Analytical model for predicting false brinelling in bearings

Journal Pre-proof Analytical model for predicting false brinelling in bearings O. Brinji, K. Fallahnezhad, P.A. Meehan PII:

S0043-1648(19)31369-9

DOI:

https://doi.org/10.1016/j.wear.2019.203135

Reference:

WEA 203135

To appear in:

Wear

Received Date: 13 September 2019 Revised Date:

12 November 2019

Accepted Date: 12 November 2019

Please cite this article as: O. Brinji, K. Fallahnezhad, P.A. Meehan, Analytical model for predicting false brinelling in bearings, Wear (2019), doi: https://doi.org/10.1016/j.wear.2019.203135. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Analytical Model for Predicting False Brinelling in Bearings O. Brinji1, K. Fallahnezhad1 and P. A. Meehan1 1

School of Mechanical & Mining Engineering The University of Queensland St Lucia, Queensland 4067

Abstract An analytical model is developed to simulate false brinelling in a bearing based on the theory of energy dissipation. This model is capable of predicting false brinelling damage when the bearings are under vibrating conditions. The bearing used in this simulation is a cylindrical roller bearing which is made of 52100 high carbon bearing quality steel (ASTM A295). The model provides a local prediction of false brinelling for the position of each roller of a cylindrical roller bearing. The model is validated using the modified 3D FE model and testrig developed by the authors in previous work. The analytical model is shown to be far more time-efficient (by more than 10000 times) compared with the false brinelling 3D FE model. The model is used to compare the false brinelling damage in the inner and outer raceways, in both the lateral and axial directions of the bearing. The results showed that the wear marks on the outer raceway were approximately 5% shallower than the ones on the inner raceway; the volume of wear on the outer raceway was about 9% greater than the wear on the inner raceway. The model is also used to investigate the influence of the amplitude of the normal load and vibration on the wear damage caused by false brinelling. It is shown that the depth of wear increases when the normal load is decreased, and the vibration amplitude is increased. This change is particularly significant when the normal load (maximum contact pressure) on the bearing is less than 1000N (800 MPa). Keywords: bearings damage, false brinelling, energy dissipation, fretting wear

Nomenclature

a* and b* CE Ed Eda Edh(x, y) Edl Edt

The minor and major of the elliptical contact area The minor and major of the elliptical stick area 1/E (E is the modulus of elasticity) The energy dissipation The energy dissipation in the axial direction The energy density The energy dissipation in the lateral direction The resultant energy dissipation

Edth Fi (ψ )

The threshold energy of wear activation The inner static load distribution

Fmax

The maximum load on the top roller

and b

Fo (ψ )

The outer static load distribution

Fr f G

The radial load Oscillation frequency The shear modulus The wear depth

H ( x, y )

Kn K(e) i Jr(ε) l ma

The deflection coefficient The load deflection factor The number of cycles The radial load integral The effective roller length. The semi-major axis coefficient of contact ellipse The total number of cycles The traction distribution

N q ( x, y )

T

φ

The tangential load during the loading and unloading processes Exponents The maximum tangential load The radial clearance The volume of wear Points inside the contact area The number of rollers The energy density wear coefficient The angle of the lateral force applied to each roller The energy wear coefficient The tangential displacement during loading δ*/2. The tangential displacement during unloading The total elastic deflection along the radial loading based on contact deformation The maximum tangential displacement during oscillation The load distribution factor The friction coefficient The Poisson ratio The curvature sums of the inner and outer contacts The correction factor

