Simulation of Tribological Contacts in Hydraulic Machines

Transcription

1 Simulation of Tribological Contacts in Hydraulic Machines Fredrik Roselin Mechanical Engineering, master's level 2018 Luleå University of Technology Department of Engineering Sciences and Mathematics

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3 Preface This master thesis ends my years as a Mechanical Engineering student at the Luleå University of Technology. This work has been carried out at the support engineer group at Parker Hannifin Pump and Motor Division Europe in Trollhättan, Sweden. I would like to thank my supervisor at LTU, Professor Andreas Almqvist for the support you given me during this work. I also want to thank my supervisor at Parker Hannifin, Per-Ola Vallebrant for all the discussions we had and the support you have given me. I have really enjoyed the time I spent here, so I also want to thank my colleagues. I would also like to express my appreciation to Björn Bragée at Comsol, Jonas Norlin and Tobias Berg at Ansys with all the help regarding those softwares. Since this is my last work as a student at LTU, I also want to thank all the new friends I have met during my studies that made my time there enjoyable. Last but not least, I want to thank my family and old friends for all the support you given me during my studies. Fredrik Roselin Trollhättan, May 2018 i

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5 Abstract Axial piston machines are operating at high pressure and varying speeds, which requires high reliability of the components. The machine components are separated by a fluid film, but sometimes this film gets penetrated by the surface asperities causing the machine to operate with metal-to-metal contact. In order to improve the design and predict the operating conditions might numerical tools be used. The goal with this thesis was to describe how the tribological contacts in Parkers machines can be modelled, considering oil and surface roughness. The so called Luleå Mixed Lubrication Model have therefore been investigated and it has been described how the model can be used in Parkers machines. The model uses a two-scale method to include the influence of real surface topographies of the components, it gives information about how the surface roughness affect the lubrication conditions. Different options of softwares have been investigated and compared to get an understanding of what possibilities and shortcomings they might have when it comes to this kind of simulations. Results show that all interesting contacts can t be treated the same, therefore must each contact be investigated separetly even though the Luleå Mixed Lubrication Model is used in all cases. iii

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7 Contents 1 Introduction Background Purpose and objectives Delimitations Literature review Lubrication theory Valve plate / Cylinder barrel Piston / Cylinder barrel Slipper / Swash plate Theory Surface topography Lubrication Reynolds equation Flow factors Stationary/non-stationary contact Two-scale Homogenization Contact mechanics Half-space theory Hertzian theory Stiffness curve Load sharing Friction Hydrodynamic friction Asperity friction Machine dynamics Wear Abrasion v

8 vi CONTENTS Adhesion Cavitation erosion Archard s wear equation Method Softwares Micro-hydrodynamics The LMLM The local scale The global scale Modelling wear Boundary lubricated regime Mixed lubricated regime Software capabilities LMLM Wear Results Parameters Implementation of the LMLM Simulating wear Commercial softwares Comsol Multiphysics Ansys Discussion 57 6 Concluding remarks 59 7 Future work 61 Bibliography 62

9 Chapter 1 Introduction This master thesis was performed in collaboration with Parker Hannifin Trollhättan, and the Division of Machine Elements at Luleå University of Technology. Parker Hannifin is a global leader in motion and control technologies, active in more than 50 countries with approximately 60,000 employees. Almost 250 pople work at Parker Pump and Motor Division Europe in Trollhättan where they develop and manufacture bent-axis and in-line hydraulic pumps and motors. 1.1 Background Hydraulic machines are exposed to high pressure which put high demands of the components to withstand high loads during operation. Therefore is high reliability required of the components. Parkers method to elaborate new ideas, in the tribology area, are mostly by full scale testing. This is both cost- and time consuming and in the end it might not even give any change in endurance. Therefore may numerical analyzes be required to discard some ideas which wont give any improvements of the design. The main reason for failure of the components is high loads and low or varying speeds which force the contact to be partly carried by the asperities instead of only the fluid. Parkers machines have some sliding interfaces i.e machine components sliding against each other while they are separated by a fluid film. The gap between the surfaces serves both as a sliding bearing and as a sealing. The geometry of the gap depends on the machines operating parameters such as: pressure, speed, fluid viscosity and 1

10 2 CHAPTER 1. INTRODUCTION displacement volume if the machine is of a variable type [1]. Fig. 1.1 show Parkers VP1-045/-075 pump, of swash plate type, and names of its containing components. Figure 1.1: Cross section of a Parker piston pump with variable displacement (VP1-045/-075) [2]. The most critical contacts, marked red in figure, are valve plate/cylinder barrel, piston/cylinder barrel and slipper/swash plate. But other contacts of interest are slipper/piston (ball-bearing) and bearing shell/bearing housing. The piston shoe is called slipper in most research papers and will therefore be called that in this thesis. Another variant of Parkers hydraulic machines is shown in Fig The valve plate/cylinder barrel interface looks as the pump in Fig. 1.1, otherwise is the design of the piston different, there is no swash plate and and there is a synchronizing shaft which have some interesting interfaces.

11 1.1. BACKGROUND 3 Figure 1.2: Cross section of a Parker piston motor with variable displacement (V14) [3]. A lot of research about the piston pumps and motors have been performed to improve the life and reliability during the last 40 years. One important aspect, which is of main importance in this thesis, have been friction. Higher friction means lower mechanical efficiency. Therefore have research about build up of a fluid film between components been of grate importance. A fluid film protect the surfaces from coming into contact with each other. This is not an easy task and it comprises lots of parameters. The viscosity of the fluid will change during operation when the temperature varies. If the speed and temperature of the system varies during operation it will affect the probability to achieve full separation between the surfaces. The angle of the swash plate change the operating conditions in machines with variable displacement. When the system fail to achieve fully separated surfaces will there be asperity contact. It will lead to higher friction, higher temperature but also different types of wear.

