Research progress on bearing fault diagnosis with localized defects and distributed defects for rolling element bearings

Abstract

In rotating machinery, rolling elements are the key members of the rolling element bearing (REB), which has tremendous application in automobiles, aerospace, ships, machinery apparatuses, and many more industrial sectors. Bearings are a significant source of various non-linear parameters which strongly affect the whole system. In the last two decades, tremendous research and approaches have been developed, considering the different parameters and faultiness for the proper understanding of the system’s non-linear behavior. This research paper summarizes the literature review on bearing fault diagnosis with localized and distributed defects for rolling element bearings with various non-linear parameters. Artificially-generated defects on the bearing races provides the vibration analysis of the system. These factors like radial clearance, unbalanced force, preload, misalignment, etc. are also considered defects in bearings. Moreover, the literature highlights some significant gaps that should be examined for future research in the condition monitoring of the mechanism.

Keywords

rolling element bearing localized defects distributed defects condition monitoring vibration monitoring

Introduction

Roller bearings are life-threatening components in rotating machinery.¹ There is a tremendous variety of industrial applications such as the auto sector, trains, machine tools, turbines, wind turbines, steel industries, construction, mining, paper, textiles, and renewable energy, also in precise operations such as the aviation industry, space shuttles, and lots of different segments. Some of the domestic equipment like washing machines, sewing machines, juicer mixer grinder, and floor cleaner also have the bearing applications.^1–6 The wide application of the roller bearings are due to their high stiffness, different load capacity, speed, ability to work with different range of temperature, high load carrying capacity, low-friction characteristics and balancing advantage over the reciprocating machinery.⁷ Because of the wide applications of bearings in rotating machinery, great concern is needed for bearing working capacity as it is the main element of the working of various machinery so that catastrophic failure can be avoided.^8–11 The improper working of bearings can be responsible for the loss of production due to the collapse of machinery, which sometimes even leads to the loss of human.^5,12 The need for vibration analysis in rolling element bearings is increasing with the more vital collective demand on running accuracy.¹³ The measuring of vibration response is a noteworthy and beneficial technique for detecting faults in rolling element bearings.¹⁴ During operation in rotating machinery, improper assembly, lack of lubrication, improper handling of various parts can cause the generation of defects on bearings.⁵ The different industrial and functional processes are responsible for the defects generation on the bearing components.¹⁵ The localized defect and distributed defects are main classification of the defects on bearings. Various defects like cracks, pits and spalls, dents, scratches, bump flanking and fault size prediction are part of localized defects, while surface roughness, waviness, and misaligned races and off-sized rolling element comes under distributed defects.^{10,11,14,16–20} Faulty design, improper installation, lack of lubrication, and heavy fatigue loading are responsible for the generation of localized defects, while the distributed defects are caused because of faulty design, manufacturing errors.²⁰ There is a great influence on vibration response due to the occurrence of faults in different parts of bearings. As a result, detecting and diagnosing bearing is critical for system condition monitoring.^20,21

This paper provides a detailed research review on the fault diagnosis of cylindrical roller bearings considering localized defects and distributed defects. This paper also reviews non-linearity parameters like preload, unbalanced rotor, clearance, and so on.

Modes of bearing failure/bearing failure reasons/causes

Researchers' rolling elements bearings are manufactured with high precision machinery, which also passes quality inspection once manufactured. However, the rolling element bearings may fail or defects are generated in the bearings during the running mode of operation. There are a variety of causes for bearing flaws to occur in various sections of the bearing, but pinpointing the exact source of bearing faults is challenging. A large number of causes are associated with bearing failure, such as fatigue, cracks, mechanical damage, wear and tear, corrosion, insufficient lubrication, plastic deformation, brinelling, faulty installation, incorrect design, etc (Figure 1).^22,23 The factors responsible for bearing failure are the reasons for higher bearing vibration. To avoid the catastrophic failure of the bearing, condition monitoring has been used for many years to identify degrading bearings.

Figure 1.

(a) fatigue spall (b) true brinelling (c) corrosion (d) false brinelling (e) lubricant failure (f) overheating (g) over loading.²²

Common modes of bearing failure are discussed below.

Fatigue

The main cause of bearing contact surface fatigue during operation is cyclic loads. Fatigue failure is also known as flanking or spalling. Small material elements are avoided gradually as result of the breaking of surfaces. Once started, this sort of collapse is gradual and spreads across the surface and subsurface. This type of failure is caused by insufficient lubrication and overloading.^22,23

Wear

Another typical reason for bearing failure is wear. Dirt and foreign substances enter into the bearing due to poor sealing or tainted oil, which is the most common cause. Foreign elements abrasively uneven the touching sides, giving them a drab form. Adverse wear modifies the profile and dimensions of the bearing components, resulting in increased bearing clearance. The rolling friction increases dramatically, resulting in severe degrees of slip and sliding, with full breakdown as a result.²³

