New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings

Abstract

Machinery components undergo wear and tear over time due to regular usage, necessitating the establishment of a robust prognosis framework to enhance machinery health and avert catastrophic failures. This study focuses on the collection and analysis of vibration data obtained from roller bearings experiencing various fault conditions. By employing a combination of techniques sourced from existing literature, distinct configurations within vibration datasets were examined to pinpoint the primary defects in roller bearings. The significant features identified through this analysis were utilized to formulate optimized stochastic model equations. These models, developed separately for inner and outer race fault features in comparison to healthy bearing features under random conditions, offer valuable insights into machinery prognosis. The application of these models aids in effective maintenance management, optimization of machinery performance, and the minimization of catastrophic failures and downtime, thereby contributing to overall machinery reliability.

1. Introduction

Bearing rolling is an advanced forming manufacturing technology for high quality bearings, and China has become an important country in bearing rolling research and production [1]. Applications for rolling bearings include the automobile, aerospace, and machine tool industries [2]. Rolling element bearings play a pivotal role in the functionality of rotating machinery, providing essential support to the machine structure and facilitating rotational motion. The repercussions of bearing failure directly impact the overall health of the machinery, making it a critical component. Various factors contribute to bearing failure, including contamination, deterioration, poor lubrication, and improper mounting [3]. Recent research data highlights that a significant majority, approximately 80%, of rolling bearing failures in rotating machines result from inadequate lubrication (refer to Figure 1) [4]. Radial cracking and fracture may lead large rolling element bearings to fail; hoop stress, roller diameter, and Hertz pressure are important contributors to this process [5]. The peculiar circumstances surrounding a set of roller bearings’ failure exposed intriguing aspects of the rollers’ metallurgy, the failure mechanism, and the assessment of oils used for lubrication [6]. Consequently, it is imperative to monitor the condition of these bearings and implement intelligent, robust, and reliable diagnostic and prognostic methods for optimal system design and machinery maintenance. Substantial research efforts have been dedicated to exploring these crucial aspects over the past few decades [7]. The monitoring of bearing degradation stands as a crucial aspect of prognostic preservation. Traditionally, researchers rely on frequency-based vibration analysis to assess the progression of mechanical damages [8]. Predictive maintenance becomes imperative when the operational characteristics of bearings deviate significantly from their norm, aiming to prevent actual failures and subsequent machinery breakdowns.

In pursuit of this objective, researchers have employed various machine learning (ML) algorithms to construct models capable of forecasting the remaining useful life (RUL) of bearings [9,10]. While these algorithms have demonstrated effective results and substantial predictive performance, there tends to be a tendency to overlook support for censored observations in the field. Only a limited number of studies have made efforts to construct models that accommodate this crucial aspect [11,12]. Given the infrequent occurrence of ball bearing failures, encountering censored observations is common, and outright dismissal may lead to reduced efficiency and the introduction of estimation bias [13]. Survival analysis, as a form of regression, has the capacity to harness subdued information [14]. However, recent investigations in the realm of bearing prognostics, utilizing survival analysis, have shown deficiencies in predictive accuracy or ranking performance [15]. Fault diagnosis for rotating equipment can be viewed as the identification of defect patterns within machinery health data. This process typically comprises two primary steps: extracting relevant features from the data, and classifying the faults. In the initial step, various vibration-based signal processing techniques are commonly employed. These techniques involve analyzing the acquired health data in the time domain, frequency domain, or time-frequency domain [16].

Time domain indicators are extensively utilized in bearing fault detection, prized for their intuitive nature and high efficiency. These indicators fall into two main categories: dimensional and non-dimensional. Dimensional signatures hinge on factors such as loading situations, rotating speed, and machinery operating conditions. On the other hand, non-dimensional indicators are independent of loads and speeds, making them preferable for monitoring bearing health. Time-domain dimensionless indicators not only overcome the limitations associated with dimensional counterparts, but also prove effective in early fault detection in rotating machinery [17,18,19]. In contrast, dimensionless parameters derived from raw vibration signals offer insights into the overall changes in bearing health. Among these parameters, Kurtosis stands out for its sensitivity to impact signals, making it widely employed in rolling bearing fault detection. However, these indicators predominantly capture impulsiveness measures. To address the cyclic behavior in rotating machine vibration signals, an indicator based on the negentropy of the square envelope spectrum is introduced [20]. Additionally, statistical point-of-view indicators and bandpass filter techniques are proposed to identify cyclo-stationary behavior [21]. Spectral kurtosis (SK) is then presented as a statistical tool for pinpointing transients in the frequency domain [22]. Investigators further introduce a novel spectral extraction method called “in fogram”, which considers both transient characteristics using negentropy of the square envelope and its spectrum [23].

