Influence of surface form deviations on friction in mixed lubrication

Influence of surface form deviations on friction in mixed lubrication

Accepted Manuscript Influence of surface form deviations on friction in mixed lubrication S. Fricke, C. Hager, S. Solovyev, M. Wangenheim, J. Wallasch...

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Accepted Manuscript Influence of surface form deviations on friction in mixed lubrication S. Fricke, C. Hager, S. Solovyev, M. Wangenheim, J. Wallaschek PII:

S0301-679X(17)30269-4

DOI:

10.1016/j.triboint.2017.05.032

Reference:

JTRI 4751

To appear in:

Tribology International

Received Date: 4 September 2016 Revised Date:

19 May 2017

Accepted Date: 22 May 2017

Please cite this article as: Fricke S, Hager C, Solovyev S, Wangenheim M, Wallaschek J, Influence of surface form deviations on friction in mixed lubrication, Tribology International (2017), doi: 10.1016/ j.triboint.2017.05.032. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Influence of Surface Form Deviations on Friction in Mixed Lubrication S. Fricke1,2*, C. Hager1, S. Solovyev1, M. Wangenheim2, J. Wallaschek2 1

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Central Research, Robert Bosch GmbH Institute of Dynamics and Vibration Research, Leibniz Universität Hannover

Abstract

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The focus of this study is on the effect of form deviations of a partial journal bearing’s sliding surface on the friction force in mixed lubrication conditions. The measured friction varied considerably when the bearing was rotated in the opposite direction at the same speed and normal load. This unexpected observation motivated the presented simulation study. The overall form deviations of the surface of the test bearing were measured with white light interferometry and used as simulation input. The two-scale simulation approach considers the effect of the surface roughness on a microscopic and the global bearing geometry with the surface form deviations on a macroscopic scale. Simulation results show that the surface form deviations can have a noticeable effect on the pressure distribution of the lubricant and hence on the size of the asperity contact area which leads to the differences in friction. The influence of the lubricant viscosity, the clearance gap and the surface roughness were analysed in a parametric simulation study. Results show that the clearance and the surface roughness control the impact of the surface form deviations on the friction force. All in all, the results show the necessity to include all scales of surface form deviations in the simulation of journal bearings in a mixed lubrication regime. Keywords: mixed lubrication, friction force, form deviations, journal bearing *Corresponding author: Sebastian Fricke ([email protected])

INTRODUCTION

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In many applications, friction forces are relevant for the efficiency, functionality or reliability of a machine element. A considerable amount of research effort has been devoted to the determination of friction forces by simulations instead of experiments to reduce development time and cost. Simulation models have to be validated before they can be applied to product development processes. This paper presents first results of a study with the goal to validate the computation of the friction force of a journal bearing. Stribeck [1], [2] identified three friction regimes of lubricated contacts depending on the sliding speed as illustrated in figure 1: boundary friction, mixed lubrication and hydrodynamic friction. The simulation results of contacts in the hydrodynamic friction regime are reliable for most material combinations [3]. Contacts in mixed lubrication conditions are, however, more complex and include a combination of hydrodynamic and boundary friction. Many physical and chemical mechanisms appear which are very difficult to include in simulation models. Understanding complex tribological systems in mixed lubrication still requires a combination of experiments and simulations.

Figure 1: The Stribeck curve

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In experiments conducted for the presented study, the friction force varied considerably when changing the sense of rotation. The sliding surface form of the bushing was scanned using white light interferometry and used as simulation input. The simulation showed a large influence of the roundness deviations on the lubricant film pressure build-up which explains the observed dependency of the friction force on the sense of rotation. This paper consists of three parts: In chapter 2, the state of the art regarding the experimental and numerical analysis of mixed lubrication with the focus on the friction force is briefly described and discussed. Chapter 3 presents the applied experimental and simulation methods and discusses a definition of the different scales of surface form deviations. The results and comparison of experiment and simulation and are presented and discussed in chapter 4.

