A Detailed Multibody Simulation Model for Ball Bearings to Predict Friction and Electrical Capacitance

Abstract

A multibody simulation model for deep-groove ball bearings is presented. The model considers friction in both the raceway and cage contacts, resulting from radial and axial loads. The model is validated against experimental measurements for a 6319 bearing under oil-bath lubrication over a speed range of 500–3000 min⁻¹ and two load ratios ($C/P=10$ and 6.5). Predicted friction torques show good agreement with measurements, with deviations between 5.5% and 22% at moderate speeds. In addition, electrical contact capacitances are calculated for a 6208 bearing and compared with an analytical approach, showing deviations in the range of 10–14%. Beyond friction prediction, the fully dynamic approach enables time-resolved analysis of roller kinematics and the identification of instability limits under axial excitation. The developed tool therefore enables reliable bearing loss prediction, supports efficiency-oriented drivetrain design, and provides a basis for electro-tribological and stability investigations.

1. Introduction

The continuous push toward higher drivetrain efficiency has renewed attention on rolling bearing losses. In many transmissions, the bearing torque is no longer negligible compared to gear mesh losses, especially under light loads, high speeds, and transient operation [1,2,3,4]. Deep-groove ball bearings are widely used in such operating regimes because of their compactness and robustness [5,6], yet their loss mechanisms are complex: friction arises not only from classical rolling resistance, but also from spin in the contact, sliding driven by kinematic constraints, cage–ball interactions, lubricant shear, and churning [7,8,9,10,11,12,13]. These contributions are strongly influenced by clearance or preload, lubrication state, and rapid changes in load and speed, conditions that are increasingly common in modern electrified drivetrains [13,14,15,16].

Unlike roller bearings, where the load is distributed along a line contact, deep-groove ball bearings primarily transmit loads through point contacts. This geometric difference changes the underlying contact mechanics: local contact pressures are higher, the sensitivity to elastic deflection and groove conformity increases, and the proportion of spin/slide components can become significant even at moderate loads [17,18,19]. Under low load or poor lubrication, kinematic slip and intermittent film formation may occur, raising the risk of surface distress and wear-related damage mechanisms [7,12,20,21,22,23,24]. Consequently, predictive models that are accurate for line contacts cannot be transferred to point contacts without revisiting both the kinematics and the contact/lubrication formulation.

Existing approaches to bearing friction modeling can be broadly categorized into empirical, semi-analytical, and dynamic-simulation-based methods [17,25,26,27,28,29]. While these methods remain valuable for design and catalog-level estimates, their accuracy is limited when bearings operate outside nominal steady conditions, such as during start–stop cycles or load reversals. The most detailed predictions are typically achieved with dynamic simulation tools that solve the multibody motion of the bearing components while resolving local contact forces. This combination enhanced with a friction modulation allows a clear and dynamic observation of different slip effects in the contact area [30].

Well-known dynamic tools are shown in

Table 1. Proprietary dynamic software of rolling bearing manufacturers.

Software	Company	Publications
BEAST (BEAring Simulation Tool, Version 2005)	SKF company (Goteborg, Sweden)	[37,38,39,40]
BRAIN (BeaRing Analysis In NSK, Version 1997)	NSK company (Maidenhead, UK)	[41,42]
Caba3D (Computer Aided Bearing Analyzer 3D, Version 2025)	Schaeffler company (Herzogenaurach, Germany)	[43,44]
CAGEDYN (Version 2010)	Timken company (North Canton, OH, USA)	[45,46,47]
IBDAS (Integrated Bearing Dynamic Analysis System, Version 2011)	NTN (Osaka, Japan)	[48]

. However, all these high-fidelity bearing dynamics tools are proprietary and part of the company’s know-how. Published dynamic models only exist for line contacts [14,26,30,31] or simplify friction or lubrication effects, which can reduce reliability when assessing efficiency or damage-relevant operating scenarios [32,33,34,35,36]. This creates a gap for research-accessible simulation approaches that retain physical detail while remaining verifiable.

At the same time, electrification introduces an additional, increasingly important dimension: rolling bearings are not purely mechanical components but also act as electrical elements of the drivetrain [49,50,51]. Voltage excitation from power electronics can drive bearing currents; the rolling contacts may behave as time-varying capacitors separated by the lubricant film. Depending on film thickness, surface roughness, and the dielectric properties of the contact, discharge and arcing can occur, which can lead to fluting, grey frosting, white etching crack, and lubricant degradation [52,53,54,55,56,57,58,59]. Even when damaging currents are not the focus, capacitance measurements have become attractive for condition monitoring and for inferring lubricant film formation in situ [60,61,62,63,64,65,66,66]. The modeling of the electrical properties of bearings has been the subject of various works, mostly focusing on contact capacitance [67], bearing capacitance [68,69,70], or discharge prediction [71]. The model presented in this work combines the prediction of friction and kinematics with the calculation of contact capacitance. It offers the opportunity to capture electro-mechanical interactions that are relevant in modern machines.

The objectives of this paper are twofold: it develops a dynamic simulation tool for deepgroove ball bearings that (i) predicts frictional losses based on a multibody description of the bearing components combined with a point-contact formulation, and (ii) extends the model with an electrical capacitance representation of the rolling contacts. This approach resolves the time-dependent kinematics of balls and cage, computes local contact conditions (normal load, slip components, and spin), and couples these to a lubrication-informed friction description appropriate for concentrated contacts. From the same predicted film, contact area, and contact kinematics, an equivalent capacitance model is attached to estimate the electrical behavior of each ball–raceway contact and the resulting bearing-level capacitance. The resulting tool is intended to support efficiency optimization (e.g., clearance/preload selection, lubricant choice, operating strategy) and to enable model-based interpretation of capacitance measurements.