Stochastic approaches play a crucial role in analyzing bearing inner race and outer race faults, employing probabilistic techniques to account for uncertainties in the bearings’ operation and fault occurrence [24]. One commonly employed stochastic model is the Markov chain model, assuming that the bearing’s health status can be represented by discrete states such as healthy, incipient fault, and severe fault. The transition probabilities between these states are determined based on the bearing’s operating conditions and fault characteristics [25,26]. Another stochastic model in bearing fault analysis is the hidden Markov model (HMM), which posits that the bearing’s health status is not directly observable, but can only be inferred from measurable data like vibration signals or acoustic emissions. The HMM assumes that the bearing’s health status follows a hidden state process, modelled by a Markov chain, with observable data generated based on the current hidden state [27,28,29,30,31]. For machine RUL prediction, stochastic process models reduce risks and repair expenditures while improving productivity, accessibility, sustainability, and accuracy [32]. Stochastic model roll bearing testing performs well most of the time, despite perhaps minimizing signals from other components such as gears [33]. For the literature, see references [34,35,36,37].

In our study, we employ the stochastic resonance approach to extract features of faulty bearings under various loading conditions. The extracted features identify the presence of defects such as spalls, cracks, wear, and other faults. These features are then used to formulate stochastic equations describing uncertainties in the system’s working conditions. These models offer a well-defined mathematical structure for classifying non-stationary periodicities in time series, providing valuable constraints suitable for integrated auto-diagnosis. This approach proves effective in fault detection, classification, and prognosis, all within a unified methodology.

2. Experimental Setup

The bearing datasets utilized in this research were acquired through experiments conducted on a bearing failure test rig, illustrated in Figure 2.

This setup is relatively straightforward, driven by a two-horsepower (HP) variable-speed induction motor. A specially designed shaft was then supported by two bearings. Notably, the bearing on the right side of the shaft represents a healthy state, while the left-supported bearing serves as the experimental bearing, specifically the SKF 6209. This test bearing underwent both radial and axial loading under defect conditions. The experimental conditions at various loadings with different bearing defects are listed in

Table 1. Experimental condition.

S.No	Radial Loading (mm)	Defect Width (mm)	Defect Depth (mm)
1	30	38	12
2	45	38	12
3	60	383	12

. To refine feature extraction and streamline the creation of robust prognostic models, we selected the peak values of defect frequencies. This strategy was employed to mitigate the impact of noise that might obscure pertinent frequencies within the broader spectral energy. By refining the peak values, we aimed for a more precise comprehension of the distinct defect propagation. In this study, our focus was specifically directed towards the analysis on the peak values linked to inner and outer race defects.

3. Spectral Analysis

The SKF 6209 test bearing underwent various loadings as outlined in Table 1. Throughout the initial 20 h of operation, vibration levels remained stable, and no significant defect frequencies were observed. Subsequently, after approximately 10 more hours of bearing operation under a radial load of 45 N, along with variable axial loading, defect frequencies associated with inner race damage were detected, accompanied by a slight increase in vibration levels. As the bearing progressed further in its failure state, outer race defects became evident. At this point, the vibration signal exhibited a distinct periodicity due to impacts generated by metal-to-metal contact. However, in the later stages of the fault, the periodic impulses became obscured by the noise energy resulting from the propagated defect across the bearing’s outer race. Spectral results for the SKF 6209 bearing under a 45 N load at different stages are depicted in Figure 3a–c for reference.

In a normal and healthy state, the bearing displays a typical vibration pattern characterized by low amplitudes and energy. However, when an inner race defect is present, there is a noticeable increase in amplitude within the vibration pattern, accompanied by observable frequencies linked to inner race defects. Similarly, the presence of an outer race defect results in elevated amplitudes in the vibration pattern, along with frequencies that can be observed and linked to outer race defects.