12 4 CHAPTER 1. INTRODUCTION 1.2 Purpose and objectives The purpose of this thesis is to perform a pre-study of how modelling and simulation of thin film flow situations in axial piston machines can be performed. The main goal is to understand how the contact operating in mixed lubrication regime can be modelled considering real surface measurements of the machine components in contact, using Luleå Mixed Lubrication Model (LMLM). Also to get an understanding of what parameters that are interesting to model for this kind of simulations in axial piston machines. If Parker want to start performing this kind of simulations, what are the different options of softwares to use. If the best software(s) to use is not available at Parker is it important to compare them to the software(s) they already have available today. This is important to understand if they need to invest in new software(s) in the future. A summary of this master thesis objectives is given as Describe how to model mixed lubrication with the LMLM. Determine interesting parameters to include while simulating a thin film in axial piston machines. Describe what is required of Parker to start working with simulations in tribology on a daily basis i.e. software and input data. Another aspect to keep in mind is that the simulation tool will be used by engineers and will thus be an extra design tool for them to use in their construction work. The tool should therefore be as user friendly as possible. 1.3 Delimitations Only three contacts in the swash plate type of machine have been investigated. Those are the valve plate/cylinder barrel, piston/cylinder barrel and slipper/swash plate. These contacts have been chosen because research papers have shown that they are the most important when it comes to modelling the machines dynamics. This thesis have only focused on describing how Parker can model their tribological contacts, therefore have no simulations been performed. Except from some contact mechanics calculations, with purpose to give the writer an increased understanding of the problem.

13 1.4. LITERATURE REVIEW Literature review A study about the use of lubrication between surfaces in contact is presented. It is followed by different techniques to predict operating lubrication regimes when considering rough surfaces. Further, critical contacts are studied regarding simulations and testing in axial piston machines of swash plate type Lubrication theory Tribology is derived from the Greek word tribos meaning rubbing. The equivalent translation to english is friction, wear and lubrication science [4]. The word was first used just over 50 years ago but the subject have been of interest to scientists and engineers as long as mechanical devices have existed. It has long been known that it s possible to lower the friction, the oldest historical records of this is 5000 years old. Evidence from wall painting show that some sort of lubricant was used to lower the friction while transporting large stone blocks [5]. The concept of hydrodynamic lubrication was first discovered by Beuchamp Tower in 1883 [6]. A few years later in 1886 came the explanation of hydrodynamic lubrication provided by Osborne Reynolds [7], using a reduced form of the Navier-Stokes equation and the continuity equation to generate a second order differential equation for the pressure in a narrow converging gap between two surfaces [8]. It gave a good agreement between analyses and experiments for some conformal contacts where the hydrodynamic pressure was of magnitude MPa [9]. Further, research governing nonconformal surfaces, where the pressure is high enough to elastically deform the surfaces, show that it s still possible to hold a fluid film due to several orders of rise in the lubricants viscosity [8]. Ever since the 1950s have researchers studied this so called elastohydrodynamic lubrication regime where the pressure is of magnitude 1-5 GPa [9]. The working conditions of the system plays a big role in maintaining elastohydrodynamic lubrication. If the load is to high or the running speeds too low the asperities will penetrate the lubricant, causing the system to operate in mixed lubrication conditions. In the mixed lubrication regime asperity contact occur, then the classical Reynolds equation can not describe the fluid flow. Therefore is an alternative approach required. The affect of roughness have

14 6 CHAPTER 1. INTRODUCTION been included in different ways, the earlier studies used stochastic models by using statistic parameters to describe the surface characteristics. This method was limited to hydrodynamic applications because it was not possible to obtain local information. When analyzing elastohydrodynamic lubrication local information about the surfaces deformation is needed. In time the computer power have increased and the numerical approaches have been developed, enable deterministic approaches. The deterministic approach explicitly states information about the roughness, therefore can details about the interaction and asperity deformation be captured [10]. The mostly used theoretical model to simulate rough surface contact mechanics is the one developed by Greenwood and Williamson [11]. They considered a nominally flat elastic rough surface in contact with a perfectly smooth surface. The rough surface was artificially made, all asperities was spherical with a constant radii and the asperity height distribution was Gaussian. Therefore could Hertzian theory be used to calculate the elastic displacement for every asperity. Further, they also presented a model calculating the contact between two rough surfaces [12], which have been widely used for mixed lubrication models. This way of calculating contacts are not perfect, the topography of the artificial surface are unrealistic and so is the use of Hertzian theory when the interaction between the asperities is not considered. The load balance between hydrodynamic and asperity pressure have earlier been obtain by calculating them separately and then fixing the load balance by applying dry contact models to the asperity contact areas and Reynolds equation to the areas where lubricant carries the load. This stochastic approach gives no local information [13]. Christensen [14, 15, 16] made a stochastic model to solve mixed lubrication without considering surface deformation. The model was later extended to take elastic deformation into account by Lebeck et al. [17]. Jiang et al. [18] published a deterministic model for mixed lubrication and solved the hydrodynamic and asperity contact pressure simultaneously. They used a measured three-dimensional (3D) rough surface, Reynolds equation and fast fourier transformation (FFT), discrete convolution FFT (DC-FFT) to calculate the surface deformation and asperity contact pressure respectively. Hu and Zhu [13] numerical approach used Reynolds equation in both full film, mixed and boundary lubricated regimes by considering dry contact as a special case of the lubricated contact. They calculated the

15 1.4. LITERATURE REVIEW 7 hydrodynamic and asperity contact pressure as one system, which was an improvement of the earlier technique of fixing the mentioned load balance. FFT was first used in contact problems by Ju and Farris [19]. They developed an algebraic relationship between the surface displacement and the contact pressure, which could be used to find the contact pressure or displacement for both smooth and rough surfaces in one-dimension (1D). This method increase the calculation speed and decrease the required storage. A multi-level multi-summation (MLMS) method was developed by Brandst and Lubrecht [20] to make the contact problem more efficient by reducing the number of equations to solve. Stanley and Kato [21] used continuous convolution FFT (CC-FFT) to solve both 1D and two-dimension (2D) rough contact problems. Wang et al. [22] implemented DC-FFT and compared it to both direct summation and MLMS and concluded that DC-FFT is the fastest method for both periodic and non-periodic problems. Solving Reynolds equation with rough surface requires a large number of elements in the computational domain in order to take every single asperity into account. This is impossible with a deterministic approach. Patir and Cheng (PC) [23, 24] developed an averaged flow model based on flow factors which compensates for the surface roughness. A modified Reynolds equation is required where the flow factors are embedded and then solved on a smooth global domain. Both stochastic and real 3D surfaces can be dealt with in this model. Surface displacement is not accounted for and one drawback with PC is that it does not consider cross flow. This model is widely used in different applications nowadays. Another averaging technique to calculate flow factors, derived by Almqvist [25], are by using a homogenization technique. This was used in Sahlin et al. [26, 27] work to described a method, so called LMLM. LMLM consider how the rough surfaces affect both the mechanical and the fluid flow in the contact. The model assumes the interface to be described by exactly two separable scales (two-scale) which means that for the model to be valid, the largest significant wavelength must be significantly smaller than the length scale of the tribological interface. The contact mechanics used in [26] are based on Stanley and Kato [21] work on contact problem of rough surfaces where a FFT-based method was used. This method was further developed by Almqvist et al. [28] to include plastic deformation.

