A new holistic approach to investigating and estimating rolling bearing RUL based on physical grounds

Abstract

Rolling bearings are critical components in rotating machinery. Throughout their operational life, they endure periodic loading cycles that could lead to the formation of spalls. While current capabilities enable early detection of incipient spalls, which helps prevent catastrophic failure of the machine, to utilize the entire operational life of the bearings, it is essential to estimate their spall severity and remaining useful life. Using physics-based models and experimental results, this article introduces an integrative approach. We develop a new conceptual framework for monitoring bearing health, assessing defect severity by identifying physical processes that govern defect evolution, and predicting bearing failure in real-world applications. The framework incorporates four models: the dynamic model, oil debris monitoring (ODM) model, damage model, and finite element model, along with experimental work, including vibration analysis, ODM data, and strain measurements using fiber Bragg grating sensors. The integration of experimental work with these models provides condition and health indicators for both diagnosis and prognosis. By using this model, the research community can gain a deeper understanding of spall propagation mechanisms, which will result in better predictions regarding the remaining useful life of rolling bearings.

Highlights

• - Physics-based spall severity and RUL estimation

• - An investigation into the physical mechanism of spall propagation

• - An integrative methodology

Introduction

Extensive research has been conducted in regard to rolling bearings, which are vital components commonly found in rotating machinery that provide support for high loads.^1–3 Because a bearing spall is one of the primary causes of failure in such a machinery, leading to economic losses and accidents, spall prognosis is a crucial task in condition-based maintenance.^4–8 Spall prognosis involves four stages. ⁵ : (1) fault detection, (2) classification, (3) severity estimation, and (4) remaining useful life (RUL) estimation. While the first two stages are well-established,^5,9 severity and RUL estimation require a more thorough physical understanding and established approaches.

As depicted in Figure 1, in general, the failure process of a bearing spall can be divided into three phases: initiation, steady propagation, and accelerated propagation. ¹⁰ The first phase marks the initiation of subsurface cracks, which are not visible on the surface. The third phase involves very rapid spall propagation, posing an immediate threat to the bearing and the machine in which it is installed. Consequently, the threshold for spalling is preferable at the end of the second phase, just before the spall reaches its critical point. The definition of a spall can vary across different maintenance facilities. This decision will depend on the specific maintenance strategy employed.

Figure 1.

Illustration of the three spall phases: initiation, steady propagation, and accelerated propagation. In most cases, the spall initiation phase may last longer in comparison to the other two phases.

To develop a methodology for predicting the RUL of rolling bearings under spalling while learning the underlying physics to create a robust set of powerful tools, numerous physics-based models and processing techniques have been developed. Among these techniques, digital twins for bearings represent the latest approach to predicting RUL.^11,12 The area of digital twins is closely related to the suggested cohesive method. The approaches presented here will, over time, enable the development of a digital twin for a roller bearing. Unlike digital twins, which often rely on learning-based approaches, the methods demonstrated are based solely on physical principles. Moreover, digital twins are typically suited for detecting a specific type of fault at a time. If a new fault occurs, a new digital twin would be required. Feng et al. ¹³ suggested a method similar to ours for gears, incorporating dynamic and wear models with vibration data to improve gear wear prediction. However, their approach employs a smaller number of models compared to ours, which are detailed later in this section. Additionally, this research integrates sensors beyond accelerometers, such as oil debris monitoring (ODM) sensors and FBG sensors for strain measurement.

As shown in Figure 2, in this study, we present a comprehensive methodology for estimating the severity and, subsequently, the RUL of spalls. This integrated methodology extends the research community’s knowledge of spall propagation and advances rolling bearing prognostics.

Figure 2.

Flowchart for estimating severity and RUL based on the integrative, experimental, and physics-based models.

