A Statistical Prediction of Electrical Discharge Initiation and Semi-Analytical Transient Mixed Lubrication Model of a Rolling Element

Despite boasting entirely dissimilar powertrain architectures from their internal combustion predecessors, electric vehicles still rely heavily on existing mechanical technologies, such as rolling element bearings and gears. As the usage of electric vehicles continues to grow, it becomes increasingly important to improve component reliability and optimize performance. Electric vehicle (EV) power train systems convert chemical potential energy stored in high-voltage DC batteries into an appropriate alternating current (AC) form to drive an electric motor shaft. Three-phase inverters, like pulse-width modulation inverters, facilitate variable speed performance of electric motors. Their stepwise output aims to replicate perfect sinusoidal conditions with high-frequency switching rates, but subsequent common mode voltages drive unwanted currents through unintended, conductive mechanical pathways [1]. Additionally, increased switching rates within modern PWM inverters magnify the presence and the effect of shaft voltages created by uneven magnetic pole distributions from asymmetric stator windings or manufacturing tolerances [2].

The complex electrical environments characteristic of EVs and hybrid-electric vehicles (HEV) can contribute to premature electric motor failures, 40% of which are bearing related [3]. Robust lubricants protect surfaces of bearing components from excessive wear. In many lubrication systems, a relatively nonconductive mineral or synthetic hydrocarbon oil or grease creates a dielectric barrier between conductive metallic surfaces. As leakage currents naturally seek to find a grounding location across bearing components, the lubricating film can store charge, like an electrical capacitor.

When the oil film thickness exceeds a certain threshold, it enters an insulating state, leading to an infinite interface impedance. However, when the film thickness is smaller than the critical value, the interface current experiences a sudden surge because of the high enough electrical field across the oil film, causing it to breakdown and initiate discharging. It should be noted that it is difficult to confirm for certain that discharge did occur as an electrical arc. Once the charged voltage exceeds the dielectric strength of the lubricant, the potential can rapidly discharge and arc across the surfaces, destabilizing and decomposing the oil in its path. The discharge phenomenon leads to the conversion of kinetic energy from the ions and electrons into heat energy and eventually it elevates the contact temperatures exceeding the material's melting point [4]. These thermal effects can result in irreversible damage to the surfaces in the form of frosting, fluting, or pitting [2,5–10]. As a result of fluting or pitting, the fused ferrous metals get removed in the form of black spheres creating craters at the surface. Furthermore, the liberated electrically induced ferrous particles, exposed to intense heat, have contributed to the degradation and oxidation of the grease. Consequently, this results in a loss of the grease's purity. These forms of mechanical damage alter tribological contacts and increase surface roughness, heavily contributing to vibrations and bearing noise [11].

EV powertrains can incorporate electrically insulating bearing surfaces or electrically conductive lubricants to mitigate the adverse effects of electro-discharge across bearing surfaces [11]. Regardless of component configuration, specific lubricants, speeds, loads, and other performance aspects of system design can be tailored to mitigate or avoid electrical damage. If manufacturers can predict the likelihood and circumstances for electrical damage based on select parameters, future designs can avoid problematic lubricants and tribological conditions susceptible to damage. Due to this, researchers are assembling models to predict the occurrence of damage [10,12,13], such as in the work by Schneider et al. which reviews models on the full rolling element bearing and transient electrical models based on the elasto-hydrodynamic lubrication (EHL) film thickness [14]. Some previous models employ an electrical circuit analysis [15] but are only loosely tied to tribological principles. A more recent work sought to improve the modeling of capacitance in bearings to improve prediction capabilities [16]. However, more work is still required for electrified rolling contacts to include roughness, start–stop motion, and grease lubrication.