4. Methodology

Experimental data were gathered from the driver end bearing under two horsepower loads, operating at a speed of 1800 rpm. The bearing exhibited an inner race fault characterized by a groove width and depth of 38 μm and 12 μm, respectively, along with an outer race fault featuring a groove width and depth of 425 μm and 209 μm, respectively. Analysis of the collected data was conducted in both the time and frequency domains to extract various features under different operating conditions. Standard formulae were applied to extract up to fifteen features from healthy bearings, inner race faulty bearings, and outer race faulty bearings. The vibration analyzer CSI 2130 was utilized for validation of extract features at different intervals, and quantification was managed to identify faults in bearings during operating conditions. Finally, ARMA (p, q) modelling techniques were employed to establish a relationship between the extracted features and those of healthy roller bearings. The subsequent sections provide detailed discussions on the modelling procedures and simulation steps. Let ${\mathrm{x}}_{\mathrm{t}}$ represent extracted features in various conditions for t = 1, 2, 3 …, n. Then ARMA (p, q) can be represented as follows:

$${\mathrm{x}}_{\mathrm{t}}=\mathsf{\alpha }+{\mathsf{\varphi }}_1{\mathrm{x}}_{\mathrm{t}-1}+\dots +{\mathsf{\varphi }}_{\mathrm{p}}{\mathrm{x}}_{\mathrm{t}-\mathrm{p}}+{{\mathsf{\epsilon }}_{\mathrm{t}}+\mathsf{\theta }}_1{\mathsf{\epsilon }}_{\mathrm{t}-1}+\dots +{\mathsf{\theta }}_{\mathrm{q}}{\mathsf{\epsilon }}_{\mathrm{t}-\mathrm{q}}$$

(1)

with ${\epsilon }_{\mathrm{t}} ~ iid (0,{\sigma }_{\epsilon }^2)$, where $\alpha$ represents the features’ varying intercept terms, ${\varphi }_{\mathrm{p}}$ and ${\mathsf{\theta }}_{\mathrm{q}}$ represents the autoregressive and moving average parameters of the features at lag i and j in the Equation (1) for i = 1, 2, 3, …, p and j = 1, 2, 3, …, q which represents varying features for particular lag.

The appropriate model order is determined by analyzing the autocorrelation (ACF) and partial autocorrelation (PACF) functions. The truncation of the ACF reveals the moving average order, while the truncation of the PACF indicates the autoregressive order of feature representation. Additionally, suitable orders can be identified based on the lowest values of the Akaike (AIC) [38] and Schwarz (SC) [39] information criteria. The AIC is computed using the following equation:

$$AIC\left(p,q\right)=\ln \left({\widehat{\sigma }}^2\left(p,q\right)\right)+\frac{2(p+q)}{T}$$

(2)

Here, p, q, $\widehat{\sigma }$, and T represent the AR component, MA component, the standard deviation of the residuals and the number of the non-missing values in the extracted features, respectively.

The Bayesian information criterion for k parameters, L maximum likelihood for n number of extracted features is estimated by

$$BIC\left(p,q\right)=2\ln \left(L\right)+k\mathrm{l}\mathrm{n}\left(n\right)$$

(3)

The parameters of the suitable order models were assessed by utilizing the maximum likelihood method. The optimization of the models was acquired on the base of unobservable features.

5. Feature Extraction

Determining the machinery characteristics most susceptible to fault formation and propagation poses a significant challenge. Moreover, the efficacy of initial features can vary depending on the working environment. In this study, we systematically observed features under diverse conditions and loads to delineate the degradation of roller bearings in various ways. These features serve as indicators of the latent and evolving faults in the studied bearings. To extract features from both healthy and faulty bearings, we analyzed recorded vibration data using the stochastic resonance technique. This approach considers the role of random noise in fault detection, simplifying the identification of faults by enhancing the signal-to-noise ratio of the bearing’s fault signatures. The extracted features are presented in Figure 4a–h, illustrating the experimental conditions under scrutiny. Notably, skewness reveals a distinct separation between faulty and healthy states, with the outer-race fault exhibiting higher skewness. Kurtosis for faulty bearings exhibits different behavior in the initial and final stages compared to healthy conditions. Similar trends are observed in crest and impulse factors. This discrepancy arises from the stronger vibration signals in faulty bearings, more consistently compared to healthy conditions. The root mean square (RMS) of the inner race vibration signal demonstrates the maximum vibration strength, as evident from the graph. Conversely, the outer and healthy vibration indices are closely related, indicating a higher likelihood of machinery failure in inner race situations. Marginal features of faulty roller bearings deviate from the average or healthy features. Peak-to-peak features in time-domain vibration data show greater variation for faulty bearings compared to healthy ones, reflecting the changing signal behavior due to faults. Evolving shape factors in the vibration signals signify degradation in both faults, with a sharp probability of machinery failure in the inner race.