ψ

The position angle of rollers

t T* ur V x, y Z αh αi αv δd δg δi δmax δ* ε µ ʋ ∑

and ∑



1. Introduction False brinelling is a phenomenon which occurs in stationary bearings, and is due to an external oscillation force [1, 2]. More specifically, false brinelling is a fretting wear process caused by vibration loads. The false brinelling that occurs between bearing rollers and raceways is due to the oscillation of normal or lateral forces on the non-rolling contact when the normal load forces the lubricant out of the contact [3, 4]. The resultant fretting causes wear damage to the surfaces of the raceways and rolling elements that, if left unchecked, will cause fatigue and bearing failure when in service. False brinelling can occur during the transportation process [3, 5] or when inappropriately stored bearings are exposed to vibration [6]. False brinelling can also occur when the bearings are used as spares or in a back-up system [7]. The possible false brinelling loading modes are shown in Figure 1 [1, 2]. These modes are oscillation of tangential load, variation of normal load, oscillatory torsion and reciprocating rolling. The variation of normal load causes the least loss of energy; hence, there is least damage to the surface compared with the other modes of false brinelling [8]. To date, several models have been proposed to model fretting wear processes. In many studies, energy dissipation is one of the most common concepts used to simulate wear and fretting wear [9-13]. Fouvry et al. examined fretting under different conditions, and developed and validated formulas for calculating the volume and depth of wear based on the work of Mindlin et al. [14, 15]. They investigated the effect of varying tangential loads on the contact surface between two elastic bodies, where the contact forces were calculated using Hertz contact theory. The change of tangential forces was associated with the alteration of displacements during loading and unloading. Micro-slip occurred during these processes and the wear occurred as a result of energy dissipation. They calculated the energy dissipation by determining the area of the tangential load-displacement curve during the loading and unloading processes. In this energy approach, it is assumed that the wear occurs above an accumulated critical cyclic stress state [8,16 and17]. This assumption is made in the analytical model used for this study because the Archard equation does not consider an accumulated critical cyclic stress state [10]. However, many studies have used the Archard equation in FEM and validated it with experimental results [2,3,18-20]. To date, several studies have been conducted to model false brinelling [2, 18-19]. Massi et al. [3] developed a single degree of freedom (DOF) spring-mass model to predict false brinelling. They presented the local contact stress for rolling bearings of the bleed system valves under the effect of vibrations induced by aircraft engines. The inputs for a 2D finite element (FE) model of the bearing were the axial and the global forces and the respective displacements between the contact surfaces. This model calculated the local contact stress rather than the wear caused by imposed vibrations. Moreover, Gallego et al. [2, 18, 19] detailed some of the computational algorithms and FEM to analyse fretting for a sphere on a flat sheet. They developed models to solve the contact problem under the normal and tangential loads. In some of these models, they used energy dissipation to calculate the wear. They calculated the depth of wear by varying the contact pressure, shear stress and slip amplitude. However, Gallego et al. used a wear volume formula developed in [19], which used a reference-fretting test to determine the reference wear coefficient for each case. In this study, the assumption of the ideal smooth surface has been considered. This assumption has

been made in many studies [2,3,9-13]. However, other researchers [21,22,23] have proposed that rough surfaces may influence the contact pressure and traction distribution which could be one of the reasons for the error in this study. In the authors’ previous work [24], an FE model was developed to predict false brinelling between the roller and raceway of a train motor bearing. They used the Archard model [24] to simulate the wear between the roller and the raceway. A field monitoring system was also used to monitor the condition of the bearing while the train was being transported. The field measurement data showed that the bearing was under a three-dimensional dynamic load during the transportation with the dominant loading modes being in a lateral position during trucking and shipping. Profilometry results of a false brinelled bearing showed that the false brinelling marks had w-shape profiles, indicating the occurrence of partial slip along the axial direction of the roller (Figure 2) due to dominant loading in this direction. Figure 2 also shows false brinelling wear marks on the surface of a bearing’s inner raceway that was damaged during the train’s transportation [24]. As shown in the figure, because of a small curvature along the Y-axis, the marks are created just in the middle of the raceway and have elliptical shapes. Also, the authors built a test-rig to physically simulate the condition of occurrence of false brinelling [25]. The test-rig is able to create marks under different conditions (radial load, vibration amplitudes and frequencies). To date, most false brinelling models have been based on the finite element method, using commercial finite element software. These models incur large computational costs, making it difficult to use them for a parametric study. Moreover, to reduce the computational time, FE models usually model just a small (critical) part of the bearing rather than providing a full bearing model. In this research, an analytical model for predicting false brinelling in bearings was developed, based on energy dissipation theory [14, 15] [9-13]. This model is capable of predicting false brinelling damage when the bearing is under vibrating conditions. The model provides a prediction of local false brinelling for each roller on a cylindrical roller bearing, as well as wear on the outer and inner raceways. The model was validated using the modified FE model developed in [24] as well as the test-rig developed in the authors’ previous work [25]. The model was used to investigate the sensitivity of the false brinelling damage to external

a. mode I

b. mode II

parameters (radial load and vibration).

c. mode III

d. mode IV

Figure 1: Possible false brinelling mechanisms (a) oscillation of tangential force (b) oscillation of normal force (c) oscillation of torsion (d) oscillation of rolling

Z

Y (Lateral direction)

X (Axial direction)

Figure 2: False brinelling marks and 3D profiles of false brinelling wear mark on the damaged cylindrical roller bearing [24]

2. Methodology This section presents the methodology used in this research. The first part details the equations used in the analytical model. The FEM model is then presented in the second part. In the third part, the false brinelling test-rig is detailed.