Due to the rapid expansion of EV production, work on electrified mechanical contacts is still establishing itself. Despite this, the area of electrical connectors has been well researched. Likewise, there is a growing usage of lubricants in electrical connectors aimed at minimizing corrosion and fretting [17,18]. However, issues arise when lubricated conditions or the lubricant alters conductivity for powered and electrical connectors. Contact resistance can increase due to vibrations within the lubricant film, and typical extreme pressure (EP) additives can chemically form a nonconductive solid film on normally conductive surfaces [19]. Rather than risking the formation of a conductivity-altering lubricant film, conductive nanoparticles can be incorporated as a lubricant additive [19,20] to reduce friction and wear while maintaining optimal contact resistance. Similarly, ionic lubricants possess inherently higher electrical conductivity and can be incorporated to achieve comparable effects [21].

The model presented in this work predicts the damage caused by discharging across the surfaces and compares results to experimental results from reciprocating rolling contact measurements. During reciprocation, the film thickness varies as the rotational speed of the rolling element changes. From a rheological perspective of the lubricant, it is essential to acknowledge the significance of the squeeze film phenomenon, particularly caused by abrupt changes in rolling motion acceleration. As the squeezing process progresses, the film thickness gradually decreases toward a steady-state film thickness [22]. Changes in film thickness are not considered within steady-state EHL and require an additional squeeze film effect to account for changing fluid pressure resulting from lubricant squeezed out from contacting surfaces. Similarly, this approach describes a transient EHL problem.

In addition, grease is employed as the lubricant in this work. Grease is a lubricant that behaves as a soft solid until enough shear or pressure release the liquid lubricant held within it by degrading the thickener [23] or pushing it out of the areas of minimum film thickness. Grease is created by adding a thickener, such as lithium, calcium, aluminum complexes, or polyurea, to oil. In this work, the thickener is polyurea, which is often used in EV and electric motor bearings (for additional properties, see Ref. [24]). In EHL, the thickener is often assumed to not influence lubricant rheology in the contact. However, that is only sometimes the case and the base oil viscosity may be due to the lack of widely accepted alternative method [24]. Grease can be pushed out of contact and not allow for replenishment, termed starvation [23,25]. Thickener can enter EHL film and effectively increase viscosity (semisolid layer). This increase in film thickness has been observed at low [26,27] and medium speeds [28].

Electrically induced bearing damage is a growing research area, and numerical modeling is still advancing. The discharging phenomena demand that any approach bridges the disciplines of electrical conduction, heat transfer, chemistry, and multiphase mechanics. The transient nature is also essential to adequately capture the timing of the discharge phenomena. The research, therefore, proposes a simplified transient model of a lubricated electrified rolling bearing contact, which also considers grease lubrication and surface roughness.

A robust and efficient model is needed to predict the probability of surface damage resulting from electrical currents discharging across the lubricant film of rolling element bearings. In a rolling element contact, the fluid film develops when motion starts. The pressure generated during film formation is often enough to cause significant deflection of contacting surfaces. This lubrication scenario, often referred to as EHL, has been theoretically and experimentally characterized by many works [29]. When an EHL film develops, the electrical conduction between the metal raceway and the rolling element decreases. Combining EHL and rough surface contact models can produce an approximate transient model of the bearing contact. When considering primary forces at play, it is evident that the applied external force, the fluid lifting force (load support), and the solid rough surface contact force will usually not balance. Therefore, the rolling element bearing will accelerate toward or away from the raceway. This acceleration is used by Newton's method to predict the velocity and position of the bearing. The velocity normal to the surface will also cause a squeeze film mechanism between the surfaces. The details of the theories implemented to include these mechanisms are described in more detail in the following sections. The assembled model can then make predictions of film thickness and electrical performance for different materials, finishes, and operating conditions.

The Newton forward matching time scheme then employs this acceleration to predict the velocity, $z˙$⁠, and the position, z. Various values were evaluated to find the appropriate time-step for convergence.

These reduced roughness parameters are then employed in the flow factors (Eq. (7)) and other places roughness parameters are considered in the model. However, this method would surely deviate from reality or a deterministic, although time consuming, calculation.