16 8 CHAPTER 1. INTRODUCTION Valve plate / Cylinder barrel During operation there are different oscillating forces acting on the barrel, causing the barrel to tilt due to torque in x- and y-direction. Some of the dependent forces are: hydrostatic pressure forces, centrifugal forces and friction forces [29]. The converging gap, due to the tilt of the barrel, is key to achieve build up of hydrodynamic pressure. Yamaguchi [30, 31] was first to published a paper about build up of fluid film between the valve plate with conventional hydrodynamic pads and the cylinder block. The conclusions was that a fluid lubrication can be established for a certain swash plate angle, but the film is liable to brake due to change in loading. Bergada et al. [32] analyzed the pressure distribution, leakage, force and torque between the barrel and valve plate. They developed new equations to analytically solve the parameters of interest. When calculating the force and torque acting on the barrel they considered both the pressure distribution along the main- and timing groove. Ivantysynova and Baker [33] investigated the power losses in the interface by consider fluid structure interaction and micro motion of the cylinder block. Output of this model was a prediction of pressure and velocity fields in the fluid film, it also calculated the leakage, viscous friction and power losses. Bergada et al. [34] modelled flow losses and the resulting flow/pressure dynamics. Some conclusions was that leakage was grater at the external land than the internal land, larger risk for cavitation when the gap is small, output pressure is small and for higher speeds. He also gave a good introduction about research regarding leakage. Wang [35] studied cavitation by using the control volume method to analyze the flow, pressure and leakage depending on the valve plate timing. Bergada et al. [36] studied how the film thickness depends on oil pressure and temperature. They did this experimentally with a axial piston pump with nine pistons and variable displacement. Some conclusions were that elasticity are of importance in metalto-metal contacts. Increase of oil temperature decrease the average film thickness and the film thickness decrease less with higher outlet pressure. The film thickness are therefore thickest at lower pressures and temperatures. Mixed lubrication occur with increasing pressure and temperature. Zecchi et al. [37] modelled the temperature in the barrel housing and outlet port under steady state operation. The model predicted

17 1.4. LITERATURE REVIEW 9 the effective flow rate in both the valve plate/cylinder barrel, the piston/cylinder barrel and the slipper/swash plate lubricating interfaces. The predicted machine efficiency and working temperature in the barrel housing and outlet port compared well to measurements in different operating conditions. Shang and Ivantysynova [38] developed a model which captured the fluid structure interaction phenomena happening in the valve plate/cylinder barrel, the piston/cylinder barrel and the slipper/swash plate lubricating interfaces. The fluid films and the solid parts behavior are both temperature and pressure dependent. The model is able to predict the barrel housing and outlet flow temperature for different machine sizes and operating conditions. Chacon and Ivantysynova [39] calculated the thermal effects on the fluid film due to compression and expansion of the fluid. They used the non-isothermal fluid flow module in Ref. [37] to calculate the heat flux and Ref. [38] to calculate the fluid temperature. The finite volume method was used to solve the Energy equation and Reynolds equation to calculate the temperature and pressure distribution in the fluid film. The finite element method was used to calculate the elastic deformation of the sliding surfaces due to the pressure and the thermal stresses. The results was compared and validated against laboratory measurements on a 130 cc swash plate type of axial piston machine. Zhu et al. [40] studied how the surface topography influenced wear of the valve plate at low speed. They used a ring-on-disc tribometer for the experiment where two different materials on the barrel was used against a valve plate made of brass. The barrel surface was processed by: coarse grinding, fine grinding, polishing and laser texturing. They concluded that surface topography greatly influences the friction and wear. The lowest friction and wear was obtained with polished surfaces Piston / Cylinder barrel Manring [41] developed a mathematical model to investigate friction considering the Stribeck curve lubrication conditions. He found that the friction was much higher when the piston acts as a pump then a motor. Increased speed lower the friction and have greater impact on fluid film build up in the motoring stroke than the pumping stroke. Jeong [42] derived two formulas to estimate the friction force and the average piston friction moment loss. The Stribeck curve effect in

18 10 CHAPTER 1. INTRODUCTION Ref. [41] can be implemented to also consider mixed and boundary lubrication. Ivantysynova and Huang [43] introduced a method to simulate gap flow with self adjusting gaps in the piston/cylinder interface. They extended the calculations of the gap to consider elastohydrodynamic effects. They concluded that the computational accuracy is much better for a elastohydrodynamic model than a rigid model. Ivantysynova and Pelosi [44] created a new model considering the generated heat from the viscous shear and the heat transfer to the solid bodies. They calculated the thermal elastic deformation and concluded that this deformation affects the fluid film thickness Slipper / Swash plate Hooke and Li [45] studied how the tilt of the slipper, which occur due to centrifugal loads in the machine and friction in the piston/slipper contact, affect the build up of fluid film. Harris et al. [46] modelled the dynamics of the slipper taking the most important factors into account: cylinder pressure, hydrostatic lift, tilting moments due to the pressure distribution over the slipper running face, contact friction between the piston and cylinder, contact friction in the ball-joint, tilting moment due to centrifugal force and elastic deformation. Kazama [47] simulated mixed lubrication in the contact for water hydraulics. He used Greenwood and Williams theory of contact mechanics and Patir and Chengs averaged flow model to account for the surface roughness. The dynamics was included in the model due to the torque around the x- and y-axis. The cavitation conditions was that all negative pressure was replaced by zeroes. Bergada [48] modelled and validated the leakage and groove pressure for a slipper with any number of grooves. He developed some new equations to account for the grooves, tilt and rotational speed. From experiment it was found that leakage increases with increased tilt which also was predicted by the theory. Bergada et al. [49] studied the hydrostatic leakage and lift characteristics of a slipper with multiple lands. They compared a flat non-tilted slipper with a tilted slipper. The leakage increase slightly when the slipper is tilted and with increased speed. The difference in pressure between the trailing and leading edge increase with increased speed. The average pressure inside the groove increases a lot

19 1.4. LITERATURE REVIEW 11 with increased speed when the slipper is tilted. Schenk and Ivantysynova [50] investigated how elastohydrodynamic deformation affects the power losses. They concluded that deformation due to fluid pressure make a huge difference. Between the pumping and suction stroke is the squeeze pressure important to consider because it affects the deformation. Schenk and Ivantysynova [51] modelled thermo-elastohydrodynamic lubrication by considering micro-dynamics and heat transfer in the solid bodies. They compared the significance between hydrostatic and hydrodynamic pressure and found that hydrostatic pressure causes most of the deformation. The hydrodynamic pressure increase when the surface gets more deformed. This behaviour is necessary, because the hydrostatic load carrying capacity decrease with increased deformation.