STATE OF THE ART

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Patir and Cheng [5], [6], [7] introduced the expanded Reynolds equation [8] with the flow factor concept to consider the impact of the surface roughness on the pressure distribution of the fluid on a global level. Rienäcker [9] confirmed the approach of Patir and Cheng and connected it with the contact model of Greenwood and Tripp [10] which allowed the separation of the hydrodynamic and asperity contact pressure. Continuing his work, Lagemann [11] combined the half space theory by Boussinesq [12] with the flow factor theory of Patir and Cheng which allowed the asperity contact pressure to be computed for real measured surfaces. These works established the combined microscopic and macroscopic simulation approach that is used in this paper. Many studies used this simulation approach to compute friction forces in mixed lubrication conditions: Bartel [13] compared the experimentally determined friction force of a journal bearing with a steel journal and aluminium shell with simulation results. A surrogate model was used to compute the fluid pressure. The computed friction force agreed well although the computed values in the mixed lubrication regime were slightly higher than the measurement. Illner [14] analysed a sliding contact in mixed lubrication under transient conditions. His results were presented for the application in a diesel injection pump and showed a good agreement between experiments and simulation. Issakson and Nilsson [15], [16], [17] focused on the friction force of a partial journal bearing lubricated with hydraulic oil with a bushing made of cast iron and a steel roller. The bearing had an additional lubricant supply pipe to support the film build-up. The simulation results also agreed very well with the measured friction force for a certain surface roughness. The simulation over- or underestimated the friction for higher or lower surface roughness, respectively. Sahlin [18] [19] validated the simulation method with flow measurements between two rough flat plates and got an excellent agreement. Habel [19] computed the transient friction force of a partial journal bearing under reciprocating movement. The material combination was polymer-steel lubricated with poly-alpha-olefin and grease. The experimental results agreed very well with the computations, too. Umbach [21] implemented the measured 2D contour of the experimentally analysed journal and bushing which influenced the simulation results considerably. The computed friction force agreed with the measurements over some operation regimes for an extensive range of material and lubricant combinations. Comprehensive comparisons of the friction force of journal bearings in weak mixed lubrication conditions were conducted by Priestner [22], [23], Allmaier [3], [24], [25], [27] and Sander [28], [29], [29] using the simulation tool AVL Excite [30]. In all of the works, the Greenwood and Tripp model [10] was used for the determination of the asperity contact pressure. The simulation results agree very well with the measurements. In summary, the published simulation results in literature give confidence that the micro- and macroscale simulation approach with load sharing is a reliable method to predict the friction force of journal bearings in mixed lubrication conditions for different material and lubricant combinations (further information given in Bartel [31] and Larsson [32]). However, the published results also show that adaptations of the method for a better prediction quality are still needed for new material and lubricant combinations. There are alternative methods to model contacts in mixed lubrication conditions: Dobrinka [33], [34] developed a deterministic simulation model applied to a partial journal bearing in mixed lubrication conditions. The model is limited to small journal bearings in weak mixed lubrication with a low asperity contact area. Lu [35] compared the results of a mixed elastohydrodynamic lubrication model for line contacts with the measured friction force of an oillubricated journal bearing. The results agreed very well over all friction regimes and varied test temperatures. Wang [36] and Wang [37] compared the deterministic approach of Hu and Zhu for point contacts [38], [39] for experiments with a steel ball to flat plate, which revealed a very good agreement for rough surfaces. The friction for smooth surfaces was underestimated. Albers and Lorentz [40], [41], [42] analyse mixed lubrication in the microscopic scale in three dimensions, which allows analysing the impact of the surface 2

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topography on the friction coefficient and on the thermal dissipations in detail. The results agree with observations in experiments. Akchurin [43] presented a simulation method to compute the friction force of line contacts. His comparisons to friction measurements of a cylinder on a plate showed a very good agreement. All in all, the understanding of contacts in mixed lubrication regime improved a lot combining experiments and simulations. However, none of the previous studies included form deviations on all scales or analysed in detail the dependency of the friction force on surface form deviations.