Equation (13) is fit to the viscosity measurements, which finds values of $γ˙c=0.0017s−1$⁠, $μo=1.78×105Pa⋅s$⁠, and n = 0.44. The fit is shown with the rheological data presented in Fig. 2 and differs from the data by an average of 7.7%.

In the previous equation, the alleviation factor [57] is given by $Ψ=(1−Ar/An)3/2$⁠, and the real area of contact predicted by the statistical model is A_r. The alleviation factor accounts for the adjacent asperities sharing the electrical conduction once their numbers grow, and they become less isolated under higher contact forces.

Note that the total contact resistance is the reciprocal of the sum of the reciprocals of R_s and R_film since they are in electrical parallel.

Note that other height distributions could also be implemented into Eq. (23), but a closed-form equation might not be obtainable. This qualitative prediction given by Eqs. (23)–(25) merely suggests how many locations and instances the film will electrically break down, causing surface damage. In reality, the mechanisms at work are more complicated, but this simplified approach is adopted to facilitate a simple prediction. The resulting damage prediction should be proportional to the observed surface damage.

Predictions of surface damage generated by the model were compared to the observed surface damage in a reciprocating rolling ball on disk measurement to evaluate the effectiveness of the proposed model (see Ref. [9] for all details of the experiment). A schematic and photo of this setup are shown in Fig. 5. A rolling ball contact is electrified by a DC power supply at 0.5 Amps as shown. A multimeter is used to measure the voltage drop across the contact, similar to 4-wire resistance measurement. The circuits are isolated from the rest of the test rig using polymer washers and 3D-printed parts.

The experimental testing was conducted in triplicate to confirm experimental repeatability. These experiments evaluated mineral oil-based polyurea grease (NLGI Grade 2, ISO 100) without any additives under electrified conditions. The ball is 9.53 mm (3/8 in.) diameter and loaded to 50 N. An electric current is applied across the contact during operation and will be discussed in more detail later. To mimic the reciprocating motion carried out during testing, a representative function for velocity is inputted into the model. These tests produce a small wear track that might be considered “frosting” of the surface (see Fig. 6). The surface wear from this short 1.5 h test is difficult to measure via mass loss or profilometry. Therefore, this study has used scanning electron microscopy to observe the surface pitting, as shown in Fig. 3. It should be noted that the severe pitting surface damage is observed to be concentrated near the ends of the wear groove. In the middle of the wear track, the surfaces do not have significant damage, are slightly polished, and contain debris redeposited from the pitting [9]. For the first case (case 1) considered, a sawtooth velocity function is employed, matching the velocity in the experiment (see Fig. 7).

The Southwest Research Institute tested the grease sample's dielectric breakdown strength using Japanese Industrial Standard (JIS) C2101 to be approximately 6.4 × 10⁶ V/m, which is in the expected range of recent literature [58,60,61]. The previously mentioned method for predicting the probable number of asperities susceptible to breakdown and discharging is employed (see Eq. (25)). In these locations, the grease film is expected to degrade, enabling plasma formation to discharge across. The comparison experiment involved lubricating 52,100 bearing steel samples under electrified conditions with ISO 100 mineral oil-based polyurea NLGI 2 grease. Table 1 contains other relevant experimental conditions for modeling purposes.

Following the outlined methodology, film thickness as a function of time is predicted, as shown in Fig. 8. The film thickness is directly related to the rolling speed's increase and decrease. The squeeze film effect and rough surface contact decrease the magnitude of the film thickness fluctuation to a minimal amount for the current case. However, this relatively small change significantly affects the predicted contact resistance (Fig. 9). The contact resistance fluctuates by many orders of magnitude as a film of grease is built up during sliding and then squeezed out and compressed when the velocity slows and changes direction. At larger film thicknesses, the contact resistance is very high because there is a large film of grease between the surfaces. The plateau observed at these high film thicknesses is due to the film resistance dominating (Eq. (19)), and it changes linearly with film thickness.