6. Stochastic Modelling of Vibration Features

In this section, extracted features are employed to construct models using stochastic techniques. Firstly, models were created by regressing inner-race faulty features onto their corresponding healthy features. Secondly, the necessary vibration intensity factors for outer-race defects are considered as regression, with analogous healthy factors serving as repressors. The resulting equations, along with their relevant statistics, are presented below:

6.1. Stochastic Association amongst Inner-Race Faulty Features with Healthy Features

(a)

Impulse Factor

$${\tau }_t^i=4.0324+1.089{\tau }_t^h-0.882\left({\tau }_{t-1}^i+{\tau }_{t-2}^i\right)-0.99\left({\omega }_{t-1}^i+{\omega }_{t-2}^i+{\omega }_{t-3}^i\right)+{\omega }_t^i$$

(4)

(b)

Crest Factor

$${\lambda }_t^{i }={0.0056+0.986\lambda }_t^{h }-0.749\left({\lambda }_{t-1}^{i }+{\lambda }_{t-2}^{i }\right)-0.978\left({\omega }_{t-1}^{ci}+{\omega }_{t-2}^{ci}\right)+{\omega }_t^{ci}$$

(5)

(c)

Shape Factor

$${\theta }_t^{i }={1.096\theta }_t^{h }-0.953{\theta }_{t-1}^{i }+0.887{\omega }_{t-1}^{si}+{\omega }_t^{si}$$

(6)

(d)

Margin Factor

$${\xi }_t^{i }=0.012-1.165{\xi }_t^{h }-0.767{\xi }_{t-1}^{h }-0.884{(\omega }_{t-1}^{mi}+{\omega }_{t-2}^{mi})+{\omega }_t^{mi}$$

(7)

(e)

Peak–Peak Factor

$${\rho }_t^{i }=0.416-3.296{\rho }_t^{h }+0.684{\rho }_{t-1}^{h }+0.821{(\omega }_{t-1}^{pi}+{\omega }_{t-2}^{pi})+{\omega }_t^{pi}$$

(8)

(f)

RMS Value

$${\mathsf{\Phi }}_t^{i }=17.96-6.037{\mathsf{\Phi }}_t^{h }+0.84{\mathsf{\Phi }}_{t-1}^h+{\omega }_t^{ri}$$

(9)

(g)

Kurtosis

$${\mathsf{{\rm K}}}_{t-1}^{i }=2.39{\mathsf{{\rm K}}}_t^{h }-0.994{\mathsf{{\rm K}}}_{t-1}^h-0.979{(\omega }_{t-1}^{ki}{+\omega }_{t-2}^{ki}{)+\omega }_t^{ki}$$

(10)

The stability and significance of stochastic models of inner-race faulty features compared with healthy features by using the p value test and good fitness test ${\mathrm{R}}^2$ values are explained in

Table 2. Stochastic model of inner-race faulty features with healthy features.

Models	p-Values				${\mathit{R}}^2$
Stochastic model of impulse factor	0.042	0.015	0.026	0.047	0.75
Stochastic model of crest factor	0.021	0.0151	0.0136	0.057	0.73
Stochastic model of shape factor	0.001	0.001	0.001		0.77
Stochastic model of margined factor	0.005	0.014	0.056	0.075	0.75
Stochastic model of peak to peak value	0.006	0.087	0.042	0.099	0.90
Stochastic model of root mean square	0.012	0.091	0.076		0.91
Stochastic model of kurtosis	0.001	0.001	0.001		0.70

.