3.1

THEORY AND METHODS Definition of Surface Form Deviations

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The DIN standard DIN 4760 [4] differs 6 scales of surface form deviations ranging from large area surface form deviations (1st order) down to the atomic structure of the material surface (6th order). Figure 2 shows the application of this differentiation on the analysed bearing. 1st order form deviations are roundness deviations of the bearing. 2nd order surface form deviations correspond to the waviness of the surface (e.g. scratches). 3rd to 4th order form deviations relate to the microscopic surface roughness.

Figure 2: Definition of surface form deviations applied on the journal bearing according to DIN 4760

Experiment: Measurement of the Friction

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Figure 3 shows the analysed experimental setup. The focus of this study lies on the friction of the partial journal bearing consisting of the bushing and roller which is driven by an external driving shaft. All parts of the partial journal bearing are made of steel. The bushing is coated with DLC (diamond like carbon) and has an opening angle of 200°. All tribological contacts of the analysed setup are completely immersed into a bath filled with a low viscosity lubricant to ensure a sufficient lubrication.

Figure 3: Analysed partial journal bearing

A custom-built test rig measures the friction between the roller and bushing directly (see figure 4a). The normal force FN is applied with a pneumatic cylinder. The normal force FN, the rotational speed of the driving shaft ωd, the temperature T and normal load distribution over the roller fN are controllable. The measurement principle is explained with the cut view shown in figure 4b. The bearing (1) is mounted on the pendulum (3) which can rotate about the axis of the driving shaft (2). The pendulum is mounted on airlubricated bearings and can hence rotate almost without friction. 3

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Figure 4: a: photo of the test rig b: cut view with the acting forces and moments

The roller is driven by the driving shaft which transfers the roller drive force  . The equilibrium of torque about the roller including the friction moment MBR can be written as

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The influence of the friction between roller and driving shaft FDf can be neglected, since only the transferred roller drive force FTf acts on the pendulum and on the roller. The friction moment MBR is determined by measuring the moment xhFM acting on the pendulum. Equation 2 gives the equilibrium of moments of the pendulum about its rotation axis   = −  ( . 2)

Combining equation 1 and 2 allows to determine the friction moment MBR. The friction coefficient µ is defined as    = ( . 3)    

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The drawback of this test rig concept is the impossibility to measure the boundary friction regime between roller and bushing. At very low rotational speeds, the friction moment between roller and bushing MBR is higher than the transferred moment due to the roller drive force FTf. This jams the roller and the test rig hence measures the friction of the emerged sliding contact between the driving shaft and roller. The friction force is measured stepwise: The normal force FN, normal load distribution fN, temperature T and rotational speed ωd are held constant for 20 seconds. The friction coefficient  is determined from the mean value of  and  measured within this time interval. Two Stribeck-curves are measured in each turning direction successively. The testing sequence is repeated until the difference between the two Stribeck-curves in the same sense of rotation is negligible to ensure a sufficient run-in. Simulation Method

The friction force is computed with a coupled simulation on the microscopic and macroscopic scale. The microscopic simulation uses 3D-scans of the surfaces of the tested journal bearing as input. It computes the flow factors according to Patir and Cheng [5], [6], [7] and Lagemann [11] to consider the impact of the surface roughness on the hydrodynamic pressure of the fluid on the macroscopic scale. Additionally a contact pressure curve is computed to determine the load carried by the asperity contact. The macroscopic simulation computes the asperity and hydrodynamic pressure distributions of the bearing and the friction considering the impact of the 1st and 2nd order surface form deviations. Figure 5 shows the measured boundary conditions which are used as simulation input. The viscosity of the lubricant was measured over a temperature and pressure range. The temperature of the bushing and the lubricant bath was measured by thermal elements. The axial contour of the roller was measured and rotated. The resulting geometry was used in the simulation. The geometry of the bushing was measured with white light interferometry. The normal load distribution was measured with a pressure sensitive foil which changes its color under pressure. The load distribution is determined by measuring the discoloration and computing a corresponding pressure. The 4

