Spiral Groove Bearing Multiphysics Modeling

Mohamed Yousri Mohamed

Mathematics, master's level (120 credits) 2019

Luleå University of Technology Department of Engineering Sciences and Mathematics Abstract

Cone crushers are widely used in the mining, mineral processing and quarrying segments of the industry to crush ores and large rocks. In such machinery, the load to be carried is rather heavy and the motion is gyratory which creates a need for a bearing set that can withstand such severe conditions. Sandvik AB is a high-technology Swedish engineering group specialized in tools and tooling systems for metal cutting, equipment, as well as tools and services for the mining and construction industries. One of their products relevant to the mining industry is the cone crusher which utilizes a 3-piece bearing set to carry thrust load. This bearing can be classified as a Spiral Groove Bearing 1, and it has been incurred that it wears out rather quickly and is believed to be running under mixed-lubrication conditions where the interfaces in the bearing-set are not fully lubricated. The aim behind this thesis is to create a multiphysics model of this bearing in order to understand deeply how it works and the reasons why it does not perform as expected as well as to predict design improvements which can improve the performance of the bearing-set, thus increasing its operating life. It has been concluded that the bearing operates under severe mixed-lubrication conditions and that the generation of a squeeze film is the only method by which lubrication takes place due to the excessive depth of the grooves which is needed to allow for an adequate amount of cold oil to flow into the grooves and cool the interface as well as to accommodate for a considerable amount of wear particles. In light of the results and insight gathered from the simulations, possible design variations of the bearing which can be advantageous in terms of mitigating asperity friction in the interfaces of the bearing are discussed and tested.

1The abbreviation S.G.B will be used interchangeably throughout the thesis. Acknowledgment

I would like to dearly thank my supervisor Prof. Andreas Almqvist for the time and devotion he offered me, which was crucial to my development and progress in this thesis. Not only did he help a lot, but his contagious enthusiasm to this field of science made me appreciate and enjoy working tirelessly on this thesis. I would also like to thank my co-supervisors at Sandvik, namely: Patrik Sjöberg, Jan A Johansson, Sonny Ek and Tatiana Smirnova for their support and for the fruitful bi-weekly progress meetings.

To mum and dad, I owe you every success that I am living now, for you have sacriﬁced a lot and worked so hard and against all odds to put me on track to succeeding on the top-level abroad. With this thesis, and my graduation from my M.Sc., I hope to have given you back at least a fraction of the happiness and pride I want to make you feel throughout your lives for what you have sacriﬁced for my sisters and I.

To Salma, my dearest love and best friend, your presence in my life is nothing short of magical. Your love and support throughout this year were one of the main reasons I was able to make it, for I have lived this year alone in what seems like the furthest up-north a person from our side of the world can reach. You were my motivator and the person I conﬁned to in the darkest moments, and were the person I shared my happiness with when things were right. I can only imagine how diﬃcult it would have been without you, and so I dedicate this thesis to you.

Finally, I would like to thank the EACEA of the European Union for the gen- erous funding of my degree as well as every professor and colleague I was lucky to meet and work with during my time in Leeds, Ljubljana and Lule˚a. Preface

This thesis is the result of my work as an M.Sc. student in Tribology of Surface and Interfaces, as a part of the Erasmus Mundus Joint Master’s Degree TRI- BOS. It represents the closure of the degree I started in Leeds in 2017 and at the same time, the beginning of a new stage in my academic career.

The thesis has been developed under the supervision of Prof. Andreas Almqvist at the Department of Engineering Sciences and Mathematics, Division of Ma- chine Elements at Lule˚aUniversity of Technology during the period 10.10.2018 to 01.07.2019.

Mohamed Yousri Mohamed Lule˚a,July 2019. Contents Page No.

1 Introduction 5 1.1 Cone Crushers ...... 5 1.2 Sandvik’s Cone Crusher Bearing Set ...... 7 1.3 COMSOL Multiphysics ...... 8 1.4 Objectives and Motivation ...... 9 1.5 Delimitations ...... 10

2 Literature Review 11 2.1 Simulations in Tribology ...... 11 2.1.1 Analytical Methods ...... 11 2.1.2 Continuum Methods ...... 12 2.1.3 Particle-based Methods ...... 14 2.3 Spiral Groove Bearing in Literature ...... 16 2.4 Multiphysics Models of Bearings in Literatu ...... 19 2.5 Gap in Literature ...... 19

3 Theoretical Basis and Methods 20 3.1 Spiral Groove Bearing Classiﬁcation and Working Principle . . . 20 3.2 Meshing ...... 24 3.2.1 Sandvik’s Geometry Mesh ...... 25 3.2.2 Parameterized Geometry Mesh ...... 25 3.2.3 Sensitivity to Mesh Density ...... 25 3.3 Navier-Stokes Equations ...... 26 3.4 1D Reynold’s Equation and its Analytical Solution ...... 28 3.5 Homogenized Reynold’s Equation ...... 29 3.6 Applied Force ...... 31 3.7 Force Balance ...... 33 3.8 Squeeze Action ...... 35 3.9 Solid Mechanics ...... 36 3.10 Analytical Solution of Pocket Bearing in 1D ...... 37 3.11 Cavitation Model ...... 39 3.12 Validation ...... 40

4 Results 42 4.1 Modelling ladder, part 1 ...... 42 4.1.1 Analytical - Rayleigh Step-bearing ...... 42 4.1.2 Numerical - Rectangular Rayleigh Step-bearing Pad, Dif- ferent Width:Length Ratios ...... 43 4.1.3 Numerical - Cylindrical (Almost-Quadratic) Pad . . . . . 44 4.1.4 Numerical - Cylindrical Pad, Diﬀerent Circumferential Widths ...... 45 4.1.5 Numerical - Straight Groove Bearing (Shallow Grooves . 46

1 4.2 Modelling ladder, part 2 ...... 47 4.2.1 Straight Groove Bearing - Rigid ...... 48 4.2.2 Spiral Groove Bearing - Rigid ...... 49 4.2.3 Sandvik’s Piston-step - Rigid ...... 50 4.2.4 Spiral Groove Bearing - Deformed ...... 51 4.2.5 Sandvik’s Piston-step - Deformed ...... 52 4.2.6 Validating the Implementation of Solid Mechanics Physics 53

5 Possible Design 54 5.1 Shallower Grooves ...... 54 5.2 Hydrostatic Pressure ...... 54 5.3 Larger Surface Area of Ridges ...... 55 5.3.1 Decreasing groove:ridge surface area ratio ...... 55 5.3.2 Decreasing number of grooves ...... 57

6 Discussion 59 6.1 Validation ...... 59 6.2 Sandvik’s Bearing vs. Parameterzied S.G.B ...... 59 6.3 Rigid vs. Deformed Simulations ...... 59 6.4 Design Changes ...... 60

7 Conclusions 61

8 Future Work 62

9 References 63

2 Nomenclature

α Threshold constant in cavitation model β Inclination angle between grooves and velocity vector in s.g.b Threshold pressure in cavitation model [Pa] Ω Piston-step area [m2] ω Eccentric velocity [rad/s] ρ Fluid density [kg/m3] σ Stress [Pa] τ Shear stress [Pa]

υi Poission ratio ε Strain aij, bi Flow factors E Young’s Modulus [Pa] E∗ Equivalent Young’s Modulus [Pa]

Fasp Asperity load [N]

Fhyd Hydrodynamic load [N] h Separation (including surface deformation) [m] hm Separation at point of maximum pressure [m] h0 Rigid separation [m] pasp Asperity pressure [Pa] phyd Hydrodynamic pressure [Pa] r1 Inlet radius of s.g.b [m] r2 Outlet radius of s.g.b [m] t Time [s] u Velocity in x-direction (in Reynold’s eq) [m/s] Deformation in x-direction (in Solid Mechanics) [m] v Velocity in y-direction (in Reynold’s eq) [m/s] Deformation in y-direction (in Solid Mechanics) [m] w Velocity in z-direction (in Reynold’s eq) [m/s] Deformation in z-direction (in Solid Mechanics) [m]

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4 1 Introduction

In this section, the spiral groove bearing used by Sandvik as well as their cone crushers in which the bearings operate will be introduced. COMSOL Multi- physics will be shortly introduced and ﬁnally, the objectives of the thesis and its delimitations will be laid out.