The stochastic regression model of the impulse factor and crest factor shows a significant level of 5% because the p-values of all the coefficients in the model are <0.05, indicating that the model fits within a 95% confidence interval. Additionally, the $R^2$ test result also shows 75% and 73%, which are good significance levels according to the standard, where 70% to 90% are acceptable. Hence, the developed model efficiently forecasts faults in the signal data at time t. Similarly, the p values of the coefficients of the shape factor model are less than 0.01, which indicates the highest significance at a 99% level of significance, and the $R^2$ value is 77%, which is a good indicator of the variance of the dependent variable. The p-values of the coefficients in the model of the margined factor are as follows: 0.005, 0.014, 0.056, and 0.075. The first three values indicate strong evidence against the null hypothesis at a 95% level, while the fourth value has low evidence against the null hypothesis at a level of 92.5%. The $R^2$ value is 75%, which is a good indicator of the variance of the dependent variable. The p-values of the coefficients in the peak to peak value are as follows: 0.006, 0.087, 0.042, 0.099. These values are significant, and the $R^2$ value is 68%. The p-values of the coefficients in the model of RMS are as follows: 0.001 ≤ 0.005, 0.002 ≤ 0.005, but the third coefficient (0.089) ≥ 0.005 and < 0.1, indicating low significance at this coefficient. Overall, the $R^2$ value is 91%, showing that the model is statistically significant. The statistically significant model of Kurtosis has high significance because the p-values of the coefficients in the model are all 0.001, showing the highest level of significance at 99%. The value of $R^2$ is 70%, indicating the maximum variance in the dependent variable.

6.2. Stochastic Association amongst Outer-Race Faulty Features with Healthy Features

(a)

Impulse Factor

$${\tau }_t^o=1.731{\tau }_t^h+0.635{ \tau }_{t-1}^i+0.99({\omega }_{t-1}^i+{\omega }_{t-2}^i+{\omega }_{t-3}^i)+{\omega }_t^i$$

(11)

(b)

Crest Factor

$${\lambda }_t^{0 }={1.639\lambda }_t^{h }+0.599{\lambda }_{t-1}^{i }+0.87\left({\omega }_{t-1}^{ci}+{\omega }_{t-2}^{ci}+{\omega }_{t-3}^{ci}\right)+{\omega }_t^{ci}$$

(12)

(c)

Shape Factor

$${\theta }_{t-1}^{o }={1.0426\theta }_t^{h }+0.951{(\omega }_{t-1}^{si}+{\omega }_{t-2}^{si})+{\omega }_t^{si}$$

(13)

(d)

Margin Factor

$${\xi }_{t-1}^{o }=0.024-1.51{\xi }_t^{h }-0.045\left({\xi }_{t-1}^{h }+{\xi }_{t-2}^{h }\right)+0.712{\omega }_{t-1}^{mi}+{\omega }_t^{mi}$$

(14)

(e)

Peak-Peak Factor

$${\rho }_t^{o }=1.636{\rho }_t^{h }+0.817{(\rho }_{t-1}^{h }+{\rho }_{t-2}^{h })+0.873{(\omega }_{t-1}^{pi}+{\omega }_{t-2}^{pi}+{\omega }_{t-3}^{pi})+{\omega }_t^{pi}$$

(15)

(f)

RMS Value

$${\mathsf{\Phi }}_t^{o }=7.38-2.987{\mathsf{\Phi }}_t^{h }-0.524({\mathsf{\Phi }}_{t-1}^h+{\mathsf{\Phi }}_{t-2}^h)+{\omega }_t^{ri}$$

(16)

(g)

Kurtosis

$${\mathsf{{\rm K}}}_t^{i }=20.17-4.586{\mathsf{{\rm K}}}_t^{h }+0.718{\mathsf{{\rm K}}}_{t-1}^h+0.917{(\omega }_{t-1}^{ki}{+\omega }_{t-2}^{ki}+{\omega }_{t-3}^{ki}{)+\omega }_t^{ki}$$

(17)

The statistical stability of the extracted features models relating to outer-race faulty features and healthy features were checked by using the p-value test and $R^2$ test. In

Table 3. Stochastic model of outer-race faulty features with healthy features.