Our integrative methodology encompasses vibration analysis,^14–16 ODM,^17,18 strain measurements, for example, using fiber Bragg grating (FBG) sensors,^19,20 and metallurgical analysis. Furthermore, four distinct physics-based models were formulated to elucidate the underlying principles behind the measured signals: (1) a dynamic model based on the dynamic analysis of free bodies and their interactions, (2) a strain model to calculate the strain created by the ball’s interaction with the races using finite element analysis, (3) a spall initiation and propagation damage model that utilizes a damage rule incorporated into a finite element model, and (4) an ODM model designed to relate spall severity to its geometry. The physics-based models and experimental work reinforce each other: the analytical and experimental techniques contribute to validating and improving the physics-based models, while the physics-based models contribute to understanding the experimental results. The integration of the models and experimental results is crucial for developing a method to estimate the RUL of a bearing. This new approach provides a broader perspective on bearing condition and enhances our understanding of the physical mechanisms governing defect progression. Furthermore, it defines health indicators that can be used to estimate the RUL of bearings.

The remainder of this article is organized as follows: In section “Physics-based rolling bearing models,” we delve into the physics-based models, while section “Experimental investigation and data analysis” describes our experimental investigation and data analysis tools. In section “Spall evolution process,” we present the physical mechanisms of spall initiation, steady propagation, and accelerated propagation using the physics-based models from section “Physics-based rolling bearing models,” coupled with the measuring techniques and research tools from section “Experimental investigation and data analysis,” alongside our experimental findings. In section “Summary: Toward severity and RUL estimation,” we summarize our findings and outline how these existing capabilities can be synergistically combined to determine spall severity and estimate the RUL in rolling bearings.

Physics-based rolling bearing models

Physics-based models are constructed using equations and assumptions that aim at closely approximating real-world conditions. These models play a pivotal role in exploring a wide array of physical phenomena and in facilitating a comprehensive analysis of signals collected from actual systems. Henceforth, they play a pivotal role in the suggested integrative methodology for spall prognosis. In this work, we discuss four models: a dynamic model, strain model, spall initiation and propagation model, and ODM model.

In each section, several examples of models from the literature will be presented, along with results from models developed in our laboratory. Any validated model can be utilized in the new method we propose.

The dynamic model

Vibration models for bearings have been widely investigated. The first bearing spall models of McFadden and Smith ²¹ and Tandon and Choudhury ²² were analytical ones based on the signal pattern. More recent studies have developed dynamic models producing physical simulations.^23–25 The dynamic model used in this study, based on Epps’s groundbreaking dissertation,^26,27 was first introduced by Kogan, Bortman, and Klein. ²⁸ For further comparisons between the dynamic models, see Madar et al. ²⁹

The physical model in Kogan et al. ²⁸ simulates bearings in both healthy and faulted states, encompassing various spall severities located either on the inner or outer race. It incorporates the geometric characteristics of the bearing and simulates the interactions between the rolling elements (REs) and the raceways. These interactions are described by kinematic, dynamic, and Hertzian contact equations, assuming that the components are rigid bodies. The model enables the differentiation of various phenomena that emerge during spall propagation, categorizing different severities into three distinct groups: (i) small defects, characterized by a single RE entering the spall, resulting in levitation above the spall; (ii) medium defects, where a single RE interacts with the spall simultaneously, subsequently hitting the bottom of the spall one or more times; and (iii) large defects, distinguished by the presence of at least two REs inside the spall. Figure 3 illustrates the energy variation as a function of the defect severity. The energy behavior first decreases in the “Medium fault” category but within the “Large Fault” category, it exhibits distinct characteristics, including sharp fluctuations, in contrast to the first two categories. The RMS measurement corresponds to known severity stages of spall propagation. Although health indicators like empirical orthogonal functions (EOF) ³⁰ have been developed, they are primarily effective for small-sized spalls, whereas this energy indicator performs well for larger spalls.

Figure 3.