1.1 Cone Crushers

Throughout the most of history, crushing of rocks and other materials happened using manpower as force is concentrated on the material to be crushed from the tip of a sledge hammer or other similar tools. Initially, most ore crushing and sizing was carried out by hand and hammers at mines or by water powered trip hammers in the small charcoal ﬁred smithies and iron works typical of the Renaissance period, after which explosives came into use. Later on, during the industrial revolution of twentieth century, various forms of mechanical crushers were developed, of which the cone crusher is one of the most widespread.

A cone crusher is a compression type of machine used in industry (notably in the mining industry) that reduces material (eg. rocks) by squeezing it between a a stationary piece of steel and a moving piece of steel. Final reduction in size is determined by the gap between the two crushing members at the lowest point. As the mantle rotates to cause the compression within the chamber, the material gets smaller as it moves down through the wear liner as the opening in the cavity gets tighter [1]. The crushed material is discharged at the bottom of the machine after they pass through the cavity and if the material is not yet small enough to pass through this cavity, it is returned back to the starting point and crushed again till it can pass through the cavity. Figures 1 and 2 help imagine the working principle of the cone crusher.

5 Figure 1: Schematic showing where the rocks/gravel enter the crushing cycle.

As we see in Figure 1, the rocks/gravel is feed from the top into the interface between the mantle and the bowl which is lined with manganese alloys (14 to 22 percent manganese) that provide a certain balance between impact resistance and abrasion resistance. The higher the manganese content, the higher the abrasion resistance but the lower the impact resistance and vice-versa. The eccentric shaft rotates with a gyratory motion and causes the mantle to rotate as well, which causes the crushing action.

Figure 2: Sectioned and labelled cone crusher [2].

6 1.2 Sandvik’s Cone Crusher Bearing Set

In Figure 2, you can see the 3-piece step bearing set comprising of a mainshaft step, step washer and a piston step. This bearing serves the crucial purpose of supporting the heavy irregular thrust load and gyratory motion of the cone crusher, therefore it has to have the appropriate design which produces the hydrodynamic lift needed for a load bearing ability that is as high as possible in order to prevent mixed or boundary lubrication from happening under such demanding conditions which leads to wearing out and loss of function of the bearing and therefore damage to the cone crusher.

In Figure 3 the bearing set used by Sandvik is shown more clearly. This bearing can be classiﬁed as a logarithmic spiral groove bearing, which will be talked about in detail in the literature review Section 2.3.

The bottom piece of the bearing (piston step) is stationary while the upper piece (mainshaft step) moves in a similar manner and eccentric speed as the main/eccentric shaft. The middle piece (step washer) moves in a gyratory motion at a speed that is unknown.

Figure 3: Sandvik’s bearing set showing the piston step and step washer.

7 1.3 COMSOL Mutliphysics

COMSOL Multiphysics is a comprehensive simulation software environment for a wide array of applications, which is well-structured and user-friendly for ev- eryone to use and it allows engineers and scientists to couple related physical applications together to include all the necessary factors for a complete model, in other words to couple any number of physics together and input user-deﬁned physics and expressions directly into a model without restrictions posed by other simulation softwares [3]. Therefore it was chosen as the software to be used to simulate Sandvik’s bearing.

COMSOL uses Finite Element Methods (FEM) to numerically approximate Partial Diﬀerential Equations (PDEs) related to the physics intended to be modelled. The major advantage of COMSOL is the ability to simulate various physics coupled together (multiphysics modelling) on rather complex geometries, such as a spiral groove bearing. The FEM is described in more detail in Subsection 2.2 of this thesis.

8 1.4 Objectives and Motivation

The main objective behind this thesis is to create a multiphysics model of Sand- vik’s bearing on COMSOL in order to predict and understand how the bearing operates. The physics/equations that would be coupled and solved are:

1. Homogenized Reynold’s Equation 2. Force balance equation

3. Solid mechanics physics (Hooke’s Law)

A multiphysics model of a parameterized spiral groove bearing is also created in parallel, which can be easily edited in terms of its geometry to be able to test diﬀerent possible changes in geometry that have the potential to make the bearing work under more favourable conditions in terms of reducing asperity contact throughout the bearing’s life. In the light of this, diﬀerent possible geometries are investigated and discussed to have an idea about what geometrical changes can improve the bearing performance.

Another end goal of the thesis is to create user-friendly applications for both Sandvik’s geometry and the parameterized geometry which can be used by Sand- vik to change geometrical and other model parameters and run simulations easily.

As for the greater motivator, sustainablity has been a main motive in industry for a while now, so an optimized bearing design that can operate for a longer lifetime will certainly be more sustainable. If the design also leads to lower friction and wear rates, then unwanted material and energy loss will be eliminated as well.

9 1.5 Delimitations

While creating the models, there was a need to make a number of justiﬁed assumptions and simpliﬁcations in order to save time as well as computational complexity of the models. These assumptions are:

1. The lubricating fluid is assumed to be isothermal and Newtonian in behaviour, which means that its density and viscousity does not rely on temperature or pressure. The justification behind assuming a Newtonian fluid is that it was certain while modelling the operation of the bearing that the pressures sustained in the lubricating fluid during a loading cycle does not exceed 30 [MPa], which is believed to be a low pressure that makes the variation in fluid properties in terms of pressure negligible. The justification behind assuming an isothermal fluid is pretty much the same, since the low pressure will not cause significant viscous heating in the fluid.

2. The heating and thermal expansion of the interface due to friction as well as the cooling of the interface due to the lubricant flow in the grooves is neglected. This is, however, important to model in future work since the frictional heating of the interface is expected to be significant and would affect the performance of the bearing.

3. Only the lower interface between the piston step and the step washer is modelled, since it was believed that modelling the upper interface as well would be redundant as both interfaces are similar to a great extent.

4. The axial loading cycle that the bearing is subject to from the cone crusher was approximated and simpliﬁed by a fourier series to make it easily con- ﬁgurable and to facilitate convergence in COMSOL.

10 2 Literature Review

In Subsection 2.1, simulation methods utilised in the ﬁeld of Tribology are introduced. In Subsection 2.2, the Finite Element Method is elaborated in more detail since it is pivotal to the work done in this project. In Subsection 2.3, the various literature in which spiral groove bearings were developed are mentioned. In Subsection 2.4, a number of multiphysics models created in COMSOL for different types of bearings are mentioned and ﬁnally in Subsection 2.5, the gap in the literature is shortly spoken about.

2.1 Simulations in Tribology

Tribology is a vast and interdisciplinary field of science which examines friction, lubrication and wear encountered in machinery, human joints and many any other relevant contact where friction can be found. The field combines chem- istry, material science, solid mechanics, fluid mechanics and physics with an aim to reduce wear and friction in machine components, which have detrimental effects on their useful life. Therefore, when designing a machinery, tribological aspects are crucial to the efficiency and life of its components and have to be taken into consideration. The basis of almost any engineering component or advanced material design as of this age is theoretical results gained from computer simulation, which is why simulations have been carried out throughout the years to study lubrication regimes, friction and wear on all scales in order to understand these behaviours and therefore properly design against them. The area of simulations in tribology has developed greatly in the past 50 years, in terms of computational speed and development of computational methods that are suit- able for different scales (nanoscale, micro-scale and large scale) and applications (dry contact, lubricated contact, materials, machine components). The field is often divided into three major parts, as in the proceedings of one of the latest Simulations in Tribology workshops held in Lorentz Center in 2018 [4], namely: analytical methods, continuum methods and particle-based methods.

2.1.1 Analytical Methods

Analytical methods are most widely used in the area of contact mechanics, and the most famous theory in this area is the Hertzian theory that has been a cornerstone in contact mechanics and tribology ever since it has been described in [5]. The Hertzian theory solves the problem of two ideally smooth surfaces in

11 frictionless contact. The JKR 2 and DMT 3 models [6, 7] are two other famous models for ideally smooth surfaces, in which the latter takes only the adhesion in the area out of the contact zone into consideration, while the former does the opposite. Rough contact has been studied by Greenwood-Williamson and their model remains the most widely used, although it has a lot of inaccurate assumptions [8]. Recent advancements in this field have been made in contacts in the presence of sharp edges [9] and conformal configurations [10] as well as studying of contact in the presence of anisotropic and functionally graded materials, with varying friction coefficient along the interface in sliding and partial slip conditions [11]. Finally, Majumdar and Bhushan have created on of the most popular theories after the GW model for studying contact of rough surfaces [12]. As for the contact of lubricated surfaces, analytical solutions also exist in special cases such as infinitely-wide bearings as shown in Subsection 3.4.

