, t_{N})$ ) in L-space ( $x = (x_{1}, x_{2}, x_{3}, \dots, x_{L})$ ) is depicted in Figure 2 as explained in the studies by Bishop et al.^17,24 The matrix parameters W could be represented as weight and biases governing the mapping in the neural network.^25–27

Figure 2.

Generating topographic mapping.

The L-dimensional L-space $(x \in R^{L})$ is to a matching point in D-space $(y \in R^{D})$

where L < D

The L-space defines a probability distribution p(x) as prior distribution of x. The t-space distribution would confine to the manifold of L-dimensional as singular for L < D. The data lie in the manifold of lower dimensional is approximately included a vector t noise model. The distribution of t is chosen, for given x and W, to be a radially symmetric Gaussian centered on y(x, W) having noise variance $β^{- 1}$ so that

p (t | x, W, β) = {(\frac{β}{2 π})}^{\frac{D}{2}} e x p {- \frac{β}{2} {‖ y (x, W - t) ‖}^{2}}

(8)

The values of W distributes in t-space, obtained by integration over x-distribution, are explained as

p (t | W, β) = \int^{} p (t | x, W, β) p (x) d x

(9)

Inverse matrix $β$ and parameter matrix W, determined by maximizing the log likelihood $L (W, β)$ for data set $D = (t_{1}, t_{2}, t_{3}, \dots, t N_{N})$ , are expressed as

L (W, β) = In \prod_{n = 1}^{N} p (t_{n} | W, β)

(10)

In the latent space, the centered node of regular grid is sum of delta functions presented in specific form of p(x) as given below

p (x) = \frac{1}{K} \sum_{i = 1}^{K} δ (x - x_{i})

(11)

The $y (x_{i}, W)$ is the mapping of each point $x_{i}$ in the data space as distribution function in the form

p (t | W, β) = \frac{1}{K} \sum_{i = 1}^{K} p (t | x_{i}, W, β)

(12)

Log-likelihood function for analytical tractability uses the delta functions for a set of K weights represented by p(x)

L (W, β) = \sum_{n = 1}^{N} In {\frac{1}{K} \sum_{i = 1}^{K} p (t_{n} | x_{k} W, β)}

(13)

Gaussian mixture model²⁸ was constrained to correspond to the distribution $p (t | W, β)$ . The Gaussians center $y (x_{i}, W)$ cannot operate alone as related to the function $y (x, W)$ . The mapping function $y (x, W)$ is continuous and smooth. The formulation of the model with the expectation maximization algorithm is mentioned below.

In the algorithm, the current weight matrix W_old and inverse noise variance $β_{o l d}$ are given at some point. In the E-step, the responsibility or posterior probabilities are estimated using W_old and $β_{o l d}$ for every data point t_n of each Gaussian component i using Bayes’ theorem

R_{i n} (W_{o l d}, β_{o l d}) = p (x_{i} | t_{n}, W_{o l d}, β_{o l d})

(14)

The data log likelihood expectation in the form

\begin{array}{l} L_{c o m p} (W, β) = \sum_{n = 1}^{N} \sum_{i = 1}^{K} R_{i n} (W_{o l d}, β_{o l d}) \\ In {p (t_{n} | x_{i} W, β)} \end{array}

(15)

Maximizing equation (15) with respect to W and $β$

y_{d} (x, W) = \sum_{m}^{M} \emptyset_{m} (x) w_{m d}

(16)

The weight and bias parameters are contained in the matrix W with dimensions M × D

\emptyset_{m} (x) = {\begin{matrix} e x p {\frac{{| x - μ_{m} |}^{2}}{2 σ^{2}}} i f m \leq M_{N L} \\ x^{l} i f m = M_{N L} + l, l = 1, \dots L \\ 1 i f m = M_{N L} + L + 1 = M \end{matrix}

The Gaussian basis functions of non-normalize form is the non-linear M_NL. The linear trend data are captured with the linear basis function L. In the M-step, W_new and $β_{n e w}$ from the maximized log likelihood $ℒ (W, β)$ equation are given by W_new from