2.1 Analytical Model This section details the methods used in developing the analytical model for predicting false brinelling damage in a bearing. The method of wear used in this model is based on the energy dissipation method proposed by Mindlin [14] and Fouvry’s theories [9, 13]. This code was developed in MATLAB. The false brinelling process is depicted in Figure 3; the feedback loop that was followed in this model is based on the Mindlin and Fouvry studies. Referring to Figure 3, the false brinelling process starts with new profile surfaces for the raceways of the bearing. The normal load is then applied on the rollers from the loading on the bearing; also, the bearing is exposed to external vibration which causes load variations. As a result of these loadings, the contact mechanics causes energy dissipation between the surfaces of a roller and raceway in the contact areas. This produces wear on the contact surfaces. The wear then changes the profile during the next cycle of loading. In this study, the wear process was modelled for both lateral and axial displacements/vibrations (Figure 4). The bearing modelled in this study

is a cylindrical roller bearing. This bearing has a very small curvature in the inner race and roller along the axial direction (Figure 4) that creates an elliptical contact area. This curvature was measured using a profilometer (Talysurf i-Series). Hence, this model uses the equation for elliptical contact for a cylindrical roller. All the mechanical processes depicted in Figure 3 are described in more detail below Figures 3 and 4.

Figure 3: Feedback model diagram for the false brinelling analytical model

Outer Raceway Roller

Inner Raceway

Figure 4: Cylindrical roller bearing

2.1.1 Loading The profile that the model starts with is new and unworn. The load distribution is based on Wan’s model [26] which is an iterative process that begins with an initial approximate maximum load on the top roller Fmax, For ball bearings,

5 Fr Z

Fmax,initial =

,

(1)

For roller bearings

Fmax,initial =

4.6 Fr Z ,

(2)

where Fr is the radial load and Z is the number of rollers. The deflection coefficient Kn is then calculated as: For point contact 2 1 1  32 2 K ( e )  2K ( e ) 3 3 K n = 2.79 ×10 CE ρi ) + CE ρ0 ) 3  ( ( ∑ ∑ π ma π ma   , −4

(3)

For line contact K n = 7.66 × 10−5

CE 0.9 Fmax 0.9 l 0.8 ,

(4)

where CE is 1/E (E is the modulus of elasticity), K(e) is the load deflection factor, ma is the semi-major axis coefficient of contact ellipse, values for which can be obtained from Table 3.1 in Analysis of Rolling Element Bearings [26] (p.59), ∑ and ∑ are the sums of the curvature of inner and outer contacts, the values for which can be obtained from Table 2.6 in Analysis of Rolling Element Bearings [26] (p.43), and l is the effective roller length. Then, the approximate value of total elastic deflection along the radial loading based on contact deformation is calculated as: (5)

δ max = K n Fmax t , where t = exponents: 2/3 for point contact and 0.9 for line contact.

Then, the value of the load distribution factor ε is calculated by using the approximate value of and the radial clearance ur as:

1 2

ε = 1 −

  2δ max + ur  . ur

(6)

Then, the value of the radial load integral Jr(ε) is found from Table 4.2 [26]. Therefore, improved approximate of Fmax,new is then calculated as:

Fmax,new =

Fr ZJ r (ε )

(7) ,

Equations 1-7 are then solved with this new estimate of Fmax,new as Fmax,initial until the difference between these is less than 1%. Afterwards, the inner static load distribution is calculated based on the roller location and its number (Figure 5a) as,

1   Fi (ψ ) = Fmax 1 − (1 − cos (ψ ) )   2ε  (8)

n

,

where ε is the load distribution factor, ψ is position angle and n is 1.11 for the roller bearing. According to [27], the outer load distribution is equal to the inner load distribution plus centrifugal force when the bearing is rotating. As there is no rotation in false brinelling, there is no centrifugal force and the outer load distribution is equal to the inner load distribution as seen in figure 5a. 2.1.2 External vibration In this model, the assumption is that there are oscillatory displacements in lateral and axial directions [24]. Figure 5b shows the direction of lateral and axial displacements on the rollers.

Figure 5: a) The distribution of inner and outer loads b) Applying lateral and axial displacements on the bearing

According to the feedback shown in Figure 3, the external vibration might be either displacement input or force input. The angle of the lateral force applied to each roller is calculated as: = 90 − (9)

where ψ0=0, ψi=360i/Z, i=1,2, 3…., n and Z is the roller number. In the axial direction, all rollers are affected by the same amplitude (Figure 5b). 2.1.3 Contact mechanics In the case of displacement input, the maximum tangential load (T* ) can be calculated as [28], 3   2   16 aG δ   * T* = µ Fi 1 − 1 −    µ φ 3 F (2 − v ) i      

(10) ,

where δ* is the maximum tangential displacement during oscillation (it can be used as width of the wear), and Fi is the normal load in the outer race and inner race. T* is the maximum tangential load, µ is the friction coefficient, ʋ is the Poisson ratio, G is the shear modulus and ϕ is the correction factor that can be calculated as [28],  1   b φ = 1 + (1.4 − 0.8ν ) log   a   b 1 + (1.4 + 0.8ν ) log   a 

a=b for lateral direction for axial direction,

(11) where

and b are the minor and major of the elliptical contact area, respectively.