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21 Chapter 2 Theory Basic theory used in the LMLM is now presented together with lubrication and wear theory. Some dynamics affects and aspects which enable hydrodynamic lift in the axial piston machines are also shown. 2.1 Surface topography The surface roughness is an important aspect to consider in thin film modelling. In order to model all three lubrication regimes is information about the surface a necessity. Fig. 2.1 show how a measurement of a components surface looks like in micro scale. Figure 2.1: Surface measurement. Engineering surfaces usually have directional patterns which is a result of manufacturing processes or running-in. These patterns are either longitudinally or transversely oriented, as can be seen in Fig A isotropic surface is also shown, but it can only be made 13

22 14 CHAPTER 2. THEORY artificially. The magnitude of γ is a way of explaining the orientation of the surface roughness. The roughness orientation will affect the flow through the gap differently depending on the flow direction. Figure 2.2: Direction of the surface roughness; longitudinal (γ > 1), isotropic (γ = 1) and transversal (γ < 1) [23]. 2.2 Lubrication To reduce friction and wear between two surfaces in contact is an easily sheared material added to separate them. This material can be a solid, liquid and in some applications even a gas [52]. The ideally conditions is when the asperities height is lower than the fluid film thickness. This operating conditions leads to very low friction coefficient, only attributed to shearing of the lubricant, and approximately no wear at all [52]. The geometry of the surfaces in contact make a huge difference for the operating working conditions. A simple way of describing contacts is by calling them conformal and non-conformal, as illustrated in Fig Figure 2.3: Difference between conformal (left) and non-conformal (right) contacts. The main difference between these geometries is the area of con-

23 2.2. LUBRICATION 15 tact. As seen in the conformal contact, the surfaces fits very well into each other which leads to a larger area of contact to distribute the load. Further, this leads to a lower contact pressure. During these working condition, hydrodynamic lubrication will occur when the pressure rises because of a converging gap. The fluid gets dragged into the gap and is therefore pressurized and able to carry high loads. The definition of hydrodynamic lubrication; the contact pressure is not high enough to cause elastic deformation of the surfaces, normally operating in the range of 1-5 MPa [52]. In the non-conformal contact, the area of contact is significantly smaller. This rises the contact pressure enough to elastically deform the surfaces, GPa. This conditions is called elastohydrodynamic lubrication [52]. A machine is often operating in different conditions i.e. varying load and speed. The lubricant separating the surfaces can be penetrated by the asperities if the speed is to low or the contact pressure gets to high. It s common to divide the lubricant system into different regimes; boundary lubrication (BL), mixed lubrication (ML) and full film lubrication (FF) [52], see Fig Figure 2.4: Lubrication regimes [53]. Boundary lubrication: is the regime where all load is carried by the asperities and therefore have the highest friction of the three regimes. This happens when either the contact pressure is to high or the speed to low, which means that the hydrodynamic contribution is almost non existing [52]. The lubricant still play a role in this regime because it often contain additives to decrease friction and wear [54]. Mixed lubrication: this regime is a combination of both boundary lubrication and full film lubrication. The regime occur

24 16 CHAPTER 2. THEORY when the speed increases or the load decreases, which is equivalent to an increase in hydrodynamic pressure. There is still mechanical contact which makes the additives important to minimize wear [54]. Full film lubrication: in this regime are there no asperities in contact with each other and the contact load is therefore carried by the fluid film. The friction is very low and appears due to the lubricant and thermal properties [52]. Full film lubrication can also be called hydrodynamic lubrication or elastohydrodynamic lubrication if the pressure is high enough to elastically deform the asperities [4]. To specify which regime the system is running in, a film parameter Λ is commonly used. The parameter is the ratio between the minimum film thickness and the surface roughness such as Λ = h min. (2.1) Rq,1 2 + R2 q,2 Approximately, the system is running in BL when Λ < 1, ML when 1 < Λ < 3 and FF when Λ > 3 [53]. This behaviour can be described with the Stribeck curve which illustrates how the friction coefficient change depending on the ratio between velocity, viscosity and load as ηu/p, see Fig Figure 2.5: Stribeck curve [53]. The separation is maintained by keeping the Λ-ratio above three. This is accomplished by designing the machine element and the lubrication depending on the operation conditions in the machine.

25 2.3. REYNOLDS EQUATION Reynolds equation The solution of elastohydrodynamic pressure for the full film lubrication was first derived by O. Reynolds in 1886 [7]. Reynolds equation can be derived from the Navier-Stokes equation, laws of viscous flow and from the principles of mass conservation. Bhushan [4] solved the Reynolds equation for smooth surfaces, but in this thesis is rough surfaces considered. Therefore are Reynolds Eq. (2.2) derived with some of Bhushans assumptions such as Laminar flow Newtonian fluid Dimensions of the machine components are much larger than the fluid film thickness. Therefore can the curvature of the fluid film be ignored Inertia when accelerate the liquid and body forces are much smaller than the viscous forces and thus may be neglected Density and viscosity are constant The fluid velocity at the moving surface is equal to the surface velocity The surfaces only move relative each other in the x-direction ( h 3 ) T p + ( h 3 ) T p = u 1 u 2 h T x 12η x y 12η y 2 x + h T t. (2.2) Where x and y are cartesian coordinates, η is the dynamic viscosity, p is the lubricant pressure, h T is the true separation between the surfaces, t is time, u 1 and u 2 is the surfaces velocity in the x-direction. The definition of the true separation is illustrated in Fig. 2.6 as h T (x, y) = h + δ 1 (x, y) + δ 2 (x, y) (2.3) where h is the separation between the surfaces mean lines, so called nominal separation. The roughness of the surfaces are described by δ 1 (x, y) and δ 2 (x, y) such as A δ i da = 0. In the full film regime are the average separation h T equal to the nominal separation. In the mixed lubrication regime is the average separation h T stated as the distance between the mean lines of the deformed surfaces. Fig. 2.6

