Modeling and Dynamic Analysis of Spherical Roller Bearing with Localized Defects : Analytical Formulation to Calculate Defect Depth and Stiffness

Since spherical roller bearings can carry high load in both axial and radial direction, they are increasingly used in industrial machineries and it is becoming important to understand the dynamic behavior of SRBs, especially when they are affected by internal imperfections.This paper introduces a dynamicmodel for an SRB that includes an inner and outer race surface defect.The proposed model shows the behavior of the bearing as a function of defect location and size. The new dynamic model describes the contact forces between bearing rolling elements and race surfaces as nonlinear Hertzian contact deformations, taking radial clearance into account. Two defect cases were simulated: an elliptical surface on the inner and outer races. In elliptical surface concavity, it is assumed that roller-to-race-surface contact is continuous as each roller passes over the defect. Contact stiffness in the defect area varies as a function of the defect contact geometry. Compared to measurement data, the results obtained using the simulation are highly accurate.


Introduction
Bearings have an essential effect in dynamic behavior of rotating machinery and a simple dynamic model is important for analysis purposes.Predicting any type of bearing fault might protect the whole system from further failure and damage.Defects cause early fatigue in bearings.In practice, bearing imperfections are categorized in two major groups, distributed defects (i.e., waviness) and local defects (i.e., spalls on the raceways) [1,2].
A large amount of literature has been published dealing with mathematical models for bearing local defect diagnosis.One of the early researches in bearing with defects was conducted by McFadden and Smith [3,4].Vibration model shows impulses equal to rolling element passing frequency caused by the interaction of rolling elements and defects.Tandon and Choudhury [1] could detect characteristic frequencies when the defect is located on the roller.
Nakhaeinejad and Bryant [5] introduced a multibody dynamic model to study vibration response in rolling element bearings with a local fault.They compared the effects of defect type and size with various radial loads in dynamic behavior of the bearing.Wang et al. [6] used a multibody dynamic approach for simulation of cylindrical roller bearing with local faults.Contact forces are calculated by combining Hertzian contact theory and the slice method.Local defects deflections are defined by the time-varying excitation.Vibration simulation of ball bearings for both local and distributed imperfections is introduced by Sopanen and Mikkola [7,8].Local defects were categorized by length and height of defects.
Kiral and Karagülle [9] applied finite element method in vibration analysis of rolling bearing.They could detect multiple defects on bearing components using frequency and time factors.They proposed the loading mechanism to generate the nodal excitation which can be used in finite element method.Rafsanjani et al. [10] studied the nonlinear model of roller bearing with local defects.Equations of motion were linearized using the modified Newmark time integration technique.Stability of the bearing with local defects was explored using the classical Floquet theory.

Shock and Vibration
Kankar et al. [11] used response surface technique to diagnose local faults in high speed rolling element bearings.Discrete spectrum shows peaks at the characteristic frequencies of the bearing.They could also predict accurate spectrum amplitude height compared to experimental results.Tadina and Boltežar [12] developed the dynamic analysis of a faulty roller bearing during run-up.This model considered the effect of centrifugal force.Defect is located on the inner and outer raceways and defined as part of ellipsoidal depression and it causes contact stiffness variation in the defect region.Liu et al. [13] combined Hertzian contact theory with the piecewise function to evaluate the vibration behavior of a roller bearing with the local defect.Kankar et al. [14] analyzed the small-size local defect's effects on the dynamic behavior of ball bearings.Behzad et al. [15] introduced a new model to estimate the vibration which originates from faulty roller bearings.They showed that when a defect grows in the bearing, stochastic excitation becomes stronger in the defective area.Sawalhi and Randall [16,17] studied the vibration of combined gear bearing system with localized bearing faults.They used the same diagnostic techniques for simulated and experimental data and could observe similar behavior for both.Vibration response of spalled rolling element bearings is also demonstrated by Sawalhi and Randall [18].They found that acceleration response for entry into a spall-like fault is different from the response for exit.Niu et al. [19] proposed the model for a local defect at high speed ball bearings.They showed that, for a high speed rolling bearing, both high speed effects and influences of defects should be considered.In this study defect properties (size and type) have not been taken into account.
Recently, the authors of [21] compared the vibration signals of a faulty bearing at different operational speeds.They studied the effect of the path of roller in the defect region for developing an algorithm to estimate the size of the defect.Petersen and Howard [22] studied a faulty bearing where defect zone is larger than the angular distance between rollers.Results illustrated the effect of defect size on rigid body modes of system which is originated from bearing stiffness variations.Unbalanced rotor-coaxial water pump with a local defect is studied in [23].The local defect is modeled as periodic excitations which are caused by unbalance force.Liu et al. [24] proposed a new model to find out the effect of defect size on vibration characteristics of system caused by timevarying contact stiffness which is not taken into account in earlier studies.
One of the first studies on measurement techniques for bearing health monitoring was done by Tandon and Choudhury [25].Defect detection methods are continuously developed and several more studies have been published [26][27][28][29][30][31][32].
A wide range of studies has been conducted to study defects in ball bearings.However, they are limited in case of spherical roller bearings.Cao and Xiao [33] introduced a vibration analysis of spherical roller bearing.Considering the effect of local imperfections on the dynamic behavior of the bearing is promising in this paper.This model is suffering from low computational efficiency which results from the system's high number of degrees of freedom.Recently, the authors of [34] studied the effects of offsized rollers on behavior of spherical roller bearings.They found that error in roller diameter might change the force acting on each roller.
This paper introduces the vibration analysis of the spherical roller bearing with local imperfections in the inner and outer raceways.Nonlinear Hertzian contact theory is used to describe the rollers' force in contact with inner and outer raceways.Radial clearance is another nonlinear term which has been considered in calculation.It is assumed that the defect located on the inner and outer rings has an elliptical cross section.With this assumption, roller-to-race-surface contact is continuous as each bearing roller passes over the defect.Variation of radii of curvature in the contact area in defect region causes the different contact stiffness.Numerical examples consider the effect of the elliptical defect's size and location on dynamic behavior of rotor-bearing system.The introduced mathematical model is verified by measurement data.