Models	p-Values				${\mathit{R}}^2$
Stochastic model of impulse factor	0.001	0.057	0.056		0.701
Stochastic model of crest factor	0.001	0.063	0.044		0.699
Stochastic model of shape factor	0.001	0.081			0.68
Stochastic model of margined factor	0.046	0.049	0.089	0.091	0.66
Stochastic model of peak to peak value	0.050	0.069	0.092		0.68
Stochastic model of root mean square	0.001	0.002	0.089		0.84
Stochastic model of kurtosis	0.071	0.022	0.078	0.088	0.76

, the p-values of the coefficient of the impulse factor model are (0.001, 0.057, 0.056) and crest factor model are (0.001, 0.063,0.044).This shows that the models are fit in the 94% to 95% confidence interval, and the $R^2$ test result also shows 70% variance between the outer race fault at time t and the independent variables. Hence, the developed model is efficient at forecasting faults in the signal data at time t.

The p values of the coefficient in the shape factor model are (0.01, 0.81), the first value indicates excellent stability at a 99% confidence level and the second value shows good stability at a 92% level, with $R^2$ value of 68%. The values in the margined factor model are (0.046, 0.49, 0.089, 0.091). The first two values are less than 0.05, which indicates strong stability of the model at a 95% level, and the next two values have low stability, at a level of 92% with $R^2$ value of 66%. The peak to peak value model has p values at (0.05, 0.069,0.092). The first value indicates high stability, at a 95% level, the second value has good stability at a level of 93%, and the third coefficient value indicates 91%, with an $R^2$ value of 68%,

The p values of the coefficient in the root mean square model are (0.001, 0.002, 0.89). The first two values are less than or equal to 0.02, which indicates the strong stability of the model at a 98% level, and the third value has low stability at a level of 91%, with an $R^2$ value of 84%. This is a good indicator for the variance of the dependent variable. The kurtosis model has p-values at (0.071, 0.022, 0.078, 0.088). These values indicate a stability between 92% to 98%, with $R^2$ value of 76%, which is a good indicator for the variance of the dependent variable.

Overall, the p-values show a 92% to 99% probability, which indicates that the developed models will predict the fault in a bearing before failure with high accuracy. The $R^2$ lies between 68% and 84%. To improve the results, we further established a robust relationship between the signals of faulty and healthy bearings, ARMA modelling techniques are employed to determine optimized parameters.

7. Optimization Result and Discussion

The vibration signatures of roller element bearings exhibit intricate patterns in the presence of inner-race and outer-race faults when compared to a healthy state. To establish a robust relationship between the signals of faulty and healthy bearings, ARMA modelling techniques were employed to determine optimized parameters. Initially, Equations (1) and (2) were employed to ascertain the appropriate model orders.

In

Table 4. Order of suitable stochastic models with assessed statistics of selection criteria.

Models	Order	AIC	BIC
Impulse Factor Inner Race Fault	ARMA (2,3)	2.6516	2.8029
Impulse Factor Outer Race Fault	ARMA (1,3)	4.3337	4.4547
Crest Factor Inner Race Fault	ARMA (2,2)	−10.6074	−10.4561
Crest Factor Outer Race Fault	ARMA (1,3)	−8.8708	−8.7498
Shape Factor Inner Race Fault	ARMA (1,1)	5.6182	5.7392
Shape Factor Outer Race Fault	ARMA (0,2)	7.9342	7.9987
Margin Factor Inner Race Fault	ARMA (1,2)	−12.4488	−12.2974
Margin Factor Outer Race Fault	ARMA (2,1)	−10.4902	−10.3806
Peak–Peak Factor Inner Race Fault	ARMA (2,3)	2.6516	2.4857
Peak–Peak Factor Outer Race Fault	ARMA (1,3)	4.3337	4.4547
RMS Inner Race Fault	ARMA (1,0)	1.7768	1.8978
RMS Outer Race Fault	ARMA (2,0)	−0.4051	−0.2341
Kurtosis Inner Race Fault	ARMA (1,2)	1.1191	1.2067
Kurtosis Outer Race Fault	ARMA (1,3)	4.2627	4.4141

the estimated statistics, including the Akaike Information Criterion (AIC) and Schwarz Criterion (SC), are presented alongside their corresponding probability values for suitable ARMA (p, q) orders. These computations span a range from one to twelve. Subsequently, the stability of the constructed models is ensured through standard statistical techniques, as evidenced by the necessary statistics provided in the parentheses of the estimated equations.