2. Materials and Methods

2.1. Multibody Simulation Model

The multibody simulation model, named LaMBDA (Lager MehrkörperBerechnung und DynamikAnalyse), is based on a modular and parameterized modeling approach established at the Chair of Machine Elements, Gears, and Tribology (MEGT). The model combines the MBS software SIMPACK with self-developed calculation routines that have been continuously extended, validated over the past two decades, and published for line contacts [14,30,72,73].

Depending on the required modeling depth, the bearing components can be represented with different geometric resolutions. In the present work, the most detailed model configuration is applied, in which all rolling bearing components are described using their real geometries. Model generation (including the definition of bodies, markers, force elements, and boundary conditions) is performed using a graphical user interface. This interface provides access to a database of bearing types and lubricants and enables efficient and user-friendly model setup.

Figure 1 illustrates the computational workflow used to evaluate contact forces within the MBS model. At each time step $\Delta t$, the solver provides the current state parameters, namely the relative position vector $\overrightarrow{s}\left(t\right)$, its velocity $\dot{\overrightarrow{s}}\left(t\right)$, and acceleration $\ddot{\overrightarrow{s}}\left(t\right)$, as well as its angular equivalents $\overrightarrow{\omega }\left(t\right)$, $\dot{\overrightarrow{\omega }}\left(t\right)$, and $\ddot{\overrightarrow{\omega }}\left(t\right)$. Together with geometric data, material properties, lubrication parameters, friction coefficients, and damping constants, these values are passed to the calculation core.

Based on the current state, contact identification is performed. This yields the contact point $\overrightarrow{p}$, the contact normal vector $\overrightarrow{n}$, and the local penetration depth $\delta$. Using a load–deformation relationship, the penetration $\delta$ is converted into the normal contact force ${\overrightarrow{F}}_{\mathrm{N}}$ and the corresponding normal moment ${\overrightarrow{M}}_{\mathrm{N}}$ [30].

In a subsequent step, the time-dependent contact state parameters are determined. These include the relative velocity at the contact points ${\overrightarrow{u}}_{\mathrm{rel}}$ and the sum velocity ${\overrightarrow{u}}_{\mathrm{sum}}$, the lubricant film thickness h, the specific lubricant film parameter $\Lambda$, the solid load bearing ratio $\phi$, and the electric capacitance C.

The normal force ${\overrightarrow{F}}_{\mathrm{N}}$, the kinematic quantities, and the lubricant properties are then used to evaluate the friction force ${\overrightarrow{F}}_{\mathrm{T}}$ and the resulting friction moment ${\overrightarrow{M}}_{\mathrm{T}}$. The detailed formulation of the friction model is described in Section 2.2. The relative velocity ${\overrightarrow{u}}_{\mathrm{rel}}$ is used to calculate the damping force ${\overrightarrow{F}}_{\mathrm{D}}$ and the associated moment ${\overrightarrow{M}}_{\mathrm{D}}$ (see Section 2.3).

Finally, the individual force contributions are superposed to obtain the total contact force

$${\overrightarrow{F}}_{\Sigma }={\overrightarrow{F}}_{\mathrm{N}}+{\overrightarrow{F}}_{\mathrm{D}}+{\overrightarrow{F}}_{\mathrm{T}}$$

and the corresponding contact moment ${\overrightarrow{M}}_{\Sigma }$. These quantities are returned to the Simpack MBS Solver, where they are used in the iterative solution of the force equilibrium for the next time step.

2.2. Contact Calculation

For the calculation of friction and the bearing’s electrical load capacitance, the contact force and the dimensions of the contact ellipse between the rolling element and the raceway are required. The contact is modeled as a point contact according to Hertz. The contact deformation and the semi-axes of the contact ellipse are calculated using the analytical approximations proposed by Hamrock and Brewe [74]. These formulations approximate the ellipticity parameter and the elliptical integrals as functions of the radii ratios of the contacting bodies [72]. The load–deformation relationship is described by Equation (1).

$$\delta =\mathcal{F}·\sqrt[3]{{\displaystyle \frac{4.5}{ϵ·R^{\prime }}}·{({\displaystyle \frac{F_N}{\pi ·k·E^{\prime }}})}^2}$$

(1)

The contact ellipse is characterized by the semi-axes a and b, given by Equations (2) and (3).

$$a=\sqrt[3]{{\displaystyle \frac{6·F_N·k^2·ϵ·R^{\prime }}{\pi ·E^{\prime }}}}$$

(2)

$$b=\sqrt[3]{{\displaystyle \frac{6·F_N·ϵ·R^{\prime }}{\pi ·E^{\prime }·k}}}$$

(3)

The ellipse parameter k appearing in Equations (1)–(3) is estimated accordingly as a function of the radius ratio $a_H$ using the following relationship:

$$k=a_H^{2/\pi }$$

(4)

with

$$a_H=\left\{\begin{array}{cc}{\displaystyle \frac{R_x}{R_y}},\hfill & R_x>R_y\hfill \\ {\displaystyle \frac{R_y}{R_x}},\hfill & R_x<R_y\hfill \end{array}\right.$$

(5)