When the external vibration is force input, the maximum tangential displacement (δ* ) can be calculated as [28], 2 ( )    3µ Fi (2 − v)  T 3 δ* = 1 − 1 − *   φ 16aG   µ Fi     (12)

,

The energy dissipation is calculated based on the Mindlin et al. studies [14, 15]. Figure 6 shows traction force versus slip for one cycle of tangential fretting (false brinelling).

Figure 6: Load-displacement cycle

The energy dissipation is the area of the tangential load-displacement graph during loading and unloading, which can be calculated per cycle as shown in Figure 6 [20], Ed = ∫

T*

− T*

(δ d − δ i )dT

,

(13) where δd is the tangential displacement during loading and it can be calculated as [28], 2 2 ( ) ( )   3µ Fi (2 − v)   T* − T  3  T*  3  δd = 2 1 −  − 1 −  − 1 φ  16aG   2µ Fi   µ Fi    

,

(14)

and δi is the tangential displacement during unloading and it can be determined as [28], 2 2 ( ) ( )   3     3µ Fi (2 − v)  T* + T T* 3  δi = − 2 1 −  − 1 −  − 1 φ  16aG   2µ Fi   µ Fi    

,

(15)

where T is the tangential load during the loading and unloading processes

The energy dissipation in the lateral and axial directions is calculated separately based on Equation 14. The resultant energy dissipation can be calculated as,

Edt = Edl 2 + Ed a 2

,

(16)

where Edt is the total energy dissipation, Edl is the energy dissipation in the lateral direction and Eda is the energy dissipation in the axial direction. 2.1.4 Wear process The volume of wear can be calculated as [13],

0  V =  α v ( ∑ Ed − Edth )

N

If

∑ ∫ (δ i =1

N

If

T*

− T*

∑∫ ( δ i =1

d

T*

−T*

d

− δ i )dT < Ed th

− δi )dT > Edth

(17)

where αv is the energy wear coefficient (µm3 J-1), i is the number of cycles, N is the total number of cycles and Edth is the threshold energy of wear activation. The wear depth can be calculated as [13], H ( x, y ) = α h

∑ Ed ( x, y ) h

(18)

where αh is the energy density wear coefficient (µm3 J-1), Edh(x,y) is the energy density (J/m2), which can be calculated as[13], Ed h ( x, y ) =

x +δ g

∫δ q ( x, y ) dx

x−

(19)

g

where, q (x, y) is traction distribution and δg is δ*/2. The energy density is calculated separately in both directions and the results are used to calculate the total energy density, similar to energy dissipation (Eq (15)). The traction distribution used in this case is for elliptical contact which can be calculated as [29],

 2 2  3µ F 1 −  x  −  y       2π ab a b  q ( x, y ) =   2 2 2 2  x y  3µ F T*  x  y  1 −   −   − 1 −   1−   −   2 ab a b F π µ        a*   b*   (20)

x a   

2

*

y −  b 

2

x y  a  − b      2

*

>1

*

2

≤ 1.

*

This model is valid for elliptical and circular contacts and can be used for any type of bearing that carries the radial load. It depends on the distribution of traction forces for wear depth and correction factor for wear volume Equation (10,19). This model is valid for both forced or displacement-controlled processes. In the simulation, a cylindrical roller bearing (NU1018M1-F1-J20AAC3) is modelled according to the parameters listed in Table 1. The bearing is made of 52100 high carbon bearing quality steel (ASTM A295) [30], with Vickers Hardness of 848 [HV], elastic modulus of 230 GPa and Poisson ratio of 0.3. The chemical composition of this steel is shown in Table 2. The friction coefficient used in this model is decreasing when the contact pressure is increased according to the curve for the steel as shown in Figure 7 [24]. The explanation for this is that the lower contact pressure causes insufficient friction force to break the surface asperities which produces a higher friction coefficient, while greater contact pressure generates enough friction force to cause plastic deformation and breakage of some of the surface asperities to particle, resulting in a lower fiction coefficient [31]. Also, according to Eqs (13,14,15 and 20), the energy dissipation and the energy density dissipation decrease when the friction coefficient is decreased, which reduces the amount and depth of wear. The results of the inner race obtained from this model are compared to the results for the FE model (Fallahnezhad et al. [24]). Details of this FE model are provided below in section 3.1. MATLAB was installed in workstation (Surface book 2, Microsoft Corporation, i7-8650 U CPU @ 1.90GHz, 2110 Mhz, 4 Core(s), 8 Logical Processor(s) Intel,16 GB RAM)

Parameter

Table 1:

Radial load Shear Modulus Elasticity Modulus Poisson’s Ratio Outer race radius Inner race radius Inner race diameter (transverse) Roller diameter Roller width Number of rollers Tangential vibration (lateral and axial) Oscillation frequency Energy wear coefficient Density Energy wear coefficient Number of days

Symbol Fr G E ν

f αv αh

Value 1500 79000 230000 0.3 61.5 51.5 2000 12 11.8 25 2 24 5.2 5.2 x10-8 8

parameters for the models

Table 2. Chemical composition of material (mass %) [30]