2.1.2 Continuum Methods

Continuum methods are the basis of most commercial simulation softwares available, and are widely used in modeling large-scale components due to their accuracy and small computational time [8]. The most notable and widely used continuum methods are the Finite Element Method (FEM), Finite Diﬀerence Method (FDM) and Finite Volume Method (FVM).

FEM is a continuum method which divides a CAD model or a specified geometry into finite-sized elements of simple geometrical shapes (meshes). A system of field equations are then approximated within each element as a simple function, such as a linear or quadratic polynomial, with a finite number of degrees of free- dom. This gives a local estimated description of the physics by a set of simple linear (and sometimes nonlinear) equations. When the contributions from all elements are added together, a large sparse matrix equation system is resulted which can be solved by any of a number of well-known sparse matrix solvers [13]. The main advantages of this method is that it is widely used in commercial software, allows for mixed-formulation or multiphysics modelling where more than one PDE or physical phenomena is approximated on each mesh, and therefore on the whole geometry. Also, it works very well on complex geometries. How- ever, mathematical formulations behind FEM are quite advanced and need an adequate amount of expertise. In comparison to FDM and FVM, its application is less straight-forward.

2To introduce the effect of adhesion in Hertzian contact, Johnson, Kendall, and Roberts created their theory of adhesive contact using a balance between the stored elastic energy and the loss in surface energy. This model considers the effect of contact pressure and adhesion only inside the area of contact. 3This model, named after Derjuagin-Muller-Toporov is an alternative model for adhesive contact where the contact profile is assumed to remain the same as in Hertzian contact but with the addition of attractive interactions only outside the area of contact.

12 FDM is one of the simplest and oldest methods of solving differential equations, of which its one-dimensional formulation was known to the famous math- ematician L.Euler as early as 1768. It was then extended to two dimensions by C.Runge in 1908. However, serious development and practical application of this method has started after the 1950s when computers have emerged which allowed solving more complex problems in engineering and science using this method. FDM is a method used more commonly on regular grids such as rectangular ones that are not complex in shape, it takes the continuous form of a PDE and replaces it with a set of discrete equations and then solves the equations. This is usually done on regular grids and is more efficient, fast and accurate than FEM in large scale computations such as astrophyiscal and meteorological simulations given that they can fit into regular grids. However, FDM runs into problems when having to deal with curved boundaries or complex geometries.

FVM is a method that is best applicable to CFD 4 simulations (ex. fluid-flow). It is similar to FEM in the sense that the domain is divided into finite-sized elements called cells, rather than meshes. It is different however, in the sense that it is based on the fact that many physical laws are conservation laws ie. what goes into one cell on one side needs to leave the same cell on another side. A disadvantage of this method is that the functions that approximate a solution when using FVM cannot be easily made of higher-order as in FDM and FEM, therefore higher accuracies are not as easily attainable.

BEM is the most effective method for simulating contact problems in contact mechanics or any problem in which we are a lot more interested in what is happening on the boundary (surface) than in the volume of the geometry. When using BEM, one can mesh only the boundary and solve on it which save computational resources. The main difference between BEM and the previously mentioned continuum methods is that BEM only needs to solve for unknowns on the boundaries, whereas FEM, FDM and FVM solve for unknowns in the volume [14]. However, to formulate a BEM, a fundamental solution linking pressure and vertical displacement in two orthogonal directions is needed, and such solution exists for a limited number of cases and mainly under the assumption that the solid can be locally considered as a flat half-space, making the BEM more restrictive to use [15].

4Computational fluid dynamics (CFD) is a branch of fluid mechanics which uses numerical analysis and simulation to solve and analyze problems that involve fluid flow.

13 2.1.3 Particle-based Methods

Particle-based methods are used for simulation mainly on the atomic and mi- croscales and most of them are based on Molecular Dynamics, they work by discretizing the domain into particles (atoms) rather than meshes, as is the case in continuum methods. The DEM 5 and MCA 6 methods may be the most notable particle-based methods, the latter being a more recent method that has proven to be effective and accurate in mesoscale simulations. This MCA method combines the advantages of classical cellular automata, discrete element methods and molecular dynamics. It assumes that the simulated system is dis- cretised into a series of small elements of a finite size called movable cellular automata [16]. The movements of those elements are simulated using Newton- Euler equations of motion while including their pair relationship, many body forces and bond forces and was created by Psakhie et al [17]. However, if a machine component is to be simulated, it would be made of billions of automata making it computationally very expensive to simulate and this the most recent problem that needs to be faced in this area. Currently, scientists in this field want to examine the possibility of parallel-computing of MCA simulations, that is to compute on multiple-processors to speed up the computational time and be able to simulate larger scale components.

2.2 The Finite Element Method

The Finite Element Method can be traced back to the appendix of a paper by Courant in 1943 [18] in which the piecewise linear approximation of the Dirich- let problem over a network of triangles was discussed. Many additions have been made throughout the following years to include partitioning of domains, boundary conditions and local polynomial approximations to come to the ﬁrst mature version of FEM in 1956 [19] in which an attempt was made at both a local approximation of the PDE of linear elasticity and the use of assembly strategies essential to FEM.

5The Discrete Element Method (DEM) is a particle-scale numerical method for modeling the bulk behavior of granular materials and numerous geomaterials 6The Movable cellular automaton (MCA) method is a computational solid mechanics method based on the discrete concept. In the MCA approach, an object being modeled is considered a set of interacting elements/automata. The dynamics of the set of automata are deﬁned by their mutual forces and rules for their interaction. This system exists and operates in time and space and its evolution in time and space is governed by the equations of motion.

14 The working principle of the Finite Element Method (FEM) is that it numerically approximates a given Partial Diﬀerential Equation related to some physical phenomena intended to be studied on a domain (eg. viscous heating of a lubricant ﬁlm) by discretizing the domain into meshes (discussed in Subsection 3.2) and numerically approximating the PDE locally on each element (mesh) and adding them up to obtain the global approximation of the PDE on the whole domain.

The FEM has gained wide popularity and approval in engineering during the twentieth century and has been used in every conceivable area of engineering that makes use of models of nature characterized by partial diﬀerential equations. The reason behind the popularity of the FEM is its ﬂexibility which is derived from these intrinsic characteristics of the FEM [20]:

1. Unstructured meshes: FEM by its nature leads to unstructured meshes which allows for placing ﬁnite elements anywhere in the domain which in turn allows the simulation of the most complex types of geometries in nature and physics.

2. Arbitrary geometries: FEM is geometry-independent and can applied to a wide range of arbitrary geometries with arbitrary boundary conditions.

3. Robustness: Stability and insensitivity to singularities and mesh distor- tion, especially when using Petrov-Galerkin method to derive the Finite Element Method which leads to more stable algorithms.

These advantages make FEM the best numerical method to be used for approximating the pressure distributions and surface deformation in Sandvik’s bearing during this project. Out of all FEM-based tools, COMSOL was the best to use due to its capability of carrying out multiphysics simulations on complex geometries, as mentioned previously.

15 2.3 Spiral Groove Bearing in Literature

The first paper to give an account of a spiral groove bearing in the usual form, with two opposing surfaces, one flat and the other grooved, was one from 1957, written by Harris, Ford and Pantall [21]. However, grooved bearings are believed to have been the subject of an earlier paper written by Whipple in 1949 [22]. Using a theoretical approach, Whipple derived a formula for predicting the pressure profile between horizontal strips, one of which has a parallel array of straight grooves as shown in Figure 4, while employing various assumptions to assure that the pressure distribution across the grooves can be regarded as linear. In this paper, Whipple makes no attempt to apply his formula directly to spiral groove bearings, but the formula was later used by other researchers to obtain rough but useful estimates of the operating characteristics of spiral groove bearings. Whipple’s analysis was then improved in accuracy and extended by researchers to become better applicable to the spiral groove bearing. One of the most important pieces of literature that laid down a solid theory of equations governing the operation of spiral groove bearings without employing much of the assumptions used in [22] is a thesis by Muijderman produced in Philips Research Laboratories in The Netherlands in 1964 [23].

Figure 4: The recurrent pattern of parallel grooves investigated by Whipple in [22], where the pattern is considered to be inﬁnite in the direction of the velocity. Figure is taken from [38].

