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 quarryingsegments of the industry to crush ores and large rocks. In such machinery, theload to be carried is rather heavy and the motion is gyratory which creates aneed for a bearing set that can withstand such severe conditions. Sandvik ABis a high-technology Swedish engineering group specialized in tools and toolingsystems for metal cutting, equipment, as well as tools and services for the min-ing and construction industries. One of their products relevant to the miningindustry is the cone crusher which utilizes a 3-piece bearing set to carry thrustload. This bearing can be classified as a Spiral Groove Bearing 1, and it has beenincurred that it wears out rather quickly and is believed to be running undermixed-lubrication conditions where the interfaces in the bearing-set are not fullylubricated. The aim behind this thesis is to create a multiphysics model of thisbearing in order to understand deeply how it works and the reasons why it doesnot perform as expected as well as to predict design improvements which canimprove the performance of the bearing-set, thus increasing its operating life.It has been concluded that the bearing operates under severe mixed-lubricationconditions and that the generation of a squeeze film is the only method bywhich lubrication takes place due to the excessive depth of the grooves whichis needed to allow for an adequate amount of cold oil to flow into the groovesand cool the interface as well as to accommodate for a considerable amount ofwear particles. In light of the results and insight gathered from the simulations,possible design variations of the bearing which can be advantageous in termsof mitigating asperity friction in the interfaces of the bearing are discussed andtested.

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 timeand devotion he offered me, which was crucial to my development and progressin this thesis. Not only did he help a lot, but his contagious enthusiasm to thisfield of science made me appreciate and enjoy working tirelessly on this thesis. Iwould also like to thank my co-supervisors at Sandvik, namely: Patrik Sjoberg,Jan A Johansson, Sonny Ek and Tatiana Smirnova for their support and for thefruitful bi-weekly progress meetings.

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

To Salma, my dearest love and best friend, your presence in my life is nothingshort of magical. Your love and support throughout this year were one of themain reasons I was able to make it, for I have lived this year alone in whatseems 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 canonly imagine how difficult it would have been without you, and so I dedicatethis 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 luckyto meet and work with during my time in Leeds, Ljubljana and Lulea.

Preface

This thesis is the result of my work as an M.Sc. student in Tribology of Surfaceand 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 atthe same time, the beginning of a new stage in my academic career.

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

Mohamed Yousri MohamedLulea, July 2019.

Contents Page No.

1 Introduction 51.1 Cone Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Sandvik’s Cone Crusher Bearing Set . . . . . . . . . . . . . . . . 71.3 COMSOL Multiphysics . . . . . . . . . . . . . . . . . . . . . . . 81.4 Objectives and Motivation . . . . . . . . . . . . . . . . . . . . . . 91.5 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Literature Review 112.1 Simulations in Tribology . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Analytical Methods . . . . . . . . . . . . . . . . . . . . . 112.1.2 Continuum Methods . . . . . . . . . . . . . . . . . . . . . 122.1.3 Particle-based Methods . . . . . . . . . . . . . . . . . . . 14

2.3 Spiral Groove Bearing in Literature . . . . . . . . . . . . . . . . . 162.4 Multiphysics Models of Bearings in Literatu . . . . . . . . . . . . 192.5 Gap in Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Theoretical Basis and Methods 203.1 Spiral Groove Bearing Classification and Working Principle . . . 203.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Sandvik’s Geometry Mesh . . . . . . . . . . . . . . . . . 253.2.2 Parameterized Geometry Mesh . . . . . . . . . . . . . . . 253.2.3 Sensitivity to Mesh Density . . . . . . . . . . . . . . . . 25

3.3 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 263.4 1D Reynold’s Equation and its Analytical Solution . . . . . . . . 283.5 Homogenized Reynold’s Equation . . . . . . . . . . . . . . . . . 293.6 Applied Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.7 Force Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 Squeeze Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.9 Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.10 Analytical Solution of Pocket Bearing in 1D . . . . . . . . . . . 373.11 Cavitation Model . . . . . . . . . . . . . . . . . . . . . . . . . . 393.12 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Results 424.1 Modelling ladder, part 1 . . . . . . . . . . . . . . . . . . . . . . 42