Corrosion

In bearing assemblies with different components, the impurities that enter into the assembly can cause corrosion, which occurs as a chemical reaction. The bearing is working at a higher temperature. Because of the humid air, higher temperatures get rapidly cooled, which results in impaired seals, change of property of lubricants and condensation, all of which lead to corrosion. As a result of the rust elements particles interfering with the lubrication and working of the bearing, the running surfaces rust, resulting in an uneven and noisy operation.²³

Electrical erosion

Electrical damage is mostly caused by the continual passage of A.C. and D.C. currents. Due to current leakages, a small current occasionally travels through the bearing. Between the races and the rolling element, an arc is created where the current jumps. At the arc point, burning or welding would cause damage to the contact surfaces. Fluting is another name for this occurrence.²²

Plastic deformation

Excessive loading of bearing contacting surfaces while stationary or in modest movements might result in plastic deformation of bearing contacting surfaces. Excessive loads cause localised plastic deformation, resulting in raceway indentation. The deformed bearing would rotate very unevenly in operation, causing significant vibration, and it would no longer be fit for service.²³

True brinelling

During the running operation of a bearing, when races come into contact with a roller or ball, they distort unvaryingly. Brinelling happens when rolling bearing parts are overloaded or frozen. In this type of failure of bearing, the metal from the contact surfaces is removed without being cut.²²

False brinelling

It’s the result of a combination of chemical and mechanical forces. False brinelling was induced by tiny impact motion or vibration in the presence of oxygen. During lubrication, the quick motion of the bearing balls in the profile in the idle state of the mechanism wears the utmost. Due to the absence of rotation of the bearing, the fresh lubrication cannot return to the spot. These are the two most common causes of false brinelling.²²

Lubricant failure

The surface of the rolling element and race is damaged as a result of insufficient lubrication, and the material is transferred from one level to the next. Insufficient lubrication between the meeting profiles is a source of increased stress in that region, which can result in the contact surfaces being welded together. The most common roots of premature bearing breakage is insufficient lubrication, which causes sliding, increased friction, heat generation, and sticking.²³

Overheating

Overheating in the bearing is caused by improper lubrication and an unrestricted handling temperature. Extreme heat developed or insufficient heat exclusion (lubrication) from the bearing causes the temperature of the bearing to rise. Surface cracks or rings appear in both contact parts in a direction perpendicular to the movement’s direction. High temperatures weaken the hardness of bearing components, causing early bearing failure.²²

Improper installation

Undue radial or axial preloading, misalignment, sloppy fittings, or destruction due to unwanted force employed in fitting of the bearing parts are all examples of faulty installation.²²

Excessive loads

Too much stress on the bearing is a conventional reasons for bearing breakdown.²²

Improper handling and storage

Due to inappropriate handling and storage, bearings are subjected to clamminess and dust. The shelf life of grease is shortened when a bearing is kept at an excessively high temperature. As a result, it’s critical to double-check the manufacturer’s grease storage recommendations. When bearing boxes are opened incorrectly and wrappings are torn incorrectly, dirt and corrosive substances can enter the bearing.²²

Vibration signature of a bearing

Because failure of rolling element bearings (REBs) is the prevailing reasons for rotational mechanism downtime, a lot of study has gone into fault identification and condition monitoring (CM).

Surface fatigue, faulty installation, and poor maintenance caused undesirable vibration in rolling element bearings because of the different types of defects that evolve on the bearing component surfaces.⁵ The main classification of bearing defects are localized and distributed. Cracks, pits and spalls on the rolling sides are examples of localized defects whereas surface roughness, waviness, and misaligned races and off sized rolling element are part of distributed defects.²⁰ During the working conditions, bearing races and rolling parts come into contact. As bearing parts come into contact, the stress are induced in the components, which results in tinny cracks due to improper lubrication between meeting elements.²⁴ Distributed defects are the result of manufacturing errors.^25–27

Bearing characteristic frequency refers to the vibration spectrum created when a rolling element bearing rotates with the occurrence of frequency in the bearing at that time. A bearing characteristic frequency is assigned to each bearing element.

The highest amplitude of vibration is observed at varying compliance frequency for the rotor bearing mechanism, which has no defects on the surfaces as well as no eccentricity. The mechanism is excites at the component’s faulty frequency when a localized or distributed flaw is produced in the bearing component.^28,29

Figure 2²⁹shows the visual appearance of the bearing. The sectional view of the bearing is also shown in Figure 2.²⁹ A pictorial view of the bearing gives us the basic understanding of its components, namely the outer race, inner race, rollers/balls, and cage.

Figure 2.

(a) A graphical illustration of a roller element bearing and (b) a cross-sectional view of the roller element bearing.²⁹

All the bearing components have their own characteristic defect frequencies. The characteristic defect frequency of the inner race is BPFI, also called ball pass frequency at the inner race. The characteristic defect frequency of the outer race is BPFO, or ball pass frequency at the outer race. The characteristic defect frequency of a roller is BSF, also called ball spin frequency, and the characteristic defect frequency of a cage is called fundamental train frequency (FTF). There are some basic fundamental equations for identifying the defect frequencies as mentioned below.