Energy of simulated vibration as a function of fault severity. Reproduced from Madar et al. ³¹

The integration of vibration signals with the dynamic model enables the investigation of CIs and HIs to assess the bearing’s condition (Figure 2). Furthermore, the dynamic model establishes essential boundary and initial conditions for various models, aiding in their development. In this way, the dynamics are implemented through the boundary and initial conditions, facilitating a comprehensive analysis of the bearing’s behavior and an understanding of the spall propagation mechanism. Furthermore, the dynamic model has great potential to serve as the foundation for a digital twin for bearing spalls, an evolving field that has been investigated in several recent studies.^32,33

Strain model

The suggested methodology promotes the integration of multiple sensors into the system to enhance the development of both CIs and HIs. In the last decade, the use of FBG sensors for condition-based maintenance has evolved because these can be installed close to the tested bearing, resulting in a high signal-to-noise ratio.

To understand the physical behavior of the strain signal of bearings, a physics-based model was developed. ³⁴ Two complementary models were integrated into the strain model (Figure 2): a quasistatic finite element (FE), and the dynamic model previously described in section “The dynamic model.” The dynamic model provides the contact force between the RE and the fault, while the FE model provides the relation between the contact force and the strain on the bearing house, where the sensor is located.

The strain model enables a wide range of spall lengths to be stimulated and describes the behavior of the strain signals as a result of various phenomena. One of the insights derived from the model and its correlation with experimental data is the depiction of the phenomenon wherein the ball hovers above the defect. This phenomenon is characterized by the presence of pulses in the signal. A comparison between the simulation results and the experimental measurements is presented in Figure 4. As can be seen in the figure, there is strong agreement between the strain signal produced by the model and the measured signal, with the predicted square-shaped signal evident in the experiment. According to the model, it is understood that the square area represents the interaction of the RE with the spall.

Figure 4.

Comparison between: (a) the simulation and (b) the experimental endurance test (healthy—green and 4.8 mm spall—red). Based on Medvedovsky et al. 34.

Spall initiation and propagation models

One of the most common reasons for bearing failure is rolling contact fatigue. ¹ The race of the bearing endures periodic cycles of load by the REs, which produce local damage that accumulates to form microcracks. The growth and coalescence of these microcracks generate a crack. When a crack reaches the surface, a spall is generated.

Understanding the driving mechanism of spall initiation and propagation is a step toward performing physical-bearing prognostics and estimating the RUL. In this section, we describe two FE models that were developed: (1) the spall initiation model and (2) the spall propagation model. Both models have integrated continuous damage (CDM) that simulates the deterioration of the spall, and the results from the dynamic model (such as contact force and impact location) are used as an input to the FE models (see Figure 2 in the “prognostic” block).

Spall initiation model

The simulation of the initiation of the spall in Gazizulin et al. ³⁵ comprises three stages: (1) separation of the metallic material to grains and microstructures,^36–39 (2) calculation of the accumulated damage $D$ in each microstructure as a result of cyclic loading, ⁴⁰ and (3) generation of microcracks.

The accumulated damage $D$ of each microstructure was modeled using an effective stress CDM. In this model, when the accumulated damage surpasses a designated upper bound, $D_{\max }$ , microcracks are generated in the microstructure. The model stops when the initiated spall reaches the surface of the race. Thus, it enables estimation of the time until a first spall is generated, and the statistical information to be extended by estimating the distribution of spall initiation times for different loading profiles.

Spall propagation model

Some of the underlying assumptions of the spall initiation model do not hold after the cracks reach the surface, and the spall has formed. For example, the assumption of continuous contact between every RE and the outer race in the initiation phase does not hold when a RE enters the spall during the propagation phase. A FE model is created to simulate the interaction between the RE and the spall’s edge in order to investigate the damage propagation in the vicinity of the spall edge.

The simulation of the propagation of the spall is composed from three complementary models: (1) the FE model, which represents the RE–spall edge interaction; (2) the dynamic model, described in section “The dynamic model,” which calculates the boundary conditions for the FE model; and (3) CDMs and fracture mechanics tools^41–47 that investigate the mechanism of spall propagation.