4.1.1 Analytical - Rayleigh Step-bearing . . . . . . . . . . . . . 424.1.2 Numerical - Rectangular Rayleigh Step-bearing Pad, Dif-

ferent Width:Length Ratios . . . . . . . . . . . . . . . . . 434.1.3 Numerical - Cylindrical (Almost-Quadratic) Pad . . . . . 444.1.4 Numerical - Cylindrical Pad, Different Circumferential

Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.5 Numerical - Straight Groove Bearing (Shallow Grooves . 46

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4.2 Modelling ladder, part 2 . . . . . . . . . . . . . . . . . . . . . . 474.2.1 Straight Groove Bearing - Rigid . . . . . . . . . . . . . . 484.2.2 Spiral Groove Bearing - Rigid . . . . . . . . . . . . . . . 494.2.3 Sandvik’s Piston-step - Rigid . . . . . . . . . . . . . . . . 504.2.4 Spiral Groove Bearing - Deformed . . . . . . . . . . . . . 514.2.5 Sandvik’s Piston-step - Deformed . . . . . . . . . . . . . 524.2.6 Validating the Implementation of Solid Mechanics Physics 53

5 Possible Design 545.1 Shallower Grooves . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Hydrostatic Pressure . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Larger Surface Area of Ridges . . . . . . . . . . . . . . . . . . . 55

5.3.1 Decreasing groove:ridge surface area ratio . . . . . . . . . 555.3.2 Decreasing number of grooves . . . . . . . . . . . . . . . 57

6 Discussion 596.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 Sandvik’s Bearing vs. Parameterzied S.G.B . . . . . . . . . . . . 596.3 Rigid vs. Deformed Simulations . . . . . . . . . . . . . . . . . . 596.4 Design Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Conclusions 61

8 Future Work 62

9 References 63

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Nomenclature

α Threshold constant in cavitation model

β Inclination angle between grooves and velocity vector in s.g.bThreshold 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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1 Introduction

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

1.1 Cone Crushers

Throughout the most of history, crushing of rocks and other materials happenedusing manpower as force is concentrated on the material to be crushed from thetip of a sledge hammer or other similar tools. Initially, most ore crushing andsizing was carried out by hand and hammers at mines or by water poweredtrip hammers in the small charcoal fired smithies and iron works typical of theRenaissance period, after which explosives came into use. Later on, during theindustrial revolution of twentieth century, various forms of mechanical crusherswere 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 inthe mining industry) that reduces material (eg. rocks) by squeezing it betweena a stationary piece of steel and a moving piece of steel. Final reduction insize is determined by the gap between the two crushing members at the lowestpoint. As the mantle rotates to cause the compression within the chamber, thematerial gets smaller as it moves down through the wear liner as the openingin the cavity gets tighter [1]. The crushed material is discharged at the bottomof the machine after they pass through the cavity and if the material is notyet small enough to pass through this cavity, it is returned back to the startingpoint and crushed again till it can pass through the cavity. Figures 1 and 2 helpimagine the working principle of the cone crusher.

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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 interfacebetween the mantle and the bowl which is lined with manganese alloys (14 to22 percent manganese) that provide a certain balance between impact resis-tance and abrasion resistance. The higher the manganese content, the higherthe abrasion resistance but the lower the impact resistance and vice-versa. Theeccentric shaft rotates with a gyratory motion and causes the mantle to rotateas well, which causes the crushing action.

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

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1.2 Sandvik’s Cone Crusher Bearing Set

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

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

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

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

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1.3 COMSOL Mutliphysics

COMSOL Multiphysics is a comprehensive simulation software environment fora 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 physicalapplications together to include all the necessary factors for a complete model,in other words to couple any number of physics together and input user-definedphysics and expressions directly into a model without restrictions posed by othersimulation softwares [3]. Therefore it was chosen as the software to be used tosimulate Sandvik’s bearing.

COMSOL uses Finite Element Methods (FEM) to numerically approximatePartial Differential Equations (PDEs) related to the physics intended to bemodelled. The major advantage of COMSOL is the ability to simulate variousphysics coupled together (multiphysics modelling) on rather complex geome-tries, such as a spiral groove bearing. The FEM is described in more detail inSubsection 2.2 of this thesis.