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Figure 5: Measured boundary conditions used as simulation input

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surfaces of the bushing and roller were scanned by a confocal microscope. A nanoindenter was used to determine the Young’s modulus of the surface. Finally the boundary friction coefficient was measured on an additional test rig which directly drives the roller to allow low rotational speeds.

(, ) =

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Microscopic Simulation The contact pressure curve gives the relation of the contact pressure  and the film thickness ℎ. The computation of the deformation of the discretized measured surfaces of the roller and bushing in contact is based on the half space theory of Boussinesq [12]. The displacement (, ) of a point in the half space due to the normal force FN at the position x’,y’ is computed by 1− !  ( $ ,  $ ) # ( . 4) " %( −  $ )! + ( − ′)!

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with the measured Young’s modulus . Ideal elastic-plastic behaviour of the material is assumed. The deformation of the surfaces is computed by applying equation 4 on all points in contact. A more detailed description of the computation method can be found in [11] and [31]. The flow factors )(ℎ) give the ratio of a flow through a gap with surface roughness to the flow through a smooth gap. They are computed solving the Reynolds equation on the deformed microscopic surfaces. The pressure flow factor Φ+,, describes the impact of the roughness on the Poiseulle-term of the Reynolds-equation. It is defined for a flow in xdirection as

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8 1 1 5ℎ67 : Φ+ = ./ 2- 3 # ;Ω< ( . 5) Ω 129 :

The shear flow factor describes the impact of the roughness on the Couette-term. It is defined for a flow in xdirection as 8 2 1 5ℎ67 : 3 # ;Ω< ( . 6) ?5 Ω 129 :

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Φ+> =

Macroscopic simulation The pressure distribution of the fluid  is computed with the Reynolds equation including the pressure flow > and shear flow factors )+,, factors )+,, 8 8 : : ?0 + ?! :(5ℎ) A!! − A0! u! − u0 :(5ℎ)+> ) :(5h) - 5ℎ : - 5ℎ : 3)+ <+ 3), <= − + ( . 7) ! : :D : 129 : 129 : 2 : 2 : A0!

The equilibrium of forces is computed with the pressure distribution of the fluid  and the local contact pressure  derived from the contact pressure curve  (ℎ)  = # ( +  )F GGGGH ;Ω ( . 8) , I

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The total friction force  is the sum of the asperity friction , and the hydrodynamic friction , . The measured boundary friction coefficient K independent from the sense of rotation is used to determine the asperity friction force. The shear stress L of the fluid is determined according to Newton’s law of viscosity.  = M,N + M,ℎ = # K  (, );Ω + # L(, );Ω ( . 9) I

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Coupling of the Micro- and Macroscopic Simulation The micro- and macroscopic simulation are coupled via the definition of the film thickness ℎ. In the microscopic simulation ℎ is defined as the distance between the mean height of the deformed surfaces, illustrated in figure 6a. This film thickness is used in the macroscopic simulation as the distance between the roller and bushing surface to include the contact pressure curve  (ℎ) and the flow factors )(ℎ). 1st and 2nd order surface form deviations are considered in the macroscopic simulation by adding the measured roundness and waviness PQ of the bushing on the clearance QK of the bearing, as shown in figure 6b. The clearance QK is difference between the diameter of the roller and bushing without roundness deviations. A final clearance distribution is hence computed by Q(, ) = QK + PQ(, ) RSDℎ PQ < QK ( . 11)

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Due to the asperity and hydrodynamic pressure distribution the roller axis is deflected by the eccentricity e. This deflection leads to the converging gap required for the hydrodynamic pressure build up as shown in figure 6b.