The calculation of the reduced radii $R^{\prime }$ is described by Equations (6)–(8).

$${\displaystyle \frac{1}{R_x}}={\displaystyle \frac{1}{R_{x_1}}}+{\displaystyle \frac{1}{R_{x_2}}}$$

(6)

$${\displaystyle \frac{1}{R_y}}={\displaystyle \frac{1}{R_{y_1}}}+{\displaystyle \frac{1}{R_{y_2}}}$$

(7)

$${\displaystyle \frac{1}{R^{\prime }}}={\displaystyle \frac{1}{R_x}}+{\displaystyle \frac{1}{R_y}}$$

(8)

According to [74], the approximation of the elliptical integrals is given by the following equations:

$$\mathcal{F}={\displaystyle \frac{\pi }{2}}+q_H·\ln a_H$$

(9)

$$q_H={\displaystyle \frac{\pi }{2}}-1$$

(10)

$$ϵ=1+{\displaystyle \frac{q_H}{a_H}}$$

(11)

2.3. Damping

According to Zeillinger [75], the damping effect of rolling bearings is largely determined by the contact points inside. The following factors significantly influence the damping behavior [75,76]:

• Material damping (hysteresis) due to deformation and the imperfect elastic properties of the contacting surfaces.
• Lubricating film damping in the entry zone and the inlet zone of the elastohydrodynamic contact between the rolling elements and the raceways.

Zeillinger also examined the dynamic behavior of the annular joint between the bearing outer ring and the housing. The joint was found to exhibit high damping capacity. However, when the joint clearance was small and the interfaces were lubricated, the resulting high stiffness limited relative motion between the contacting surfaces, preventing the damping mechanism from becoming effective, which is why this damping effect was disregarded. It is difficult to determine exact damping values. For this reason, parameter models are more commonly used in dynamic simulations. LaMBDA (Ver. 3.0) uses a parametric damping model adapted from Teutsch [72] that can be adjusted using two parameters (${\delta }_{\max }$ and $d_{\max }$) to calculate the damping force in contact based on bearing element penetration.

The roller raceway damping function calculates the damping coefficient d between these two boundary values using a continuous cubic function (see Figure 2). Here, $v_{\mathrm{N}}$ is the penetration velocity along the contact normal vector. The damping force $F_{\mathrm{D}}=-v_{\mathrm{N}}·d$ is then determined as follows [72]:

$$F_{\mathrm{D}}(v_{\mathrm{N}},{\delta }_{\max },d_{\max })=-v_{\mathrm{N}}·\left\{\begin{array}{cc}0,\hfill & \delta <0,\hfill \\ f_{\mathrm{k}}({\delta }_{\max },d_{\max },\delta ),\hfill & 0\le \delta <{\delta }_{\max },\hfill \\ d_{\max },\hfill & \delta \ge {\delta }_{\max }.\hfill \end{array}\right.$$

(12)

This function $f_{\mathrm{k}}$, which depends on penetration $\delta$, is uniquely determined by selecting the two parameters ${\delta }_{\max }$, representing the penetration at maximum damping and $d_{\max }$, representing the maximum damping. The function also depends on the penetration velocity and is uniquely determined by the choice of the two parameters. The slope is zero for both $\delta =0$ and ${\delta }_{\max }$.:

$$f_{\mathrm{k}}({\delta }_{\max },d_{\max },\delta )=-{\displaystyle \frac{2d_{\max }}{{\delta }_{\max }^3}}{\delta }^3+{\displaystyle \frac{3d_{\max }}{{\delta }_{\max }^2}}{\delta }^2$$

(13)

This damping model is used for both the integral and slice contact model because discretising the contact area does not affect the calculated damping force.

2.4. Friction Components

In EHL contact, the contact partners usually do not roll perfectly against each other, resulting in a superposition of rolling and sliding. Both forms of movement are counteracted by frictional forces. The total raceway friction force $F_{\mathrm{T}}$ is described by Equation (14).

$$F_{\mathrm{T}}=-F_{\mathrm{roll}}\pm F_{\mathrm{slide}}$$

(14)

The rolling friction $F_{\mathrm{roll}}$ counteracts the rolling motion of the body and is caused by the compression of the fluid at the entry of the contact. Rolling friction can be calculated according to Crook [77] by solving the following integral for the contacting area.

$$F_{\mathrm{roll}}={\displaystyle \frac{1}{2}}·{\int }_{x_e}^{x_a}h·l·{\displaystyle \frac{dp}{dx}}dx$$

(15)

The portion from the load zone and the outlet contribute only slightly to the rolling resistance and are therefore neglected in [78]. The resulting sliding friction force $F_{\mathrm{slide}}$ (see Equation (16)) can be decomposed into a solid-body sliding friction component $F_{\mathrm{slide},\mathrm{s}}$ and a fluid sliding friction component $F_{\mathrm{slide},\mathrm{f}}$.

$$F_{\mathrm{slide}}=\psi ·F_{\mathrm{slide},\mathrm{s}}+(1-\psi )·F_{\mathrm{slide},\mathrm{f}}$$

(16)

The weighting is taken into account by the asperity load ratio $\psi$, which describes the fraction of the total load carried by solid-body contacts.

$$\psi ={\displaystyle \frac{F_{\mathrm{s}}}{F_{\mathrm{N}}}}$$

(17)