26 18 CHAPTER 2. THEORY show the surface parameters where the dotted lines are the surfaces mean lines. Figure 2.6: Geometry parameters of the rough surfaces. The left hand side in Reynolds Eq. (2.2) describe the pressure driven flow, also called Poiseuille flow and the first term on the right hand side describe the shear driven flow also called Couette flow, see Fig The last term in Reynolds equation represent the time dependent flow, also called the squeeze term. Figure 2.7: (Left) Poiseuille flow, (right) Couette flow. The lubricants viscosity and density is pressure dependent, but density normally has little affect compared to the other factors and is therefore often neglected. However, the pressure affect on density can be described with Dowson-Higginsons relationship [55] ( ρ = ρ p ) p (2.4) where ρ 0 is the atmospheric pressure. The viscosity on other hand is important to model as a function of pressure, η = f(p), in order to obtain accurate results. The viscosity increase several orders of

27 2.3. REYNOLDS EQUATION 19 magnitude with increased pressure. One commonly used equation for this in EHL analysis is the Barus pressure-viscosity relationship η = η 0 e αp (2.5) where η 0 is the fluids dynamic viscosity at a reference temperature and αp can be calculated with Roelands expression ( ( αp = (ln η ) 1 + p ) Z 1) (2.6) p 0 where p 0 = Pa and Z is obtained according to [8] Z = α (ln η ) (2.7) where α is a pressure-viscosity parameter for the lubricant which normally is known for oils. Generally in engineering applications is heat generated as a result of shearing of the fluid. Since the fluid viscosity is highly dependent of the temperature is it not realistic to assume isothermal conditions. The viscosity decrease with increasing temperature, which results in a thinner film thickness. The temperature of the modelled components can be used to calculate the viscosity of the fluid if the fluids temperature is assumed equal to the modelled components temperature. Otherwise must the fluid temperature be computed. Then Reynolds temperature-viscosity relationship can be used [56] η = η 0 e β(t T 0) (2.8) where β is the Reynolds temperature-viscosity coefficient, T is the lubricants temperature, which is the same as the modelled components if that is assumed. And last T 0 is the lubricants reference temperature. Eqs. (2.5) and (2.8) was expressed by Barus as a pressure and temperature relationship [55], η = η 0 e αp β(t T 0). (2.9)

28 20 CHAPTER 2. THEORY 2.4 Flow factors Why flow factors are used is briefly explained here, in Section 3.2 is this further explained and compared to other methods. When considering rough surfaces in contact with each other is a very fine mesh required to account for every asperity. This is very time consuming because of the computer power which is available today. Therefore, an technique have been developed which involves the use of so called flow factors, which is embedded into Reynolds equation. What the flow factors do is to account for how the lubricant flow behave between the surfaces, both in contact and out of contact. When building a model one surface might be considered smooth to decrease the complexity, the consequences of this assumption with respect to lubricant transport is explained in Section To solve the flow factors the problem is divided in two scales, see Section In this thesis homogenized flow factors have been used and is introduced in Section Stationary/non-stationary contact If a surface roughness is only applied to one of the interacting surfaces the lubricant transport will depend on which surface is rough. This is usually called stationary and non-stationary contacts, see Fig Let s consider the interaction between the valve plate/cylinder barrel and let the solution domain be the valve plates circumference. Then, if a topography is applied to the valve plate, the contact will be stationary since the valve plate is our solution domain and it s the cylinder barrel which is rotating against the solution domain. But if the topography is applied to the cylinder barrel instead, the contact will be non-stationary since the roughness is moving relative the solution domain. In the non-stationary case the lubricant is dragged by the rough surface, meaning that a lot of lubricant is transferred through the contact compared to the stationary case where the lubricant is trapped in in the roughness and dragged by the smooth surface.

29 2.4. FLOW FACTORS 21 Figure 2.8: On the (Left) is the non-stationary contact illustrate while on the (right) is the stationary contact illustrated. This is good to keep in mind, if the contact is stationary or nonstationary depends on the contact which is modelled in the machine and which surface is the solution domain. But if both surfaces is rough, then the contact always is non-stationary Two-scale The contact of interest is divided into two scales, a local and a global scale. The roughness is included in the local scale and the geometry of the components in the global scale. The separated scales build the lubricated contact problem when they are put together, see Fig These denominations will be used through out this thesis. This partition of scales is further explained in Section 3.2. Figure 2.9: Difference between local and global scale Homogenization These flow factors are based on Almqvist [25] work on homogenization theories of the Reynolds equation. The mathematical derivation include a lot of details and is complex, therefore is only theory regarding the implementation explained. Homogenization is an mathematical averaging technique which can be applied to the Reynolds equation with oscillating film thickness. The homogenized Reynolds equation is solved on a smooth global scale by assuming that the wavelength tends to zero. The flow factors replaces the film thick-

30 22 CHAPTER 2. THEORY ness component in the Reynolds Eq. (2.2) and rewrites as ( a11 x 12η ) p + ( a12 x x 12η ) p + ( a21 y y 12η = u 1 2 ) p + ( a22 x y 12η b 1 x + u 2 b 2 2 ) p y x + h T t (2.10) The flow factors are calculated on the local scale for a small periodic window of the roughness. The solutions of the local scale problem must be solved using periodic boundary conditions on the domain A and must be a non-stationary problem. The local scale problems to be solved are A (h T A χ 1 ) = h3 T x A (h T A χ 2 ) = h3 T y A (h T A χ 3 ) = ± h T x (2.11) where χ 1, χ 2, χ 3 are the solution variables and h T defines the separation between the surfaces according to Eq. (2.3). The local scale problem is solved for a range of separation where the surfaces are both in contact and out of contact. Integrating the local scale variables over the domain A gives the pressure flow factors a i,j as a 11 ( h T ) = 1 l x l y A a 12 ( h T ) = 1 l x l y a 21 ( h T ) = 1 ( h T 1 + χ 1 x A ) da h T χ 2 x da χ 1 h T y da ) da l x l y A a 22 ( h T ) = 1 ( h T 1 + χ 2 l x l y A y (2.12) where l 1 and l 2 are the length of the domain A in the x- and y- direction respectively. The flow factors a 11 and a 22 represent the flow in x- and y-direction respectively and a 12 and a 21 represent the cross flow. These cross flow terms are not included in PC method.