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4.1

RESULTS AND DISCUSSION

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Figure 6: Definition of the clearance c0, roundness deviation of the bushing ∆c, resulting clearance c, film thickness h in the micro- and macroscopic simulation and the eccentricity of the roller axis e

Validation

Figure 7 shows the comparison of the computed and measured Stribeck-curves. As it can be seen, the dependency of the measured and computed friction on the turning direction coincides qualitatively. The simulation predicts both Stribeck-curves with an acceptable accuracy. Three rotational speeds ω are marked in figure 7. ω1 marks the end of the boundary friction regime. Due to the jamming of the roller (see description of the test rig in chapter 3.2) the simulation cannot be validated in this regime. ω2 lies in the middle of the mixed lubrication regime. However, the turning dependency of the computed friction is quantitatively less than the experiment. The hydrodynamic regime begins at the rotation speed ω3. The simulation predicts a slightly higher friction. However the experiment and the simulation show no dependency on the sense of rotation.

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Figure 7: Comparison between simulation and experiment (experimental and simulation results)

Physical Mechanism: Explanation for the Dependency of the Friction on the Sense of Rotation

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The computed asperity and hydrodynamic pressure plots of the tested bearing reveal the reason for the dependency of the friction on the sense of rotation. Figure 8 shows the plots of the asperity contact pressure distributions of the bearing for both rotation directions for the three rotational speeds ω1, ω2 and ω3 marked in figure 7. Accordingly, figure 9 shows the hydrodynamic pressure distributions.

Figure 8: Plots of the asperity contact pressure of the bearing in opposite rotating directions with the same normal load; The scratches due to wear are visible. upper row: clockwise rotation, lower row: counterclockwise rotation, meaning of the scale: red – high pressure, blue - low pressure (simulation results)

At rotational speed ω1, the bearing operates at the end of the boundary friction regime where the normal load is almost completely supported by the asperity contact. As it can be seen, the asperity contact pressure distribution is very similar in both rotating directions, which leads to a similar friction force. The hydrodynamic pressure distribution clearly differs. In the mixed lubrication regime at the rotation speed ω2 the load is carried by the asperities and by the lubricant film. The hydrodynamic pressure distribution still differs considerably. This leads to a different total asperity contact area shown in figure 8, which is the reason for the considerable difference of the friction force in opposite rotating directions. 7

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At the start of the hydrodynamic regime at the rotation speed ω3, the asperity contact in both turning directs differs only slightly. Although the lubricant film is almost fully built up, some differences of the hydrodynamic pressure distribution can be observed. Obviously the hydrodynamic pressure build-up is responsible for the turning-dependency of the computed friction. The reason why the hydrodynamic pressure is build-up better in counterclockwise turning direction is explained with figure 10. The roundness deviations of the bushing and the hydrodynamic pressure at the axial position x=0.5b in both rotating directions at the rotational speed ω2 is plotted. In counterclockwise rotating direction, the roller surface slides towards an ascending bushing surface which leads to a converging gap. This supports the pressure build-up. Vice versa the gap expands in clockwise turning direction and the pressure build-up is disturbed.

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Figure 9: Plots of the hydrodynamic pressure of the bearing in opposite rotating directions with the same normal load; upper row: clockwise rotation, lower row: counterclockwise rotation, meaning of the scale: red – high pressure, blue low pressure (simulation results)

Figure 10: Ratio of hydrodynamic pressure and local surface form deviation ∆c at x=0.5b of the bearing in opposite rotating directions at the rotational speed ω2 and the same load (simulation results)

4.3

Simulation Parameter Study

The influence of the clearance c0, the viscosity η0 and the roughness of the bushing (3rd and 4th order surface form deviations) on the dependency of the friction on the turning direction was studied with simulations. The 8