Here, $F_{\mathrm{s}}$ describes the proportion of the normal force $F_N$ transmitted via the solid contacts [79]. The solid-body sliding friction force $F_{slide,s}$ can be determined from the normal force acting in the contact and the solid-body friction coefficient $\mu$, which is modeled as a function of the relative velocity between the rolling element and the raceway using the cubic approach proposed in [72].

$$F_{slide,s}=\mu ·F_N$$

(18)

The sliding component in the lubricant builds up due to the shearing of the lubricant. Due to the low pressures in the inlet and outlet, the viscosity decreases, which is why the shear stress for these areas can be neglected and the main influence comes from the pressure directly acting in the contact [72]. The sliding component $F_{\mathrm{slide},\mathrm{f}}$ is calculated according to Equation (19).

$$F_{\mathrm{slide},\mathrm{f}}=\int \tau dA$$

(19)

For the calculation of the shear stress $\tau$, the Ree–Eyring model is used, considering only the viscous component of the shear rate $\dot{\gamma }$ [80]. The viscous shear rate $\dot{{\gamma }_v}$ depends on the Eyring shear stress ${\tau }_e$ and the dynamic viscosity $\eta$ and is given by the following equation:

$${\dot{\gamma }}_v={\displaystyle \frac{{\tau }_e}{\eta }}\sinh \left({\displaystyle \frac{\tau }{{\tau }_e}}\right)$$

(20)

The Eyring shear stress represents the shear stress at which deviations from Newtonian behavior become significant [72]. Assuming a linear velocity distribution across the lubricant film thickness, the viscous component of the shear rate can be approximated by Equation (21) according to Lubenow [81].

$$\dot{{\gamma }_v}={\displaystyle \frac{v_{\mathrm{rel}}}{h_{\mathrm{cen}}}}$$

(21)

In Equation (21), $v_{\mathrm{rel}}$ represents the relative sliding velocity of the contacting surfaces. The thermally corrected central lubricant film thickness $h_{\mathrm{cen}}$ is given by the following equation:

$$h_{\mathrm{cen}}={\Phi }_{\mathrm{T},\mathrm{MuWi}75}·h_{\mathrm{cen},\mathrm{iso}}$$

(22)

The isothermal central film thickness $h_{\mathrm{cen},\mathrm{iso}}$ (given in Equation (22)) is calculated using the approximation proposed by Chittenden et al., who distinguish between the radii in the direction of the flow and transverse to it, thereby accounting for arbitrary lubricant entrainment directions. Since this formulation assumes isothermal conditions within the contact zone, thermal effects are incorporated through the thermal correction factor ${\Phi }_{\mathrm{T},\mathrm{MuWi}75}$ proposed by Murch and Wilson in [82] (Equation (23)).

$${\Phi }_{\mathrm{T},\mathrm{MuWi}75}={\displaystyle \frac{3.94}{3.94+L^{0.62}}}$$

(23)

The thermal load parameter L is definied as

$$L={\displaystyle \frac{{\eta }_0·\beta ·u_m^2}{{\lambda }_{\mathrm{F}}}}=\left(-{\displaystyle \frac{\partial \eta }{\partial T}}·{\displaystyle \frac{u^2}{{\lambda }_{\mathrm{F}}}}\right)$$

(24)

where ${\eta }_0$ is the dynamic viscosity at transmission pressure, $u_{\mathrm{m}}$ the mean entrainment velocity, ${\lambda }_{\mathrm{F}}$ the thermal conductivity of the oil, and $\beta$ the temperature coefficient. The calculation of $h_{\mathrm{cen},\mathrm{iso}}$ in Equation (25) is based on the dimensionless parameters introduced by Dowson and Higginson, as listed in Table 2.

$$h_{\mathrm{cen},\mathrm{iso}}=4.31·R^{\prime }·G^{0.49}·U^{0.67}·W^{-0.073}·\left(1-\exp \left(-1.23·{\left({\displaystyle \frac{R_{\mathrm{s}}}{R_{\mathrm{e}}}}\right)}^{{\displaystyle \frac{2}{3}}}\right)\right)$$

(25)

$${\displaystyle \frac{R_{\mathrm{s}}}{R_{\mathrm{e}}}}={\displaystyle \frac{\frac{R_{\mathrm{y}}}{R_{\mathrm{x}}}·co{s}^2\xi +si{n}^2\xi }{co{s}^2\xi +\frac{R_{\mathrm{y}}}{R_{\mathrm{x}}}·si{n}^2\xi }}$$

(26)

Table 2. Dimensionless parameters for film thickness calculation.

-	Equation
Velocity parameter	$$U={\displaystyle \frac{{\eta }_0·v_{\sum }}{E^{\prime }·R_e^{\prime }}}$$ (27)
Load parameter	$$W={\displaystyle \frac{F_{\mathrm{N}}}{E^{\prime }·{R_e^{\prime }}^2}}$$ (28)
Material parameter	$$G={\alpha }_{\mathrm{p}}·E^{\prime }$$ (29)

Table 2. Dimensionless parameters for film thickness calculation.

-	Equation
Velocity parameter	$$U={\displaystyle \frac{{\eta }_0·v_{\sum }}{E^{\prime }·R_e^{\prime }}}$$ (27)
Load parameter	$$W={\displaystyle \frac{F_{\mathrm{N}}}{E^{\prime }·{R_e^{\prime }}^2}}$$ (28)
Material parameter	$$G={\alpha }_{\mathrm{p}}·E^{\prime }$$ (29)

Cage Contact

The total frictional torque is only slightly affected by the friction between the rolling elements and the cage. The friction force $F_{T,C}$ acting at the contact is modelled as

$$F_{\mathrm{T},\mathrm{Cage}}={\mu }_{\mathrm{c}}·F_{\mathrm{N},\mathrm{C}}$$

(30)

where $F_{N,C}$ denotes normal load between the rolling elements and the cage and ${\mu }_c$ the coefficient of friction, modeled as a function of the relative velocity between the rolling element and the cage according to the cubic approach from [72].