31 2.5. CONTACT MECHANICS 23 The shear flow factors b i are calculated as b 1 ( h T ) = 1 ( h T h 3 T l x l y A b 2 ( h T ) = 1 l x l y A ) χ 3 da x ( h 3 T ) χ 3 x da. (2.13) The flow factors are calculated for a range of separations, both in contact and fully separated. As can be seen in Fig. 2.10, the different flow factors will compensate the flow differently when the separation change. The flow factors compensate the most when the separation is small, when the separation increase they has less and less impact of the flow, until the surface roughness doesn t have any impact at all. Note that the flow factors will change with different surfaces and the relative position of there interaction, so the figure is only an example. Figure 2.10: Flow factors in Reynolds equation [56]. 2.5 Contact mechanics The theory when two surfaces interact with each other under a normal load is now presented. This is of interest when it comes to study operating conditions in a contact i.e. friction, wear and lubrication. The surface deformation is numerically calculated according to the Half-space theory in Section An analytic solution is given with the Hertzian theory in Section How the surface stiffness depends on the surface topography is explained in Section

32 24 CHAPTER 2. THEORY Half-space theory If a rough surface is loaded against a smooth surface can the separation between the surfaces be calculated according to g i = h i + u i g 0 (2.14) where h i is the initial gap between the surfaces, u i is the total normal deformation and g 0 is the surface displacement. If the bodies is considered linear elastic and in the case of line loading of an elastic half-space, with known pressure distribution, then the deflection of the surface can be calculated in any point in the x-direction, see Fig [57]. Figure 2.11: Line loading of an elastic half-space, 1D [57]. The deflection of the surface is given by u z = 2(1 ν2 ) πe a b (1 2ν)(1 + ν) p(s) ln x s ds E ( x a ) (2.15) q(s)ds q(s)ds b x where p(s) is the pressure distribution and q(s) the shearing in the tangential direction. Frictionless interface is assumed by neglecting the shearing components in Eq. (2.15), it gives u z = 2(1 ν2 ) πe a b p(s) ln x s ds (2.16) where the Young s modulus for the combined surfaces is stated as E = 1 (2.17) 1 ν1 2 E ν2 2 E 2

33 2.5. CONTACT MECHANICS 25 or E = E 1 ν 2 (2.18) if the material parameters for the two surfaces are assumed equal. For the combined surfaces can the Boussineq s equation be used if the domain is infinite long and the the elastic deformation under normal load is assumed frictionless. Boussineq s equation is stated as u(x, y) = where the so-called kernal is K(x x )p(x )dx (2.19) K(x x ) = 2 πe 1 (x x ) 2 + (y y ) 2 (2.20) in 2D. Eq. (2.20) is discretized by specifying a discrete domain such as Ω := x [cd 1, cd 2 ]. The length of the domain in x-direction is L = cd 2 cd 1 (2.21) where the calculation domains start and end point are given by cd 1 and cd 2 respectively. Adding N number of nodes in the domain gives the element length as x = L N 1. (2.22) Each node in the domain can now be described as x i = cd 1 + i x where i = 0...N 1. If the pressure inside each iterval [ x i x 2, x i + x ] 2 are assumed constant, Eq. (2.19) can therefore be discretized according to u(x i ) = K(x x )p(x )dx N 1 j=1 p j x j x Then Boussineq s Eq. (2.23) can be rewritten as u(x i ) = N 1 j=1 K(x i x )dx (2.23) x j 1 2 x p j K i,j (2.24)

34 26 CHAPTER 2. THEORY where K i,j is the discrete form of K(x x ) according to K i,j = 2 [( πe x i x j + x ) ln 2 x i x j + x 2 ( x i x j x ) ln 2 x i x j x ] (2.25) 2 x where the kernal is the influence coefficient relating to the normal deformation at point x i owing to a unit load acting on position x. From Eq. (2.24) is it shown that two steps are needed to calculate the normal surface deformation: 1. Determine the influence coefficients K i,j 2. Calculate the multisummation. Several numerical approaches can be used to obtain the kernal and the multisummation in Eq. (2.24) e.g. discrete convolution method and the fast fourier transform [58]. These two methods are of most importance to make an efficient solver for Eq. (2.24) in 3D [59] u i,j = M k=1 k=1 N = K ijkl P kl. (2.26) In order to solve Eq. (2.14) are frictionless contact, no adhesion and no penetration of the contacting surfaces assumed. In Fig is the rough surface deformation shown when a smooth rigid surface is loaded against a rough surface. g 0 is the smooth surface displacement due to the applied load. Figure 2.12: Deformed surface when loaded in 1D.

35 2.5. CONTACT MECHANICS Hertzian theory An analytic solution to the contact mechanic problem for a cylindercylinder contact or sphere-sphere contact is the Hertzian theory, the later case will be considered in this section. The Hertzian theory is a special case in half-space theory. Solutions for contact pressure and dimensions of the concentrated contact were first derived by Hertz in 1882 on the basis of [54]: Elastic deformation Both solids can be regarded as an elastic half-space, which means that the contact area is small enough for the global geometry to not be relevant. Infinitesimal strains Frictionless surfaces in contact. To obtain the analytic solution, first an equivalent rigid body against an elastic body are constructed as in Fig Figure 2.13: Equivalent contact mechanic problems. The equivalent Young s modulus are calculated according to Eq. (2.17), which for convenience is restated here E = 1 (2.17) 1 ν1 2 E ν2 2 E 2 and the equivalent radius is calculated according to 1 R = 1 R R 2. (2.27)

36 28 CHAPTER 2. THEORY If the two spheres have the same radius then the equivalent radius can be stated as [54] R = R 4. (2.28) For two spheres in contact the contact area will be circular. The half width of the contact is calculated as ( ) 3F R 1/3 a = (2.29) E and the Hertzian pressure according to p h = 3F 2πa 2 (2.30) where F is the applied load. The Hertzian theory can be used to verify the numerical results when solving the contact mechanics problem. Thus, the Hertzian pressure and half width is compared to the numerical pressure and radius Stiffness curve The interfacial separation between two surfaces in contact depends on the applied nominal pressure. Different applied loads results in change of the average interfacial separation. This average separation depends on the magnitude of the surface deformation and is very important because it is this volume which potentially could hold lubricant. The relationship between the average interfacial separation and the applied nominal pressure is often referred to as the surfaces contact stiffness and is expressed as [60] f( h T ) = p nom (2.31) where h T is the average separation, as in Fig. 2.12, of the deformed gap between the two surfaces and p nom is the applied pressure. This relationship change with different surfaces in contact and the relative position between the surfaces. The surfaces contact stiffness Eq. (2.31) can be illustrated as in Fig