It should be noted that at the times when the electrical resistance is high in the contact, it is difficult for the electrical current (electrons) to flow across the lubricant film. This allows charge to build and discharge. This build and discharge are also governed in time by the capacitance properties of the film and are equally important for predicting the performance of the bearing [14–16]. Using the same general framework presented in this work, the calculation of capacitance is straightforward.

Since the average film thickness changes with time, so does the average dielectric voltage of the film. The predicted dielectric voltage is shown in Fig. 10, but clearly it never falls below the average applied voltage of 1.15 V. This voltage is obtained by measuring the voltage drop across the rolling element in the test. A sample voltage reading during testing is shown in Fig. 11. The voltage does not follow the predicted variations in resistance over this range. This is due to the limiting breakdown voltage of the lubricant film. Therefore, it is more helpful to show it over a smaller time scale. Changes in the voltage can be observed over a smaller time scale that could be the discharges. Note that 31 V is applied to the entire circuit in the test setup, but other resistances account for most of that voltage. The heating element that is in series has a resistance of $58Ω$ and at 0.5 Amps contributes to most of the voltage drop. Again, it is shown that the voltage across the contact is much smaller than the dielectric voltage predicted by the average film thickness, but this calculation does not yet account for the surface roughness.

Since the dielectric voltage predicted by the average film thickness is not low enough to allow for discharging, the model also numerically predicts the number of asperities larger than the critical height (Eq. (25)). These asperities may facilitate dielectric breakdown below the applied voltage (1.15 V). These predictions are shown in Fig. 12 for different locations along the track of the rolling ball. The model predicts electrical erosion along the outer ends of the wear track, which is observed in the form of pits during experimental tests. As demonstrated by the images, little to no pitting damage appeared in the track's mid-section, where the velocity was highest. However, raised surfaces in the track's mid-section indicate a deposition of material removed during the pit formation process. All three repetitions of the test confirmed this trend.

Next, the numerical model was used to predict the damage for two other cases. Case 2 is for a full rotation before the rolling direction is switched and the velocity changes direction. These results are obtained by using the velocity profile given in Fig. 13. The velocity reaches a nearly constant value of 0.01 m/s in between the direction changes. The effective track length of one rotation is 0.14 m. The predicted film thickness is shown in Fig. 14, and the number of asperities above the critical height is shown in Fig. 15. The number of asperities is nearly an order of magnitude smaller than predicted for case 1. Little to no damage is observed for case 2. Therefore, the model is in qualitative agreement with experimental observations. This is also repeated for a third case (case 3), where the disk is rotated ten times at a speed of 0.01 m/s before the direction is changed. This also predicts little to no damage and agrees with experimental observations.

The theory outlined in the current work suggests a “sweet spot” or finite range of film thickness that is more conducive to the electrical discharging and perhaps arcing phenomena. When the film thickness is relatively large, the electric voltage cannot overcome the dielectric voltage and insulative strength of the lubricating film. In contrast, when the film thickness is minimal, the solid asperity contacts provide a direct pathway for the electric current. In this case, solid contacts facilitate the flow of charge across the bearing surfaces without discharging or arcing across the lubricant. Although beneficial for electrical performance under electrified conditions, the case of a very thin lubricating film introduces the possibility of increased mechanical wear, such as adhesion and abrasion, between the surfaces.

The proposed model still makes many assumptions that may oversimplify the real situation. For instance, it does not consider how the electrical field influences on fluid and material properties and behavior [62,63]. Many of the properties in the model, such as dielectric strength and conductivity, are also dependent on changes in pressure and temperature, which is neglected by the model. This work does not consider how the relatively high pressures in the EHL contact will change the properties of the solids and fluids (except for viscosity, which is included in the employed EHL theory). The current work does not consider how wear of the surfaces will change the roughness and change the contact resistance. There are also oxides and tribo-films that will influence the contact resistance. In addition, it does not consider how electrical damage to the grease would alter its behavior during the test, or how surface damage would also influence the film thickness. In future iterations of the model, the influence of various lubricant base oils and additives could also be considered. Additives often increase conductivity and can influence degradation mechanism, such as nanoparticles [9].