16 The working principle of the spiral groove bearing is described in more detail in Subsection 3.1. The pumping eﬀect added to the restriction eﬀect gives an advantage to spiral groove bearings over other types of self-acting bearings in that they produce a pressure distribution from the edge of the bearing to the center while pressure at the center of the tilting pad thrust bearing, for example, is zero. This contributes greatly to a higher load bearing ability for the same size of the bearing. Therefore, air can be used as a lubricating medium and still produce an acceptable load carrying capacity [24]. Since air is more environ- mentally friendly than oil and grease and of course incurs less cost, it gives this type of bearing a large prospect of being more widely applied in industry. A quotation from [23] states the following:

“If the grooved disc, of 100 mm diameter is placed on the smooth one and ro- tated in the right direction, a rotational speed of about 1 revolution per second will be enough to separate the two discs by a layer of air about 11 micron thick, which will then support the weight, 200 grf (1,96 Newtons), of the grooved disc. This combination thus works as an air-lubricated thrust bearing.”

The logarithmic spiral groove bearing can also produce relatively high load carrying capacities for a smaller size of a bearing, which makes it widely used in electrical appliances and since spiral groove bearings in general can produce very high load carrying capacities for a given size of bearing compared to other types of self-acting bearings, they can sometimes be the only design solution in extreme cases such as huge thrust bearings under extreme load conditions in turbines, or in very small bearings (a few mm in diameter) with large rotational speeds.

The reason why grooves in this type of bearing are logarithmic, is that there exists a certain optimum angle β between the tangent to the grooves and the local velocity vector which produces the highest load carrying capacity and the grooves being logarithmic ensures that this angle is kept constant throughout the bearing surface. This optimum angle was found to be 20◦ for inward-pumping logarithmic spiral groove bearings, according to recent calculations made in [39].

17 Figure 5 shows the piston step of Sandvik’s bearing as an example of a logarithmic spiral groove bearing. The grooves in the piston step of Sandvik’s bearing are excessively deep as mentioned previously, which prevents the formation of a full-ﬁlm of lubricant due to restriction action and thus detrimentally aﬀects the load-carrying capacity of the bearing.

Figure 5: Piston step of Sandvik’s bearing set.

18 2.4 Multiphysics Models of Bearings in Literature

A few relevant multiphysics models created using COMSOL for diﬀerent types of bearings which I have found to be very useful are shown below:

1. A fully-coupled model of a spring-supported thrust bearing used in hydropower applications was developed [25]. 2. An isothermal multiphysics model is created for journal bearings where geometric parameters can be easily altered [26].

3. A multiphysics model was created to aid in design optimization of aerostatic thrust bearings [27].

2.5 Gap in Literature

Multiphysics models of bearings on commercial software such as COMSOL Mul- tiphysics are available, but for speciﬁc types of bearing such as the spiral groove bearing, they are hard to come across. This necessitates the creation of a multiphysics model with which one can understand deeply how the bearing operates and which acts as a design tool to help improve the design of the bearing to create more favourable lubrication conditions, which would be of great use for Sandvik in particular.

19 3 Theoretical Basis and Methods

In this section, the theoretical basis of concepts and equations needed for simulation of the physics in Sandvik’s bearing is laid out including the Navier-Stokes equations, Homogenized Reynold’s equation, meshing, Solid Mechanics physics and the mass-conserving cavitation model used to account for cavitation.

3.1 Spiral Groove Bearing Classiﬁcation and Working Prin- ciple

Spiral groove bearings are categorized as self-acting bearings whose geometry leads to hydrodynamic pressure build-up. To understand this more, let’s look at Figure 6 which shows the classiﬁcation of bearings. Indeed, in Figure 6, a classiﬁcation of bearings can be seen, of which the following observations can be made:

1. Self-acting bearings is a sub-class of no-contact bearings where a lubricating medium is present to prevent contact between bearing surfaces.

2. Bearings in which pressure build-up is a result of a wedge effect/action are a subclass of self-acting bearings. Other effects that can induce pressure build-up are the stretch effect and buffer effect. From now on, the wedge effect will be replaced by the restriction effect or restriction action interchangeably since we will not talking about the tilting pad thrust bearing (also generically named the wedge bearing) to avoid confusion.

3. Thrust bearings and radial bearings are sub-classes of bearings that operate using the restriction eﬀect/action. Thrust bearings are those which support mainly or exclusively thrust (axial) loads.

4. Spiral groove bearings are one sub-class of thrust bearings, while stepped bearings (ex. Rayleigh step-bearing) and Mitchell bearings (ex. tilting pad thrust bearings) are other sub-classes of thrust bearings.

5. Under the sub-class of flat spiral groove bearings, bearings with transverse flow are the logarithmic spiral groove bearings and those with no transverse flow are the herringbone spiral groove bearings.

20 If one follows the green boxes from the very top till the bottom of the ﬁgure, the inward pumping logarithmic spiral groove bearing will be reached, which is the type of bearing used by Sandvik in their cone crushers but with excessively deep grooves.

Figure 6: Classiﬁcation of bearings, re-constructed from [24].

21 Now to understand the restriction effect/action, let’s imagine the simple Rayleigh step-bearing shown in Figure 7, when the lubricating oil is pushed in the direction shown in the figure it has to pass through a narrower gap (restriction) right after the step which causes pressure to be build up in order to open up this gap to balance the flow at the inlet and outlet. If the two surfaces are restrained by an external force, then the pressure build-up will restrain the amount of fluid trying to enter at the inlet and will push more fluid out at the outlet so that flow is equal at the inlet and outlet, this is called the restriction action.

What causes pressure build-up in the logarithmic spiral groove bearing, for example, is the restriction action and pumping action together, in which the pumping action happens due to the angle between the tangent line to the grooves and the local velocity vector of the opposing surface which is rotating. The direction of rotation determines if the pumping will be inward or outward [28]. As the lubricating ﬂuid is pumped over the grooves, a narrow gap is faced similar to the Rayleigh step-bearing which leads to pressure build-up. Inward pumping is required to produce a positive pressure distribution and load carrying capacity.

It is very important to note that the grooves in the piston step and mainshaft step of Sandvik’s bearing are excessively deep in order to allow for a larger volume of cold lubricant to be pumped into the grooves to prevent overheating of the interface as well as to accommodate for a large amount of wear that happens on the bearing surface during its operation, where the worn out material falls into the grooves. Such deep grooves do not allow for hydrodynamic pressure build-up by a mixture of restriction action and inward pumping as explained in this subsection, but rather only by squeeze action.

Figure 7: Rayleigh step-bearing working principle.

22 In Figure 8, diﬀerent types of self-acting bearings can be seen, of which the logarithmic spiral groove bearing and herringbone spiral groove bearing are sub-classes of the spiral groove bearing.

Figure 8: Types of self-acting bearings [29].

23 3.2 Meshing

Meshing is used in COMSOL to represent the geometrical domain in a dis- cretized manner in order to numerically approximate any give PDE on the domain using the Finite Element Method. Meshes come in diﬀerent sizes and shapes, with sizes ranging in COMSOL from ‘Extremely Coarse’ to ‘Extremely Fine’ and shapes including: tetrahedra (tets), hexahedra (bricks), triangular prisms (prisms), and pyramids. Mesh sizes may also be numerically speciﬁed. The hexahedral shape is the best if accuracy is of a high concern. The shapes are shown below in Figure 9.

Figure 9: Diﬀerent shapes of meshes [36].

Meshing requires a balance between the desired accuracy, rate of convergence of the simulation and the computational time needed to run the simulation. If the mesh is very fine, the results will be more accurate but computational time can increase by orders of magnitude which is undesirable. Therefore, in COM- SOL, meshing of a geometry is very customizable. One can choose the best shapes, sizes and combinations of shapes and sizes to produce fairly accurate results (small error, better rate of convergence) with an acceptable computational time. It is then necessary to have a mesh size fine enough to capture all the important flow phenomena and especially in the important areas in the 3D geometry (eg. at the step in the Rayleigh step-bearing or in the boundary between ridges and grooves in Sandvik’s bearing), but it should be made coarser in areas of the geometry where there is no important details to be captured as to not burden the CPU and cause unnecessary increases in computational time.