The formulae for these four characteristic bearing defect frequencies are listed as below.

Shaft Frequency $f_{s} = \frac{2 Π N}{60}$

Varying compliance frequency $V C = f_{c} \times N$

Cage Frequency (FTF) $f_{c} = \frac{f_{s}}{2} [1 - (\frac{d}{D}) \cos α]$

Outer Race Defect Frequency (BPFO) $f_{o r} = \frac{N f_{s}}{2} [1 - (\frac{d}{D}) \cos α]$

Inner Race Defect Frequency (BPFI) $f_{i r} = \frac{N f_{s}}{2} [1 + (\frac{d}{D}) \cos α]$

Rolling Element Defect Frequency (BSF) $f_{r} = \frac{D f_{s}}{d} [1 - (\frac{d^{2}}{D^{2}}) \cos α^{2}]$

where N is the number of rolling elements, $f_{s}$ is the shaft rotational frequency, d is the rolling element diameter, D is the pitch circle diameter, and $α$ is the contact angle.^29,28

Dynamics of bearing

The forces acting on the different parts of the bearings during working conditions give a glimpse of the dynamics of bearings. Different applications of components have different modelling of the bearing under various parameter considerations. Under extreme situations, non-linear bearing modelling will be more realistic. In the last two decades, there has been tremendous research going on to analyse the dynamic behaviour of bearings under external force excitation.

There are various stages of output based on the system’s behaviour. A linear system has periodic and predictable output, while a nonlinear system is unpredictable. The possible output of linear and non-linear systems is shown in Figure 3.²⁹ Through the use of modern computational methods, most of the non-linear problems can be solved with moderate computation effort. Non-linearity of a system is a certain factor that could contribute to causing chaotic behaviour of a system. For the analysis of non-linear systems, tools like phase portrait, poincare map, orbit analysis, and stability of motion can be used.³⁰ In actual industries working with different parameters, it is difficult to find linear performance of the bearing. Researchers have developed dynamic models to understand the static and dynamic nature of bearing vibrations. The nonlinearity is responsible because of the different parameters like friction, geometry, forces acting on the body and so on.³¹ The different behaviour of the mechanism is because of these different nonlinear parameters.

Figure 3.

Input output possiblities for linear and non linear system.²⁹

This article reviews the literature on nonlinear analyses of rolling element bearings. It mainly focuses on nonlinearity generated through localized defects, distributed defects, and combined localized and distributed defects. The factors like radial clearance, unbalanced force, preload, misalignment, etc. are also considered with defects in bearings, which are discussed in subsequent sections.

Rolling element bearing

Rolling element bearing with localized defects

McFadden and Smit^32,33 have proposed a dynamic model for single and multiple localized faults considering constant radial load for the inner race of a rolling element bearing. They have also studied the influence of loading with the Stribeck equation and transmission parameters with delta function, and the results are validated by the run of experiments. Su et al.³⁴ have extended McFadden’s work for the study of tapered roller bearings to characterize of the vibration response under varying loads and with local defects on varying components of the bearing. Patil et al.³⁵ constructed a 2-DOF lumped parameter theoretical model. Localized surface defects are modelled by the half-sine function. Authors have noticed that the value of vibration spectra rises with the higher value of defect dimension for both outer and inner race faults. Rafsanjani et al.³¹ looked at a methodical model to anticipate the impact of localised defects on the dynamic response and stability of a mechanism. The effect of internal radial clearance and bearing loading zone is considered in this model. The author observed that chaotic behaviour is present and its value rises for ball defects, whereas there is not much effect for outer race flaws. Ghafari et al.³⁶ have inspected the effect of localized defects on rolling element bearings. Lyapunov exponent, and correlation dimension are employed to compute the chaotic signature’s influence of the breakdown. The study concluded that chaotic factors have pronounced potential, which has a great impact on studying the behaviour of bearings. Kiral and Karagülle^37,38 investigate a force model to model the localized defects in the rolling element bearings. A finite element vibration analysis method is employed for the single and multiple defect detection of different components of bearings through the time domain and frequency domain. Authors have investigated the statistical parameters like crest factor, kurtosis, and RMS values for simulated vibration responses.

Tandon et al.³⁹ have proposed the extended work done by McFadden and Smith for analyzing the behavior of REBs with localized defects on different components of bearings with radial and axial load. The defects on bearing are modeled as a sequence of regular-shaped impulses, such as rectangular, triangle, and half-sine pulses, as seen in Figure 4.³⁹ Here ∆ is the impulse width, K is the impulse height, and T is the time period. The author reported that as the load varies the level of amplitude varies while the vibration level is exaggerated by the pulse shape.

Figure 4.