The corresponding impulse response functions (IRFs) for various features of faulty bearings are displayed in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. The features of outer race faults show impacts at the ball pass frequency and its harmonics, as evidenced by spikes in the IRFs at these frequencies. The presence of sidebands further confirms the occurrence of outer faults. In contrast, for inner race faults, the impulse response produces impulses at the shaft rotation frequency and its harmonics. The IRFs exhibit synchronized irregularities with the shaft rotation, and multiple peaks of the shaft rotation frequency and sidebands are visible. These observations highlight distinctive patterns in the impulse response functions associated with different bearing faults, providing valuable insights for the diagnosis and analysis of bearing health.

The statistical features of a healthy bearing exhibit stability over time, in stark contrast to the variations observed in extracted features when inner or outer race faults are present. The magnitude of these statistical features intensifies with the rate of inner or outer race defect propagation and damage growth, showcasing both linear and non-linear tendencies. Specifically, the impulse factor for the outer race shows a more pronounced deviation from a healthy bearing compared to the inner race, as evidenced by the estimated coefficients. This discrepancy arises from variations in crest factor information for healthy and faulty bearings, extracted from both vibration and acoustic emission signals. A low crest factor, typical for a healthy bearing, signifies a smooth and steady vibration signal. Conversely, in the presence of a fault, the vibration signal becomes more complex, leading to an increased crest factor. Consequently, the coefficients of the crest factor exhibit more significant variations in the case of outer race faults than in inner race faults. Examining the shape factor reveals that a healthy bearing typically has a shape factor close to unity, indicating a smooth and steady vibration signal with minimal amplitude variation. The presence of a fault, however, induces changes in the vibration signal’s complexity, resulting in a shift in the shape factor. The estimated shape factor equation proves valuable for enhancing the accuracy and robustness of fault diagnostics in condition monitoring of rotating machinery. The margin factor, representing the difference between the peak and RMS value of vibration signals, is typically high for a healthy bearing. However, the development of inner or outer race defects alters these signals, leading to variations in the margin factor and adjusted coefficients in the estimated models. Conversely, the peak-to-peak factor is generally low in a healthy bearing, indicating a continuous and smooth vibration signal. In the presence of inner or outer race faults, this factor escalates, resulting in fluctuating computed quantities. The RMS value, a widely-used statistical measure of vibration signal amplitude, remains relatively stable over time for a healthy bearing, serving as an indicator of the overall vibration level generated by the bearing. However, in the presence of a fault, the vibration signal complexity increases, causing changes in the RMS value and deviations in the predicted coefficients. Lastly, kurtosis, a statistical measure of the shape of a probability distribution, plays a crucial role in vibration analysis and condition monitoring of rotating machinery, including bearings. A healthy bearing typically exhibits a low kurtosis value due to the smooth functioning of the roller bearing. In Contrast, inner and outer race faulty bearings show fluctuations in kurtosis due to impulsive vibration signals, prompting corresponding adjustments in the coefficients.

8. Conclusions

This study seeks to quantify the behavior of a healthy roller bearing in comparison to inner and outer race faulty bearings, especially under increasing loads, utilizing vibration techniques. Vibrant data were systematically collected under various experimental conditions, and signal processing techniques were applied to extract significant features. While these features exhibit smooth performance in a healthy bearing, inner or outer faulty roller bearings manifest a distinctive peak or notch at the fault frequency or its harmonic, indicative of the presence of a fault. Stochastic model equations have been meticulously developed by discretely regressing inner and outer race faulty bearing features against healthy bearing features across diverse experimental conditions. The optimized and suitable model is rigorously assessed using stochastic modelling techniques. The computed stochastic coefficients offer crucial insights into the association of healthy bearing features with the structures of inner and outer race faulty bearings. These estimated models prove valuable in predicting the remaining useful life of roller element bearings affected by inner and outer race faults. Additionally, they serve as a support tool for maintainers, aiding in the development of effective maintenance strategies to minimize the risk of failure and maximize the overall lifespan of the machinery.