37 2.6. LOAD SHARING 29 Figure 2.14: Stiffness curve. 2.6 Load sharing When the system fail to obtain full film lubrication the asperities will penetrate the lubricating film and cause metal-to-metal contact. The load sharing concept is used to calculate how much of the total load is carried by the asperities and the lubricant respectively, when operating in the mixed lubrication regime. The total load F t is both dependent on the contact mechanics part and the lubricant part, such as [60] F t = F a + F l (2.32) where F a is the load carried by the asperities and F l the load carried by the lubricant. Johnson et al. [57] developed the load sharing concept. He modelled the elasticity of the lubricant and the asperities as two coupled springs which was dependent on each other. Other authors [56], [60] and [58] used a similar approach like Johnson. They considered the asperities and the lubricant as an uncoupled system. They assumed that the average interfacial separation is equal to the minimum film thickness as 2.7 Friction h = h min. (2.33) Depending on which lubrication regime the system is operating in the friction will vary, as seen in the Stribeck curve Fig When operating in the full film lubrication regime the entire load is carried

38 30 CHAPTER 2. THEORY by the hydrodynamic pressure and the friction will therefore be equal to the hydrodynamic friction, f = f hyd. In the mixed lubrication regime the load is distributed on both the asperities and the fluid, the friction is thereby the summation of the hydrodynamic friction and the asperity friction, f = f hyd + f a Hydrodynamic friction When operating in full film lubrication there are hydrodynamic friction which can be calculated according to (( T h 2 + d 11 f hyd = A ( ) 1 ηu + 6c 11 h T ) dx ) phyd x + d 12 p hyd y (2.34) where c 11, d 11 and d 12 are flow factors, p hyd is the average hydrodynamic pressure or film pressure and U is the velocity of the moving surface. The flow factors and 1 h T are calculated by solving the following equations 1 ( h T ) = 1 1 da h T l x l y A h T c 11 ( h T ) = 1 h T χ 3 l x l y A 2 x da d 11 ( h T ) = 1 h T χ 1 l x l y A 2 x da d 12 ( h T ) = 1 h T χ 2 l x l y 2 x da. A (2.35) An example of how these flow factors can look like is illustrated in Fig

39 2.8. MACHINE DYNAMICS 31 Figure 2.15: Friction flow factors [56] Asperity friction The asperity friction is calculated when the system is working in the boundary lubricated regime by multiplying the average asperity pressure and the friction coefficient as f a = µ p nom da (2.36) Ω where µ is the friction coefficient and p nom is the average asperity contact pressure for a given average interfacial separation in Fig The friction coefficient is experimentally obtained. 2.8 Machine dynamics The dynamics of the machine is not trivial, as many components are working together and influence each other in different ways. The three main factors which affects the working condition in a swash plate machine is: speed, oil pressure and the angle of the swash plate. In hydraulic machines both hydrostatic and hydrodynamic pressure is used to create lift, as a result the surfaces get separated. The high operating pressure in the machine creates the hydrostatic lift. Restrictors in both the piston and the slipper is used to transfer the fluid into the contacts, see Fig The interaction between the

40 32 CHAPTER 2. THEORY ball on the piston and the slipper works as a ball-joint (spherical bearing) and the interaction between the slipper and swash plate as a pad bearing [41, 61, 62]. Figure 2.16: Configuration of the investigated contacts [61]. Looking at the piston/cylinder barrel interface, there are many forces acting on the piston which causes the piston to tilt, see Fig The fluid pressure p acts on the piston and the opposite force R from the slipper reacts with the angle α of the slipper/swash plate. F 1 and F 2 are the reaction forces from the cylinder wall and f 1 and f 2 are the friction forces against the cylinder wall. The machine often operate at varying speeds, at higher speed the centrifugal force gets more significant. The centrifugal force is calculated according to F cf = mωr 2 (2.37) where m is the piston mass, ω is the rotational speed and r is the radius of the circle which the pistons follow. The force increase with the rotational speed and could therefore be one of the reasons for brake down of the fluid film.

41 2.8. MACHINE DYNAMICS 33 Figure 2.17: (Left) forces acting on the piston [41], (right) piston centrifugal force. The result of the tilted piston is critical when looking at the build up of hydrodynamic pressure. Fig show where it is possible to create hydrodynamic pressure which for the pumping piston is at point (1) and for the motoring piston at point (2). This means that there is high risk for metal-to-metal contact at the opposite end of the piston. Figure 2.18: Piston/Cylinder barrel, pumping and motoring stroke [41]. The interaction between the valve plate and cylinder barrel depends on many factors. The barrel will tilt due to the high and low pressure side. Since the pressure is oscillating, it causes the barrel to oscillate as well, which results in a varying gap between the surfaces. The oscillating movement of the barrel also depends on the pistons dynamics. This is a very complex system and the forces and torque in the machine must be determined iteratively with varying opening position of the valve plate, different swash plate angles and varying pressure.

42 34 CHAPTER 2. THEORY Another aspect to take into account in the interaction valve plate/ cylinder barrel is the deformation of the barrel due to high hydrostatic pressure. The barrel will deform at the flanks of the valve ports, these areas are marked red in Fig. 2.19a and as a cross view of one of the valve ports in Fig. 2.19b. These behaviour decrease the gap between the surfaces and therefore increases the probability for metal-to-metal contact. (a) Areas where the the cylinder barrel might deform. (b) Valve port before and after deformation. Figure 2.19: Deformation of cylinder barrel. The contact between the slipper and swash plate is crucial. The slipper must be able to follow the swash plate movement smoothly. Therefore is often the contact between the slipper and piston a spherical bearing to get low friction. The hydrostatic pressure is the reason of the majority of the deformation. This non-flatness is necessary in order to build up a fluid film in the contact. High centrifugal forces appear due to the slipper, similar as for the piston at higher speeds. Therefore does the risk of fluid brake down increase between the slipper/swash plate at higher speeds [45, 46, 51].