Three typical impulse forms (a) rectangular form (b) trangular form (c) Half- sine form.³⁹

Choudhury and Tandon have discussed the effectiveness of acoustic emission (AE) measurements in the recognition of flaws in bearings. In the analysis of the inner race and roller of the bearing with small defects, ring down counts are used for finding flaws with acoustic emission signal.⁴⁰ Niu et al.⁴¹ presented a dynamic model to analyse the vibration of bearing at higher speeds for localized surface defects on raceway based on the Gupta’s model. The different parameters are also considered along with additional deflection owing to variation in the applied forces, variation in Hertzian contact coefficient, and material absence when modelling local faults. The author concluded the speed of the shaft and the different dimensions of the flaws give a relationship because of the performance of the vibration responses. Niu et al.⁴² prolonged the work by considering the effects of cage, slippage, and three-dimensional movement to forecast the fluctuations in characteristic frequency with flaws in the raceway. Kankar et al.⁴³ have demonstrated the nonlinear performance of ball bearing with spall as a localized fault on all the parts of the bearing i.e. inner race, outer race, and ball. The author also considered the radial clearance and the number of balls as sources of non-linearity. Bifurcation graphs, Fast Fourier Transformation (FFT), and Poincare plots are plot for the result analysis of the individual defects of bearing components with constant radial load. Uadhyaya et al.⁴⁴ have study the dynamics of shaft as timoshenko beam, and considered the non linearity with localized defects, unbalanced and radial clearance. The author has analysed the state of the system with bifurcation plots and the frequency spectrum and phase trajectory maps are used for system response. Patel et al.⁴⁵ developed a seven- degree of freedom model considering the collective flaws on the inner race and outer race. The defect is modelled as a sinusoidal wave. The author analysed the system performance from periodic to chaotic in nature for variable speed. Patel et al.²⁰ have modeled modelled a 9-degree of freedom model with consideration of joint inner and outer race faults as sinusoidal waves. The results are plotted on the Poincare plot, time domain plot, and frequency domain plot and are proven by performing the experiments with the same parameter considerations.

Sassi et al.⁴⁶ established a theoretical model for dynamic investigation with 3 degree of freedom for the ball bearing having a point fault. The defect is modelled with various parameters such as rotation of the bearing, load acting on the bearing, elasticity of the bearing, oil film features, and transfer path between the bearing and transducer. Behzad et al.⁴⁷ consider the stochastic impact force model to forecast the phenomena of induced vibration because of the localized defect. The author compares the stochastic vibration generation model to the classical impulse train model with healthy and defective bearings. Aktürk et al.¹² scrutinise the dynamic vibration of shaft bearing models with and without flaws. In their model, the authors considered the mass of the shaft and the mass of rolling elements, but ignored the mass of the housing. The results are investigated with time and frequency domain plots. Randal et al.^48,49 presented a combined dynamic model of the bearing and gear with a wide range of bearing defects with slippage taken into account. The author also considers the extended faults in the analysis of the dynamic model. The simulation results are compared with experimental outcomes to authenticate the model. Patel et al. have established a 6 degree of freedom dynamic model of the bearing with different loading conditions having circular localized defects on races. The author accounts for the mass of the shaft, and mass of bearing components with housing in the model. The results are plotted in the time and frequency domains, which concludes that multiple defects' vibration is greater as compared to single defects.²⁶ Kogan et al.⁵⁰ developed a new model for the evolution of spell size for the outer race and ball with periods of contact and without contact of the rolling element and outer race. The author has considered radial load, speed, and gravity for the modelling of expressions. Results are analysed with the radial load and gravitational parameters, which conclude that the Time to Impact is effective with varying the value of load. Khanam et al.⁸ investigated force modelling based on engineering mechanics fundamentals to calculate the behaviour of flaw bearings. The function is designed with the bearing dimension, speed, load, and fault dimension in mind. Excitation forces are generated primarily at three points: when the ball enters an inner race defect, when it collides with the defect, and when it exits the defect’s edge. Based on time variance between the entry of the ball and the happening of impact, the defect dimensions have been predicted by the author.

Ahmadi et al.²⁷ analysed the nonlinear dynamic model through the finite size of rollers, taking into consideration the path of the rolling element. The authors validate the results with previously published literature with rectangular defects and sharp-edged defects and conclude that the model is authenticating for different sizes of bearing defect geometries. Peterson et al.⁵¹ presented a dynamic model with different load distribution and stiffness for the bearing races defects with different sizes under radial and axial load with different degrees of rotation. The author concluded that the changes in amplitude of stiffness for defective bearings are considerably larger and found significant changes at the entrance and leaving of the bearing faults. Ahmadi et al.⁵² investigate the method to find out the size which is biassed by operational speed by considering the inertia and centrifugal forces acting on the bearing. The results of simulations and experimental observations demonstrate that the distinctive actions that are produced at the angular extent of the fault have a notable speed dependency. Patel et al.⁵³ deliberate the deep groove ball bearing with varying frequency of applied loads by considering the shaft, housing, and balls masses for the mathematical model. For the inner and outer raceways, a 600-m circular defect is considered. Patel et al.⁵⁴ investigate the deep groove ball bearing with circular faults on the inner race and outer race with different sizes under different dynamic constrains. The author remarks that with an increase in defect sizes, the visibility of BPFI and BPFO is better in the outer race compared to the inner race.