\emptyset^{T} G_{o l d} \emptyset W_{n e w}^{T} = \emptyset^{T} R_{o l d} T

(17)

$β_{n e w}$ from

\frac{1}{β_{n e w}} = \frac{1}{N D} \sum_{n = 1}^{N} \sum_{i = 1}^{K} R_{i n} (W_{o l d} β_{o l d}) ∥ W_{n e w} \emptyset (x_{i}) - t_{n} ∥^{2}

(18)

where ∅ is the K × M matrix with element $\emptyset_{i j} = \emptyset_{j} (x_{i}),$ T is the N × D matrix with element $t_{n k}$ , R is the K × N matrix with element $R_{i n}$ , And G is the K × K diagonal matrix with element

G_{i i} = \sum_{n = 1}^{N} R_{i n} (W, β)

(19)

The E-step and M-step is applied in the EM algorithm. The E-step is related to the evaluation of the posterior probabilities in equation (14). The M-step is presented by the solution of equations (17) and (18), as discussed in the study by Bishop.²⁹

Feature extraction

Collected bearing signals using data acquisition (DAQ) system are explored for the extraction of fault features. In this, the characteristic equations of time and frequency domain as listed in Table 1 are used to evaluate the fault features such as RMS, kurtosis, crest factor, frequency center, power spectrum, and so on. All the feature vectors are categorized on the basis of monotonicity. Monotonic fault feature vectors are selected for further processing of bearing. When the impulses are significant, the crest factor is enough to show the bearing fault accurately than the other features.³⁰ Furthermore, relevant trending characteristics are shown in the other features. The highly monotonic features are used for the detection of fault in bearing as it is treated as an effective feature for calculation of bearing degradation. Recent publications^31–34 have suggested the frequency domain feature for the prediction of degradation in gear and bearings. The selected fault feature vectors are being applied as an input vector to the GTM.

The fault feature vectors used for the estimation of HDI are applied for the assessment of bearing degradation. Initially, the GTM network is trained with the exponential value of HDI. The trained GTM network is used to pass the test vector of fault feature. As HDI increases, the degradation becomes more severe.^2,32,34

The HDI shows the early stage of bearing condition and propagation of fault. With the help of the traditional Poisson exponential control chart, a more accurate position of the bearing condition is given as suggested in an earlier study.³⁵ Therefore, the following procedure of exponential control chart is applied for calculation of exponential values of HDI (represented HDIE). The flow chart of HDIE is given in Figure 3.

Table 1.

Computational formulae of features.

Time-domain features	Equation
Mean	$\bar{X} = \frac{1}{N} \sum_{t = 1}^{N} X (t)$
Root mean square (RMS)	$X_{R M S} = \sqrt{\frac{1}{N} \sum_{t = 1}^{N} X {(t)}^{2}}$
Maximum amplitude	$X_{m a x} = m a x (X (t))$
Minimum amplitude	$X_{m i n} = m i n (X (t))$
Standard deviation	$σ = \sqrt{\frac{1}{N} \sum_{t = 1}^{N} {(X (t) - \bar{X})}^{2}}$
Skewness	$X_{s k e w} = \frac{1}{N σ^{3}} \sum_{t = 1}^{N} {(X (t) - \bar{X})}^{3}$
Kurtosis	$X_{k u r t o} = \frac{1}{N σ^{4}} \sum_{t = 1}^{N} {(X (t) - \bar{X})}^{4}$
Crest factor	$C F = \frac{m a x \| X (t) \|}{R M S}$
Impulse factor	$I F = \frac{m a x \| X (t) \|}{\frac{1}{N} \sum_{t = 1}^{N} \| X (t) \|}$
Shape factor	$S F = \frac{R M S}{\frac{1}{N} \sum_{t = 1}^{N} \| X (t) \|}$
Margin factor	$M F = \frac{m a x \| X (t) \|}{{(\frac{1}{N} \sum_{t = 1}^{N} \sqrt{\| X (t) \|})}^{2}}$
Peak to peak	$P P = X_{m a x} - X_{m i n}$
Frequency-domain features	Equation
Mean frequency	$f_{m} = \frac{1}{N} \sum_{n = 1}^{N} S (n)$
Frequency center	$f_{c} = \frac{\sum_{n = 1}^{N} fn (S (n))}{\sum_{n = 1}^{N} S (n)}$
RMS frequency	$f_{R M S} = \sqrt{\frac{\sum_{n = 1}^{N} f^{2} n (S (n))}{\sum_{n = 1}^{N} S (n)}}$
Power spectrum	$P o w e r (n) = {\| X (n) \|}^{2}$
Maximum power spectrum	$X_{m a x f} = m a x (P o w e r (n))$
Maximum envelope	$X_{m a x e n} = m a x (E n v)$