3. Electrical Capacitance

In this work, the bearing capacitance is calculated based on the contact capacitance. The conversion of an elastohydrodynamic lubricated contact to a capacitive network is depicted in Figure 3. The contact is divided into three different regions, each of which is assigned a capacitance: the capacitance of the Hertzian area $C_{\mathrm{Hertz}}$, the capacitance of the inlet region $C_{\mathrm{Inlet}}$, and the capacitance of the outlet region $C_{\mathrm{Outlet}}$.

The three different capacitance are assumed to be in parallel. Therefore, the overall contact capacitance $C_{\mathrm{Con}}$ can be calculated according to Equation (31).

$$C_{\mathrm{Con}}=C_{\mathrm{Inlet}}+C_{\mathrm{Hertz}}+C_{\mathrm{Outlet}}$$

(31)

The Hertzian contact area is commonly regarded as a plate capacitor. Under this assumption, its capacitance $C_{\mathrm{Hertz}}$ can be calculated by Equation (32)

$$C_{\mathrm{Hertz}}={ϵ}_0·{ϵ}_{\mathrm{r}}·{\displaystyle \frac{A_{\mathrm{Hertz}}}{h_{\mathrm{cen}}}}$$

(32)

where the Hertzian contact area $A_{\mathrm{Hertz}}$ is the plate area of the capacitor and the central gap height $h_{\mathrm{cen}}$ is its plate distance. The permittivity of this model capacitor ${ϵ}_0·{ϵ}_{\mathrm{r}}$ consists of the permittivity of vacuum ${ϵ}_0$ and the relative permittivity of the lubricant ${ϵ}_{\mathrm{r}}$, which acts as a dielectric.

Equation (32) assumes the surfaces of the rolling element and the raceway to be in parallel. This approximation is not valid for the inlet and the outlet zone. The capacitance of these regions can be determined numerically [68], semi-analytically [70], or analytically by using a correction factor $k_{\mathrm{c}}$ that is applied to $C_{\mathrm{Hertz}}$. The correction factor $k_{\mathrm{c}}$ is defined by the ratio between the overall contact capacitance $C_{\mathrm{Con}}$ and the capacitance of the Hertzian contact area:

$$k_{\mathrm{c}}={\displaystyle \frac{C_{\mathrm{Inlet}}+C_{\mathrm{Hertz}}+C_{\mathrm{Outlet}}}{C_{\mathrm{Hertz}}}}={\displaystyle \frac{C_{\mathrm{Con}}}{C_{\mathrm{Hertz}}}}$$

(33)

With knowledge of $k_{\mathrm{c}}$, $C_{\mathrm{Con}}$ can be approximated fully analytically based on $C_{\mathrm{Hertz}}$ using Equation (32). The correction factor method is therefore well suited for a quick approximation of contact capacitance but has lower accuracy compared to more complex calculation methods [69,83]. This is why there are several empirically determined estimates for $k_{\mathrm{c}}$, including constant factors [84] or linear equations as a function of $h_{\mathrm{cen}}$ [65,85,86]. A more recent approach was published by [67] and calculates $k_{\mathrm{c}}$ using the dimensionless factors by Hamrock and Dowson [87]:

$$k_{\mathrm{c}}=9.5923·U^{0.3305}·G^{0.3431}·W^{-0.3342}·k_e^{0.1391}$$

(34)

It was derived based on numerical calculations of the contact capacitance of radial deep-groove ball bearing and directly takes into account the operating conditions.

Due to its simplicity, the correction factor method is easy to implement into existing contact simulations. Therefore, it is applied in this work for the calculation of $C_{\mathrm{Con}}$. In the first step, the capacitance of the Hertzian contact area $C_{\mathrm{Hertz}}$ is determined based on the previously calculated contact properties. For this purpose, the discrete elements of the contact simulation are regarded as small plate capacitors that are connected in parallel. The plate area of each capacitor is given by the size of the discrete element. The plate distance is given by the local gap height value. In this way, the minimum gap height can be considered in the capacitance calculation. The relative permittivity of each small capacitor is calculated by Bode’s equation (given in Equations (35)–(37)) in dependency of the inlet temperature of the oil and the local pressure inside the small capacitor.

$${ϵ}_{\mathrm{r}}(p,T)={ϵ}_{\mathrm{r}\mathrm{T}}+k_3·(\rho (p,T)-{\rho }_{\mathrm{T}}\left(T\right))$$

(35)

where ${ϵ}_{\mathrm{r}\mathrm{T}}$ is the temperature-dependent relative permittivity given by

$${ϵ}_{\mathrm{r}\mathrm{T}}=k_1·\beta {\left(T\right)}^{-{\displaystyle \frac{1}{2}}}·{\rho }_{\mathrm{T}}{\left(T\right)}^{{\displaystyle \frac{1}{6}}}+k_2·T$$

(36)

and $\beta \left(T\right)$ is the isothermal compressibility

$$\beta \left(T\right)=-{\displaystyle \frac{1}{V}}{\displaystyle \frac{dV}{dp}}={\displaystyle \frac{a_1}{a_2+a_3·T+a_4·T^2+a_5·T^3}}$$

(37)

The parameters $k_1,k_2$, and $k_3$ are characteristic values of the lubricant that were experimentally determined. In this way, the capacitance $C_{\mathrm{Hertz}}$ is determined taking into account local pressure changes and their effect on permittivity. The surrounding capacitance [88] of the bearing is included in the applied approach. The focus is on contact capacitance, as the dynamic effects examined via the multibody simulation affect the contact area.