The simulation of crack evolution at the spall edge comprises three stages: (1) preloading—applying the contact force between the RE and the spall edge, which was calculated by the dynamic model, in the FE model; (2) investigating the damage and crack propagation at the spall edge using CDMs and fracture mechanics tools; and (3) re-loading and unloading of the RE on the spall edge to obtain updated stress fields. Spall propagation can be conceptualized as a sequential three-stage process, as visually represented in Figure 5: (1) subsurface cracks emerge beneath the trailing edge of the spall, (2) surface cracks initiate at the trailing edge of the spall, and (3) finally, cracks propagate until a fragment detaches from the raceway.

Figure 5.

Schematic description of the propagation stages of a spall: subsurface cracks appear underneath the spall trailing edge, surface cracks appear in front of the trailing edge of the spall, and cracks propagate and coalesce until a fragment is released from the raceway.^48,49

In addition, the model enables simulation of the overall process of the fragment release from the spall trailing edge, considering the severity and shape of the fragments. This model may facilitate calculation of the spall growth rate and determination of the number of cycles required for fragment release. It enables the estimation of each spall’s severity and load to determine when the spall will increase. Consequently, it allows for the investigation of spall propagation as a function of the number of cycles.

ODM model

The ODM sensor detects and classifies particles within the oil system based on their type and size. This classification is determined by the electromagnetic induction generated by the particles in a coil system. Consequently, the sensor allows monitoring of the accumulated quantity of particles categorized by their size, and the total cumulative mass loss. The accumulated mass loss has a direct physical correlation with the volume of defects developing within the bearing.

Advancement of the ODM model permits the evaluation of spall severity within the bearing and enables prediction of the bearing’s health and RUL. ⁵⁰ The model is designed to describe the geometry of spalling within bearings. This is established based on the correlation between the cumulative mass loss measured by the ODM sensor, the defect volume, and the material density, presuming the presence of a single defect on the bearing race. The ODM model is subdivided into two geometric models: the first model portrays the shape of a spall during its initial phase until it spans across the race, while the second model illustrates the spall’s shape during its propagation along the circumference of the track.

The model facilitates diagnostic procedures by computing the length of the defect. The depth of the spall is considered to be at the point where the shear stress is at its maximum, and while the width of the spall is known, the length can be calculated, as it is relative to the accumulated mass loss. Moreover, this model enables prediction of the mass loss threshold at which the transition from stable to accelerated spall propagation occurs.

Experimental investigation and data analysis

Spalls manifest through various phenomena. Additionally, alterations in the microstructure within the spall region can be analyzed using metallurgical tools. This section outlines four diagnostic and research tools: vibration analysis, strain signals (measured by FBG sensors), ODM, and metallurgical analysis. As depicted in Figure 2, these tools play a pivotal role in the integrative methodology, aiding physical understanding of spall propagation phenomena and development of new approaches for spall severity and RUL estimation. The endurance and seeded fault experiments discussed in this chapter were conducted either in our laboratory or in collaboration with us. In this research, we integrated the results from all the conducted experiments.

Analysis of vibration signals

Vibration analysis of rolling bearings, measured using vibration sensors, has a long history. Most of the techniques for estimating spall severity are based on detecting events that occur during the RE–spall interaction or to other phenomena related to this interaction.^51–55 The analysis of the vibration signal can be divided into three main stages: suppression of interferences (such as the transfer function and effects of other rotating components),^56,57 preprocessing to emphasize the spall signature in the signal (like envelope estimation), ⁵⁸ and estimation of the spall’s state (involving calculation of various CIs and utilizing machine-learning techniques). ⁵

In most cases, the bearing signal is weaker compared to signals from other components in the system, necessitating signal processing techniques to enhance the spall signature in the vibration signal. Various approaches can be employed, with one common method being calculation of the envelope signal.^9,21 Additionally, several machine-learning^59–62 and hybrid approaches^63,64 have been proposed for bearing diagnosis.