Modeling of the Spherical Roller Bearing
A modeling approach for the basic spherical roller bearing model is presented earlier in [20,35].For the completeness of this paper, this model is briefly reviewed in this section.In this case, to reduce the computational complexity of the system, some simplifications are used in the model.It is assumed that no slipping occurs during bearing operation.Since in typical operating conditions of SRBs the centrifugal forces that arise from rollers are significantly smaller than the bearing contact force, the centrifugal forces are neglected in this study.It is also defined that inner ring is connected to the rotor rigidly.Also bearing damping has not been taken into account in this study.Figure 1 [20] shows important parameters of spherical roller bearing.
In this model, summations of individual roller contact forces lead to the bearing load calculation [35].When  spherical roller bearings are loaded moderately, the roller's contact surface can be considered elliptical [36].Therefore, the th roller contact force in row  can be calculated as [35] where  tot con is the contact stiffness coefficient of roller in contact with inner and outer race and     is the rollers' and the races' elastic deformation, which is is determined from relative displacements among the inner and outer raceways.The initial distance, between centers of two raceways (O-1, O-2), is [35] where  out  and  in  are the radii of curvature for the inner and outer raceways as shown in Figure 2 [20].  and   are roller diameter and bearing internal clearance, respectively.
The equivalent loaded distance can be formulated as where    and    are roller axial and radial displacements: where   ,   , and   are displacements in the global coordinate and    is the roller's angular position (see Figure 3).Contact direction is negative in the 1st row and positive in the 2nd row of rollers.
Displacement among inner and outer races is given by Elastic compression becomes And the contact direction for each roller can be calculated as Lastly, the bearing force in the global coordinate can be calculated as where  is the number of rollers in one row.It should be noted that only negative values in the rollers contact force are not taken into account (   = 0).

Local Defects Mathematical Modeling
In general, imperfections are the main vibration source in the roller bearing.This paper studied the vibration response that occurs from local imperfections on the inner and outer raceways of spherical roller bearing.In most of the earlier studies of local defects in roller bearings, it was assumed that the rollers lose their connections in the district of the defect, and it is modeled with a large impulsive force on rotor bearing systems [1].Recently, Tadina and Boltežar [12] have proposed the model of the defect on raceways as an ellipsoidal shape and it leads to change in contact stiffness in the defect district.This study uses the same approach for modeling defects in spherical roller bearings.
3.1.Defect on Outer Race. Figure 4 shows the elliptical defect on outer race.Midpoint of the defect is positioned at .The angular width of the defect is   and the maximum depth is   .When a roller goes over the defect, the elastic deformation in defect district can be written as In ( 6),  + (   ) is the extra displacement in the defect region, which should be withdrawn from general displacement (see Appendix A) in comparison with (9).This additional displacement is defined only when the roller passes a defect district,  −   /2 <    <  +   /2.Variation in radii of curvature in the defect region leads to different contact stiffness.Consequently, new contact stiffness can be calculated according to Hertzian contact theory [35].Nevertheless, the calculation of new stiffness where   and   are ellipse principal axes and  is a polar constant, calculated as From ( 1) and ( 9) finally the roller contact force in defect district can be expressed as follows: where  indicates the defect region,  tot con, is the contact stiffness, and  con is contact angle of roller in defect region (Figure 4).