Figure 3.

Flow chart of assessment of HDIE.

Poisson distribution, $X_{t}, t = 1, 2, \dots n$ , is observed as an independent process. When mean µ of process control is µ = µ₀, it means that the process is in control. When the mean values change, say, µ = µ₁, then the process is out of control. The Poisson exponential control chart is defined as

Q_{t} = λ * X_{t} + (1 - λ) * Q_{t - 1}

(20)

where λ is the smoothing constant, 0 < λ ⩽ 1; Q₀ = µ₀ is initial value set for target; head start value Q₀ is for fast initial response (FIR).

Experiments and data analysis

The performance of the proposed methodology is validated using the collected experimental data. The bearing test rig is used to conduct the experiment in the laboratory as the normal bearing fitted with bearing housing. The detail of bearing test rig is given in Figure 4(a), which comprises a motor, speed controller, hydraulic loading system, and a bearing housing and loading. The synchronous motor, bearing set, coupling, and speed controller are the assembled components of the bearing test rig. The hydraulic jack with arm lever is used to apply the radial load on bearing housing as the details are shown in Figure 4(a).

Figure 4.

(a) Bearing test rig. (b) Dimensions of the tested bearing.

The test bearing is mounted on the bearing housing. Run-to-failure test is conducted to collect the data over the lifespan of bearing. Acquired data are used for detection of fault and degradation analysis. Uncertainty in RMS data indicates the fault initiation and propagation in bearing. Once failure is reported, the experiment is stopped to remove the faulty bearing and the next experiment is conducted with a fresh bearing. The running time of each bearing is recorded for further processing of the collected data.

The bearing vibration signal is collected through piezoelectric accelerometer IMI 608 A11 model sensor and friction torque transducer. Friction torque transducer consists of signal conditioner, junction box, and two beams (RH Beam and LH Beam), which are made of Spectra Quest. Two different stiffness sets of beams are used to adjust the extensive range of experiments. Their transverse and torsional stiffness are 2500 lbs/in., 5700 lbs/in. and 72.8e3 lb-in./radian, 166e3 lb-in./radian, respectively. Strain gage fatigue limit rated output is 2.2 V relative to the free unloaded voltage for both pairs, and their torque calibrations, at ∆V = 3.00, are 55 lb-in./V and 125 lb-in./V, respectively. Both are mounted on the bearing housing and eight channel National Instrument’s PCI-4474 dynamic signal acquisition card. The DAQ control parameters are sampling frequency 5.12 kHz, frequency lines 1600, and time duration 0.8 s.

The length of the vibration signal for a period of 0.8 s is captured and stored after every 10 min throughout the lifespan of the bearing. The SKF 6205 deep groove ball bearing is tested. The geometric details and dimensions of the tested bearing are given in Figure 4(b). In this study, the normal bearings are used to conduct the run-to-failure test. The details of tested bearing information are given in Table 2. The characteristic fault frequencies at input frequency of 50 Hz are given in Table 3.

Table 2.

Test bearings information.