24 3.2.1 Sandvik’s Geometry Mesh

In Figure 10 (a), the meshing of Sandvik’s piston-step is shown where the mesh type is tetrahydral, mesh size is set to ’ﬁne’ in COMSOL with 231514 domain elements, 49966 boundary elements, and 8815 edge elements. It is expected that setting ’ﬁne’ mesh on COMSOL for this geometry would produce a much larger number of meshes than for the parameterized geometry, which is needed for greater detail in Sandvik’s geometry.

3.2.2 Parameterized Geometry Mesh

In Figure 10 (b), the meshing of the parameterized piston-step is shown where the mesh type is tetrahydral, mesh size is set to ’extra ﬁne’ in COMSOL since there is little detail in the geometry and so using extra ﬁne meshes would not be overly expensive in terms of computation time. with 231514 domain elements, 49966 boundary elements, and 8815 edge elements.

3.2.3 Sensitivity to Mesh Density

I have tried running both models with varying physics-controlled element sizes in COMSOL ranging from ’fine’ to ’extremely fine’ and I found that the results in the load sharing graphs converge to certain values as the element size is smaller but without significant error when using larger element sizes in the case of Sandvik’s geometry, therefore the physics-controlled mesh size ’fine’ was chosen for Sandvik’s geometry to significantly reduce the computation time.

25 (a) Meshing on Sandvik’s geome- (b) Meshing on the parameterized try geometry.

Figure 10: Meshing of surfaces

3.3 Navier-Stokes Equations

The Navier-Stokes Equations were derived by Navier, Poisson, Saint-Vernant and Stokes between 1827 and 1845 [30]. They govern the motion of fluids and can be seen as Newton’s second law of motion of fluids. Of the most comprehensive forms of these equation is the three-dimensional unsteady form, which is comprised of a set of coupled differential equations that are used to describe how velocity, pressure, temperature and viscousity are realted and how they vary in a given fluid flow problem. These equations are at the heart of fluid-flow modelling and can be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing as well as almost any other case where we have a fluid flow. The term Navier-Stokes equations is commonly used to name the equations of momentum conservation which pertain to Newton’s second law for fluids, but one can also include the continuity equation (conservation of mass) and energy equation under that term because they are commonly used together to describe the flow of a fluid.

Since these equations are considerably complex, they have a very small number of analytical solutions. It is relatively easy to solve these equations analytically for laminar flows between two parallel plates or for flows in circular pipes but for more complex geometries the equations ought to be approximated numerically using finite element, finite volume, finite difference as well as spectral methods. The Navier-Stokes set of equations are shown below in equation (2) and are solved together with equation (1).

26 Continuity equation: ∂ρ ∂ρu ∂ρv ∂ρw + + + = 0 (1) ∂t ∂x ∂y ∂z

X-Momentum:

∂ρu ∂ρu2 ∂ρuv ∂ρuw ∂p 1 ∂τ ∂τ ∂τ + + + = − + xx + xy + xz (2a) ∂t ∂x ∂y ∂z ∂x Rer ∂x ∂y ∂z

Y-Momentum:

∂ρv ∂ρuv ∂ρv2 ∂ρwv ∂p 1 ∂τ ∂τ ∂τ + + + = − + xy + yy + yz (2b) ∂t ∂x ∂y ∂z ∂y Rer ∂x ∂y ∂z

Z-Momentum:

∂ρw ∂ρuw ∂ρvw ∂ρw2 ∂p 1 ∂τ ∂τ ∂τ + + + = − + xz + yz + zz (2c) ∂t ∂x ∂y ∂z ∂z Rer ∂x ∂y ∂z

The Navier-Stokes equations and the continuity equation are also often solved with the Energy equation if the ﬂuid ﬂow is non-isothermal. However, since an isothermal solution is assumed in this thesis, the Energy equation is not written here.

It is important to note that solving the Navier-Stokes equation for the case of Sandvik’s bearing or any other flow problem would give more accurate and comprehensive numerical simulations than solving the Reynold’s equation because then any turbulent flow can be modelled and also in the case of the Rayleigh step-bearing, the flow in the gaps between successive pads can be modelled as well, however it is very computationally expensive to do so and in most cases parallel-computing on super computers is required which is not available to me, and is also unnecessary given that the Reynold’s equation will give sufficient and accurate simulations as well. The homogenized form of Reynold’s equation used in this thesis to model mixed lubrication conditions in Sandvik’s bearing is discussed in Subsection 3.5.

27 3.4 1D Reynold’s Equation and its Analytical Solution

The analytical solution of Reynold’s equation is used to compute the pressure distribution in the inﬁnitely wide Rayleigh step-bearing.

Since the Rayleigh step-bearing in this case is inﬁnitely wide, the pressure distribution is assumed to be constant in the direction of width since there will be virtually no side-leakage, and thus the pressure gradient in this direction is negligible. Equation (2) is the Reynold’s equation applicable to the inﬁnitely wide Rayleigh step-bearing in which the pressure gradient and velocity are negligible in the y and z directions and are accounted for only in the direction of the length of the bearing (x-direction).

d h3 dp u dh = (3) dx 12η dx 2 dx

The analytical solution of this equation applicable to the 2D inﬁnitely wide Rayleigh step-bearing pad is:

dp h − h = 6ηu m (4) dx h3

where hm is the height at the maximum pressure point and h is the ﬁlm thickness at any given point along the bearing length. This equation should be integrated to produce the pressure distribution.

28 3.5 Homogenized Reynold’s Equation

From the Navier-Stokes equations, the 2D Reynold’s equation can be derived under a number of assumptions to work as a less computationally expensive method for numerically simulating fluid-flow and producing accurate, meaning- ful pressure distributions and fluid-velocity fields for 2D (particularly needed for non infinitely-wide cases) and 3D geometries. The assumptions made in order to derive this form of the Reynold’s equation are:

1. Newtonian ﬂuid 2. No boundary slip 3. Incompressible ﬂow

As a consequence of these assumptions, inertia terms and pressure gradient across ﬁlm thickness direction will be negligible. The resulted equation (2) derived under these assumptions is the Reynold’s equation in 2D.

∂ ρh3 ∂p ∂ ρh3 ∂p ω ∂ρh ∂ρh ∂(ρh) + = + + (5) ∂x 12η ∂x ∂z 12η ∂z 2 ∂x ∂z ∂t where ω is the vector of the eccentric motion in the x and z directions. Finally, ∂(ρh) ∂t is the squeeze term. The left hand side of the equation pertains to the Poiseuille ﬂow and the right hand side to the Couette ﬂow and the squeeze term.

The 2D Reynold’s equation works well in solving full-film lubrication problems, however when solving mixed lubrication problems where the oil film thickness is lower than the average surface roughness height and therefore asperity contact comes into play, it would need very fine meshes in order to compute the flow of the lubricant between individual asperities. Therefore, a deterministic solution of mixed lubrication problems is not possible as of now due to limited computational resources, and so one of the most common ways of solving mixed lubrication problems is to compute the homogenized form of Reynold’s equation on the global scale while having the flow factors which are computed on a local scale roughness window obtained from a representative topology as an input to the equation.

29 The time-dependent 2D homogenized form of Reynold’s equation with assumption of symmetric roughness is:

∂ ρa (h) ∂p ∂ ρa (h) ∂p ω ∂(ρb (h)) ∂ρb (h) ∂(ρh) 11 + 22 = 1 + 2 + (6) ∂x 12η ∂x ∂z 12η ∂z 2 ∂x ∂z ∂t

In Figure 11, the ﬂow factors a11 can be seen interpolated against the separation h, and as mentioned before they are obtained in a parameterized fashion from a local scale roughness taken from a representative topology as in [31].

Figure 11: Interpolation of ﬂow factors a11 against separation h [31,32].

30 3.6 Applied Force

The thrust force which Sandvik’s bearing set is subject to is highly fluctuating in a stochastic manner, and so a two-minute signal of the hydroset pressure from the cone crusher was captured. When multiplying this pressure with the hydroset piston area, the thrust force that the bearing set is subject to is obtained. This force however, with its high fluctuations, will make it difficult for COMSOL to capture it and will cause convergence error. To solve this problem, a simple force function was selected, adapted and periodised from the force data as shown in Figure 12.

Figure 12: Selection, adaptation and periodisation of pressure data.

31 The periodised force function is shown in Figure 13 for 1 [s], where the force function is initialized by starting from a lower force to aid in convergence.