4. Results and Discussion

In the following, the simulation results are presented and discussed. First, the friction torque is used as a validation quantity for the model implementation. Subsequently, the capability of the fully dynamic model to predict cage instability under axial excitation is examined. Finally, the applicability of the LaMBDA tool to the calculation of the Hertzian contact capacitance is demonstrated.

4.1. Friction Torque

For validation purposes, the friction torque has proven to be a particularly suitable resultant quantity. It directly arises from the shear stresses in the bearing contacts. This allows a comprehensive comparison with established state-of-the-art methods used in both academia and industry. The SKF catalogue method is an analytical model, based on empirically fitted formulas and averaged operating conditions. FVA-Workbench by FVA (Forschungsvereinigung Antriebstechnik e.V., Frankfurt, Germany) and Bearinx by Schaeffler are quasi-static simulation tools, which resolve load distribution and contact conditions under steady-state assumptions.

Figure 4a presents a comparison of the friction moment ($M_{\mathrm{R}}$) of a 6319 deep-groove ball bearing subjected to a radial load ratio of ($C/P=10$) for three operating speeds of 500 min⁻¹, 1500min⁻¹ and 3000 min⁻¹. The geometry data of the bearing are given in

Table 3. Geometric and material data of the used deep-groove ball bearing 6319.

Parameter	Unit	Bearing 6319
Outer diameter	mm	200
Inner diameter	mm	95
Pitch diameter	mm	147.5
Raceway diameter inner ring	mm	117.4
Raceway diameter outer ring	mm	177.5
Rolling element diameter	mm	30
Groove radius inner ring	mm	15.375
Groove radius outer ring	mm	15.75
Number of rolling elements	-	9
Nominal contact angle	°	0
Basic dynamic load rating (C)	kN	160
Poisson’s ratio	-	0.3
Elastic modulus	GPa	210

.

The results obtained with the proposed multibody simulation tool LaMBDA (purple) are compared against experimental friction torque measurements from the FVA Project 364 IV (red) [89], as well as against established quasi-static calculation approaches, namely the FVA Workbench, (Ver. 11.0) (blue), BearinX (Ver. 2026) by Schaeffler (green), and the catalogue-based method published by SKF (yellow). The detailed results are given in

Table 4. Friction torque of the DGBB 6319.

Load Case	Method	500 min−1	1500 min−1	3000 min−1
$C/P=10$	LaMBDA (Ver. 3.0)	828 Nmm	1194 Nmm	1285 Nmm
	Measurement/FVA364IV	925 Nmm	1306 Nmm	1564 Nmm
	FVA-Workbench (Ver. 11.0)	685 Nmm	1158 Nmm	1312 Nmm
	Bearinx (Ver. 2025)	653 Nmm	1124 Nmm	1270 Nmm
	SKF-Catalogue	814 Nmm	1190 Nmm	1380 Nmm
$C/P=6.5$	LaMBDA (Ver. 3.0)	1192 Nmm	1547 Nmm	1750 Nmm
	Measurement/FVA364IV	1456 Nmm	1750 Nmm	1847 Nmm
	FVA-Workbench (Ver. 11.0)	1141 Nmm	1599 Nmm	1844 Nmm
	Bearinx (Ver. 2025)	1078 Nmm	1511 Nmm	1736 Nmm
	SKF-Catalogue	1330 Nmm	1720 Nmm	2000 Nmm

.

The experimental bearing test series was lubricated using the ISO VG 100 mineral oil FVA3A, a reference oil with additives. The oil level was adjusted to half the height of the lowest rolling element, in accordance with the lubrication specifications reported for FVA3A [89]. The measured oil bath temperature $\theta$ was explicitly considered in all simulation and calculation methods to ensure a consistent thermal boundary condition.

A corresponding set of experiments and simulations was conducted for a higher radial load ratio of $C/P=6.5$. The results of this second load case are shown in Figure 4b, enabling an assessment of the influence of load level and lubricant formulation on the predicted friction behavior. The total friction torque increases with rising radial load due to changes in the contact conditions, as described by Hertzian contact theory. As the load increases from $C/P=10$ to $C/P=6.5$, the elliptical contact area between the rolling elements expands. This leads to an expansion of local relative speed within the contact. Due to elastic deformation and kinematic constraints in the rolling contact, differential slip occurs and is distributed over the contact area. As the contact size increases, the regions experiencing micro-slip grow, resulting in higher friction force and, consequently, increased total friction torque. Furthermore, the contact is of point type, in contrast to line contact. Owing to the doubly curved geometry of the contacting bodies, point contacts inherently exhibit higher slip gradients and greater local relative velocities. As a result, differential slip is more pronounced than in line contacts, further contributing to the increase in friction torque. All this leads to an increasing total friction torque of 57.4% at 500 min⁻¹ for the measurements, when increasing the radial load from $C/P=10$ to $C/P=6.5$. LaMBDA gives a total friction torque difference of 44.0%.