Units N N/mm2 N/mm2 mm mm mm mm mm g Hz MPa-1 MPa-1 days

Input

C 0.93-1.05

Mn 0.25-0.45

P 0.025

S 0.015

Si 0.15-0.35

Ni 0.25

Cr 1.35-1.60

Cu 0.30

Mo 0.1

0.72 0.7

COF

0.68 0.66 0.64 0.62 COF = 2.5413N-0.267

0.6 0.58 120

140

160

180

200

220

240

Normal Stress (MPa) Figure 7: Friction coefficient versus normal stress [24]

2.2 FEM for False Brinelling A three-dimensional FE model [24] was used to simulate the false brinelling and is redescribed here for convenience. The FEM model is modified and validated with full-bearing experimental results in [25]. A FORTRAN code is developed to model false brinelling. It can control the position of the contact nodes through the ABAQUS UMESHMOTION subroutine within an adaptive meshing constraint. The process of FE simulation and the algorithm for the FORTRAN code were presented in the authors’ previous work [24] and validated by numerical and experimental results. In this study, the same simulation process is used to develop a three-dimensional FE model. To reduce the computational time, the model includes just one roller and a small part of the inner race. The FE model is a modified version of the one in the previous study [24], intended to enhance the precision of the simulation and reduce the computational time. In this model, the small curvature along the inner race axis and the roller (Y-direction) (Figure 4) is considered in the model (≈ 2000 mm). To reduce the computational time, the meshing structure is changed. As seen in Figure 8, the small contact patches of the inner race and roller are meshed with small-structured break elements. These elements need to be small enough to appropriately model the contact pressure and relative displacement at the contact area. The rest of the model is meshed by tetrahedral elements which allows increased element sizes away from the contact area [32]. The simulation process comprises two operations[8]. First, the normal load is applied to the roller. Second, the oscillating loading is applied to the roller, and the bottom surface of the inner race is constrained to have no motion. In the second part, one cycle of reciprocating load is applied incrementally to the roller. With each increment, the positions of the contact nodes were updated in the ABAQUS/CAE model, based on the new wear depth, calculated by the UMESHMOTION code. Accordingly, the values of contact pressure and relative displacement changed with each increment. The required data is extracted by means of a

PYTHON code. This code provides the new coordination of the surface nodes and calculates the volume of wear for each case. The code uses ABAQUS/CAE implicit solver. ABAQUS was installed in workstation (OptiPlex 9020, Dell Inc, i7-4790 CPU @ 3.60GHz, 3601 Mhz, 4 Core(s), 8 Logical Processor(s) Intel,16 GB RAM).

Figure 8: Updated FE model and its meshing structure

In the analytical model, the energy approach is used to model the false brinelling to more conveniently take into account an energy threshold for wear activation. In the FE model, Archard’s equation is used to simulate the false brinelling process. Although in some studies Archard’s equation validity has been questioned [10], it has been widely used by researchers investigating the different applications of FE models [33-36].

2.3 False Brinelling Test-rig The false brinelling test-rig that was presented in the authors’ previous work [25], is used to physically simulate the condition of false brinelling occurrence and is redescribed for convenience. The test-rig has three main parts as shown in Figure 9: housing and loading system to hold the bearing and apply radial load, shaker table to provide required vibrations and rotational displacement and monitoring system including accelerometers, load cell, encoder and data acquisition system to monitor and record the experiment’s outputs.

Figure 9: Schematic view of the false brinelling test-rig [25]

The false brinelling test-rig can simulate lateral vibrations by altering the housing mount to the shaker platform. Figure 10 shows the lateral vibration set-up.

a Radial force

b

Figure 10: (a) Schematic view of the lateral vibration set-up (b) Actual lateral vibration set-up

The experiment begins with the mounting of the shaft and the bearing using the housing system shown in Figure 10. The loading system consists of load screws and load bars that are used to apply the required static (radial) load to the bearing [25]. Table 3 presents the parameters of the lateral vibration test. Table 3. The parameters of the lateral vibration test

Test Number

Frequency

1 2

24 24

Static radial load 800 N 800 N

Amplitude of acceleration

Test duration

2g 1g

192 hours 192 hours

Then, the test results for the wear marks on the inner race surface are scanned using a Taylor Hobson Talysurf i5 surface profiler to measure the maximum depth of wear.

3. Results and Discussion In the first part of this section, the analytical model is verified by the FE model. The second part presents the validation results of the analytical model, FEM and test-rig. In the third part, the case study results of the analytical model are presented.