Ashtekar et al.⁵⁵ investigate the motion ball bearing elements with considering the local defects as bump or dent. The superposition principle was taken into account for analyzing defects dynamics. Author concluded that there is a great impact of defects and irregularities in bearing motion and forces. The piecewise function model was developed by Shao et al.⁵⁶ to define the fault in a cylindrical roller bearing. The time-varying deflection excitation and the time-varying contact stiffness excitation induced by the defect are both considered in a 2-degree of freedom dynamic model for a cylindrical roller bearing with a localized surface imperfection on its races. Pandya et al.⁵⁷ investigate a high speed rotating shaft with combined bearing element defects. The localized defects are introduced as spall of 300 × 300 × 100 microns on bearing elements from laser machining process. The results are shown as envelope analyses, bifurcation diagrams, and Poincare’s maps, which demonstrate the advent of nonlinearity in the dynamic response due to rotor speed and the presence of a coupled localized defect. Nakhaeinejad et al.⁵⁸ present localized defects of dents and pits on the inner and outer races and balls, which were modelled through a vector bond graph. The author analysed the different defects with size and shape for various clearances and radial loads. The dynamic model is validated by experimental results. Leblanc et al.⁵⁹ investigated the effects of unbalanced rotor forces on cage materials, counter-rotating motions, and structural deformations of the inner and outer rings. The researchers found that the cage material’s inertia is discovered to have an effect on the cage trajectory, while when the ring flexibility is taken into account, the ball pass harmonics are considerably altered. Kulkarni et al.⁶⁰ developed a dynamic model to anticipate how a bearing with a localized fault on the outer race would vibrate. The fault was formulated using a time-varying displacement excitation model based on a cubic hermite mixing function. This model predicts the influence of changes in the defect’s angular position, defect size on the outer race, numerous defects on the outer race, and load variation on the vibration amplitude. Wang et al.⁶¹ present a dynamic model of cylindrical roller bearing with localized defects as dents on raceways while considering Gupta’s model. The Newton-Euler method is used for forming the equations and solving them by the fourth-order runge-kutta integration method with variable steps. Khadam et al.⁶² represent engineering mechanics based force mechanisms at different points of bearing. The author concluded that the increasing amplitude of force at the distinctive outer race defect frequency is observed to rise with defect, suggesting that this could be a sign of increasing defect. Shah et al.⁶³ investigate the vibration generated by healthy and defective bearings, considering lubrication film and non-linear Hertzian contact. The purpose of this work is to investigate the effects of radial load, lubricant, defect size, and position on vibration amplitude at bearing defect frequency. The presence of lubricant, shaft speed, load, defect position, and defect size have all been found to affect the vibration amplitude of the characteristic defect frequency.

Rolling element bearing with distributed defects

Rolling element bearings may produce vibrations because of varying compliance or contact forces that fluctuate over time between the bearing’s numerous components.

Gustafson et al.⁶⁴ calculated the effect of rolling element components for waviness defects and concluded that inner race waviness disturbs the amplitude of vibrations at ball passage frequency. Tallian et al.⁶⁵ studied analytical and experimental vibration induction due to geometric imperfection. The author concluded that there is a remarkable amount of vibration found in the occurrence of faults. Meyer et al.⁶⁶ forecasts the frequency character of vibration generated from rolling element bearings due to distributed defects considering constant trust load. Wardle^67,68 has studied the low frequency vibration forces due to trust load for ball bearings. The author has demonstrated experimentally along with theoretically that distributed defects on the outer race cause vibrations at the harmonics of the outer race ball passage frequency. Sunnersjo⁶⁹ examined the vibration appearances of the bearings due to surface irregularities. The author has concentrated on distributed defects on the inner race as waviness and on the ball as varying ball diameter. The author concluded that cage speed harmonics occur due to varying ball diameters. Waviness in the inner ring occurs at shaft harmonics, with a sideband spaced with the roller passage frequency at greater harmonics. Akturk et al.⁷⁰ have chosen the parameter preload and the number of balls in an analysis of dynamic behavior of ball bearings. The author concluded that proper selection of the number of rolling elements and preload on bearing could reduce vibrations considerably. Akturk⁷¹ prepared a computer-generated platform to analyse the influence of waviness on bearing races. The author plots the outcomes in the time domain and frequency domains. The result concluded that severe vibration occurred because of the ball’s pass frequency and its harmonics matched the natural frequency. Akrut et al.⁷² considered the influence of changes in the diameter of the rolling element with three DOF systems for shaft bearing mechanisms. According to the author, the peak radial vibration induced in the system due to changes in the diameter, occurs at a speed equal to half the balls count multiplied by the speed of the cage. Lynagh et al.⁷³ recommended an in-depth model with consideration of the influence of nonlinear connection of springs in balls with race contacts. The off sized balls, surface waviness, and internal radial clearance are the parameters that account for the analysis. The model can be used to forecast the complex spectra of real-time vibrations. Tiwari et al.⁷⁴ proposed the effect of an unbalanced rotor considering Hertzian contact and radial internal clearance. The author analysed the appearance of sub harmonics and super harmonic frequency components in the system due to higher values of clearance and unbalanced forces. Chaudhary et al.⁷⁵ analysed a theoretical model to predict the vibration of rolling element bearings considering surface waviness of bearing races as well as off-size rolling elements under radial load. The author explained that because of numerous forms of distributed faults in rolling element bearings, flexural vibration on races occurs. Chaudhary et al.⁷⁶ extended his work by considering effects of shaft and housing. The author concluded that every number of waviness is responsible for an increase in the specific frequency of components. The spectral component amplitudes were found to be higher in outer race waviness than in inner race waviness. Discrete spectra with large modules at multiples of cage frequency are also predicted by the model where the distributed defect as off-size rolling elements is considered.