Figure 13: Periodised force function over a period of 1 [s].

32 3.7 Force Balance

Since Sandvik’s bearing operates under mixed lubrication conditions, part of the applied force is carried by the hydrodynamic pressure generated in the lubricant and the rest is carried by the pressure caused due to asperity contact. In order to calculate this force partitioning (load sharing), a force-balance equation is posed as a Global ODE physics in COMSOL and solved. The equation is as follows:

F − Fhyd − Fasp = 0 (7) where

Z Fhyd = 0.9 phyd (8) Ω and

Z Fasp = 0.9 pasp (9) Ω

The hydrodynamic and asperity pressures are integrated over the surface area of the piston step and then multiplied by 0.9 to obtain the hydrodynamic and asperity forces. Multiplying by 0.9 is for the reason that during eccentric motion of the step washer relative to the piston step, between only 83 % to 92 % of the piston step area is in contact due to the eccentric shift between the step washer and the piston step and depending on the magnitude of the crusher stroke. The percentage of area in contact area for the maximum and minimum crusher stroke are shown in Table 1 below. The piston step area and area in contact are calculated using COMSOL’s ’measure’ feature which is able to measure the surface area of boundaries and objects. The multiplication factor 0.9 (pertaining to 90 %) is thus chosen as an intermediate value within the range.

Crusher stroke Piston step area Area in contact Max: 0.036 [m] 0.05334 [m2] 0.0442722 [m2] − 83% Intermediate: 0.025 [m] 0.05334 [m2] 0.048006 [m2] − 90% Min: 0.016 [m] 0.05334 [m2] 0.0490728 [m2] − 92%

Table 1: Area in contact for maximum and minimum crusher strokes.

33 In Figure 14, the area in contact between the piston step of the straight groove bearing and the step washer can be seen shaded in purple while the area out of contact is not shaded.

Figure 14: Area in contact for the straight groove bearing.

34 3.8 Squeeze Action

There are two main methods by which hydrodynamic pressure is generated, one is the wedge (restriction) action explained previously and illustrated for a Rayleigh step-bearing in Figure 7, the other method is the squeeze action which occurs when one surface moves with a velocity towards the other as illustrated in Figure 15, which can be caused by a force that is increasing in time, for example.

Figure 15: Squeeze action illustration [33].

When looking at the homogenized form of Reynold’s equation (5), we can see ∂(ρh) the squeeze term is ∂t . This term can be easily posed in COMSOL’s Co- eﬃcient Form Boundary PDE physics which is used to solve the equation and leads to COMSOL being able to capture this squeeze eﬀect in time-dependent simulations.

35 3.9 Solid Mechanics

It is then necessary to couple the homogenized Reynold’s equation with the Solid Mechanics physics in order to take into account the surface deformation caused by the combination of hydrodynamic and asperity pressures at the interface and the counter-eﬀect that this deformation has on both the hydrodynamic and asperity pressures. This is done in the following steps:

1. The step washer is assumed to be rigid and the elastic properties of the piston step are taken as the equivalent Young’s modulus (E∗) and equivalent Poisson ratio (ν∗) of the combined material of the step washer and piston step, where:

2 2 1 1 − ν1 1 − ν2 ∗ = + (10) E E1 E2

and

2. The equation needed to calculate surface deformation in the x, y and z coordinates is the isotropic elastic material behaviour equation (Hooke’s law), and is written as follows:

σ = E.ε (11)

3. To ﬁnally achieve the coupling, the deformed separation between the step washer and piston step at the interface is:

h = h0 + v (12)

where h0 is the rigid separation and v is the surface deformation in the y-direction (direction of ﬁln-thickness/separation). The ﬂow factors which are input to the homogenized Reynold’s equation are taken from an interpolation against h and used to solve for the hydrodynamic pressure at the interface. This is how the coupling between the homogenized Reynold’s equation and Solid Mechanics physics is achieved.

36 3.10 Analytical Solution of Pocket Bearing in 1D

One of the analytical solutions that would be largely useful and relevant to the case of spiral groove bearings is the pocket bearing which can be seen as a much more simplistic form of a spiral groove bearing. Why this solution will be helpful is because each groove can be seen as a pocket if sectioned, as shown in Figure 16.

Figure 16: Schematic showing section of a pocket-bearing [34].

The solution is then shown in Figure 17. It is important to note that a cavitation model has been included in order to omit the negative pressures caused by cavitation in the area a to z shown in Figure 16.

Figure 17: Analytical and FDM solutions of the pocket bearing in 1D [34].

37 It can be noted that the pressure drops from 0 to a, then is constant at zero (where cavitation should be happening) between a and z, then the pressure rises sharply after z where it reaches maximum at b (at the step), then drops again after the step in a non-linear manner. In a spiral groove bearing, there are typically more than one of these pockets (grooves) in a given radial direction so the pressure distribution should look like that in Figure 17 but repeated with the same number of times as there is grooves while the magnitude of maximum pressure drops with every groove starting from the inner edge till reaching the outer edge of the bearing. However, this is for the case when grooves are shallow enough to allow for hydrodynamic pressure build-up due to wedge/restriction action.

38 3.11 Cavitation Model

In the simulations carried out in this project, a mass-conserving cavitation model was needed to omit non-physical negative pressures theoretically resulting from solving Reynold’s equation in a case where there is a diverging gap. A cavitation model found in [35] is used, where the density and viscousity of the lubricant are non-constant and change according to the following equations:

  1, p - pa > 0 f =  0, p - pa <-β  (13) p−pa 3 p−pa 2 1 − 2( β ) − 3( β ) , -β ≤ p − pa ≤ 0

where

f + α ρ = ρ (14) 0 1 + a

f + α η = η (15) 0 1 + a where f is the degree of saturation of the ﬂuid and it varies with the ﬂuid pressure as seen in equation (12), β is the threshold pressure and α is a threshold constant that takes a value approaching zero. Then in the areas where pressure is less than −β the value f is set to zero and in the area where pressure is between −β and zero this is the transient and so on. The threshold pressure β is set to 9000 [Pa] and α is set to 0.0001.

39 3.12 Validation

To validate the models experimentally is diﬃcult, expensive and time consum- ing since recreating the conditions of the cone crusher in a bearing test rig is not an easy task. Also, as far as my knowledge, similar simulations in literature to be compared against are not present for the time being. Therefore it was necessary to validate the numerical models against some analytical solution that we are certain of. A modelling ladder was made starting from the most simple 2D analytical solution of a Rayleigh step-bearing to make sure that the results we see for every step in the ladder are sensible. The modelling ladder was divided into two parts,

1. Modelling ladder, part 1: In this part, the grooves are shallow enough to allow for hydrodynamic pressure build-up due to restriction action and is shown in Figure 18. Pure rotational motion of the counter-surface is also assumed rather than eccentric motion.

Figure 18: Validation modelling ladder, part 1.

40 2. Modelling ladder, part 2: In this part, the grooves are excessively deep as in Sandvik’s piston step geometry and is shown in Figure 19. The eccentric motion of the counter-surface (ie. step-washer) is also considered.

Figure 19: Validation modelling ladder, part 2.

41 4 Results

In this section, the results from the simulations of the Rayleigh step-bearing and the spiral groove bearing according to their respective modelling ladders are shown and discussed.

4.1 Modelling ladder, part 1

In this subsection the solutions following the validation modelling ladder shown in Figure 18 are laid out.

4.1.1 Analytical - Rayleigh Step-bearing

The analytical solution was computed on MATLAB using the script found in the Appendix, assuming an inﬁnitely wide 2D Rayleigh step-bearing. The pressure distribution (showing the maximum pressure as the analytical solution) is shown in Figure 20.

Figure 20: Analytical pressure distribution.

42 4.1.2 Numerical - Rectangular Rayleigh Step-bearing Pad, Diﬀerent Width:Length Ratios

The pressure distribution schematics produced for diﬀerent width:length ratios are shown in Figure 21. The maximum pressure value for each variant is shown in Table 2.

Figure 21: Pressure distribution schematics for diﬀerent width:length ratios.

Width:length ratio Max. pressure [MP a] 1:1 2.8 1.5:1 3.6 6:1 3.9 20:1 4.0

Table 2: Maximum pressure values for diﬀerent width:length ratios.