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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 bearingoperates. 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 createdin parallel, which can be easily edited in terms of its geometry to be able totest different possible changes in geometry that have the potential to make thebearing work under more favourable conditions in terms of reducing asperitycontact throughout the bearing’s life. In the light of this, different possible ge-ometries are investigated and discussed to have an idea about what geometricalchanges can improve the bearing performance.

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

As for the greater motivator, sustainablity has been a main motive in industryfor a while now, so an optimized bearing design that can operate for a longerlifetime will certainly be more sustainable. If the design also leads to lower fric-tion and wear rates, then unwanted material and energy loss will be eliminatedas well.

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1.5 Delimitations

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

1. The lubricating fluid is assumed to be isothermal and Newtonian in be-haviour, which means that its density and viscousity does not rely ontemperature or pressure. The justification behind assuming a Newtonianfluid is that it was certain while modelling the operation of the bearingthat the pressures sustained in the lubricating fluid during a loading cy-cle does not exceed 30 [MPa], which is believed to be a low pressure thatmakes the variation in fluid properties in terms of pressure negligible. Thejustification 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 wellas the cooling of the interface due to the lubricant flow in the grooves isneglected. This is, however, important to model in future work since thefrictional heating of the interface is expected to be significant and wouldaffect the performance of the bearing.

3. Only the lower interface between the piston step and the step washer ismodelled, since it was believed that modelling the upper interface as wellwould 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 crusherwas approximated and simplified by a fourier series to make it easily con-figurable and to facilitate convergence in COMSOL.

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2 Literature Review

In Subsection 2.1, simulation methods utilised in the field of Tribology are in-troduced. In Subsection 2.2, the Finite Element Method is elaborated in moredetail since it is pivotal to the work done in this project. In Subsection 2.3, thevarious literature in which spiral groove bearings were developed are mentioned.In Subsection 2.4, a number of multiphysics models created in COMSOL for dif-ferent types of bearings are mentioned and finally in Subsection 2.5, the gap inthe 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 anyother relevant contact where friction can be found. The field combines chem-istry, material science, solid mechanics, fluid mechanics and physics with an aimto reduce wear and friction in machine components, which have detrimental ef-fects on their useful life. Therefore, when designing a machinery, tribologicalaspects are crucial to the efficiency and life of its components and have to betaken into consideration. The basis of almost any engineering component or ad-vanced material design as of this age is theoretical results gained from computersimulation, which is why simulations have been carried out throughout the yearsto study lubrication regimes, friction and wear on all scales in order to under-stand these behaviours and therefore properly design against them. The area ofsimulations in tribology has developed greatly in the past 50 years, in terms ofcomputational 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 isoften divided into three major parts, as in the proceedings of one of the latestSimulations 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, andthe most famous theory in this area is the Hertzian theory that has been acornerstone in contact mechanics and tribology ever since it has been describedin [5]. The Hertzian theory solves the problem of two ideally smooth surfaces in

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frictionless contact. The JKR 2 and DMT 3 models [6, 7] are two other famousmodels for ideally smooth surfaces, in which the latter takes only the adhesionin the area out of the contact zone into consideration, while the former doesthe opposite. Rough contact has been studied by Greenwood-Williamson andtheir model remains the most widely used, although it has a lot of inaccurateassumptions [8]. Recent advancements in this field have been made in con-tacts in the presence of sharp edges [9] and conformal configurations [10] as wellas studying of contact in the presence of anisotropic and functionally gradedmaterials, with varying friction coefficient along the interface in sliding andpartial slip conditions [11]. Finally, Majumdar and Bhushan have created onof the most popular theories after the GW model for studying contact of roughsurfaces [12]. As for the contact of lubricated surfaces, analytical solutions alsoexist 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 avail-able, and are widely used in modeling large-scale components due to their ac-curacy and small computational time [8]. The most notable and widely usedcontinuum methods are the Finite Element Method (FEM), Finite DifferenceMethod (FDM) and Finite Volume Method (FVM).

FEM is a continuum method which divides a CAD model or a specified geome-try into finite-sized elements of simple geometrical shapes (meshes). A system offield 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 simplelinear (and sometimes nonlinear) equations. When the contributions from allelements are added together, a large sparse matrix equation system is resultedwhich 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 soft-ware, allows for mixed-formulation or multiphysics modelling where more thanone PDE or physical phenomena is approximated on each mesh, and thereforeon the whole geometry. Also, it works very well on complex geometries. How-ever, mathematical formulations behind FEM are quite advanced and need anadequate amount of expertise. In comparison to FDM and FVM, its applicationis less straight-forward.