For both load cases, all approaches exhibit a clear and monotonic increase in friction torque with increasing rotational speed (e.g., the total friction torque increases by 69.1% from 500 min⁻¹ to 3000 min⁻¹ for the measurements and 55.2% for LaMBDA). This trend reflects the growing contribution of lubricant shear losses, rolling–sliding effects in the contacts, and churning losses at higher speeds. Across the investigated speed range, most numerical and analytical methods tend to underestimate the experimentally measured friction torques, with the deviation generally increasing at higher rotational speeds. It should be noted, however, that no information on measurement scatter or standard deviation is provided in the FVA 364 IV dataset, which limits a quantitative assessment of the statistical significance of these deviations.

For the quasi-static load case, all models show good agreement between the calculated and experimental results. This demonstrates that the quality of the developed LaMBDA model is at a comparable level to that of the other models. The main advantage of the approach presented here lies in its coupling with the MBS. The following section therefore describes a load case investigating the dynamic behaviour of the roller speed when excited in axial direction. Load cases like this can only be investigated by multibody simulations.

4.2. Roller Speed Instability

While friction torque is a useful validation quantity, it can also be predicted with established quasi-static approaches. In contrast, a central motivation for employing a fully dynamic multibody model is the prediction of stability limits of the bearing kinematics under external excitation. Deep-groove ball bearings may exhibit a transition from a stable running state to an altered, more unstable regime when the system is subjected to axial vibration. Such transitions are governed by the interplay of excitation frequency, excitation amplitude, and static load and are directly linked to changes in roller kinematics as well as to intermittent contacts outside the nominal load zone. These phenomena are inherently time dependent. They cannot be captured by quasi-static load distribution methods.

To demonstrate this use case, Figure 5 summarizes an experimentally and numerically derived instability boundary for a 6220 deep-groove ball bearing under axial system excitation from the FVA project 589 I [90]. The diagram maps the onset of an altered (unstable) running behavior by the excitation frequency f and the excitation speed $v_{\mathrm{exc}}$ for three different static radial loads $F_{\mathrm{stat}}$. Here, $v_{\mathrm{exc}}$ denotes the vibration velocity amplitude of the imposed harmonic axial displacement of the bearing system. The acceleration of the excitation speed directly determines the inertial forces acting on the raceways and the rollers, which are critical for the stability of the internal bearing kinematics. Figure 6 shows the operation conditions for this bearing.

Two characteristic trends are evident. Firstly, the instability boundary exhibits monotonic decay: comparatively high excitation frequencies are required to trigger the transition at low excitation speeds (small vibration amplitudes), whereas instability already occurs at substantially lower frequencies at high excitation speeds. This behaviour suggests that the governing parameter is related to acceleration during excitation rather than to a single resonant frequency. Accordingly, the boundary shows a continuous course without pronounced peaks, which would otherwise suggest resonance-dominated amplification.

Secondly, increasing the static load shifts the instability boundary towards higher excitation levels. For the highest static load, substantially higher combinations of f and $v_{\mathrm{exc}}$ are needed to disturb the bearing motion compared to the lightly loaded case. Physically, a higher static load increases the confinement of rollers within the load zone. Conversely, under sufficiently large excitation accelerations, additional contact forces can occur outside the nominal load zone, keeping the rolling elements closer to their kinematic speeds around the circumference. In this regime, the change in the distribution of rolling element speeds is significant, which constitutes a relevant stability concern.

4.3. Capacitance

The tool LaMBDA is not limited to the calculation of bearing friction. Furthermore, it can be used to calculate the capacitance of the Hertzian contact area. This is depicted in Figure 7 for a pure axially loaded DGBB 6208. As under pure axial load, all of the rolling elements are equally loaded, equal contact condition and therefore equal contact capacitances are generated. Figure 7 therefore gives the capacitance $C_{\mathrm{Hertz}}$ of the Hertzian contact area for a single rolling element for both the inner ring contact and the outer ring contact. Three different axial loads are investigated (700 N, 1300 N, 2000 N) at constant rotating speed (2750 rpm) and constant oil temperature (40 °C). The calculation is carried out for a mineral oil.

Table 5. Bode-parameters for the calculation of density, viscosity, and relative permittivity of the mineral oil used for capacitance calculation.

Parameter	Mineral Oil	Parameter	Mineral Oil
${\rho }_{\mathrm{s}}$	1073.465942	$A_1$	0.03380703
${\alpha }_{\mathrm{s}}$	0.0005911	$A_2$	2.82632938
$a_1$	0.08970567	$A_3$	785.285992
$a_2$	6348.426753	$A_4$	0.00162619
$a_3$	−27.52010404	$k_1$	0.00379557
$a_4$	0.04547532	$k_2$	0.00258151
$a_5$	−0.00002769	$k_3$	0.00130612

gives the parameters to calculate the oils’ permittivity according to Equation (35). The geometry data of the bearing is given in

Table 6. Geometric and material data of the used deep-groove ball bearing 6208.

Parameter	Unit	Bearing 6208
Outer diameter	mm	80
Inner diameter	mm	40
Pitch diameter	mm	60
Raceway diameter inner ring	mm	48.083
Raceway diameter outer ring	mm	71.917
Rolling element diameter	mm	11.91
Groove radius inner ring	mm	6.19
Groove radius outer ring	mm	6.31
Number of rolling elements	-	9
Nominal contact angle	°	0
Poisson’s ratio	-	0.3
Elastic modulus	GPa	210

.