3.1 Comparison with FE Model The results for the analytical model were compared with the results for the FE model in terms of the false brinelling wear depth and the shape of the false brinelling wear profile. Several simulations are performed for a range of normal loads (110,150,1100 and 5000N) and the lateral vibration of 2g. In both simulations, the threshold energy of wear activation Edth is assumed as zero, meaning that the wear process starts from the beginning. The maximum wear depths of different cases, obtained from the analytical and FE models, are compared (Table 4). The duration of the simulation is one cycle. Table 4 shows that the FE and analytical results are well-aligned (the maximum difference is 10 %). Table 4:

The

Normal Load (N) 110 150 1100 5000

The maximum wear depth from the analytical model (µm) (wear shape) 4.78x10-8 4.17 x10-8 6.87 x10-9 1.57 x10-9

The maximum wear Difference depth from FEM (µm) (%) (wear shape) 4.35 x10-8 3.91 x10-8 5.48 x10-9 1.39 x10-9

10 6 10 10

comparison results

The comparison of false brinelling wear profiles of the analytical and FE models is shown in Figure 11. The simulations are performed for the normal load of 500 N and a lateral vibration of 2g for one cycle. It can be seen that the analytical model can well predict the depth of the wear. Despite the wear depth of the analytical model being around 10 % higher than that of the FE model, it predicts a smaller slip area. A possible explanation for this phenomenon could be that during the loading oscillation, the stick area varies which causes an intermediate area where there is not a complete stick or slip that the analytical model does not consider. The tractions in this area change during loading and unloading. This area is considered when calculating the traction distribution in the circular contact area according to [8,29]. For the elliptical contact, the intermediate area is not considered in the equation proposed in [29]. On the other hand, the FE model is capable of simulating the interaction between each node, so the traction forces in the stick area are not equal.

Figure 11: Wear Profile for the analytical model and the FE model

Figure 12: Wear volume for the analytical model and the FE model

Figure 12 shows the volume of wear obtained from the FE and the analytical models. The results from both models show the linear behaviour of the wear volume. The volume of wear predicted during 16.5 million cycles from the FEM is 5.14x104 µm3, while the volume of wear of 4.43x104 µm3 is predicted from the analytical. This could be a result of the interaction between the elemental component considered in the FE model, while the analytical model calculates the energy dissipation for the whole area (Equation 13). Table 5 compares the solution time of the FE and analytical models, for one cycle. It is clear that the analytical model is approximately 10000 times more time-efficient than the FE model. Table 5: Solution time of one cycle, for the FE and the analytical model

Solution time of one cycle

FE model 184 minutes

Analytical model 1 second

3.2 Experimental Validation Details of the experimental validation are provided in [25]; for convenience, a summary is provided here. The profile scanning shows that the two wear marks on the inner race outer surface of the test 1 shown in Table 6. These two marks were under the maximum lateral vibration (profiles of the marks are presented in the authors’ previous work). Table 6 shows the maximum wear depth, in the experimental tests, based on the position of the rollers. The analytical model and FEM then, are used to simulate the wear marks and are compared to the wear marks from the test-rig. This verifies how the variation of radial force and the lateral amplitude of lateral forcing affect the false brinelling wear damage. The simulation is conducted based on the parameters in Table 1 and 3. Figure 13 shows the comparisons of the maximum depth of wear obtained from experimental tests, the analytical and FE models. The analytical model predicted the wear depth within 30% error. Also, the analytical model predicts wear depth 10 -50 % higher than the FEM. This could be due to the assumption and simplification of determining the threshold energy wear activation, wear and friction coefficients. Also, the results show that the wear decreases when the normal load for experimental computational and simulation is increased. Table 6: Loads, amplitudes and wear depth of each wear mark, for the lateral forcing specimens [25]

Case

Static load on the roller (N)

Test 1- mark 1 Test 1- mark 2 Test 2- mark 1 Test 2- mark 2

150 110 150 110

Amplitude of acceleration (g) 2 2 1 1

The maximum wear depth (μ μm) 3 6 1.6 3

Figure 13: Comparison between the experimental and modelling maximum wear depth lateral forcing

In this study, the material properties (friction and wear coefficient) are obtained from the authors’ previous paper [24], which were measured at a constant frequency and contact

geometry. This approach has been used in many previous studies [2-4, 10-13]. However, more investigation to determine these coefficients is required as they may change as a result of changing parameters such as frequency, normal load and sliding amplitude, all of which are outside the scope of this work.