Bearing	Radial load (kN)	Speed (r/min)	Running time (min)
Bearing 1, test 1 (B1-1)	6.5	3000	19,900
Bearing 2, test 2 (B2-2)	8.7	3000	6750

Table 3.

Characteristics fault frequencies at input frequency (50 Hz).

Types of faults	Characteristic fault frequency (Hz)
Inner race fault	270.75
Outer race fault	179.25
Ball fault	235.73
Cage fault	19.91

Results and discussion

As discussed in the above section, the diagnosis and prognosis of the bearing are presented together in this article. The collected data of bearings B1-1 and B2-2 are taken for the verification of the proposed methodology. The envelope spectra are applied to the diagnosis of bearings B1-1 and B2-2. The run-to-failure history of bearings B1-1 and B2-2 is depicted in Figure 5(a). In Figure 5(a), the bearing condition is presented over the lifetime. It signifies that the bearing is almost normal up to 60% of running time. Suddenly, a high impact is seen as a result of the initiation of fault. The fault is propagated, indicating the beginning of degradation in the bearing. In view of the above, the bearing condition is categorized into three zones as normal, degradation, and failure zones. These zones are also termed as normal, slight, and severe. The waveforms of bearings B1-1 and B2-2 are shown in Figure 5(b) at three zones namely normal, slight, and severe. It clearly indicated that the effect of impulsive force due to the fault in bearing is clearly seen in the vibration signature as a peak value of impact. The performance of bearing degradation over the lifetime, evaluated in three zones, is presented in this article. The waveform and spectrum of normal bearing B1-1 are depicted in Figure 6(a) and (b), respectively. The frequency bandwidth taken as 0.5 kHz to 1.5 kHz for an analysis of envelope spectra indicated the highest energy of resonance frequency. The resulting values of envelope spectrum are shown in Figure 6(c). In Figure 6(c), it clearly indicated that the peak values are the periodic frequency of shaft and there are no other peak values in the spectra. It means that the bearing is normal and no defect occurred in the bearing. At the severe stage, the waveform and spectrum of signal are shown in Figure 7(a) and (b) and bandwidth of frequency band is applied to the spectrum of signal. The envelope spectrum at the severe stage is shown in Figure 7(c). In Figure 7(c), the highest peak value (179 Hz) is clearly matched with the characteristic fault frequency of outer race of bearing (see Table 3). It signifies that the fault occurs in the outer race of the bearing. Similar procedure is followed for the other bearing B2-2. The waveform, spectrum of signal, and envelope spectrum of normal bearing B2-2 are shown in Figure 8(a)–(c). The bearing condition at severe stage is depicted in Figure 9(a)–(c). In Figure 9(c), the peak values of spectrum (235 Hz and 270 Hz) are same as the characteristic fault frequency of inner race and ball of bearing (see Table 3). It means that the fault occurred in ball and inner race of bearing. It is clarified from the above procedure that the envelope spectrum is a powerful tool for the diagnosis of fault in the bearing. It has been verified from the other results that are published in the literature.^36,37

Figure 5.

(a) Run-to-failure history of bearings (i) bearing B1-1 and (ii) bearing B2-2. (b) Vibration signatures of bearings (i) bearing B1-1 and (ii) bearing B2-2.

Figure 6.

Normal bearing B1-1: (a) waveform, (b) spectrum, and (c) envelope spectrum.

Figure 7.

Severe bearing B1-1: (a) waveform, (b) spectrum, and (c) envelope spectrum.

Figure 8.

Normal bearing B2-2: (a)waveform, (b)spectrum, and (c) envelope spectrum.

Figure 9.

Severe bearing B2-2: (a) waveform, (b) spectrum, and (c) envelope spectrum.