The first stage may involve suppressing transfer function effects and interference from other rotating components like gears. ⁶⁵ Transfer function suppression can be achieved through background estimation using methods such as cepstrum-liftering,^66–68 adaptive clutter separation,^69,70 or autoregression,^15,71,72 followed by phase estimation through the original phase process^69,73 or by extracting the minimal phase.^54,74 Interference from other components can be suppressed using techniques like Dephase ⁶⁹ and adaptive/self-adaptive noise cancellation. ²

In Sol et al. ⁷⁵ the dynamic model was combined with signal processing for a new CI, correlating with the severity of the spall. As depicted in Figure 6, the severity was estimated based on the shift in the bearing tone. This new CI, relevant for angular contact bearings, is very important as it is not dependent on the transfer function and, therefore, has great potential to improve spall severity estimation regardless of the sensor’s location. For more details about the experimental set up and results illustrated in Figure 6, see Sol et al. ⁷⁵

Figure 6.

Investigation of the relationship between the bearing tone and fault severity. Reproduced from Sol et al. ⁷⁵

Strain analysis

An algorithm for spall severity estimation was presented in Medvedovsky et al. ³⁴ The block diagram of the algorithm is depicted in Figure 7(c). In the first stage, the algorithm analyzes the squared strain signal to emphasize the pulses in the signal. We then define a threshold, which is a chosen percentage of the highest peak value in the signal. The peaks above this threshold, corresponding to $\approx 1/\mathrm{BPFO}$ , enable the algorithm to detect which of the peaks in the signal [the periodical segment in Figure 7(a) and (b)] represent the hovering of the RE, which is used to estimate the spall’s severity. In the second stage, the algorithm estimates the spall severity based on the average of the estimated values of each segment alone. For each segment, the spall severity is estimated by the distance between the two negative minima that precede and follow the maximum value, corresponding to the RE–spall interaction. This distance is correlated to the severity of the spall, as indicated by the model described in section “ODM model.”

Figure 7.

(a) Examples of measured strains of a spall with a severity of 3 mm. (b) Examples of measured strains of a spall with a severity of 0.8 mm. (c) A block diagram of the new algorithm, presented by and based on Medvedovsky et al. ³⁴

The automatic algorithm was tested by an endurance experiment whose results are depicted in Figure 8. As can be seen from the figure, the technique very accurately estimates the spall’s severities at the beginning and end of the experiment. Furthermore, the technique estimates a monotonic increment of the spall’s severity as a function of the time as expected, because this severity can only increase, and not decrease.

Figure 8.

The estimated spall’s severities during the endurance test, together with the measured spall’s severity at the beginning and end of the test. The missing points in the middle of the test are due to a recording problem. Reproduced from Medvedovsky et al. ³⁴

Strain analysis by FBG sensors has two main advantages over the analysis of vibration signals: (1) the fiber can be installed very close to the bearing, hence, mitigating some of the transfer function effects, and (2) it is unaffected by electrical interferences as the measurement is implemented by optical means. However, this method also has several drawbacks, including, for example, a higher price and less industrial experience. Still, together with vibration analysis and ODM, it can improve the ability to diagnose the bearing state. For more details about the experimental set up and results illustrated in Figure 8, see Medvedovsky et al. ³⁴

Oil debris monitoring

ODM is a traditional technique for monitoring the state of bearings.^2,76 The ODM sensor measures the quantity and sizes of particles in the oil line. Through this analysis, the severity of a spall can be estimated.^17,77

However, ODM has several drawbacks. For instance, it can suffer from the mixing of particles from different rotating components, which can obscure the detection of spall propagation in a single bearing. ² Nevertheless, the method can still aid in estimating the spall state either independently or in conjunction with other techniques (e.g., vibration analysis).

Furthermore, ODM can also facilitate investigation of the physical phenomena underlying spall propagation. For example, in Madar et al., ¹⁰ an ODM sensor monitored the state of spalls in multiple bearings. The analysis revealed two phenomena: the existence of a transition point at which spall propagation begins to accelerate, and the observation that the particle severity distribution remains constant beyond a certain state.

Figure 9 presents an example of measured oil debris from six different experiments. The measured mass loss allows the progression of spall propagation to be tracked. As discussed in section “Introduction” and evident in Figure 9, spall propagation can be divided into three distinct phases: initiation, steady propagation, and accelerated propagation. Additionally, ODM measurements demonstrate that there is a notable randomness in the initiation time of spalls. However, once generated, the randomness in propagation time diminishes. For more details about the experimental set up and results illustrated in Figure 9, see Madar et al. ¹⁰

Figure 9.