43 2.9. WEAR Wear Wear occur in all three lubrication regimes, although it s through fatigue only in the full film lubrication regime. The most severe wear appear in the boundary and the mixed lubrication regime, where there is direct contact between the surfaces. The most common wear mechanism of all is abrasion, in this section is abrasion, adhesion and cavitation erosion presented and so is theory about calculating wear Abrasion Abrasion can be caused by a hard rough surface in contact with a less hard one or by hard particles between two mating surfaces, resulting in removal or displacement of material. Abrasion can be divided in two groups i.e. two-body and three-body abrasive wear, see Fig Figure 2.20: Two- and three-body abrasive wear mechanism [63]. Two-body abrasive wear appear when the asperity plough the softer surface. Three-body abrasive wear occur when a hard particle, larger then the separation of the surfaces, appear between two surfaces sliding on each other and plough at least one of them. The particles removed due to two-body abrasive wear might be the reason for three-body abrasive wear to start [54]. The two-body abrasive wear only occur when the asperities penetrates the fluid film but three-body abrasive wear can occur even in full film lubrication regime, if the particle is larger than the film thickness Adhesion At places where asperities of two sliding surfaces are in contact with each other and when the adhesive bonding becomes strong enough, it can cause micro welding. The junction is then ripped of from one of the surfaces and may be transferred to the mating surface or it

44 36 CHAPTER 2. THEORY becomes a free particle. These can be seen as four steps i.e. (1) asperities comes into contact when the asperities penetrate the fluid film, (2) micro-welding due to strong adhesive bonding and plastic deformation, (3) formation of adhesive junction in the plastic zone and last (4) the adhered fragment is transferred from the soft to the hard surface. These steps are shown in Fig The more severe type of adhesion is smearing and galling. They both appear when the velocity is low and the load is high. The softer material might smear to the harder surface i.e material is transferred from one surface to the other. It s like putting butter on a piece of bred, if the softer material is the butter and the harder material the bred. Galling is like cold welding between the surfaces due to contamination in the fluid. Big parts of the surfaces adhere, causing large fragments of material to be removed and it might ruin the surfaces in only one pass. Figure 2.21: Adehesive wear process [64] Cavitation erosion It s usually hard to determine that the cause of wear is caused by cavitation, often does it not even show while testing a machine. And even if it shows is it hard to determine that cavitation did occur. This is because it happens over time and might take a very long time before it s actually noticeable. Cavitation occur when the pressure drop below the vapor pressure, causing formation of vapor bubbles within the liquid. The bubbles then collapse when high pressure is encountered again, see the critical zone for cavitation in Fig

45 2.9. WEAR 37 The collapse of the bubbles creates a shock wave which can be strong enough to damage the components [35]. The chock waves cause high stresses on the surfaces and it can lead to pitts. Pitting is when cracks are formed under the surface, in time they propagate and flakes are removed. Figure 2.22: Region where cavitation erosion are most prone to occur Archard s wear equation Archard s wear equation is a simple and accurate way of calculating wear. But the concept was first introduced by Holm [65], who said that the wear volume per sliding distance is proportional to the normal force for each material pair according to [66] Q = K F N H (2.38) where H is the material hardness, K is the wear constant which is the number of abraded atoms per atomic collision. Archard, however, further developed the concept and said that the wear constant is the probability of particles to wear off when the surfaces asperities collide with each other. The wear depth w d is of interest in order to model wear and can be calculated by computing the pressure. In this termonology, Eq. (2.38) can now be rewritten as w d = kps (2.39) where Achard s dimensional wear coefficient is k = K/H and s is the sliding distance. As can be noticed, the pressure and wear coefficient is constant over the sliding distance which results in constant wear rate during sliding.

46 38 CHAPTER 2. THEORY The wear coefficient must be experimentally found and can be calculated according to [66] k = W v F N s (2.40) where W v is the worn volume, F N is the load and s is the sliding distance. An approximate wear and friction coefficient for all three lubrication regimes are shown in Table 2.1 to give an understanding of its magnitude. Table 2.1: Typical values of wear and friction coefficient at different operating conditions [64]. Operating conditions k values µ values Unlubricated 1 to Boundary lubricated 10 6 to Full film lubrication <

47 Chapter 3 Method Simulation of thin film lubrication in a axial piston machine can be performed with different softwares. The way of solving the necessary equations will be compared between: Matlab, Comsol Multiphysics, IgorPro and Ansys. Where the last two mentioned softwares are available at Parker today. First, the compared softwares is presented in Section 3.1, in Section 3.2 different modelling techniques is presented and in Section 3.3 the method to simulate thin film lubrication, LMLM, is presented in total, independent on used software(s). How wear can be implemented in the LMLM is presented in Section 3.4. Last is it described what is required of the softwares in order to solve the LMLM in Section Softwares Similar problems have been solved using Matlab with a live linker to Comsol Multiphysics [56, 59, 67]. They used Matlab to solve the contact mechanics problem and the flow factors, Comsol to solve the Reynolds equation. Neither of these softwares are available at Parker today. Comparable softwares they do have are Ansys and IgorPro, this does not mean that these softwares should (or could) be used in a similar connected way as Matlab and Comsol was in the mentioned papers. Therefore, these softwares are compared in the perspective of how they can solve Parkers contact problems, either by using only one of these softwares or a combination of them. From an engineering point of view, it might be an advantage to have a good visualization over view of the problem. Both Ansys 39

48 40 CHAPTER 3. METHOD and Comsol can give this to the user because both are able to import files from Computer Aided Design (CAD) softwares or neutral format files exported from CAD programs. The user can therefore see the modelled component and add working steps e.g. choose boundaries etc. by clicking instead of doing it mathematically as in IgorPro or Matlab. 3.2 Micro-hydrodynamics When looking at a surface with the eye, it might look smooth but in micro scale it s very rough. This huge difference in scale between the components design and the surface roughness is an issue when solving the operating conditions for a system numerically, see Fig In order to solve the Reynolds equation on these different scales must the domain be discretized, using e.g. the finite difference method (FDM) or the finite element method (FEM) [68]. Figure 3.1: Different scales of a tribological contact [68]. There are two methods to solve these kind of problem. The first way of including the surface topography in the calculations is called the deterministic approach or the direct approach. The roughness is then included directly on the entire component, so for thin film calculations the film thickness is directly included. The elements is therefore made small enough to capture the roughness while dis-