Degradation of bearing is evaluated from the collected data of tested bearings B1-1 and B2-2 over the lifetime of the bearing. The run-to-failure history of the bearings B1-1 and B2-2 is depicted in Figure 5(a) and (b). These datasets are taken for the calculation of HDIE of bearing. Table 1 listed the characteristic equations of features that are used for the estimation of features. The extracted features of bearings B1-1 and B2-2 are shown in the Figures 10(a)–(h) and 11(a)–(h), respectively. In Figures 10(a)–(h) and 11(a)–(h), the bearing degradation pattern indicates the damage initiation and propagation in the bearing. It signifies that the initial impact on bearing is generating large disturbing forces that initiate an incipient fault in the bearing. As presented, at point (A) the propagation of fault is gradual over a lifespan as shown at points (B) to (D). The faults suddenly reached a higher value showing the failure of bearing at point (E). To construct the assessment models and training of GTM, the initial 700 feature vectors from the bearing normal values without deterioration are used.

Figure 10.

(a)–(h) Features of time and frequency domain throughout bearing B1-1 lifetime.

Figure 11.

(a)–(h) Features of time and frequency domain throughout bearing B2-2 lifetime.

The assessment model of bearing is used to test the feature vector and the trained GTM identifies the input feature vectors. The HDI is improved exponentially to get the degradation as HDIE. The bearings B1-1 and B2-2 HDIE plots are depicted in Figures 12(a) and (b) and 13(a) and (b), respectively.

Figure 12.

(a) Bearing degradation HDIE for the bearing B1-1. (b) HDIE expanded view for the bearing B1-1.

Figure 13.

(a) Bearing degradation HDIE for the bearing B2-2. (b) HDIE expanded view for the bearing B2-2.

Figures 12(a) and 13(a) show the health degradation curve of bearing in terms of HDIE. The values of HDIE in Figure 12(a), remaining the same from the beginning, indicate the healthy condition of bearing. Suddenly, the level of HDIE at the time step 1600 increases slightly to indicate the development of incipient fault in the bearing. After the time step 1600, the HDIE values show significant changes than the normal condition. At times, the HDIE plunges to the normal stage because the damage grows fast and the crack is smoothened. The bearing degradation is seen in the severe stage at the time step 1900 as the HDIE level moves rapidly. Such behavior indicates a sufficient growth of crack, resulting in the damages in the bearing material. The result was verified with the published literature.^31–33 The crack propagation effect was verified with the published article on the same.³⁸ The bearing B2-2 HDIE curve is shown in Figure 13(a) and (b). It is observed from the degradation curve that the nature of the degradation is nearly same as shown in the previous case; only the values at the early and severe stages are different, namely 340 and 395, respectively. Hence, the HDIE is a more accurate indicator of the essence of bearing degradation. The noise content in the extracted fault feature is very negligible.

Therefore, the proposed HDIE method is most efficient to capture the damage progress in the bearings. This process is helpful to the industries for the purpose of maintenance, for observing the progress of bearing damage, and to take precautionary measures before a catastrophic failure of components.

Conclusion

Envelope spectra for diagnosis of fault in the bearings and an assessment of the performance of bearing degradation using GTM has shown a satisfactory result as the proposed methodology in this article. Bearing results indicated that the frequency band encompassed one or more resonance in the envelope spectra. Resonance condition of the bearing, estimated from the spectrum of signal, is clearly shown in the result of bearing diagnosis and verified from the two tested bearings. Evidently, the highest resonance energy would be the best for the frequency bandwidth of band-pass filter. The factors affecting characteristic fault frequency are more accurately observed in the envelope spectra. The experimental verification of the proposed method using the two bearings presented accurate values of characteristic frequency as bearing B1-1 generated fault in the outer race and bearing B2-2 in the inner race and ball of the bearings.

It is proposed that the HDI is evaluated from the feature set vectors of time and frequency domains. More refinement is done to the indicator using the evolution technique to quantify a proper value of degradation with the new health indicator as HDIE. Trained GTM model, developed using HDIE, presented satisfactory results. The trend of propagation of fault is effectively presented using the HDIE over the complete lifetime of bearings, and it is observed to be capable of predicting damage at an early stage. Diagnostics and prognostics aspects of bearing, taken with this proposed methodology, can be applied as the maintenance strategy in industries.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

VM Nistane

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