The accumulated ODM mass of several bearings in an endurance test experiment. Reproduced from Madar et al. ¹⁰

Metallurgical analysis

While metallurgical analysis cannot be used to monitor the bearing condition, it does enable investigation of the spall condition between working periods of the bearing with nondestructive metallurgical tools, such as with different microscopes. In addition, a destructive metallurgical tool, such as cut sections in the spall area and metallographic etching, facilitates in-depth analysis of the spall state “postmortem.” Moreover, this type of analysis allows investigation of the exact spall state and, thus, validation and improvement in modeling spall propagation.

Metallurgical analysis can be implemented by several techniques, including analyses of scanning electron microscope images,^78–80 optical microscope images, and computed tomography scanning.

Scanning electron microscope images are measured by a beam of electrons that interact with the atoms of the material. These techniques can reach a resolution of less than 1 nanometer. Optical microscope images are measured with the basic principle of visible light, while metallographic etching is used to observe the metal features at microscopic levels with chemical techniques. Computed tomography scanning creates a 3D image of the spall utilizing X-rays to examine cracks in the vicinity of the spall. In Figure 10(a), the image of bearing, and in Figure 10(b), the image is from a scanning electron microscope. As can be seen, this figure illustrates the surface and subsurface cracks, which eventually will release fragments from the trailing edge of the spall.

Figure 10.

(a) Image of the bearing. (b) is from a scanning electron microscope. Reproduced from Alves et al. ⁴⁷

Spall evolution process

In this section, based upon the suggested integrative methodology illustrated in Figure 2, we delve into the spall propagation process and propose approaches to address spall severity and RUL estimation. We present outcomes related to the three phases within the evolutionary process of spalling in bearings, covering both models and experimental findings. Additionally, we introduce a selection of proposals regarding CIs and methods for estimating spall severity and RUL.

Spall initiation phase

The duration of the initial phase of a bearing’s life, that is, the time until a spall is initiated, depends on the bearing properties and loading profile. For instance, a larger load is expected to initiate a spall faster than a smaller one. The physical model described in section “Spall initiation model” explains the underlying physical phenomena behind spall initiation, as well as the inherent randomness in the duration of the first phase. Random variations in grain shapes, positions, and material properties lead to generation of the various positions of microcracks. Differences in damage generation across various locations yield a corresponding variation in the rate of microcrack propagation until their emergence at the surface.

Estimating the severity and RUL is not relevant at this stage, prior to generation of the spall. The primary objective in this stage is to identify whether a spall has been generated. This can be achieved through vibration, strain, and ODM analysis by extracting CIs from the measured data, calculating an integrated score, and identifying significant deviations above a specified threshold as an anomaly indicating the transition to the second phase of steady spall propagation.

Spall steady propagation phase

In the second phase, the spall steadily propagates under constant operating conditions. The RE impact on the edge of the spall generates residual tension stresses on the surface of the RE impact location and residual compression stresses below the surface. The cyclic stresses, combined with the overlay of residual tensile stresses induced by plastic deformation and external loading, advance the initiation and propagation of fatigue cracks ahead of the spall.

The fracture mechanic model, together with the CDM based on effective stress, indicates that cracks are propagated at the spall edges until a breakage occurs, as demonstrated in Figure 5(a). Using fracture mechanics tools, it becomes evident that regions with residual compressive stresses withdraw due to tensile stresses at the crack tip. Consequently, cracks propagate at the spall edge until the detachment of fragments occurs. An interesting property evident by the ODM measurements is that the severity distribution of the emitted particles during the steady propagation phase is constant. The FE model with the CDM based on accumulated inelastic hysteresis energy predicts this phenomenon. Figure 11 presents a comparison between the simulation results (Figure 11(e)) and the metallurgical analysis conducted on an actual spall edge (Figure 11(a) and (d)).

Figure 11.