Defect on Inner Race.
Corresponding to the outer race defect, for an inner race defect (Figure 5), the elastic deformation at defect region will be evaluated as and the roller contact force can be expressed as follows: , =  tot con, ( (   ) , ) where  tot con, is the contact stiffness which follows the same calculation routine as in (10) and (11).Figure 4: Defect on outer race [20].
the bearing with outer race defect.Figure 6 shows the defect which is located on one side of the outer race.The defect parameters are presented in Table 2 and it is located in 280 degrees from the -axis and maximum depth of the defect is 0.05 mm.Initial displacement for center of bearing in the vertical direction is assumed to be equal to   = −0.0324mm and inner race rotates with rate of  in = 78.54rad/s (750 rpm).Calculation routine for characteristics frequencies of SRB is introduced in Appendix B, and results for 750 rpm are listed in Table 6.With assumption of elliptical defect, connection loss between the roller and the rings does not happen immediately.So, in this case, size of the defect becomes important.Therefore, in the following example, effect of different elliptical defect size is considered.Figure 7 shows the three elliptical defects in outer ring of bearing.It is assumed that all these faults have the same depth, and the only difference is in their size.Table 2 lists the dimensions of the outer ring defects.
Figure 8 shows the system response for the three different size defects which are introduced in Figure 7. Frequency domain spectrum could detect the roller pass outer race frequency and second harmonic (Table 6).Furthermore, it is found that force amplitude is risen significantly by increasing the elliptical defect dimensions.
Figure 9 considers the effect of defect depth, where increasing the defect depth can rise the force amplitude.Single elliptical defect ( = 20 mm and  = 10 mm) is chosen and maximum depth of the defect is increased from 0.05 mm to 0.15 mm and the defect is located in 280 degrees from the axis.Initial displacement for center of bearing in the vertical direction is assumed to be equal to   = −0.0324mm and inner race rotates with rate of  in = 78.54rad/s (750 rpm).Force profile in time domain for elliptical defect is analyzed in Figure 10 for defect type II from Table 2 with depth 0.05 mm.It is clearly shown that bearing force is decreased when roller enters defect area.

Load Analysis for
Bearing with Inner Ring Defects.This section investigates the vibration response of defect on inner ring.The same as outer race case, initial displacement for center of bearing in the vertical direction is assumed to be equal to   = −0.0324mm and inner race rotates with a rate of  in = 78.54rad/s (750 rpm).Figure 11 shows the defect located on one side of the inner race.Figure 12 shows the three elliptical defects in inner ring of bearing.It is assumed that all these faults have the same depth, and the only difference is in their size.Table 3 lists the parameters of the inner ring defect.
The roller pass inner ring frequency and its harmonics, with shaft rotating frequency harmonics (Table 6), can be detected in Figure 13.Compared to defect in outer ring, size of the elliptical defect does not have significant effect in force amplitude spectrum.
Figure 14 considers the effect of defect depth, where increasing the defect depth can rise the force amplitude.A single elliptical defect ( = 20 mm and  = 10 mm) is chosen, and maximum depth of the defect is increased from 0.05 mm to 0.15 mm and the defect is located in 280  from the axis.Initial displacement for center of bearing in the vertical direction is assumed to be equal to   = −0.0324mm and inner race rotates with rate of  in = 78.54rad/s (750 rpm).
Force profile in time domain for elliptical defect is analyzed in Figure 15 for defect type II from Table 3 with depth 0.05 mm.It is clearly shown that bearing force drops down when roller enters defect area.

