Aus Wissenschaft und Forschung 54 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 Using exact macroscopic geometry in elastohydrodynamic simulations of point and elliptical contacts Sven Wirsching, Marcel Bartz* Eingereicht: 25.8.2022 Nach Begutachtung angenommen: 25.1.2023 Dieser Beitrag wurde im Rahmen der 63. Tribologie-Fachtagung 2022 der Gesellschaft für Tribologie (GfT) eingereicht. * Sven Wirsching, M.Sc. Dr.-Ing. Marcel Bartz Lehrstuhl für Konstruktionstechnik KTmfk, Friedrich- Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen Bei bordgeführten Rollenlagern existiert eine Vielzahl von unterschiedlichen tribologischen Kontaktformen. So sind nicht nur Linienkontakte an den Laufbahnen, dem Käfig sowie den Wälzkörpern, sondern auch Punkt- und Ellipsenkontakte zwischen Wälzkörperstirnfläche und Ringbord vorhanden. Die Kraftübertragung erfolgt über diese geschmierten, konzentrierten Wälz- und Wälz-Gleit-Kontakte. Abhängig von der Belastungssituation tragen diese Kontakte unterschiedlich zum Betriebsverhalten des Rollenlagers bei. Axiale Lasten auf bordgeführte Rollenlager werden vorwiegend über die Punkt- und Ellipsenkontakte zwischen Rollenstirn und Ringbord übertragen. Diese ölgeschmierten Punkt- und Ellipsenkontakte können mit Hilfe von Thermo-Elastohydrodynamik (TEHD) Simulationen berechnet und ausgelegt werden. Bei bestehenden Methoden der TEHD Berechnung von Punkt- und Ellipsenkontakten werden die makroskopischen Geometrien der Kontaktpartner vereinfacht, ähnlich der Theorie nach H ERTZ , über Ellipsoide beschrieben. Kontakte realer, komplexer Geometriepaarungen von Wälzkörper und Ringbord, wie sie zur Optimierung der axialen Belastbarkeit oder des Reibungsmomentes von Rollenlager eingesetzt werden, lassen sich damit allerdings nur ungenau ermitteln. Im Vergleich Kurzfassung zur exakten Berücksichtigung der makroskopischen Geometrie sind größere Diskrepanzen in der Schmierfilmhöhe, des Kontaktdrucks und der Reibung zu verzeichnen. Aus diesem Grund wird in diesem Beitrag eine TEHD Simulation vorgestellt, welche die exakte makroskopische Geometrie des Punktbzw. Ellipsenkontaktes berücksichtigt. Die makroskopische Geometrie wird mit Hilfe von mathematischen Funktionen erzeugt und über ein Ray-Tracing Verfahren der Ersatzkörper für die TEHD Simulation generiert. Verschiedene Geometriepaarungen aus Kugel, Ebene, Kegel und Torus werden untersucht. Die Ergebnisse für Schmierfilmhöhe, Kontaktdruck und Reibung werden mit den Ergebnissen aus herkömmlichen TEHD Simulationen, welche eine Geometriebeschreibung über Ellipsoide verwendet, verglichen. Durch den Vergleich der berechneten Geometriepaarungen werden die Möglichkeiten und Grenzen der veränderten Geometriebeschreibung beurteilt. Schlüsselwörter Elastohydrodynamik, Kontakt, Reibsysteme, Berechnungs- und Simulationsmethoden In rib-guided roller bearings, there are a large number of different tribological contact forms. These include not only line contacts on the raceways, the cage and the rolling elements, but also point and elliptical contacts between the rolling element end face and the ring rib. Load is transmitted via these lubricated, concentrated rolling and rolling-sliding contacts. Depending on the Abstract load situation, these contacts contribute differently to the operating behavior of the roller bearing. Axial loads on rib-guided roller bearings are mainly transmitted via the point and elliptical contacts between the roller end and the ring rib. These oil-lubricated point and elliptical contacts can be calculated and designed using thermo-elastohydrodynamic (TEHD) simulations. TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 54 1 Introduction In view of dwindling resources, increasing environmental awareness and stricter legal requirements regarding climate protection, the energy-efficient design of tribological contacts is becoming increasingly important. In many machine elements, friction and wear occur in tribological contacts, which contributes significantly to the energy consumption of technical systems [5]. Recent investigations show that in highly loaded contacts the hydrodynamic lubricating film formation has to be considered together with the superimposed elastic deformation and therefore the theory of hydrodynamic (HD) lubrication alone is not sufficient [1], which is why the numerical modeling of thermo-elastohydrodynamic (TEHD) contacts is of high importance. Basically, TEHD contacts differ in geometry, kinematics, and lubrication condition, and especially the effects due to geometric pairing have to be considered [10]. In existing methods for the TEHD calculation of point and elliptical contacts, the macroscopic geometries of the contact partners are described in a simplified manner, similar to the theory according to H ERTZ , using ellipsoids. In order to successfully minimize friction, an accurate representation of the geometry pairing is necessary. A typical machine element that can benefit from the potential minimization of friction through a more accurate description of the geometry is, for example, the roller bearing. The secondary frictional sliding contacts between the roller end face and ring rib in roller bearings significantly influence the operational behavior and friction. This contact generates high frictional forces. Therefore, the use of an exact geometry description due to the so called P IM P [14] in TEHD simulations using T RIBO FEM [12] - both developed by the Institute of Engineering Design (KTmfk) - is investigated in this paper for rib contacts in a tapered roller bearing. By extending the geometry description to complex geometry pairings, these frictional secondary contacts can be calculated more accurate. In the following, the physical and mathematical relationships as well as the mode of operation of the TEHD simulation are first presented (section 2), in order to subsequently show the application in rolling bearing technology, using the example of a tapered roller bearing (section 3). Afterwards, the results of these simulations are discussed and summed up in section 4. Finally, an outlook is given in section 5 2 Materials and Methods In the following section, the used TEHD model and the underlying formulas as well as the simulation are explained at first. Subsequently, the calculation of friction and the generation of the geometry and the velocity field is described. At the end, the material and lubricant parameters are specified and the investigated load cases are presented. 2.1 Thermo-Elastohydrodynamic Simulation Within T RIBO FEM, the TEHD simulation is implemented based on an approach by H ABCHI [3] using the commercial FEM software COMSOL M ULTIPHYSICS . Here, the R EYNOLDS differential equation (1) Aus Wissenschaft und Forschung 55 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 In existing methods for the TEHD calculation of point and elliptical contacts, the macroscopic geometries of the contact partners are described in a simplified manner, similar to the theory according to H ERTZ , using ellipsoids. However, contacts of real, complex geometry pairings of rolling elements and ribs, as used to optimize the axial load capacity or the frictional torque of roller bearings, can only be determined inaccurately with this method. Compared to the exact consideration of the macroscopic geometry, larger discrepancies in the lubricant film height, contact pressure and friction can be observed. For this reason, this paper presents a TEHD simulation that considers the exact macroscopic geometry of point or elliptical contacts. The macroscopic geometry is generated using mathematical functions and a ray-tracing method is used to generate the equivalent body for the TEHD simulation. Different geometry pairings of sphere, plane, cone and torus are investigated. The results for lubricant film height, contact pressure and friction are compared with the results from conventional TEHD simulations, which use a geometry description via ellipsoids. By comparing the calculated geometry pairings, the possibilities and limitations of the modified geometry description are assessed. Keywords elastohydrodynamics, contact, friction systems, calculation and simulation methods, bearings pressure term ( ) ( , ) 12 ( ) + ( ) ( , ) 12 ( ) = ( ) ( , ) + 2 + ( ) ( , ) + 2 velocity term TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 55 The numerical solution scheme of the TEHD simulation is shown in Figure 1 (a). After reading in all the required input variables and functions, initial values are determined on the basis of H ERTZ ian theory. These are used as initial solutions for the calculation of the deformation and the lubrication gap. In the next step, the fully coupled, stationary, isothermal N EWTON ’s model is solved in the FE domain (P, H). For this purpose, a tetrahedral mesh with refinement in the contact center is used, see Figure 1 (b). [9] Without thermal or non-N EWTON ian effects, the calculation would be completed at this point. Taking these effects into account, a sequential solver calculates the integral terms of the generalized R EYNOLDS equation, followed by the fully coupled system of pressure and deformation in the FE domain (P, H) and the velocity distribution in the FE domain (U), see Figure 1 (c). For the latter, a triangular mesh with regular distribution in the gap direction is used. If thermal effects are considered, the last step of the sequential solver is the calculation Aus Wissenschaft und Forschung 56 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 together with the lubricant film height equation (2) is applied while satisfying the equilibrium of forces (3) and taking into account the pressure and temperature dependence of lubricant density and viscosity due to the C ARREAU model. A mass-conserving cavitation model is used. The temperature distribution within the lubricant film is described using the energy equations. In these equations, heat sources due to shear and compression of the lubricant are taken into account: (4) ( , ) = + ( , ) + ( , ) = ( , ) + + + + = 0 Figure 1: Numerical solution scheme of the TEHD simulation T RIBO FEM in COMSOL M ULTIPHYSICS (a) with the computational domains for elastic deformation (b), fluid velocity (c) and temperature (d) according to [8, 9] TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 56 of the temperature distribution in the FE domain (ϑ) using a tetrahedral mesh, see Figure 1 (d). These steps are repeated until the solution converges for all solution variables. Further basic aspects on FEM for TEHD contacts can be found in H ABCHI [3] and for more information on the implementation in COMSOL M ULTIPHYSICS software, the reader is referred to TAN et al [11] and L OHNER et al [7]. 2.2 Friction Calculation Depending on the lubrication condition of the contact, the friction is composed of the fluid friction and solid friction components. In this model, a fully flooded lubrication gap, which completely separates the two bodies, is assumed. Thus, only fluid friction is present. The fluid friction force according to H ABCHI [2] is determined via the integral of the shear stresses on the center plane of the area Ω u . (5) 2.3 Geometry and Velocity Calculation While the geometry of the roller face can be described by the face radius r R and the eccentricity e, the ring rib is described by the rib angle γ between the raceway and the ring rib as well as the rib radius r B , as shown in Figure 2 (a, b). The location of the contact point is another boundary condition. In the further investigations, the assumption is made that the contact point is ideally centered on the ring rib due to the axis of rotation of the rolling bearing. By varying the radii and the eccentricity, different rib angles can be obtained. Based on the parameters mentioned above, both bodies can be created using the P IM P method developed at KTmfk by W IRSCHING et al [14] and transferred to the TEHD simulation. The P IM P method generates a substitute geometry using a ray-tracing procedure. This method first requires the relative positions of the two bodies in contact and their local coordinate system within the global coordinate system. Using = | . position vectors q(x, y, z) and normal vectors n(x, y, z), as given in Figure 2 (a), both geometries are generated on a projection plane and then a substitute geometry g 0 is calculated by subtraction. The surface speeds of the secondary contact of the roller face and ring rib are determined with the help of rolling bearing kinematics. Assuming a static outer ring, the rolling element speed can be calculated according to [4] (6) using the pitch diameter D pw , the mean diameter of the rolling element d w and the operating contact angle β. The rolling element speed can be determined by (7) according to [4]. With the help of the rib angle γ, the speeds and the position of the contact point to the axis of rotation D KP , the surface velocities in xand y-direction of the inner ring rib (8) and the roller face (9) can then be determined. The velocity components in xand y-direction of both contacting bodies can be calculated at each node of the FE area Ωu. = with = 2 + , = 2 1 cos = 2 2 + cos , = 2 = 2 + cos and = 2 ( ) Aus Wissenschaft und Forschung 57 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 Figure 2: Exemplary tapered roller bearing with its dimension and the position as well as the normal vectors of the rib contact (a), parameters describing the rib geometry (b), and the calculation of the velocity field (c) according to [13] with and TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 57 3 Results of the TEHD Simulation In the following section, the results of the investigations of the two geometry pairings torus-cone and torus-torus are shown. For each of the two cases, TEHD simulations were performed using geometry generation via ellipsoids similar to H ERTZ as well as geometry generation using the P IM P method with the parameters from Sections 2.4 and 2.5. First, the results for the pressure distribution and lubricant film thickness distribution in the xz-section plane are shown. As can be seen in Figure 3 (a) and (c), the pressure as well as the P ETRUSEVICH peak decreases with increasing radius and the distribution flattens. Additionally, there are major differences in the pressure distribution. The geometry generation similar to H ERTZ leads to higher pressures compared to the P IM P method. This can be observed for both cases and all contact radii. In case 2, see Figure 3 (c), greater deviation can be investigated compared to case 1, see Figure 3 (a). The results of the lubricant film thickness distribution show higher lubricant films for larger radii and small deviations for both cases, as can be seen in Figure 3 (b) and (d). Here, the geometry generation similar to H ERTZ results in slightly thicker lubricant films compared to the P IM P method. In case 2, see Figure 3 (d), there is a greater difference in the lubricant film thickness distributions of the two geometry generations compared to case 1, see Figure 3 (b). Furthermore, a larger deviation in the pressures compared to the lubricant film heights can be observed. In the next step, the reference result variables maximum pressure, minimum lubrication gap and the coefficient of friction are evaluated for the two cases toruscone and torus-torus. The behavior of the maximum pressure is similar to the previously observed pressure distribution in the xz-section plane, as can be seen in Figure 4 (b) and (e). In both cases the maximum pressure decreases with an increase of the radii and it is lower for the geometry generation by the P IM P method compared to H ERTZ . For both cases, the difference of the geometry generations is similar. The reverse is true for the minimum lubrication gap. As can be seen Aus Wissenschaft und Forschung 58 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 2.4 Material and Lubrication Properties Knowledge of the material and lubricant properties is still required to set up the simulation. A standard tapered roller bearing of the type 30207 with rolling bearing steel (100Cr6) is studied and for the lubricant a mineral oil with no additives is used. The data for the material and lubrication properties can be taken according to Table 1. Exemplary, a harm operation condition of the tapered roller bearing with the rotation speed of the inner ring of 500 rpm is set to be constant. The rib contact is loaded with a normal force of 100 N. The outer ring of the tapered roller bearing is fixed and it is supplied with sufficient lubricant to guarantee full lubrication. 2.5 Investigated Geometry Cases Since the influence of geometry generation on the results of TEHD simulations is investigated in this work, two different cases with complex geometry pairings of the secondary rib contact in the tapered roller bearing are studied. First, a torus-cone and second, a torus-torus pairing is used. In the first case, a torus is maped on the roller end face (radius with eccentricity) and a cone on the ring rib (no radius). In the second case, a torus geometry is present on the roller end face (radius with eccentricity) and the ring rib (radius), respectively. In addition, the radii are varied for both cases and all other geometry parameters are obtained. The exact data can be taken from Table 2. On the one hand, the geometries are described with the help of ellipsoids, similar to the method according to H ERTZ . On the other hand, the geometries are generated with the P IM P method according to W IRSCHING [14]. parameter unit value Y OUNG ’s Modulus E N/ mm 2 210 000 P OISSON ratio - 0.3 density solid S kg/ m 3 7 850 thermal conductivity solid k S W/ (m K) 46 heat capacity solid c S J/ kg K 470 viscosity Pa s 0.1 density lubricant L kg/ m 3 750 density temperature coefficient 1/ K 0.00075 thermal conductivity lubricant k L W/ (m K) 0.1 heat capacity lubricant c L J/ (kg K) 1 500 temperature viscosity coefficient p 1/ Pa 15.0 pressure viscosity coefficient p - 0.05 parameter unit case 1 (torus-cone) case 2 (torus-torus) roller radius r R mm 25 50 75 50 75 100 eccentricity e mm 1.5 rib radius r B mm 0 0 0 75 50 50 pitch diameter D pw mm 82.0 mean roller diameter d w mm 8.1 Table 1: Material and lubricant properties for all simulations Table 2: Geometry parameters for the two different cases torus-cone and torus-torus TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 58 in Figure 4 (c) and (f), smaller minimum lubrication gaps result for the geometry generation similar to H ERTZ compared to the P IM P method. This behavior is opposite to the previously shown behavior of the lubricant film height distribution in the xz-section plane. Tendentially, the difference in minimum lubrication gap is more significant in case 2, see Figure 4 (f). But one can observe the same behavior for an increasing lubricant gap with an increase of the radii. When calculating the coefficient of friction, it is noticeable that it decreases with an increase of the radii in both cases, as can be seen in Figure 4 (a) and (d). The geometry generation similar to H ERTZ produces larger coef- Aus Wissenschaft und Forschung 59 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 Figure 3: Comparison of the results for the pressure distribution and the lubricant film thickness distribution in the xz-section plane for case 1 torus-cone (a), (b) and case 2 torus-torus (c), (d) Figure 4: Comparison of the calculation results of friction coefficient (a), maximum pressure (b) and minimum lubricant gap (c) for case 1 torus-cone and (d), (e), (f) for case 2 torus-torus TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 59 bricant film lead to larger coefficients of friction in the fluid friction, resulting in the deviations of the coefficient of friction in Figure 4 (a) and (d). In summary, it was shown that the generation of the geometry has an impact on the results of the TEHD simulation. For this purpose, two cases with complex torus geometries for different radii were considered. The geometries have been generated with the simplified method according to H ERTZ and with the P IM P method and the results for pressure, lubricant film thickness and coefficients of friction have been analyzed. The deviations were addressed. In conclusion, the geometry generation with ellipsoids leads to an overestimation of the pressure and the coefficient of friction and an underestimation of the lubricant film thickness. For more accurate TEHD simulations a more accurate geometry generation like the P IM P method should be used. 5 Outlook The presented TEHD calculations for secondary contacts shall be used in further steps for the design of rolling bearings. By the use of machine learning methods, an optimization of the contacts is possible, as shown in the work of M ARIAN and W IRSCHING [8, 13]. Thus, secondary contacts can be optimized with respect to load capacity as well as friction. This allows to support the design of frictionand load-optimized machine elements and to develop energy-efficient technical drive systems, which is becoming more and more important especially against the background of current challenges, such as the scarcity of resources as well as environmental and climate protection. Aus Wissenschaft und Forschung 60 Tribologie + Schmierungstechnik · 69. Jahrgang · 5-6/ 2022 DOI 10.24053/ TuS-2022-0045 ficients of friction in both cases and for all radii. In addition, a larger difference in the coefficient of friction can be seen in case 2. In case 2, see Figure 4 (d), the difference is much larger for small radii. 4 Conclusion and Summary The results of the two considered cases, torus-cone and torus-torus from Figure 3, show plausible and expected relationships for all the investigated radii. Larger radii result in a larger contact area, which reduces the pressure in the contact and allows the lubricant to remain longer in the contact, which increases the lubricant film thickness, as can be observed in Figure 5. The deviations in the pressure distribution result due to the different accuracy of the used geometry description. With the method similar to H ERTZ , the contact pairing is mapped using an ellipsoid, resulting in a symmetrical, elliptical contact surface. The use of the P IM P method generates the real geometries, resulting in curved, non-symmetric ellipses, as already shown by K ELLEY and W IRSCHING in [6, 14]. The contact area thus differs in shape and size for the two geometry generation methods used, as can be seen in Figure 5. As seen in Figure 5, there is a greater difference in the shape and size of the contact surface when an additional torus geometry is used. The real geometry thus deviates more strongly from the simplified description using ellipsoids similar to H ERTZ , which in turn affects the calculation of the reference result variables maximum pressure and minimum lubrication gap. The greater the deviation of the contact area, the greater the deviation in the maximum pressure as well as in the minimum lubrication gap, as can be seen in Figure 4 (b, c, e, f) in comparison with Figure 5. A higher pressure and a lower lu- Figure 5: Comparison of exemplary pressure surface distribution of the different geometry generations (H ERTZ , P IM P) for case 1 torus-cone with the radius of 75 mm (a, c) as well as case 2 torus-torus with the radius 50 mm (b, d) TuS_5_6_2022.qxp_TuS_5_6_2022 09.02.23 16: 31 Seite 60 References [1] A RAMAKI , H., C HENG , H. S. u. Z HU , D.: Film Thickness, Friction, and Scuffing Failure of Rib/ Roller End Contacts in Cylindrical Roller Bearings. Journal of Tribology 114 (1992) 2, 311-316. 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