Comparison between the simulated fragment release based on the accumulated inelastic hysteresis energy CDM and spall images from an endurance test of bearings, and from the swashplate bearing of a Sikorsky CH-53 helicopter. (a–c) Micro-CT of the spall trailing edge from an endurance test. (b, and c) are cross sections of image (a). (d) Optical microscope image of the Sikorsky CH-53 swashplate. (e) The estimated accumulated damage in the spall by the CDM based on the accumulated inelastic hysteresis energy. The dashed lines represent the fragments that have detached or are about to detach from the raceway.

As can be seen in this figure, the accumulated damage generates surface and subsurface cracks in several locations. Those cracks may release fragments of different severities, for example, small fragments that detach from the large fragment (Figure 11(e)) or large fragment detachment (Figure 11(b)].

The diagnosis of the spall severity by the ODM is straightforward. As can be seen in Figure 9, the accumulated emitted particles indicate the spall’s severity. As explained in Madar, ¹⁰ the spall’s severity $L$ can be estimated by Equation (1).

$$L=\frac{M}{k·\rho ·A}$$

(1)

where $A$ is the spall’s cross-sectional area, $k$ is a dimensionless calibration factor for the ODM sensor, and $M$ is the mass loss measured by the ODM sensor.

Estimation of the RUL can be carried out through three procedures: While it can be estimated by calculating the expected remaining time based on statistical information, this approach does not allow for RUL estimation when different bearings or conditions are tested that were not used to calculate the statistical information.

Another option for estimating the RUL in the steady phase is to extrapolate its remaining time in this phase. The transition point between steady propagation and accelerated propagation occurs when the spall’s severity reaches the distance between two REs. ¹⁰ The severity of the estimated spall up to the current test point can be modeled using a polynomial, and the remaining time can then be estimated by predicting when the spall will reach a severity corresponding to the distance between two REs.

The third option is to combine the two previous approaches for RUL estimation with a physical model. The physical model enables the extension of the RUL estimation to new operating conditions and bearing geometries that were not previously tested.

Estimation of the RUL in the steady phase is important as operation under the corresponding conditions could be still useful and might significantly extend the overall operation time.

Accelerated propagation phase

The empirical results clearly show that the steady propagation phase is followed by a final phase of accelerated propagation. The transition point between these phases occurs when the spall reaches a severity corresponding to the distance between two REs, as explained in Madar et al. ¹⁰

In this phase, the spall is significant and is also progressing at a very high rate. Therefore, from a maintenance perspective, when the spall reaches this stage, immediate maintenance action is required. Generally, the RUL prediction should estimate how much time remains before the spall reaches this phase of rapid deterioration.

Summary: toward severity and RUL estimation

In this study, we introduced an integrative methodology that extends current capabilities toward prognosis by exploring the phenomena of rolling bearing spall propagation. The focus is on advancing the current capabilities of spall detection and classification in the last two stages of prognosis: severity and RUL estimation. The integrative methodology utilized in this research incorporates various techniques, including vibration analysis, ODM, strain measurements and metallurgical analysis.

Additionally, four distinct physics-based models were developed to elucidate the fundamental principles underlying the measured signals. These models include a dynamic model, a strain model, a spall initiation and propagation model, and an ODM model. The synergy between these physics-based models and the employed measuring techniques enhances the overall research and investigation process. The measuring techniques play a crucial role in validating and improving the physics-based models, while the models contribute to a more profound understanding of the physical characteristics associated with the measurements. Consequently, this collaborative approach facilitates advancements in both measuring techniques and the comprehension of physical characteristics.

As a result, the integration of these physics-based models and measuring techniques contributes to the refinement and enhancement of methods used for understanding the underlying phenomena governing the evolution of defects in bearings and prognosis of rolling bearing spall propagation in real-world applications. Additional sensors can be employed to aid in the diagnosis and prognosis of bearings, including temperature measurements, oil composition analysis, and imaging techniques. Combining these with the approaches presented in this research may lead to improved results. Future work could involve incorporating these measurement techniques and analyzing their impact.