2To introduce the effect of adhesion in Hertzian contact, Johnson, Kendall, and Robertscreated their theory of adhesive contact using a balance between the stored elastic energy andthe loss in surface energy. This model considers the effect of contact pressure and adhesiononly inside the area of contact.

3This model, named after Derjuagin-Muller-Toporov is an alternative model for adhesivecontact where the contact profile is assumed to remain the same as in Hertzian contact butwith the addition of attractive interactions only outside the area of contact.

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FDM is one of the simplest and oldest methods of solving differential equa-tions, 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 byC.Runge in 1908. However, serious development and practical application of thismethod has started after the 1950s when computers have emerged which allowedsolving more complex problems in engineering and science using this method.FDM is a method used more commonly on regular grids such as rectangularones that are not complex in shape, it takes the continuous form of a PDEand 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 accuratethan FEM in large scale computations such as astrophyiscal and meteorologicalsimulations given that they can fit into regular grids. However, FDM runs intoproblems 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-sizedelements called cells, rather than meshes. It is different however, in the sensethat 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 solutionwhen 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 contactmechanics or any problem in which we are a lot more interested in what is hap-pening on the boundary (surface) than in the volume of the geometry. Whenusing BEM, one can mesh only the boundary and solve on it which save com-putational resources. The main difference between BEM and the previouslymentioned continuum methods is that BEM only needs to solve for unknownson the boundaries, whereas FEM, FDM and FVM solve for unknowns in thevolume [14]. However, to formulate a BEM, a fundamental solution linking pres-sure and vertical displacement in two orthogonal directions is needed, and suchsolution exists for a limited number of cases and mainly under the assumptionthat the solid can be locally considered as a flat half-space, making the BEMmore restrictive to use [15].

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

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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 bydiscretizing the domain into particles (atoms) rather than meshes, as is the casein continuum methods. The DEM 5 and MCA 6 methods may be the mostnotable particle-based methods, the latter being a more recent method thathas proven to be effective and accurate in mesoscale simulations. This MCAmethod combines the advantages of classical cellular automata, discrete elementmethods and molecular dynamics. It assumes that the simulated system is dis-cretised into a series of small elements of a finite size called movable cellularautomata [16]. The movements of those elements are simulated using Newton-Euler equations of motion while including their pair relationship, many bodyforces and bond forces and was created by Psakhie et al [17]. However, if a ma-chine component is to be simulated, it would be made of billions of automatamaking it computationally very expensive to simulate and this the most recentproblem that needs to be faced in this area. Currently, scientists in this fieldwant to examine the possibility of parallel-computing of MCA simulations, thatis to compute on multiple-processors to speed up the computational time andbe 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 byCourant in 1943 [18] in which the piecewise linear approximation of the Dirich-let problem over a network of triangles was discussed. Many additions havebeen made throughout the following years to include partitioning of domains,boundary conditions and local polynomial approximations to come to the firstmature version of FEM in 1956 [19] in which an attempt was made at botha local approximation of the PDE of linear elasticity and the use of assemblystrategies essential to FEM.

5The Discrete Element Method (DEM) is a particle-scale numerical method for modelingthe bulk behavior of granular materials and numerous geomaterials

6The Movable cellular automaton (MCA) method is a computational solid mechanicsmethod based on the discrete concept. In the MCA approach, an object being modeledis considered a set of interacting elements/automata. The dynamics of the set of automataare defined by their mutual forces and rules for their interaction. This system exists andoperates in time and space and its evolution in time and space is governed by the equationsof motion.

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The working principle of the Finite Element Method (FEM) is that it numeri-cally approximates a given Partial Differential Equation related to some physicalphenomena intended to be studied on a domain (eg. viscous heating of a lubri-cant film) by discretizing the domain into meshes (discussed in Subsection 3.2)and numerically approximating the PDE locally on each element (mesh) andadding them up to obtain the global approximation of the PDE on the wholedomain.

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

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

2. Arbitrary geometries: FEM is geometry-independent and can appliedto a wide range of arbitrary geometries with arbitrary boundary condi-tions.