It is clear here that the closer the width:length ratio is from being inﬁnitely wide, the closer the maximum pressure value becomes of that of the analytical maximum pressure solution shown in Figure 20 and the closer the width becomes to the length, the smaller this maximum pressure value becomes due to side leakage. This proves that the analytical solution does not suﬃce when we add another dimension (width) and that the numerical solution of some form of the 2D Reynold’s equation is required for more accurate approximations of the pressure distribution. It is also clear that the pressure maximum is at the step, as seen in the analytical solution.

43 4.1.3 Numerical - Cylindrical (Almost-Quadratic) Pad

This pad is built in polar coordinates, using the simple assumption that the further away the pad is from the center of the bearing and the smaller the angle is between the left and right sides of the pad, the more rectangular it will look like..The pressure distribution schematic is shown in Figure 22 and the maximum pressure value is shown in Table 3.

Figure 22: Pressure distribution schematic of the 3D cylindrical (almost- quadratic) pad.

Parameter Value Max. pressure 2.78 [MP a]

Table 3: Maximum pressure value of the 3D cylindrical (almost-quadratic) pad.

It is notable that the maximum pressure value here pertains to the maximum pressure value of the rectangular Rayleigh step-bearing pad (1:1 width:length ratio) since the width:length ratio in this cylindrical (almost-quadratic) pad is almost 1:1.

44 4.1.4 Numerical - Cylindrical Pad, Diﬀerent Circumferential Widths

Next, the cylindrical Rayleigh step-bearing pad was modelled while varying the circumferential width and keeping the surface area constant at 0.002722 m2. The schematics of the pressure distributions are shown in Figure 23 and the maximum pressure values are shown in Table 4.

Figure 23: Circumferentially wide (top), typical (bottom left), circumferentially narrow (bottom right).

Parameter Wide Typical Narrow Max. pressure [MP a] 0.7 2.7 2.9 LCC [kN] 0.39 2.36 2.7

Table 4: Maximum pressure and load carrying capacity values for the three variants.

It is notable that the more circumferentially wide the bearing pad is, the more side leakage will be present which will lead to a lower maximum pressure value and a lower load carrying capacity but also for the typical width, the maximum pressure value pertains approximately to the 1:1 width:length ratio rectangular pad solution.

45 4.1.5 Numerical - Straight Groove Bearing (Shallow Grooves)

In this subsection, the time-dependent results from the parameterized straight groove bearing with shallow grooves are shown. The load sharing graph can be seen in Figure 24 and the graphic representation of the hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure 25.

Figure 24: Load sharing in the rigid straight groove bearing (shallow grooves) - load in [N].

(a) Hydrodynamic pressure at 0.14 [s] (b) Asperity pressure at 0.14 [s] - pres- - pressure in [Pa]. sure in [Pa].

Figure 25: Rigid straight groove bearing (shallow grooves) pressure schematics.

46 If one section (groove + ridge) is taken out of this straight groove bearing surface, the pressure distribution can be seen to resemble that of the cylindrical Rayleigh step-bearing pad shown in Figure 23 in that the maximum pressure occurs on the step (groove-ridge boundary) due to the restriction/wedge action but it is most similar to the pressure distribution expected from a pocket bearing as shown in Figure 17 where the pressure is zero in the grooves but spikes up rapidly to reach a maximum at the groove-ridge boundary then falls down non-linearly over the ridge, and thus the pressure distribution obtained in this simulation can be deemed as sensible.

4.2 Modelling Ladder - Part 2

In this subsection the solutions following the validation modelling ladder shown in Figure 19 are laid out. In this modelling ladder, the grooves are deep just like in Sandvik’s piston step. Due to the depth of the grooves, a hydrodynamic pressure build-up due to the restriction action will not be present, but rather only due to a squeeze action over the ridges as explained previously. For each variant, the load sharing graph will be shown for 0.2 [s] of the loading cycle and also a graphic representation of the hydrodynamic pressure build-up due to squeeze action as well as a graphic representation of the asperity pressure.

47 4.2.1 Straight Groove Bearing - Rigid

The ﬁrst variant in this modelling ladder is the most simple one, with parameterized straight grooves and a rigid surface (ie. no coupling of Solid Mechanics physics). The load sharing graph can be seen in Figure 26 and the graphic representation of the hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure 27.

Figure 26: Load sharing in the rigid straight groove bearing - load in [N].

(a) Hydrodynamic pressure at 0.14 [s] (b) Asperity pressure at 0.14 [s] - pres- - pressure in [Pa]. sure in [Pa]

Figure 27: Rigid straight groove bearing pressure schematics.

48 4.2.2 Spiral Groove Bearing - Rigid

The second variant in this modelling ladder is the parameterized spiral groove bearing with a rigid surface (ie. no coupling of Solid Mechanics physics). The load sharing graph can be seen in Figure 28 and the graphic representation of the hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure 29.

Figure 28: Load sharing in the rigid spiral groove bearing - load in [N].

(a) Hydrodynamic pressure at 0.14 [s] - (b) Asperity pressure at 0.14 [s] - pres- pressure in [Pa]. sure in [Pa].

Figure 29: Rigid spiral groove bearing pressure schematics.

49 4.2.3 Sandvik’s Piston-step - Rigid

The third variant in this modelling ladder is Sandvik’s actual piston-step geometry with a rigid surface (ie. no coupling of Solid Mechanics physics). The load sharing graph can be seen in Figure 30 and the graphic representation of the hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure 31.

Figure 30: Load sharing in Sandvik’s bearing - load in [N].

(a) Hydrodynamic pressure at 0.14 [s] - pres- (b) Asperity pressure at 0.14 [s] - pressure in [Pa]. sure in [Pa].

Figure 31: Rigid Sandvik’s bearing pressure schematics.

50 4.2.4 Spiral Groove Bearing - Deformed

The fourth variant in this modelling ladder is the parameterized spiral groove bearing with a deformed surface (ie. Solid Mechanics physics coupled). The load sharing graph can be seen in Figure 32 and the graphic representation of the hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure 33.

Figure 32: Load sharing in deformed spiral groove bearing - load in [N].

(a) Hydrodynamic pressure at 0.14 [s] - (b) Asperity pressure at 0.14 [s] - pressure pressure in [Pa]. in [Pa].

Figure 33: Deformed spiral groove bearing pressure schematics.

51 4.2.5 Sandvik’s Piston-step - Deformed

The ﬁfth variant in this modelling ladder is Sandvik’s actual piston-step geometry with a deformed surface (ie. Solid Mechanics physics coupled). The load sharing graph can be seen in Figure 34 and the graphic representation of the hydrodynamic and asperity pressure taken at 0.14 [s] are shown in Figure 35.

Figure 34: Load sharing in deformed Sandvik’s bearing - load in [N].

(a) Hydrodynamic pressure at 0.14 [s] - (b) Asperity pressure at 0.14 [s] - pres- pressure in [Pa]. sure in [Pa].

Figure 35: Deformed Sandvik’s piston-step pressure schematics.

52 4.2.6 Validating the Implementation of Solid Mechanics Physics

In order to validate the implementation of the Solid Mechanics physics in the models, a series of simulations were run for the parameterized spiral groove bearing with diﬀerent degrees of elasticity of the bearing surface measured with the equivalent Young’s modulus. The goal is to make sure that both the hydrodynamic and asperity load in the load sharing graph would converge towards the rigid load sharing solution as the bearing surface is set to be more and more rigid. The convergence of hydrodynamic and asperity load shares towards the rigid solution are shown in Figure 36.

(a) Hydrodynamic load share.

(b) Asperity load share.

Figure 36: Load sharing for diﬀerent Young’s Modulus of bearing material.

53 5 Possible Design Changes

In this section, the insight gathered from the simulation results in the previous section is used to predict and test possible advantageous design changes which can mitigate undesirable asperity contact in Sandvik’s bearing-set during its operation.

5.1 Shallower Grooves

If we were to compare, for example, the results for the Straight Groove Bearing in 4.1.5 where the grooves are shallow and in 4.2.1 where the grooves are excessively deep, it would be immediately clear that shallow grooves (in order of a few multiples the oil film thickness over the ridges) allow for full-film generation due to the restriction action discussed previously and this leads to complete mitigation of undesirable asperity contact. Such full-film generation is not possible when the grooves are excessively deep as in the current design Sandvik’s bearing-set. The reason why the grooves were chosen to be excessively deep in the first place is to allow for a larger volume of cold oil to flow in the grooves and cool down the heat generated due to asperity friction over the ridge interface, and thus it may be wise to think of ways to enhance the squeeze action over the ridges while keeping the depth of the grooves. However, if shallow grooves were used, will the full-film generation of the lubricant reduce heating of the interfaces to an extent that a large volume of cold oil flow in the grooves will not be needed any more? This is an important question to investigate in future work where the thermal effects will be included in the model. Also, it is important to note that in cone crushers, the operating conditions in terms of load and motion are extreme and may make impossible for a full-film of lubricant to be generated, in which case shallow grooves will be a great disadvantage.