Harsha et al.⁷⁷ have examined the non-linear vibration of a balanced rotor for ball bearings, considering nonlinearity which can be caused by Hertzian contact force, distributed fault, variable compliance, and internal radial clearance. The author analysed the system behaviour from periodic, quasi-periodic, and chaotic behaviour performances from the result plots of Poincare maps, frequency spectra, and time displacement responses. Harsha et al.⁷⁸ have deliberated the influence of surface waviness and number of balls for the stability of the rotor. According to the author, the ball count is a crucial factor for vibration investigation of the rotor bearing mechanism at the design stage. Harsha et al.⁷⁹ investigated a theoretical model of a rotor bearing mechanism having waviness to analysed the dynamic behaviour of the system. The outcome of the research is compared with previous literature and validated. Harsha et al.⁸⁰ evaluated the nonlinear vibration characteristics due to the influence of ball size variation on a rotor-bearing system. The maximum radial vibrations generated in the mechanism when ball size variations are at the ball count multiplied by the cage speed. Harsha et al. analysed a balanced stiff rotor with rolling element bearings’ response, considering the effect of radial internal clearance and speed as non-linear parameters. The author analysed the system behaviour from periodic, quasi-periodic, and chaotic behaviour performances from the result plots of Poincare frequency bands. Harsha et al.⁸¹ have explored the outcome of an unbalanced rotor considering radial clearance and Hertzian contact as non-linearity for dynamic analysis of the mechanism. The results are plotted on Poincare plots, phase plans, time displacement responses and FFT. The author analysed the system behaviour from periodic, quasi-periodic, and chaotic behaviour performance from the result plots. Harsha et al.⁸² explore nonlinear theoretical models of bearing due to radial clearance with Hertzian contact as a non-linearity. Results are plotted in time response, rotor trajectories, and Poincare charts to examine the system behaviour with periodic, quasi-periodic, and chaotic features. The author concluded that the highest value of vibration was shown at the range of ball passage frequency (BPF). Harsha et al.⁸³ presented a non-linear analytical model to explore the rolling element bearings with cage run out, and number of balls. The author concluded that due to cage run out, the effect of ball pass frequency will be decreased as the number of balls increases. The number of balls is an important parameter to consider at the design stage. Cage run-out causes the most axial vibrations when a speed equal to the number of balls multiplied by the cage speed. Harsha et al.⁸⁴ evaluated a theoretical model for the study of the effect of distributed defects and unbalanced rotor. The modified Newton β mark method is applied to resolve the nonlinear equations. The mechanism is responsible for the excitement at ball passage frequency with even number of waves count while due to an unbalanced rotor, it excites at a mixture of ball passage frequency and rotational frequency. Natrajan et al.⁸⁵ explore an analytical nonlinear dynamic model of bearing owing to cage run out for an unbalanced rotor. The highest peaks are observed in the frequency of vibrations caused by cage run-out, which is equal to the number of balls multiplied by the cage speed. The author analysed the system’s response in three regimes: periodic motion, quasi-periodic oscillations, and chaotic response.