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parameters were simply varied as specified in table 1. The validation simulation case was used as a reference and apart from the changed parameter the same boundary conditions such as the measured geometry of the bushing were used. Table 1: Overview of the computed cases

Standard deviation surface bushing

Clearance

Viscosity of lubricant

reference 1 2 3

σ2,0 2σ2,0 σ2,0 σ2,0

c0 c0 c0 c0/2

η0 η0 η0/2 η0

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Case

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Figure 11 shows the ratio of the computed friction in clock- and counterclockwise sense of rotation over the dimensionless rotation speed for the reference case and the cases 1-3. The reasons for the impact of the parameter change on the turning dependency is discussed in the following.

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Figure 11: Ratio between the computed friction in clockwise (cw) and counterclockwise (ccw) rotation direction (simulation results)

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Case 1: Surface Roughness In case 1 the standard deviation (σ=Rrms) of the surface of the bushing σ2,0 was doubled. The corresponding flow factors and contact pressure curve for the rougher surface were computed with the microscopic simulation and considered in the macroscopic simulation. As it can be seen in figure 11 the ratio between the friction in clock- and counterclockwise turning direction and hence the impact of the roundness deviations on the friction is shifted to higher sliding speeds and reduced. It is assumed that higher asperity peaks interact at higher film thicknesses, but also lead to more deformations which compensates to a certain degree the surface form deviation. Case 2: Viscosity For case 2 the viscosity η0 was divided in half. As it can be seen in figure 11 the viscosity has no effect on the influence of the roundness deviations on the friction. This is expected, since changing the viscosity only changes the load capacity of the lubricant film. The only effect is that the hydrodynamic pressure build-up shifts to lower respectively higher sliding speeds. Case 3: Clearance The clearance is c0 divided in half for case 3. Reducing the clearance between roller and bushing shifts the Stribeck curves to lower sliding speeds. The influence of surface form deviations is increased as illustrated in figure 11 and figure 12. In case 3 the friction is nearly doubled in clockwise turning direction. Decreasing the clearance leads to a higher converging gap as illustrated in the lower part of figure 12. This improves respectively worsens the hydrodynamic pressure build up even more.

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CONCLUSION AND OUTLOOK

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Figure 12: Impact of the clearance on the convergence of the gap

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The impact of form deviations of a partial journal bearing’s sliding surface on the friction force in mixed lubrication conditions is studied by simulations. The study was motivated by an experiment where the friction force of a partial journal bearing varied considerably when rotating in opposite directions. The complete surface of the tested journal bearing bushing was scanned using white light interferometry and was considered in the simulation. The main goal of this study was to understand under which circumstances surface form deviations have a considerable influence on the friction force. A parametric study varying clearance gap, lubricant viscosity and surface roughness was conducted. The following results were received: • Surface form deviations of the bearing surface can considerably influence the friction force of journal bearings, since they have a pronounced effect on the lubricant film pressure build-up. In mixed lubrication conditions, this reduces or increases the asperity contact area which dominates the friction. Due to this effect, the asperity contact area depends on the direction of rotation at the same sliding speed and load. It can have the same impact on the emerging friction force as other contact properties, such as the surface roughness or the lubricant viscosity, and should hence be included in the simulation. • Experiment and simulation agree well. • Increasing the surface roughness of the bearing surface reduces the influence of the roundness deviations on the friction. • Lubricant viscosity has no impact on the influence of the roundness deviations on the friction. • Increasing the journal bearing clearance reduces the influence of the surface form deviations. It has to be underlined that the presented study and all conclusions are mainly based on simulation results. It is also important to note that all results received are only valid for the measured surface form deviation analysed here.