As electrical powertrain architectures continue to replace traditional combustion precursors, the effect of electrified conditions on bearing technologies remains paramount. Until now, limited modeling work has quantified the damaging effect of electrical conduction across lubricated bearing surfaces. The model presented in this work predicts a simple mixed lubrication numerical model of the electrically induced damage of a rolling element contact. The established EHL equation for spherical rolling contact is modified with an effective viscosity and flow factor term and includes a statistical model of elastic–plastic rough surface contact. A new statistical method is used to consider how roughness can locally cause the surface to be close enough for the film to dielectrically break down and allow for discharging and perhaps arcing. The model predicted surface pitting of electrically induced bearing damage at locations and during conditions similar to that observed in testing.

The authors thank for the support of the National Lubricating Grease Institute (NLGI) Research Grant. The authors also thank Carlos Sanchez and Peter Lee of the Southwest Research Institute for measuring the dielectric properties of the grease, Chad Chichester of Dupont for the guidance throughout the project, Paul Slade for his insights into the microscale arcing process, and Anton-Paar and Bruker for their support via equipment.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

a =

elastic contact radius of rolling element predicted by Hertz model

d =

distance between the mean of the surface asperities or peaks

h =

film thickness, separation of mean surface height

m =

mass of rolling element assembly

t =

time

u =

entrainment velocity

z =

location of rolling element

E =

elastic modulus

G =

EHL normalized elasticity term

L =

Moes dimensionless material parameter

M =

Moes dimensionless load parameter

N =

Carreau model constant

R =

radius of rolling element

U =

EHL normalized velocity term

W =

EHL normalized force term

$P¯$ =

single asperity contact force

d_crit =

asperity height susceptible to breakdown or discharging

h₁ =

film thickness at time, t₁

h₂ =

film thickness at time, t₂

h_crit =

film thickness which is susceptible to discharging

h_o =

initial film thickness

r_asp =

single asperity contact radius

A_d =

deformed asperity amplitude

A_hertz =

Hertz area of contact

A_i =

initial or undeformed asperity amplitude

A_n =

nominal or apparent contact area

F_c =

rough surface contact force on rolling element

F_ehl =

lubricant hydrodynamic force predicted by EHL

F_E =

externally applied force on the rolling element

F_L =

lubricant hydrodynamic force from sliding and squeeze film effects

F_SF =

fluid squeeze film force

N_ca =

number of asperities above critical height for discharging

R_c =

electrical spreading contact resistance

R_film =

electrical lubricant film resistance

R_peak =

asperity radius of curvature

S_d =

dielectric breakdown strength

V_applied =

voltage applied across the contact in model

V_d =

dielectric breakdown voltage

V_m =

voltage measured during test across the contact

E′ =

equivalent modulus of elasticity: $1E′=(1−ν12)E1+(1−ν22)E2$

$γ˙$ =

shear rate

$γ˙c$ =

critical shear rate for Carreau equation

η =

areal asperity density

λ =

wavelength of the asperities

μ =

dynamic fluid viscosity

μ_eff =

effective dynamic fluid viscosity

μ_o =

base dynamic fluid viscosity for Carreau equation

ν =

Poisson's ratio

ρ_oil =

electrical resistivity of oil

ρ_steel =

electrical resistivity of steel

σ =

combined RMS surface roughness

σ_d =

deformed RMS surface roughness

σ_i =

initial or undeformed RMS surface roughness

σ_s =

combined RMS asperity height

ϕ_x =

flow factor for modified Reynolds equation in x direction

ϕ_y =

flow factor for modified Reynolds equation in y-direction

Φ =

asperity height distribution

Ψ =

alleviation factor for rough surface conduction

$∇$ =

dimensionless asperity flattening parameter