5.2 Hydrostatic Pressure

If the lubricating oil is pumped into the interface through the center of the piston-step using an adequate amount hydrostatic pressure, this can separate the piston-step and step-washer surfaces from coming into contact. However, if we rely only on hydrostatic pressure and it ceases to exist momentarily due to a technical fault, the piston-step and step-washer surfaces will crash in a way that can destroy the interface.

54 5.3 Larger Surface Area of Ridges

If we were to choose to only enhance the squeeze action over the ridges, then increasing the surface area of the ridges on which squeeze action occurs while decreasing the surface area of the grooves can be a good choice. However, the total volume of oil that can ﬂow in the grooves has to be kept constant in order to not reduce the cooling capability in the bearing. The increase in surface area of the ridges can be done by editing geometric parameters in the model in the following manner:

1. Decreasing groove:ridge surface area ratio + increasing groove depth to keep total groove volume constant. 2. Decreasing number of grooves + increasing groove depth to keep total groove volume constant.

And therefore, these two possibilities were tested and the results are shown in 5.3.1 and 5.3.2.

5.3.1 Decreasing groove:ridge surface area ratio

Herein, 5 different variants of the parameterized spiral groove bearing are modelled, with different groove depths and groove:ridge surface area ratios. The geometric parameters of the different variants are shown in Table 5 where G:R is the groove:ridge surface area ratio, GD is the groove depth. The current (de- fault) design parameters are as in variant 2. The hydrodynamic and asperity load shares for all variants are shown in Figure 37. The results were truncated at 0.18 [s] because some of the simulations faced trouble in convergence beyond this point in time.

Variants G:R GD [m] Total groove volume [m3] 1 1:2 5.7E-4 1E-4 2 1:3 6E-3 1E-4 3 1:4 8E-3 1E-4 4 1:5 1E-2 1E-4 5 1:6 1.7E-2 1E-4

Table 5: Geometric parameters of variants of the parameterized spiral groove bearing with diﬀerent groove:ridge surface area ratios.

55 (a) Hydrodynamic load share.

(b) Asperity load share.

Figure 37: Load sharing for diﬀerent groove:ridge surface area ratios.

56 5.3.2 Decreasing number of grooves

Herein, 5 different variants of the parameterized spiral groove bearing are modelled, with different groove depths and number of grooves. The geometric parameters of the different variants are shown in Table 6 where GN is the number of grooves and GD is the groove depth and are set such that the total groove volume is kept constant.

Variants GN G:R GD [m] Total groove volume [m3] 1 7 1/3 6E-3 1E-4 2 6 0.28 7E-3 1E-4 3 5 0.235 8.4E-3 1E-4 4 4 0.19 8.8E-3 1E-4 5 3 0.14 1.4E-2 1E-4

Table 6: Geometric parameters of variants of the parameterized spiral groove bearing with diﬀerent numbers of grooves.

57 (a) Hydrodynamic load share.

’ (b) Asperity load share.

Figure 38: Load sharing for diﬀerent groove numbers.

58 6 Discussion

In this section, the results shown in the previous section as well as the insight gained about the behaviour of Sandvik’s bearing will be discussed.

6.1 Validation

If the modelling ladder is followed step by step, it would be clear that the simulation results are logical such that every simulation result behaves as expected in light of the simulations that comes before it in the ladder. However, the logical links between successive simulations may not be clear by simply looking at the results and so they will be discussed in more detail in Subsections 6.2 and 6.3.

As mentioned previously, further validation against experimental results in a test rig would be very useful, although the operating conditions that Sandvik’s bearing-set is subject to may be diﬃcult to recreate in a test-rig.

6.2 Sandvik’s Bearing vs. Parameterzied S.G.B

Comparing the load sharing graph for Sandvik’s bearing (deformed) in Figure 34 and that of the parameterized spiral groove bearing in Figure 30, it is notable that the squeeze action in Sandvik’s bearing is more prominent and thus asperity contact is less. This is simply due to a larger area of ridges in Sandvik’s bearing.

6.3 Rigid vs. Deformed Simulations

A hypothesis for why the the deformed bearings seem to perform worse (more asperity contact), is that as the ridges deform, they face maximum deformation at the center of the ridge and the step washer (opposite surface) moves towards the ridge with an amount that is proportional to the deformation. It was noticed that as the step washer moves towards the deformed ridges, its surface gets really close to the sides of the ridge, and so higher asperity pressure is noticed on the sides of each ridge. This can be clear when looking at Figure 33 (b) and Figure 35 (b).

59 6.4 Design Changes

The design changes proposed in Section 5 may well and truly be useful in mitigating asperity friction on the interfaces in Sandvik’s bearing, however a number of factors are not considered in the simulations and should be considered before applying such changes. These factors include:

1. The cost of manufacturing narrower and/or deeper grooves compared to the savings made due to lower asperity friction over the lifetime of the bearing.

2. The distance between grooves in the current design of the piston-step was tailored to allow the whole surface of the piston-step to be exposed to the lubricating oil in 1 gyratory cycle of the opposing surface (step-washer), and so the eﬀect of changing this distance has to be considered and prefer- ably tested experimentally.

3. If grooves were made to be shallow, the ability of the bearing to create a full-ﬁlm of lubricant due to restriction action under the extreme conditions posed on it in the cone crusher should be investigated experimentally, as it is believed that it may be impossible for such a full-ﬁlm of lubricant to be generated, in which case the operation of the bearing may cease detrimentally.

4. For each proposed design change, the eﬀects of temperature rise of the surfaces in the interface on the performance of the bearing has to be considered. This is important to be able to determine whether excessively deep grooves are really needed, as their function is to allow for a suﬃcient volume of lubricant to be pumped through them and therefore cool the interface.

60 7 Conclusions

In this thesis, FEM-based mutliphysics models have been created on COMSOL for Sandvik’s bearing as well as for a comparable parameterized S.G.B. For Sandvik’s model, it is not possible to easily change the geometry in order to investigate the eﬀect of such changes and so the model for the parameterized S.G.B was created for this purpose. User-friendly applications have also been developed out of these two models for Sandvik to use (see Appendix).

When looking at the load sharing graphs obtained from both models in Section 4, which describe how the applied thrust load is partitioned between hydrodynamic and asperity loads in the lubricated interfaces of the bearing, it can be concluded that Sandvik’s bearing runs under severe mixed-lubrication conditions. This is believed to be, as explained previously, due to the excessive depth of the grooves which make it impossible for a full-film of lubricant to be generated due to a restriction action and therefore, the only method by which a lubricant film is generated in the interfaces of the bearing-set is the squeeze action. The pressure sustained in the lubricant film due to squeeze action is not sufficient to fully support the thrust load that the bearing-set is subject to and so asperities come into contact.

In Section 5, several possible design changes which can either enhance the squeeze action or lead to full-ﬁlm lubrication have been discussed and tested. In Subsection 6.4, the factors that has to be considered before applying these design changes are mentioned. Overall, the objectives of the thesis have been met, although improvements have to be made in order to have models that produce very realistic and reliable results. These improvements to be made are discussed in the following section.

61 8 Future Work

In future work, the models for Sandvik’s bearing as well as the parameterized S.G.B are to be developed such that they include thermal eﬀects, and are able to run simulations for many loading cycles rather than just one. The simulation results may also be validated experimentally if time and resources allow. In this manner, the models will be more realistic and will take into consideration almost all physical phenomena that Sandvik’s bearing is subject to.

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66 Appendix

In order to enable Sandvik to use the models and easily test the eﬀects of design changes, user-friendly applications have been created out of the models for Sandvik’s bearing as well as for the parameterized S.G.B. The graphical interface of the application created from the parameterized S.G.B model is shown in Figure 39 as an example.

Figure 39: Graphic interface of application created from the parameterized S.G.B model.