3.3 Case Study of Full Bearing The analytical model was used to predict false brinelling for a range of normal loads and lateral and axial vibrations. The simulation parameters are listed in Table 1. The results presented in this section include the wear depth and wear volume on the inner and outer raceways, for lateral, axial and combined forcing. The wear marks are related to different positions where rollers are in contact with the races, based on the applied normal load. Figures 14 and 15 show the depths and volumes of wear caused by lateral forcing on the surface of the inner and the outer raceways, respectively. It can be seen that in all cases, the wear marks have a W-shape profile. On the inner raceway, the deepest marks occur in the position of the third roller (0.242 µm). The reason is that the normal load on rollers changes according to the position of the roller based on the load distribution on the bearing [26] (Tables 7). Based on Equation 18, the magnitude of the wear depth depends directly on the traction force and, accordingly, normal pressure (Equation 19), and relative displacement (when the wear coefficient is constant). The normal pressure increases when the normal load is increased (28% difference between roller 1 and roller 3). However, by increasing the normal load, there is a reduction in the amplitude of the relative slip displacement (by about 40% difference between roller one and roller three) which in turn reduces the depth of wear (Figure 16c). The same trend can be seen for the wear volume and the wear marks of the outer race (Figure 14 and 15). However, the wear marks on the outer raceway are about 5% shallower than those on the inner raceway, while the wear volumes on the outer raceway are about 9% greater than those on the inner raceway (Figure 16b). The reason is that the contact areas between the outer race and rollers are about 20% larger than the contact areas in the inner race (Table 7), which results in a greater volume of wear (Figure 16). However, the larger area reduces the maximum normal pressure on the surface which in turn reduces the depth of wear (Table 7).

Figure 14: The wear volume under lateral forcing on a) inner raceway and b) outer race

Figure 15: The wear profile under lateral forcing on a) the inner raceway and b) the outer raceway

Table 7: Contact parameters and wear depth and volume under the lateral forcing on the inner and outer raceways for

No. of Roller

Normal Load (N)

1 2 3

464 388 174

Lateral Contact Radius (mm) Inner 0.0853 0.0804 0.0615

Outer 0.08 0.075 0.056

Axial Contact Radius (mm) Inner 2.35 2.21 1.69

Outer 3.07 2.89 2.22

Contact Area (mm2) Inner 0.63 0.56 0.33

Outer 0.77 0.68 0.39

affected rollers

Wear Volume (μm3)

b 200000

0.25 0.2 0.15 0.1 0.05

150000 100000 50000

0

0 Inner raceway

Outer raceway

c

Inner raceway

6.00E-04 5.00E-04

(mm)

Tangential Displacement

Wear Depth (μm)

a 0.3

4.00E-04 3.00E-04 2.00E-04 1.00E-04 0.00E+00 Inner raceway Top Roller

Second Roller

Outer raceway Third Roller

Outer raceway

Figure 16: (a) wear depth, (b) wear volume, (c) tangential displacement under the lateral forcing on the inner and outer raceways for affected rollers

Under the axial forcing, the depth and volume of wear on the inner and outer raceways have the same pattern as those under the lateral forcing (Figure 17 and 18). However, the wear marks caused by axial vibration (under the same normal load and vibration) are deeper than those caused by lateral forcing (Figure 16a and 19a). This is because the tangential displacements of the axial forcing are around 30% greater than the ones in the lateral direction according to Equations 10, 14 and 15 (Figure 16c and 19c). The wear marks on the inner raceway are around 5% deeper than those on the outer raceway (Figure 17). This is due to the normal pressure in smaller contact areas on the inner raceway being greater than that in the larger contact areas on the outer raceway (Table 7), under an equal value of normal load. The wear volumes on the outer raceway are around 9 % lower than those on the inner raceway as shown in Figure 19b.

Figure 17: The wear profile under axial forcing on a) inner raceway and b) outer raceway

Figure 18: The wear volume under axial forcing on a) inner raceway and b) outer raceway

0.35 0.3 0.25 0.2 0.15 0.1 0.05

b300000 Wear Volume (μm3)

Wear Depth (μm)

a

250000 200000 150000 100000 50000

Figure 19: (a) wear depth, (b) wear volume, (c) tangential displacement under the axial forcing on the inner and outer raceways for affected rollers

Figures 20 and 21 show the maximum wear depth and wear volume under the effect of the combined forcing (Equation 16) on the inner and outer raceways, respectively. In general, the difference in volume and depth of the wear is due to changes to the normal load and the tangential displacement. The normal load decreases by around one third for the top roller and third roller respectively according to the load distribution as seen in Table 7 (Equation 8). Under the same amplitude, this causes the third roller to move more than the top roller. The tangential displacement increases by around 40 % for the top roller and third roller (Figure 16c and 19c) according to Equation 11. This variation of the normal load also produces contact areas on the top roller that are larger than those on the third roller. This causes the third roller to have greater energy dissipation than the top roller according to Equations 13 and 19. The larger contact areas on the outer raceway cause more wear than the ones on the inner raceway under the same load. On the other hand, small contact areas on the inner raceway cause wear marks that are slightly deeper than those on the outer raceway. This is because the stick areas on the inner raceway are around 23% smaller than the ones on the outer raceway under the same normal load. This causes the maximum local traction load to be up to 7% higher in the slip areas within the smaller contact areas than the one within larger contact areas (Equation 20). To sum up, the normal load has the greatest impact on the wear volume and wear depth compared to the amplitude of lateral and axial vibrations. Hence, the

normal load plays a role in determining the contact area and relative displacement. Figure 20: The wear profile under combined forcing on a) inner raceway and b) outer raceway Figure 21: The wear volume under combined forcing on a) inner raceway and b) outer raceway