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

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

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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 be-lieved to have been the subject of an earlier paper written by Whipple in 1949[22]. Using a theoretical approach, Whipple derived a formula for predictingthe pressure profile between horizontal strips, one of which has a parallel arrayof straight grooves as shown in Figure 4, while employing various assumptionsto assure that the pressure distribution across the grooves can be regarded aslinear. In this paper, Whipple makes no attempt to apply his formula directlyto spiral groove bearings, but the formula was later used by other researchersto obtain rough but useful estimates of the operating characteristics of spiralgroove bearings. Whipple’s analysis was then improved in accuracy and ex-tended 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 ofequations governing the operation of spiral groove bearings without employingmuch of the assumptions used in [22] is a thesis by Muijderman produced inPhilips 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].

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The working principle of the spiral groove bearing is described in more detailin Subsection 3.1. The pumping effect added to the restriction effect gives anadvantage to spiral groove bearings over other types of self-acting bearings inthat they produce a pressure distribution from the edge of the bearing to thecenter 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 samesize of the bearing. Therefore, air can be used as a lubricating medium and stillproduce 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 thistype of bearing a large prospect of being more widely applied in industry. Aquotation 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 secondwill 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 car-rying capacities for a smaller size of a bearing, which makes it widely used inelectrical appliances and since spiral groove bearings in general can producevery high load carrying capacities for a given size of bearing compared to othertypes of self-acting bearings, they can sometimes be the only design solution inextreme cases such as huge thrust bearings under extreme load conditions inturbines, or in very small bearings (a few mm in diameter) with large rotationalspeeds.

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

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

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

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2.4 Multiphysics Models of Bearings in Literature

A few relevant multiphysics models created using COMSOL for different typesof 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 hy-dropower applications was developed [25].

2. An isothermal multiphysics model is created for journal bearings wheregeometric parameters can be easily altered [26].

3. A multiphysics model was created to aid in design optimization of aero-static 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 groovebearing, they are hard to come across. This necessitates the creation of a multi-physics model with which one can understand deeply how the bearing operatesand which acts as a design tool to help improve the design of the bearing tocreate more favourable lubrication conditions, which would be of great use forSandvik in particular.

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3 Theoretical Basis and Methods

In this section, the theoretical basis of concepts and equations needed for simu-lation of the physics in Sandvik’s bearing is laid out including the Navier-Stokesequations, Homogenized Reynold’s equation, meshing, Solid Mechanics physicsand 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 geometryleads to hydrodynamic pressure build-up. To understand this more, let’s lookat Figure 6 which shows the classification of bearings. Indeed, in Figure 6, aclassification of bearings can be seen, of which the following observations canbe made:

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

2. Bearings in which pressure build-up is a result of a wedge effect/action area subclass of self-acting bearings. Other effects that can induce pressurebuild-up are the stretch effect and buffer effect. From now on, the wedgeeffect will be replaced by the restriction effect or restriction action inter-changeably 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 op-erate using the restriction effect/action. Thrust bearings are those whichsupport mainly or exclusively thrust (axial) loads.

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

5. Under the sub-class of flat spiral groove bearings, bearings with transverseflow are the logarithmic spiral groove bearings and those with no trans-verse 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 isthe type of bearing used by Sandvik in their cone crushers but with excessivelydeep grooves.

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

21

Now to understand the restriction effect/action, let’s imagine the simple Rayleighstep-bearing shown in Figure 7, when the lubricating oil is pushed in the direc-tion shown in the figure it has to pass through a narrower gap (restriction) rightafter the step which causes pressure to be build up in order to open up this gapto balance the flow at the inlet and outlet. If the two surfaces are restrained byan external force, then the pressure build-up will restrain the amount of fluidtrying to enter at the inlet and will push more fluid out at the outlet so thatflow 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 ex-ample, is the restriction action and pumping action together, in which the pump-ing action happens due to the angle between the tangent line to the grooves andthe local velocity vector of the opposing surface which is rotating. The directionof rotation determines if the pumping will be inward or outward [28]. As thelubricating fluid is pumped over the grooves, a narrow gap is faced similar tothe Rayleigh step-bearing which leads to pressure build-up. Inward pumping isrequired to produce a positive pressure distribution and load carrying capacity.