Changqing et al.⁸⁶ developed the five degree of freedom ball bearing dynamic model by considering the effects of waviness, radial load, preload, and clearance at high speed. Due to the high speed of the mechanism, the effects of gyroscopic and centrifugal effects are taken into consideration because of the ball for analysis. Analysis of the results concluded the waviness on the outer race has more impact compared to the waviness on other elements of the bearings. Chen et al.⁸⁷ proposed a static model for predicting the load performance of a cylindrical roller bearing due to off-sized rolling parts. Each rolling element’s dimension and order were treated arbitrarily in this model. The bearing load distribution was investigated in relation to rolling element off-size and the matching arrayed order. The bearing load distribution could be greatly changed by the rolling element off-size, according to this research. Chen et al.⁸⁸ developed an algebraic model to investigate the effects of rolling element diameter defects and race roundness on radial internal clearance. The race curves were created using the Fourier series. The rolling element diameter inaccuracies were shown to have a substantial impact on the bearing’s radial external clearance and run-out. Ji et al.⁸⁹ developed a model to investigate the effects of off-sized rolling elements on the contact stress between the rolling elements and the raceways as well as the axis orbit.Babu et al.^90,91 presented the six-degree of freedom dynamic model of an angular contact ball bearing with the impact of frictional moments and waviness on bearing elements. Different waviness orders with different amplitudes are considered for the analysis. The load dependent frictional moment has a high amplitude of vibration. Inner race radial waviness generates a large amplitude of vibration compared to other elements’ waviness. The author also expended his work considering the effect of an elastically deformed shaft along with waviness order and frictional moment. The author concluded that, compared to a rigid rotor, an elastically deformed shaft generates a high amplitude of vibrations. Kankar et al.⁹² provided a mathematical model to investigate the effect of inner and outer race waviness on rotor bearing vibration characteristics. The waviness is described as unevenness on the circumference of the races, which is further expressed by a Fourier series, rather than as a perfect sinusoidal feature. Chaudhary et al.⁹³ developed two different methods in the frequency domain, namely the Base Excitation Method and the Bearing Kinematics Augmented Base Excitation Method, both of which consider the inner race roundness profile. The author analysed the frequencies with different subcritical harmonic responses and amplitudes, considering the various profiles of waviness. The result shows that the dynamic models developed were found to be well associated with previously developed models. Tingarikar et al.⁹⁴ investigate the dynamic model of cylindrical roller bearings in the presence of different orders of waviness while considering the external dynamic load. The unbalance of shaft and static load is considered as external dynamic loading. The author has validated his results by comparing them with the results of prior researchers. Adamczak et al.⁹⁵ analysed the 6304-cylinder ball bearing for the influence of the surface waviness of the inner race and outer race on vibration amplitude. The author also considers the deviation of waviness measurement in his experiments with a different range of bandwidth. The waviness deviation is more visible in the medium frequency bandwidth and less visible in the low frequency bandwidth. Surface waviness has a greater impact on vibration amplitude in the medium frequency bandwidth. Jang et al.⁹⁶ studied the influence of ball waviness on vibration analysis of rotating bearings, taking into account centrifugal force and gyroscopic moment of the ball. He validates his research by comparing the results with previous results of other researchers. Liu et al.⁹⁷ developed a two-degree of freedom dynamic model by lumped parameter model to investigate the effect of uniform and non-uniform surface waviness. There is no impact on the fundamental frequency of the vibration

Rolling element bearing with combined localized and distributed defects

Sopanen et al.⁹⁸ suggested a six DOF analytical dynamic model of a ball bearing with an elastohydrodynamic film and non-Hertzian contact deformation. Inner and outer ring waviness, as well as localised faults in the inner and outer rings and misalignments, are considered as non-linear parameters for the analysis. Sopanen et al.⁹⁹ simulate the dynamic model with the elastohydrodynamic film and distortion due to non-Hertzian contact through profitable software (MSC.ADAMS). The consequence of the diametric clearance was studied and found to have a noticeable influence on the performance of the system. The author concluded that bearings generate the vibrations in the passage of ball frequencies of the inner race and the outer race of the ball passage for the waviness order close to number of balls while local defects generate the vibration at defect frequencies. Dolence et al.¹⁰⁰ studied that distributed flaws can be diagnosed by comparing the envelope spectra of vibration signs. The changes in the dimensions of flaws and the variation of spacing in balls are investigated. The author has derived the diagnosis rule of distributed defects and localized defects. He concluded that distributed flaws can be distinguished by noting the amplitudes of two characteristic frequencies (BPFO, FTF, and BPFI, FTF/FTFI) for the raceways. While localized flaws stimulate only BPFO or BPFI. Liu et al.¹⁰¹ investigated the dynamic model of a deep groove ball bearing under radial load with combined localized and distributed faults. The author concluded that the combined defect has higher acceleration amplitude compared to a single defect, and the frequency of the spectrum of acceleration increases with the defect size in the case of combined faults. Shah et al.¹⁰² analysed the experimental vibration of a deep groove ball bearing having combined defects on raceways. The author reviewed the influence of variation in width of local defect, waviness order, load, speed of shaft, and role of lubricant on vibration level in the existence of combined defects. The fault frequency depends on the position and category of the fault, and the amplitude of the characteristic defect frequency increases with the combined faults in the system. Table 1 shows research findings from the literature reviewed for bearing defects modeling.

Table 1.

Research findings from the literature reviewed for bearing defects modeling.

Reference	Position of defect	Type of defect	Defect Modeling
Sopanen and Mikkola^98,99	Inner race outer race	Localized defect: Line (rectangular) spall Distributed defect: Waviness	The tangent and STEP functions’ characteristics are used. The STEP function uses cubic polynomials to approximate a heaviside step function
Cao and Xio¹⁰³	Inner race outer race Rollers	Distributed defect: Waviness	The angular position of the defect, the tangential size of the defect, and the radial depth of the defect are the three parameters used to define the defect
Rafsanjani et al.³¹	Inner race outer race Rolling element	Localized defects: Point spall	Defined by the impulse operator. The applied load and the angular velocity at the point of contact determine the amplitude of the impulse, which can also be controlled by the relative motion of the load zone
Patel et al.²⁶	Inner race outer race	Localized defects: Line spall	Additional deflection at the defect of the ball when it passes through the defect is defined by the width of the defect
Patil et al.³⁵	Inner race outer race	Localized defects: Spall	Additional deflection of the defect is modelled as a circumferential half - sinusoidal wave
Tadina and Botezar¹⁰⁴	Inner race outer race Ball	Localized defects: Dent	The outer race is radially distorted and finite element simulated. The inner and outer races' dents were modelled as ellipsoidal depressions, whereas the rolling element’s dent was modelled as a flattened sphere
Yuan et al.¹⁰⁵	Inner race outer race Ball	Localized defects: Spall	A geometry and kinematics-based fault model is introduced, where the faults are defined in the model based on the surface profile change
Wang et al.⁶¹	Inner race outer race Roller	Localized defects: Different types of defects	The time-varying deflection excitation introduces local deflections owing to flaws in the created model
Patel et al.⁴⁵	Inner race outer race	Localized defects: Line spall	A localized defect is introduced in the model which is derived from a half-sinusoidal model for cylindrical roller bearing
Patel et al.¹⁰	Inner race outer race Roller	Localized defects: Line spall	Modelled defect as half-sinusoidal model. The analysis is based on the time-to-impact notion
Shah et al.⁶³	Inner race outer race Ball	Localized defects: Line spall Distributed defect: Waviness	Additional deflection at the defect of the ball when it passes through the defect
Liu et al.⁹⁷	Inner race outer race Ball	Localized defects: Line spall Distributed defect: Waviness	The localized defect is modelled as the time-varying displacement and contact stiffness excitations
Tingarikar et al.⁹⁴	Inner race outer race	Distributed defect: Waviness	Race waviness is modelled using a sinusoidal function or a superposition of sinusoidal functions of various orders

Conclusion and future research investigations

The latest research and progress in fault diagnosis bearings with defects have been briefly summarized in the present review paper. Time-domain techniques can help to indicate the fault of the bearings, while a frequency domain technique can be used to indicate the fault and its exact location in the bearings. Over the years, a numbers of researchers have worked on dynamic modeling of bearings with different system parameters like speed, load, number of rolling elements etc. Each model has its own advantages and limitations. The rotor bearing mechanism is found to be very sensitive when there is a slight change in operational parameters. The rotor bearing mechanism produces more vibration when the defect is on the inner race compared to the outer race of the bearing. According to the preceding literature survey, the majority of researchers concentrated on the detection of localized and distributed flaws in bearings, while combined localized and distributed faults with other nonlinear parameter considerations received less attention. The dynamic model can be represented by considering the mass of the shaft, mass of housing, mass of bearings, stiffness, defects, speed, damping, etc. As a result, it is critical to investigate the vibration performance of rolling element bearings using a variety of parameters in order to develop a dynamic model that can provide systematic descriptions of categorically dynamic behavior.

The following points can be used to identify research gaps that should be addressed in future studies:

The results of the review suggest an ever-growing trend in the research issues related to defect modelling of the bearing with different geometrical and operational parameters. According to the preceding review, the majority of researchers concentrated on the identification of localized flaws and distributed flaws in bearings, while combined defect analysis received less attention. Also, there is less focus on multiple defects compared to single defects on all bearing components.

Bearing defects can range from line, area, and extended area spalls with various surface roughness profiles, which can be created to seem like operational flaws seen in real-world applications. The location of raceway spalls in and out of the bearing load zone could also be changed to investigate changes in vibration responses.

Most researchers have developed the defects artificially through machining processes. But in actual reality, the faults in the bearings develop with time. The emerging stages of flaws, on the other hand, were not taken into account in prior research for defect analysis. During the development phase of the bearing, it produces noticeable signals that are useful for determining the bearings' and connected system’s health.

The shaft has been assumed to be stiff in earlier research, but at higher rotating speeds, this assumption will lead to erroneous results. For more accurate simulation, the shaft should be considered flexible.

The coupled modelling of dynamic bearing models and rotor models, as well as the comprehensive FE modelling of the entire rotor bearing system, should be explored more in the future, as modelling theory and computation efficiency improve. To broaden the industrial application of theoretical models, further research into accurate modelling and simulation is needed.

The majority of researcher work on development for qualitative and quantitative nonlinearity analyses in rolling element bearings. Techniques that are more resilient and simple should be investigated, which may be further tested on rotor-bearing systems in a variety of operating circumstances using a combination of theoretical and experimental studies. Different parameters like mass of shaft, bearing elements, housing, load on bearing, speed, clearance, linear or nonlinear bearing stiffness, preload, friction, unbalanced forces, lubrication, damping, different defects, etc. should be included for dynamic modelling and analysis of rolling element bearings.

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

Samruddhi Patel

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