Transient Analysis and Experimental Verification
Rotor with two spherical roller bearings is a case study in full rotor-bearing system simulation and experimental verification.Figure 16 shows the schematic of the rotor and corresponding dimensions are listed in Table 4.According to Newton's second law, the differential equation of motion for a rotating structure can be written as follows: where M, C, G, and K are the mass, damping, gyroscopic, and stiffness matrices, respectively.q is the displacement in generalized coordinates, Ω is the shaft rotation speed, and F is a force vector that includes the external forces, the bearing forces, and the gravity forces for the rigid rotor.Firstly, governing differential equations would be decreased to a set of first-order differential equations, and then the state space form of the equation will be solved with MATLAB ODE integrators [20].The beam element is used to model the rotor and finite element method is applied.Each element has four degrees of freedom per node.By defining 17 nodes in the length of the rotor, this rotor-bearing system has 72 dofs where 4 dofs come from bearings supports lateral displacements, and the remaining 68 dofs describe the rotor displacements.A linear spring-damper is defined for connection of bearing housing to the ground.Rotating structure properties are shown in Table 5.In this system, it is assumed that support structure has higher stiffness compared with bearing stiffness, and support damping is rough estimation value which is calculated by defining 5% as modal damping ratio.
Figure 17 shows the measurement setup where the bearing with defect is placed in one side of the rotor.Measurements were done for inner and outer ring defect separately.Defect parameters are shown in Tables 2 and 3 for defect type III.External belt drive rotates the rotor and data acquisition starts to record the data when the system settles down in constant rotational velocity.500 ± 10 N external force is applied to the rotor in the vertical direction to clarify the vibration originating from bearing defect.External force was measured by washer load cell (HC 2001, DS Europe).Furthermore, two ICP accelerometers (98 mV/g) are used to measure the vibrations, which are located on top of the bearing housing transferring data to acquisition unit (CSI 2130).4096 samples are recorded in 4 seconds at 1024 Hz.
For the outer race defect, Figure 18 shows spectrum for vertical displacement amplitude in frequency domain and compares the simulation result with measurements.Fast Fourier Transform (FFT) analysis is applied to transient analysis results to detect high peaks in frequency domain (sampled at 1024 Hz for 4 seconds).Simulation spectrum shows peaks correctly on the roller pass outer ring frequency (118.1 Hz) and its second harmonic spectra (displayed in Table 6).Similar frequencies can also be detected in measurement at 115.1 Hz.This frequency gap might originate from internal slipping between bearing components [20].Another important point is that actual rotor-bearing systems are suffering from unknown noises which can be visible in measured signals (Figure 18(b)).Analytical model has not the capability to detect all unknown vibration noises.For the inner race defect, Figure 19 shows spectrum for vertical displacement amplitude in frequency domain and compares the simulation result with measurements.
Spectrum contains peaks at rotor rotation velocity and its harmonics in simulation and measurement plots.Roller pass inner ring frequency (RIF) and its harmonics (RIFSF) are also visible in plots.

Conclusion
Vibration simulation of spherical roller bearing with local defect on inner and outer raceways was introduced and simulation predictions were compared and verified with measurement data.Rotor is modeled using finite element method, and governing equations of motion were solved by MATLAB numerical integrators.An important achievement was efficiency of the proposed bearing model, which is demonstrated by different numerical examples and can be used in actual rotor-bearing simulations.Bearing force was calculated from summation of roller contact forces using Hertzian contact theory which is the source of nonlinearity in the system.With assumption of elliptical surface concavity on inner and outer raceways, roller's connection with rings is not lost immediately and it prevents impulse force (available literatures) when roller passes over the defect district.It means that the roller is in contact with surfaces, whenever the defect depth is smaller than elastic deformation between rollers and races plus also bearing internal clearance.As a result, this roller contact force should be considered in total bearing force.Furthermore, because of changing the radii of curvature in defect region, new contact stiffness needs to be defined.Numerical examples demonstrated a set of dynamic responses according to defect dimension, type, and position.Hence, it is possible to classify an SRB defect by comparing the measured data.Numerical results also exposed that local defect would be detected in bearing characteristics frequencies.The measurements detected bearing characteristic frequencies different than those that simulation predicted once.This frequency difference between two approaches is found to be 1.7-2.7%.This is originated from internal slipping between bearing internal components which are ignored in early assumptions.Nevertheless, the actual rotor-bearing system has unknown parameters which cannot be predefined in simulation process.

A. Calculation of Roller Additional Displacement in Defect Region
Based on geometric parameters of elliptical defect which is shown in Figure 20, the following parametric equations can be written: Finally,  can be calculated to get defect depth depending on roller angular position.

Figure 6 :Figure 7 :
Figure 6: Properties of defect on outer ring.

Figure 8 :Figure 9 :
Figure 8: Defect size: force spectrum for defect on outer raceway in vertical direction.

Figure 10 :Figure 11 :Figure 12 :
Figure 10: Bearing time domain force profile plot for outer race defect in vertical direction.

Figure 13 :Figure 14 :
Figure 13: Defect size: force spectrum for defect on inner raceway in vertical direction.

Figure 15 :
Figure 15: Bearing time domain force profile plot for inner race defect in vertical direction.

Table 1 :
Geometric parameters and material properties of bearing, 22216-EK.
eters and material properties of the bearing, whereas the dimensions are depicted explicitly in Figure1.In these sets of examples, effect of defect size is considered.4.1.Load Analysis for Bearing with OuterRing Defects.This model is implemented to study the vibration response of

Table 4 :
Dimensions of the rotor-bearing system.
Cage rotational frequency can be calculated by  is the shaft frequency and   is a bearing pitch diameter which is defined as  =  out +  in .(B.4)