It is very important to note that the grooves in the piston step and mainshaftstep of Sandvik’s bearing are excessively deep in order to allow for a larger vol-ume of cold lubricant to be pumped into the grooves to prevent overheating ofthe interface as well as to accommodate for a large amount of wear that happenson the bearing surface during its operation, where the worn out material fallsinto the grooves. Such deep grooves do not allow for hydrodynamic pressurebuild-up by a mixture of restriction action and inward pumping as explained inthis 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 thelogarithmic spiral groove bearing and herringbone spiral groove bearing aresub-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 thedomain using the Finite Element Method. Meshes come in different sizes andshapes, with sizes ranging in COMSOL from ‘Extremely Coarse’ to ‘ExtremelyFine’ and shapes including: tetrahedra (tets), hexahedra (bricks), triangularprisms (prisms), and pyramids. Mesh sizes may also be numerically specified.The hexahedral shape is the best if accuracy is of a high concern. The shapesare shown below in Figure 9.

Figure 9: Different shapes of meshes [36].

Meshing requires a balance between the desired accuracy, rate of convergenceof the simulation and the computational time needed to run the simulation. Ifthe mesh is very fine, the results will be more accurate but computational timecan increase by orders of magnitude which is undesirable. Therefore, in COM-SOL, meshing of a geometry is very customizable. One can choose the bestshapes, sizes and combinations of shapes and sizes to produce fairly accurateresults (small error, better rate of convergence) with an acceptable computa-tional time. It is then necessary to have a mesh size fine enough to captureall the important flow phenomena and especially in the important areas in the3D geometry (eg. at the step in the Rayleigh step-bearing or in the boundarybetween ridges and grooves in Sandvik’s bearing), but it should be made coarserin areas of the geometry where there is no important details to be captured asto 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 meshtype is tetrahydral, mesh size is set to ’fine’ in COMSOL with 231514 domainelements, 49966 boundary elements, and 8815 edge elements. It is expectedthat setting ’fine’ mesh on COMSOL for this geometry would produce a muchlarger number of meshes than for the parameterized geometry, which is neededfor 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 wherethe mesh type is tetrahydral, mesh size is set to ’extra fine’ in COMSOL sincethere is little detail in the geometry and so using extra fine meshes would not beoverly 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 sizesin COMSOL ranging from ’fine’ to ’extremely fine’ and I found that the resultsin the load sharing graphs converge to certain values as the element size issmaller but without significant error when using larger element sizes in thecase of Sandvik’s geometry, therefore the physics-controlled mesh size ’fine’ waschosen for Sandvik’s geometry to significantly reduce the computation time.

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(a) Meshing on Sandvik’s geome-try

(b) Meshing on the parameterizedgeometry.

Figure 10: Meshing of surfaces

3.3 Navier-Stokes Equations

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

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

26

Continuity equation:

∂ρ

∂t+∂ρu

∂x+∂ρv

∂y+∂ρw

∂z= 0 (1)

X-Momentum:

∂ρu

∂t+∂ρu2

∂x+∂ρuv

∂y+∂ρuw

∂z= −∂p

∂x+

1

Rer

[∂τxx∂x

+∂τxy∂y

+∂τxz∂z

](2a)

Y-Momentum:

∂ρv

∂t+∂ρuv

∂x+∂ρv2

∂y+∂ρwv

∂z= −∂p

∂y+

1

Rer

[∂τxy∂x

+∂τyy∂y

+∂τyz∂z

](2b)

Z-Momentum:

∂ρw

∂t+∂ρuw

∂x+∂ρvw

∂y+∂ρw2

∂z= −∂p

∂z+

1

Rer

[∂τxz∂x

+∂τyz∂y

+∂τzz∂z

](2c)

The Navier-Stokes equations and the continuity equation are also often solvedwith the Energy equation if the fluid flow is non-isothermal. However, since anisothermal solution is assumed in this thesis, the Energy equation is not writtenhere.

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

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3.4 1D Reynold’s Equation and its Analytical Solution

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

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

d

dx

(h3

12η

dp

dx

)=u

2

dh

dx(3)

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

dp

dx= 6ηu

h− hmh3

(4)

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

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3.5 Homogenized Reynold’s Equation

From the Navier-Stokes equations, the 2D Reynold’s equation can be derivedunder a number of assumptions to work as a less computationally expensivemethod for numerically simulating fluid-flow and producing accurate, meaning-ful pressure distributions and fluid-velocity fields for 2D (particularly needed fornon infinitely-wide cases) and 3D geometries. The assumptions made in orderto 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 gradientacross film thickness direction will be negligible. The resulted equation (2) de-rived under these assumptions is the Reynold’s equation in 2D.