One of the major benefits of LaMBDA is the advanced calcuation of bearing kinematics. Therefore, the capacitance values calculated by LaMBDA are compared to capacitance values calculated using an analytical method. For this comparison, the hydrodynamic speed of the roller contacts was calculated solely based on the bearing geometry. The mathematical description of the hydrodynamic speed is given in [84]. As the calculation is based on geometrical relations inside the bearing, it gives equal hydrodynamic speed for each contact. With help of the hydrodynamic speed, the central gap height is calculated according to Chittenden et al. [91]. The Hertzian contact area $A_{\mathrm{Hertz}}=\pi ab$ of both contacts is calculated based on the mechanical load on the rolling element using Equations (2) and (3). The relative permittivity is calculated for the Hertz pressure $p_0$. Subsequently, the capacitance values to compare with the results of LaMBDA are calculated using Equation (32).

As depicted in Figure 7, the contact capacitances in both models are in the same order of magnitude. At the inner contacts, the capacitance values by LaMBDA are around 14% higher compared to the analytical method; at the outer ring, the deviation is in the range of 10% to 13%. The contact capacitance rises with rising axial load according to both calculations. This is due to the enlargement of the Hertzian contact, which overweighs the lowering of the gap height. A rise of axial load from 700 N to 2000 N leads to a reduction of 9% in gap height, while the contact area increases by 78%. The capacitances of the outer ring contacts are generally lower than the capacitances of the corresponding inner ring contacts. Due to the radii ratios of ball and raceway, the outer ring contacts can build about 20% higher lubricating films and about 10% larger contact areas compared to the inner ring contacts. In combination, the effects lead to 10% to 14% lower contact capacitance. Compared to the analytical calculation method, LaMBDA gives higher contact capacitances for all investigated loads. The main difference between the two models is the consideration of dynamic effects in LaMBDA. They lead to lower gap heights due to shear thinning of the oil or heating due to the high pressure. These lower gap heights lead to a rise of contact capacitance. The results show that LaMBDA is capable of calculating the capacitance of the Hertzian contact area. Furthermore, it can consider complex bearing dynamics in the lubricating film formation and their effect on contact capacitance.

5. Conclusions

The presented friction model for deep-groove ball bearings demonstrates very good agreement with measured friction torques and accurately captures the experimentally observed trends under varying operating conditions. Owing to its fully dynamic formulation, the model enables a time-dependent evaluation of the bearing kinematics, allowing transient effects and changes in the motion of individual rolling elements to be resolved in detail.

Beyond friction prediction, the model provides access to phenomena that cannot be addressed by quasi-static approaches. In particular, it allows the prediction of roller speed instability limits under axial excitation, where changes in roller kinematics govern the transition from stable to instable conditions. Furthermore, the implemented framework enables the calculation of Hertzian contact capacitances, thereby extending the applicability of the model to electro-tribological analyses of rolling bearings under transient conditions.

Despite these positive results, the current model still exhibits limitations that should be addressed in future work. In particular, the systematic inclusion of churning and drag losses is required, as these effects influence rolling element kinematics and, consequently, local contact friction. In existing engineering approaches, churning and drag losses are typically described using empirical catalogue methods provided by bearing manufacturers. For example, the Schaeffler catalogue provides in [6] equations for load-independent losses in terms of the friction torque $M_0$:

$$\begin{array}{c}\hfill M_0=f_0·{10}^{-7}·d_{\mathrm{m}}^3·{\left(\nu ·n\right)}^{{\displaystyle \frac{2}{3}}},\mathrm{for}\nu ·n\ge 2000{\mathrm{mm}}^2/\left(\mathrm{s}\min \right)\end{array}$$

(38)

$$\begin{array}{c}\hfill M_0=160·{10}^{-7}·f_0·d_{\mathrm{m}}^3,\mathrm{for}\nu ·n<2000{\mathrm{mm}}^2/\left(\mathrm{s}\min \right)\end{array}$$

(39)

These equations include churning and drag losses. However, rolling resistance is also contained in $M_0$ and needs to be separated, as rolling resistance is already considered within the contact friction forces and is inherently load dependent. The SKF approach directly provides a formulation for the churning and drag losses in ball bearings in terms of a drag torque:

$$\begin{array}{c}\hfill M_{\mathrm{drag}}=V_M·K_{\mathrm{Ball}}·d_m^5·n^2\end{array}$$

(40)

where $V_M$ and $K_{\mathrm{Ball}}$ are determined from tabulated values given in [92]. The resulting drag torque is defined with respect to the entire bearing. However, these approaches describe the losses on a global bearing level. A more detailed description requires the allocation of churning losses to individual rolling elements. Churning acts in the form of a resisting moment $M_{\mathrm{ch}}$ on the rolling elements (see Figure 8a), while drag results in hydrodynamic force $F_{\mathrm{drag}}$ acting on the rolling elements (see Figure 8b). This allows the influence of fluid-induced forces on the rolling element kinematics to be considered. The resulting changes in kinematics affect the behaviour in the acceleration zones and thereby influence the contribution to the friction torque.

In summary, the presented model provides a solid and versatile basis for the analysis of friction behaviour and dynamic stability in deep-groove ball bearings. Its dynamic nature enables the investigation of time-dependent effects and offers the potential for the early prediction of critical operating conditions, including smearing-related damage mechanisms.