3.4 Sensitivity Analysis The analytical model was used to investigate the influence of a normal load and vibration on the wear damage caused by false brinelling. To this end, the bearing was tested under a range of acceleration amplitude (0.1-3g) and normal load (up to 5000N). The duration of the simulation was 192 hours with 24 Hz. Figure 22a shows that the wear depth increases nonlinearly when the acceleration amplitude is increased. In addition, Figure 22b shows that the depth of wear decreases when the normal load is increased, and decreases more sharply when the normal load is below 1000 N. Under the vibration amplitude of 3g, the wear depth

decreases from 5.6×10-8µm to 3×10-8µm, when the normal load (maximum contact pressure) is increased from 100 N (358 MPa) to 700 N(600 MPa). When the normal load is more than 1000 N (800 MPa), the change of the wear depth is insignificant, particularly under the smaller vibrations. This may be due to the normal pressure and the magnitude of vibration changing the relative displacement and the traction distribution. This plays a role in changing the wear depth as seen in Equations 19 and 20. Figure 23 shows the maximum normal pressure, maximum traction, maximum energy density and sliding displacement under lateral vibration of 2g and normal load (up to 5000N). It can be seen that the maximum normal pressure increases with an increase in the normal load. According to Equation 11, this decreases the relative displacement, although the traction increases with increasing the normal under constant tangential force. This shows that the sliding displacement decrease has higher effect than the traction increase in reducing the maximum energy density, thereby causing the reduction of the wear depth when the normal load is increased. An increase in vibration amplitude under constant normal load also increases the tangential displacement and traction loads of the roller which leads to greater energy density and the wear depth according to Equation 19. Also, the experimental results of maximum wear depth under a redial load of 800 N and vibration amplitude of 2 and 1g as shown in Figure 22 indicate that the wear depth decreases when the normal load is increased. a.

b.

Figure 22: a. Wear depth versus normal load b. Wear depth versus acceleration amplitude

Figure 23: Maximum normal pressure, maximum traction maximum energy density and sliding displacement versus normal load for one cycle of fretting wear, under the lateral vibration of 2 g.

4. Conclusions An analytical model was developed to simulate the false brinelling process in bearings. This model includes three main parts to determine the load distribution, model the contact mechanics and simulate the wear process. This model was verified by a 3D FE false brinelling model, in terms of the maximum wear depth and the wear profile shape. The analytical model was approximately 10000 times more time-efficient compared with the false brinelling 3D FE model, achieving an error within 10 %. Also, the analytical model was validated using a false brinelling test-rig. The analytical model predicts the wear depth within 30% error of the experimental results. This could be due to the simplification of determining the threshold energy wear activation, wear and friction coefficient. The results obtained from the experiments and the models show that the depth of wear decreases when the normal load is increased under constant tangential forcing. It was shown that an increase of the normal load increases the maximum normal pressure. This results in a decrease of the relative sliding displacement, although the traction increases as a result of increasing the normal load. The effect of relative sliding displacement decrease dominates

over the effect of traction increase causing the maximum energy density to drop, thereby reducing the depth of false brinelling wear. The results showed that the wear marks on the outer raceway were shallower than those on the inner raceway by approximately 5%, while the volume of wear on the outer raceway were about 9% greater than the wear volume on the inner raceway. This is because the contact areas between the outer race and rollers were about 20% larger than the contact areas in the inner race, which can result in a greater amount of wear. Moreover, the larger area reduces the maximum normal pressure on the surface which in turn reduces the depth of wear. Under the axial forcing, the depth and volume of the wear on the inner and outer raceways was the same as those under the lateral forcing. However, the wear depths caused by axial vibration (under the same normal load and vibration) were greater than those caused by lateral forcing. The reason is that the relative sliding displacements of the axial oscillations were about 30% greater than those in the lateral direction. This causes greater energy dissipation in the axial direction than the lateral direction. The analytical model was used to perform a sensitivity analysis of the influence of a normal load and vibration on the false brinelling wear damage. The results showed that the wear depth increases non-linearly when the acceleration amplitude is increased. This is because an increase of the vibration amplitude with constant normal pressure increases the tangential displacement. In addition, it was shown that the wear depth decreases when the normal load is increased. This is because increasing the normal load decreases the tangential displacement which causes a reduction of the frictional energy density. The wear depth decreased more sharply when the normal load (maximum contact pressure) was below 1000 N (800 MPa).

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The effect of radial loads on the false brinelling damage of bearings was investigated. An analytical model was developed for predicting false brinelling under radial loads. A comparison of the model’s results with the validated 3D FE model was conducted. A validation of the model’s results with experimental results was conducted. A study case was simulated using the model.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: