Aus Wissenschaft und Forschung 25 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 Use of analytically describable geometries to calculate the contact between rolling element face and rib in bearing simulations Sven Wirsching, Sebastian Schwarz, Stephan Tremmel* Eingereicht: 22. Januar 2020 Nach Begutachtung angenommen: 8. Mai 2020 Die Vielzahl der Kontaktstellen in Wälzlagern tragen abhängig von der Belastungssituation unterschiedlich zur Reibungsleistung, zur Temperaturentwicklung und damit zum Betriebsvermögen des Wälzlagers bei. Unter axialer Last beeinflusst vor allem der Kontakt zwischen Rolle und Bord den Betreib und damit die Reibungsleistung eines Rollenlagers, wie es bei der neuartigen Angular Roller Unit (ARU) der Firma Schaeffler der Fall ist. Bei bestehenden Programmen zur Auslegung von Wälzlagern erfolgt die Kontaktberechnung zwischen Wälzkörperstirn und Bord in der Regel mit der H ERTZ schen Theorie, was für komplexe Geometrien jedoch eine Vereinfachung darstellt. Dies kann zu Diskrepanzen bei der Berechnung der Einfederungen, Kontaktflächen und Kontaktpressungen verglichen zu hochwertigen numerischen Berechnungen (zum Beispiel Finite Elemente Analyse) führen. In diesem Beitrag wird daher eine neue, genauere Methode zur Berechnung des Kontaktes zwischen Wälzkörperstirn und Bord vorgestellt und mit der Finite Elemente Methode verglichen. Dazu werden die Geometrien analytisch beschrieben und bilden eine Ersatzgeometrie, welche in eine zweidimensionale Bettung, dem Ansatz nach W INKLER folgend, gedrückt wird. Über das Kräftegleichgewicht im Kontakt, ähnlich den Arbeiten von P OPOV , kann das Kontaktgebiet und die daraus resultierende Pressungsverteilung berechnet werden. Schlüsselwörter Kontakt, Pressung, Reibung, Berechnungs- und Simulationsmethoden, Kegelrollenlager, Wälzlager Depending on the load situation, the large number of contact points in rolling bearings contribute differently to the frictional power, temperature development and therefore the operating capacity of the rolling bearing. Under axial load, the contact between the roller and rib in particularly influences the operation and thus the frictional performance of a roller bearing, as it is the case with the new Angular Roller Unit (ARU) from Schaeffler. In existing programs for the dimensioning of rolling bearings, the contact calculation between the rolling element face and rib is usually based on the theory of H ERTZ , although this is a simplification for complex geometries. This can lead to discrepancies in the calculation of deflections, contact surfaces and pressure compared to high quality numerical calculations (for example finite element analysis). In this paper a new, more accurate method for calculating the contact between the rolling element face and rib is presented and compared with the finite element method. The geometries are described analytically and form a substitute geometry which is pressed into a two-dimensional bedding following the approach of W INKLER . By means of the equilibrium of forces in the contact, similar to the work of P OPOV , the contact area and the resulting pressure distribution can be calculated. Keywords Contact, pressure, friction, calculation and simulation methods, tapered roller bearings, bearings Kurzfassung Abstract * Sven Wirsching, M.Sc. Sebastian Schwarz, M.Sc. Dr.-Ing. Stephan Tremmel Orcid-ID: https: / / orcid.org/ 0000-0003-1644-563X Lehrstuhl für Konstruktionstechnik KTmfk Friedrich-Alexander-Universität Erlangen-Nürnberg 91058 Erlangen, Deutschland TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 25 compared with finite element calculations in order to check the quality of the method. 2 Description of the models An essential objective in the dimensioning of modern roller bearings is the reduction of friction power. For this purpose, computationally intensive simulation tools are used so that the dynamic operating behavior can be recorded. Due to the already mentioned weaknesses of previously used calculation methods of contacts in rolling bearings, the new method should contribute to a more precise determination of the contact surface and pressure and thus to a more accurate and at the same time calculation-efficient determination of the frictional power. First, the geometries used in practice at the rolling element face and rib are analyzed. Second, these are generalized and the relevant parameters for describing the geometry are determined. Then, the method developed for pressure distribution in point contact (“Pressungsverteilung im Punktkontakt”) P IM P is explained. Finally, the pressures calculated with the method for both simple (ball) and complex (toroidal) geometries are compared with the results of a finite element analysis (FEA), in order to ensure a high quality of results of the presented method. 2.1 Description of the parameters of the geometry pairings to be investigated Different geometries of the rolling element face and rib, as typically used in roller bearings, are investigated. From this, the relevant parameters for their description and relevant geometry pairings are determined. The geometry of the rib can be reduced to two parameters: The so-called rib angle α between raceway and rib and the so-called rib radius r B , see Figure 2 (b). The geometry of the roller face can be described by the face radius r R and the eccentricity e, i.e. the distance of the center to the roller axis, see Figure 2 (b). In the next step, the parameters for generalized roller bearings are defined in such a way that the contact point between the roller face and rib always has the same distance to the roller axis as shown in Figure 2 (b). Aus Wissenschaft und Forschung 26 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 1 Introduction Roller bearings have a large number of different contacts, for example between the rolling elements and the raceways, the rolling elements and the cage pockets or other rolling elements, as well as between the rolling element face and the rib. Depending on the load situation, each of these contacts influences differently the frictional power, the temperature development and thus the operating capacity of the bearing. For special applications, for example in the gear sector, Schaeffler developed a new type of bearing design, the so-called Angular Roller Unit (ARU) [7]. This represents a very interesting case with regard to the rib contact situation. The ARU is geometrically based on a tapered roller bearing, but with an additional rib at the outer ring, as shown in Figure 1 (b). The rib arrangement is analogous to that of cylindrical roller bearings for support and fixed bearings ‒ thus the ARU can be used as a fixed bearing. The axial load capacity in the preferred direction is comparable to that of tapered roller bearings and in the non-preferred direction similar to that of cylindrical roller bearings. For an effective design of the ARU in order to fully exploit its performance capacity, it is necessary to calculate the frictional power for these load cases as precisely as possible. The frictional power is significantly influenced by the contact between the rolling element face and the rib. Quasistatic [6] and dynamic [8] rolling bearing simulation programs based on multi-body systems are used for the dimensioning. The state of the art for calculating the contact between the rolling element face and rib in calculation tools for dimensioning rolling bearings is the theory of H ERTZ [3] or the so-called disk model, as described by T EUTSCH in [9]. However, more complex geometries (for example torus), which are typically used to reduce friction between the rolling element face and rib, cannot be described with sufficient accuracy. This leads to more or less large discrepancies in the calculation of the deflection, contact area and shape as well as the contact pressure, compared to more sophisticated numerical calculations (e.g. finite element analysis). In addition, the discrepancies affect the calculation of the frictional power, because this is determined from the solid and fluid friction, which in turn depend on the distribution of the pressure and the sliding speed in the contact area. Therefore, this paper presents a new contact calculation method for the efficient determination of the pressure distribution, which is especially suitable for rolling bearing simulation. In addition, results obtained with this method are Figure 1: Geometrical features of (b) ARU, derived from (a) tapered and (c) cylindrical roller bearings TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 26 A tapered roller bearing with an inner ring bore diameter of 60 mm is used as an example for the investigations. Table 1 lists the determined combinations of geometry and parameters, which are used to change the roller and rib geometry of the bearing for the contact calculation using the Finite Element Method (FEM) and the P IM P method. The contact force F N for a single rolling element is varied from 100 to 400 N with increments of 50 N. These normal contact forces are derived from normal axial loads of roller bearings of mentioned size. 2.2 Structure of the P IM P method For an integration of the P IM P method in multi-body calculation tools for rolling bearing simulation a coupling of the contact calculation with the global multi-body calculation is necessary. In the following, this procedure is described in four steps, as already introduced in [13]. Step 1: Calculation of the contact point In multibody simulations, the position of the individual bodies in relation to each other must be determined first. For this purpose, the position of the body’s own coordinate systems is determined in a global, fixed coordinate system, as described by V ESSELINOV in [10]. In the following, an idealized contact point is selected based on the geometries of the bodies, in which the bodies touch each other at exactly one point in the sectional plane, defined by the axes of rotation of the rolling element and ring. The position of the contact point is described with the aid of the position vector q in the body’s coordinate systems. The normal vectors n of the surfaces of the two bodies are generated in the contact point. Step 2: Generation of the local geometry The geometry of each contact partner is described analytically in a further Cartesian coordinate system G. In this system, the forms considered here can be represented by one of the three equations mentioned in (1), see [1]. (1) The local vector q(x,y,z) describes not only the position of the contact point but also the position of a projection plane. The normal vector n(x,y,z) specifies the orientation of the projection plane in the contact point for each body. In the Cartesian coordinate system G, the geometries of the contact partners above the projection plane can be described by means of the ray tracing method with the ray equation (2) (see also G LASSNER [2]) and the vectors q(x,y,z) and n(x,y,z). (2) By inserting equation (2) in (1) and resolving to t, a function g G (x,y) can be formulated which describes the distance of each point of the geometry to the contact plane. For example, in the case of a sphere, the result is as shown in Figure 3 (a): Aus Wissenschaft und Forschung 27 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 No. geometry roller geometry rib r R in mm e in mm α in ° r B in mm 1 sphere cone 29 0 92 0 57 0 90 0 2 sphere torus 29 0 92 30 57 0 90 30 3 torus cone 14 2 90 0 28 2 89 0 4 torus torus 14 2 90 30 28 2 89 30 Figure 2: (a) Determination of the vectors for the position of the contact point; (b) Relevant parameters for the description of the geometry at the rib and the roller face Table 1: Geometry and parameter combinations for contact calculation with FEM and P IM P + − w ∙ U = 0 + + U = y ( + + U + } − y ) = 4} ∙ ( + ) cone: sphere: torus: l⃑(6) = €⃑ + 6 ∙ ‚⃑ TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 27 residual stress and frictionless, ideally smooth surface. Under these conditions the contact force (similar to P OPOV in [4]) can be calculated for each discrete element according to equation (5). This spring element is deformed by the displacement profile u z (x,y) and, together with the material properties described by the reduced Young’s modulus E* from equation (4), causes a reaction force ΔF z x/ y (x,y) for the respective discretization direction, as shown in equation (5). (5) With equation (5) the contact force ΔF z (x,y) = ΔF zx (x,y) · X (x) + ΔF zy (x,y) · Y (y) on an area element ΔA = Δl x · Δl y can be calculated. All forces ΔF z x/ y (x,y) are determined for each row of spring elements in xand y-direction, see Figure 4. Aus Wissenschaft und Forschung 28 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 At the contact point t has the value zero. In this way, the geometries of both bodies are described by the distance functions g 1 (x,y) and g 2 (x,y). The two projection planes are parallel and congruent at the contact point. In the following, you can create a substitute geometry with the equation g(x,y) = g 1 (x,y) - g 2 (x,y) by the difference of these geometry descriptions. The contact of the substitute geometry with a plane describes equivalently the contact of the two rigid bodies. A displacement of the substitute geometry g(x,y) into the contact plane corresponding to the penetration d gives the displacement profile u z (x,y) = g(x,y) - d. The intersection of the substitute geometry with the contact plane is the contact surface. The contact plane is discretized in xand y-direction as shown in Figure 3 (c). A superposition of onedimensional beddings with spring elements for xand ydiscretization forms the basis for a two-dimensional bedding, similar to what W INKLER shows in [12] for the one-dimensional case. A spring element represents a discrete one-dimensional material section Δl x/ y . The reduced Young’s modulus E* represents the material properties as described by P OPOV in [5]. This is calculated from the Young’s modulus E 1 , E 2 and the Poisson’s ratio υ 1 , υ 2 of the two bodies, see equation (4). (4) Step 3: Calculation of the contact force After the substitute geometry has been generated and the contact has been mapped via a rigid body penetration, the contact force can be determined. The following assumptions are made: isotropic, homogeneous material, purely elastic behavior, dry normal contact, freedom of Figure 3: (a) Geometry generation according to the ray tracing method using the example of a sphere; (b) displacement of the substitutive geometry in xy-plane; (c) contact calculation using the substitutive geometry and the two-dimensional bedding of W INKLER ƒ „ ( , ): 6 = −i€  + €  +€ †  † k + 9i€  + €  +€ †  † k − i€ +€ + € † y k ∆ˆ ‰ ‹/  ( , ) = ∗ ∙ 0 ‰ ( , ) ∙ ∆ / u u = ∗ ∙ [ƒ( , ) − ] ∙ ∆ / u (3) 1 ∗ = 1 − m n n − 1 − m Figure 4: Discretization of the contact surface in bands respectively spring element rows in xand y-direction Each strip is weighted and summed up with the parameters X(x), Y(y) depending on the proportion of the area of the respective direction. These parameters take into account the elliptical surface portion for each direction and are described by X(x) = 4/ (π · a) and Y(y) = 4/ (π · b). This solution is exact for H ERTZ ian and rotationally symmetric contact situations. However, TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 28 if the contact surface diverges strongly from a symmetrical elliptical shape, this is only an approximation. The more precisely the fitting parameters X(x), Y(y) are determined, the more accurate the approximation. To determine the contact force F N of the substitute geometry, Ω must be integrated over the entire contact area. This corresponds to the sum of the partial integrals of the strips for xand y-direction and thus the sum of the forces of each strip per direction. The condition applies that ΔF z (x,y) = 0, if u z (x,y) ≥ 0. Thus, the contact force F N results from equation (6). (6) The contact force F N is an essential result for the solution of the equilibrium of forces in the multi-body simulation. Thus, the determination of the penetration is an iterative process: The contact area and force change due to the continuous adaptation of the penetration. In the equilibrium of forces, the pressure distribution can be calculated from the resulting penetration d res and the resulting area. Step 4: Calculation of the pressure distribution In half-space theory, the surface displacement is calculated on the basis of the pressure, wherefrom the deformations of the contact partners and the area of the contact zone can be derived, as described by W ILLNER in [11]. In contrast, the P IM P method uses the displacement as the starting point for the calculation, which causes a contact force and in turn a pressure. Thus, similar to P OPOV in [5], the pressure can be determined for each discrete element, see equation (7). The discrete contact force ΔF z (x,y) acts on a surface element ΔA, which is represented by spring elements and leads to a discrete pressure Δp(x,y). Taking into account the weighting by the parameters X(x) and Y(y), Δp(x,y) is calculated from the reduced modulus of elasticity E* from equation (4) and the resulting displacement d res of the substitute geometry g (x,y). (7) This allows the pressure distribution to be determined for the entire contact zone. This distribution can also be used for a more precise calculation of friction. 2.3 Description of the finite element model For a verification of the P IM P method, the finite element method (FEM) is used as a higher-level simulation method. The model and its boundary conditions are briefly described at this point. For the simulation, a simplified model of a tapered roller bearing consisting of an inner ring segment and a tapered roller with the dimensions shown in Figure 5 is used. Abaqus/ CAE 6.14 is used as the calculation program. This is used to construct the geometry and mesh it with hexahedral elements type C3D8R. The areas of the two bodies around the raceway and rib as well as the roller shell and end faces are meshed more closely, as shown in Figure 6 (a) by the light blue and yellow zones. Raceway and rib contact are described using a surface-tosurface contact algorithm with the tapered roller as master and the ring as slave component. In the tangential direction, the contact is frictionless and in the direction of the surface normal, the property “Hard” contact is defined. Due to the symmetry, only half of the ring seg- Aus Wissenschaft und Forschung 29 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 ˆ ‘ = ∆ˆ ‰ ‹ ( , ) ∙ ’( ) dx • ‹ + ∆ˆ ‰  ( , ) ∙ –( ) dy •  ˜ = ∗ ∙ [ƒ( , ) − .&% ] ∙ [’( ) + –( )] = ∗ ∙ ˜ [ƒ( , ) − ] ∙ ∆  ∙ ’( ) dx • ‹ + [ƒ( , ) − ] ∙ ∆ u ∙ –( ) dy •  ™ Figure 5: Sketch of inner ring and rolling element geometry Figure 6: Setup of the FE model of rolling element and inner ring segment using symmetry: (a) areas of smaller element size; (b) defined boundary conditions Δ ( , ) = ∆ˆ † ( , ) ∆› TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 29 seen in Figure 7 (b). The contact area, in contrast, is 20 - 30 % larger, as shown in Figure 7 (d). The pressure is dependent on the contact force and the area. Accordingly, for a constant contact force and a smaller maximum pressure, the contact surface must be larger and vice versa, as shown in Figure 7 (b) and (d). If we look at the structure of the P IM P method, it is clear that the deformation of the bodies in the edge areas is insufficiently considered, since the individual spring elements are not coupled and do not influence each other. In addition, the discrete contact forces are transmitted normally to the contact plane and thus normally to the spring element and not to the deformed surface. The shapes of the contact zones according to the FEM and P IM P method in Figure 7 (c) look qualitatively very similar. In principle, the contact of a sphere with a cone has an elliptical shape, as can be seen in Figure 7 (c) for both the FEM and the P IM P method on the basis of the ratio of the half-axes a/ b. The pressure distributions of both methods also show a high degree of similarity, as illustrated in Figure 7 (c). Overall, plausible and high-quality contact results for this geometry pairing can be generated with the P IM P method in milliseconds instead of several minutes with the FEM. 3.2 Geometry pairing No. 2: ball/ torus Geometry pair No. 2 is defined by the combination of a spherical surface on the roller face and a torus on the rib. The corresponding parameters are also listed in Table 1. For this geometry pairing, the same behavior of the maximum pressures of the FEM and P IM P method is shown in Figure 8 (a), analogous to geometry pairing No. 1. The calculated pressures of the P IM P method are 15 - 20 % smaller than those of the FEM and the contact are- Aus Wissenschaft und Forschung 30 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 ment and the roller is modelled and a corresponding symmetry condition is specified. The ring segment is firmly clamped on the inside. The roller is subjected to an axial and radial force Fa and Fr as components of the rib contact force F N , as shown in Figure 6 (b).The forces are adjusted in accordance with the specified flange contact force and the geometry of the roller face and flange are changed in accordance with the values in Table 1. 3 Results from the pressure calculation In the following, the results of the contact calculations by FEM and P IM P method for the four geometry pairs defined in Table 1 are compared and discussed. Important result variables are the maximum pressure and the size of the contact area, as these are important factors for H ERTZ . In addition, the shape of the contact surface is evaluated, because this influences the distribution of the sliding speeds in a subsequent friction calculation. 3.1 Geometry pairing No. 1: ball/ cone For geometry pair No. 1, the roller face is described by a spherical surface and the rib by a conical surface. The parameters defined for this can be found in Table 1. Figure 7 (a) shows that the maximum pressures for various contact forces calculated according to the P IM P method correspond qualitatively well with the results from the FE calculations. The values for the maximum pressures for the same contact force show almost the same distance. The relative deviations in Figure 7 (b) confirm this. Compared to FEM, the maximum pressures calculated using the P IM P method are 15 - 20 % lower, as can be Figure 7: Contact results of the FEM and P IM P method for the geometry pair No. 1: ball/ cone; (a) comparison of the maximum pressures; (b) relative deviation of the maximum pressure to the FEM; (c) comparison of the shape of the contact area; (d) relative deviation of the contact area to the FEM TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 30 as are 20 - 30 % larger, see Figure 8 (a), (b) and (d). The reason is - as already described - the neglecting of coupling in the P IM P method. The similarity in the shape of the contact zone for this pairing can also be seen in Figure 8 (c). Both calculation methods create an elliptical shape of the contact zone. However, the ratio of the semi-axes a/ b shows slight differences in the ellipses, which can also be traced back to the operation of P IM P. The distribution of the pressure over the contact area shows a high degree of qualitative agreement, as can be seen from the color gradient in Figure 8 (c). Consequently, this geometric pairing is described plausibly and qualitatively well with the P IM P method too. 3.3 Geometry pairing No. 3: torus/ cone For geometry pair No. 3, a torus is used on the roller face and a cone surface on the rib. This case cannot be represented with theory of H ERTZ , because in a sectional plane of the torus the curvature radius is not constant. The corresponding geometry parameters can be taken from Table 1. In comparison to FEM, the maximum pressures calculated with the P IM P method are between 10 and 20 % lower according to Figure 9 (a). In contrast, the contact area is 10 - 25 % larger, as shown in Figure 9 (d). The deviation of the maximum pressure Aus Wissenschaft und Forschung 31 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 Figure 9: Contact results of the FEM and P IM P method for the geometry pair No. 3: torus/ cone; (a) comparison of the maximum pressures; (b) relative deviation of the maximum pressure to the FEM; (c) comparison of the shape of the contact area; (d) relative deviation of the contact area to the FEM Figure 8: Contact results of the FEM and P IM P method for the geometry pair No. 2: ball/ torus; (a) comparison of the maximum pressures; (b) relative deviation of the maximum pressure to the FEM; (c) comparison of the shape of the contact area; (d) relative deviation of the contact area to the FEM TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 31 4 Conclusion and summary In this paper the method “Pressungsverteilung im Punktkontakt” (P IM P) is presented, which allows a precise and time-efficient calculation of the contact for use in multibody simulation programs for rolling bearing simulations. The P IM P method is divided into four steps and uses analytical calculation steps, which allows the contact to be calculated in a few milliseconds, in contrast to FEM, whose calculation time can take several minutes depending on the number of nodes and bodies. In addition, the P IM P method provides more accurate results for complex geometries compared to conventional methods (H ERTZ , disk model). To check the quality of the calculation results, the results of the P IM P method are compared with the results of the FEM. The P IM P method, in comparison with FEM, shows differences in the calculation of the maximum pressures and the size of the contact area, which is due to the missing coupling of the spring elements. Thus, elastic deformations of the surfaces, i.e. the flattening of the bodies, are not exactly taken into account. Nevertheless, the shape of the contact zone and the pressure distribution can be mapped qualitatively well with the P IM P method in comparison to an FE solution. All of the geometric pairings examined in this paper provide plausible results when calculating the contact with the P IM P method. Thus, the P IM P method can help to increase the accuracy of the friction calculation mainly by the high-quality description of the shape of the contact zone. Nevertheless, further research work is still required, so that the results are more and more in line with the FEM. This could be achieved, for example, by Aus Wissenschaft und Forschung 32 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 is smaller for small contact forces, as the discrepancy in shape and size of the contact surface is smaller. This can be attributed to lower elastic deformation of the surfaces at low contact forces. For larger contact forces, the shape of the contact zone calculated according to FEM is a curved ellipse, as illustrated in Figure 9 (c). With the P IM P method, a similar shape can be generated. A larger deviation in the curvature of the ellipse leads to a larger deviation in the area, which in turn affects the pressure. The distribution of the pressure over the contact area shown in Figure 9 (c) shows that the pressure at the edge of the contact area is lower when using the P IM P method compared to FEM, since no coupling of the spring elements is taken into account. However, the P IM P method also produces plausible and high-quality results for the torus/ cone geometry pairing. 3.4 Geometry pairing No. 4: torus/ torus In geometry pair No. 4, both the roller face and the rib are toroidal shaped. The corresponding parameters are given in Table 1. The maximum pressures of the P IM P method in Figure 10 (a) show a similar behavior for r R = 14 mm as for geometry pair No. 3, which can also be explained by the curvature of the contact ellipse. For r R = 28 mm, however, the results approach those of the FEM. The deviations in the size of the contact surface are also reduced. The shape of the contact zone and the pressure distribution also show a high degree of similarity, as can be seen in Figure 10 (c). Thus, the P IM P method also allows a high-quality and time-efficient calculation of the contact for toroidal roller face and rib. Figure 10: Contact results of the FEM and P IM P method for the geometry pair No. 4: torus/ torus; (a) comparison of the maximum pressures; (b) relative deviation of the maximum pressure to the FEM; (c) comparison of the shape of the contact area; (d) relative deviation of the contact area to the FEM TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 32 coupling the spring elements so that a flattening of the bodies can be represented by the elastic deformation. It can be expected that the results will further converge with FE solutions. Acknowledgement The authors thank Schaeffler Technologies AG & Co. KG for funding the project and for permission to publish the results. References [1] B RONŠTEJN , I. N.: Taschenbuch der Mathematik. 8., vollst. überarb. Aufl. Frankfurt am Main: Harri Deutsch, 2012 [2] G LASSNER , A. S.: Graphics gems. The graphics gems series. [Nachdr.]. Boston [u.a.]: Acad. Press, 1998 [3] H ERTZ , H.: Ueber die Berührung fester elastischer Körper. 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Düsseldorf: VDI-Verlag 2019, S. 159-172 [9] T EUTSCH , R.: Kontaktmodelle und Strategien zur Simulation von Wälzlagern und Wälzführungen. Maschinenelemente- und Getriebetechnik-Berichte. Techn. Univ, 2005 [10] V ESSELINOV , V.: Dreidimensionale Simulation der Dynamik von Wälzlagern. Karlsruhe: Universität Karlsruhe (TH), 2003 [11] W ILLNER , K.: Kontinuums- und Kontaktmechanik: Synthetische und analytische Darstellung. Berlin, Heidelberg: Springer Berlin Heidelberg; Imprint; Springer, 2003 [12] W INKLER , E.: Die Lehre von der Elasticitaet und Festigkeit mit besonderer Rücksicht auf ihre Anwendung in der Technik : für polytechnische Schulen, Bauakademien, Ingenieure, Maschinenbauer, Architecten, etc. Prag: Dominicius, 1867 [13] W IRSCHING , S., B OHNERT , C., T REMMEL , S. u. W ARTZ- ACK , S.: Method for Calculating the Contact Between Roller End Face and Ring Rib of Roller Bearings in Multi-Body Simulations. 74th STLE Annual Meeting & Exhibition. 20.05. - 23.05.2019, Nashville/ USA: 2019, S.1 - 3 Aus Wissenschaft und Forschung 33 Tribologie + Schmierungstechnik · 67. Jahrgang · 3/ 2020 DOI 10.30419/ TuS-2020-0015 TuS_3_2020.qxp_TuS_3_2020 18.08.20 11: 24 Seite 33