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NOMENCLATURE FDf

Symbols b c e E

FM FN fN FR FTf

bearing width bearing clearance eccentricity Youngs modulus 10

friction between roller and driving shaft measured force normal force normal force distribution friction force transferred roller drive force

ℎ67

MBR p rd rr t u x,y,z xh µ μ0 η ρ

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σ2 σ12

film thickness local film thickness moment between roller and bushing pressure radius of the driving shaft radius of the roller time sliding speed direction of coordinate system distance of contact roller to measurement point FM friction coefficient boundary friction coefficient viscosity density

Ω ω )+,, > )+,,

standard deviation roughness surface 1 combined standard deviation roughness surface 1 and 2 area rotational speed pressure flow factor shear flow factor Possion’s ratio

Index a h 0

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normal vector

asperity hydrodynamic reference value

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REFERENCES

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1. R. Stribeck, Kugellager für beliebige Belastungen, Zeitschrift des Vereines deutscher Ingenieure, 45(3):73–9 (part I) and 45(4): 118–125 (part II), 1901 2. R. Stribeck, Die wesentlichen Eigenschaften der Gleit- und Rollenlager, Zeitschrift des Vereines deutscher Ingenieure, 46(37):1341-1348 (part I) and 46(38):1432-1438 (part II) and 46(39) 1463–1470 (part III), 1902 3. H. Allmaier, D.E. Sander, F.M. Reich, Simulating Friction Power Losses In Automotive Journal Bearings, The Malaysian International Tribology Conference 2013, MITC2013, Procedia Engineering 68 (2013) 49 – 55, 2013 4. Norm: DIN 4760:1982-06, 1982 5. N. Patir, H. Cheng, An average flow model for determining effects of three- dimensional roughness on partial hydrodynamic lubrication. ASME, Transactions. Journal of Lubrication Technology 1978;100:12– 7, 1978 6. N. Patir, H. Cheng, Application of average flow model to lubrication between rough sliding surfaces. ASME, Transactions, Journal of Lubrication Technology 1979;101:220–30, 1979 7. N. Patir, Effects of surface roughness on partial film lubrication using an average of model based on numerical simulation ,Northwestern University, PhD thesis, 1978 8. O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamps towers experiments, including an experimental determination of the viscosity of olive oil, Philospohical Transactions Royal Society 177, 1886 9. A. Rienäcker, Instationäre Elastohydrodynamik von Gleitlagern mit rauhen Oberflächen und inverse Bestimmung der Warmkontur, RWTH University, PhD thesis, 1995 10. J.A. Greenwood, J.H. Tripp, The elastic contact of rough spheres, Journal of Applied Mechanics 89 3, 153-159, 1967 11. V. Lagemann, Numerische Verfahren zur tribologischen Charakterisierung bearbeitungsbedingter rauer Oberflächen bei Mikrohydrodynamik und Mischreibung, University of Kassel, PhD thesis, 2000 12. J. Boussinesq, Application des Potentiels à l’Etude de l’Equilibre et du Mouvement des Solides élastiques, Gauthier-Villars, Paris, 1885 13. D. Bartel, Berechnung von Festkörper- und Mischreibung bei Metallpaarungen, University of Magdeburg, PhD thesis, 2000 14. T. Illner, Oszillierendes kippbewegliches Axialgleitlager bei Grenzreibung und Kraftstoffschmierung, University of Magdeburg, PhD thesis, 2010 15. P. Isaksson, D. Nilsson, R. Larsson, Elasto-hydrodynamic simulations of complex geometries in hydraulic motors, Tribology International 42 (2009) 1418–1423, 2009 16. D. Nilsson, B. Prakash, Influence of different surface modification technologies on friction of conformal, tribopair in mixed and boundary lubrication regimes, Wear 273, no.1, 2011 Nov 1, p.75(7) (ISSN: 00431648), 2011 11

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Roundness deviations can have a considerable effect on the friction force of a journal bearing in mixed lubrication conditions Simulation results show a pronounced effect of the roundness deviations on the lubricant film pressure build-up and hence on the size of the asperity contact area which leads to the differences in friction The results show that it is required to include form deviations on all scales for validation purposes

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