∂

∂x

(ρh3

12η

∂p

∂x

)+

∂

∂z

(ρh3

12η

∂p

∂z

)=ω

2

(∂ρh

∂x+∂ρh

∂z

)+∂(ρh)

∂t(5)

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 thicknessis lower than the average surface roughness height and therefore asperity con-tact comes into play, it would need very fine meshes in order to compute theflow of the lubricant between individual asperities. Therefore, a deterministicsolution of mixed lubrication problems is not possible as of now due to limitedcomputational resources, and so one of the most common ways of solving mixedlubrication problems is to compute the homogenized form of Reynold’s equationon the global scale while having the flow factors which are computed on a localscale roughness window obtained from a representative topology as an input tothe equation.

29

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

∂

∂x

(ρa11(h)

12η

∂p

∂x

)+

∂

∂z

(ρa22(h)

12η

∂p

∂z

)=ω

2

(∂(ρb1(h))

∂x+∂ρb2(h)

∂z

)+∂(ρh)

∂t(6)

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

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

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3.6 Applied Force

The thrust force which Sandvik’s bearing set is subject to is highly fluctuatingin a stochastic manner, and so a two-minute signal of the hydroset pressurefrom the cone crusher was captured. When multiplying this pressure with thehydroset piston area, the thrust force that the bearing set is subject to is ob-tained. This force however, with its high fluctuations, will make it difficult forCOMSOL to capture it and will cause convergence error. To solve this problem,a simple force function was selected, adapted and periodised from the force dataas 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 forcefunction 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 theapplied force is carried by the hydrodynamic pressure generated in the lubricantand the rest is carried by the pressure caused due to asperity contact. In orderto calculate this force partitioning (load sharing), a force-balance equation isposed as a Global ODE physics in COMSOL and solved. The equation is asfollows:

F − Fhyd − Fasp = 0 (7)

where

Fhyd = 0.9

∫Ω

phyd (8)

and

Fasp = 0.9

∫Ω

pasp (9)

The hydrodynamic and asperity pressures are integrated over the surface areaof the piston step and then multiplied by 0.9 to obtain the hydrodynamic andasperity forces. Multiplying by 0.9 is for the reason that during eccentric mo-tion 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 thestep washer and the piston step and depending on the magnitude of the crusherstroke. The percentage of area in contact area for the maximum and mini-mum crusher stroke are shown in Table 1 below. The piston step area and areain contact are calculated using COMSOL’s ’measure’ feature which is able tomeasure the surface area of boundaries and objects. The multiplication factor0.9 (pertaining to 90 %) is thus chosen as an intermediate value within the range.

Crusher stroke Piston step area Area in contactMax: 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 groovebearing and the step washer can be seen shaded in purple while the area out ofcontact 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 aRayleigh step-bearing in Figure 7, the other method is the squeeze action whichoccurs when one surface moves with a velocity towards the other as illustrated inFigure 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

the squeeze term is ∂(ρh)∂t . This term can be easily posed in COMSOL’s Co-

efficient Form Boundary PDE physics which is used to solve the equation andleads to COMSOL being able to capture this squeeze effect in time-dependentsimulations.

35

3.9 Solid Mechanics

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

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

1

E∗ =1 − ν2

1

E1+

1 − ν22

E2(10)

and

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

σ = E.ε (11)

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

h = h0 + v (12)

where h0 is the rigid separation and v is the surface deformation in they-direction (direction of filn-thickness/separation). The flow factors whichare input to the homogenized Reynold’s equation are taken from an inter-polation against h and used to solve for the hydrodynamic pressure at theinterface. This is how the coupling between the homogenized Reynold’sequation 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 thecase of spiral groove bearings is the pocket bearing which can be seen as a muchmore simplistic form of a spiral groove bearing. Why this solution will be helpfulis 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 cavita-tion model has been included in order to omit the negative pressures caused bycavitation 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 risessharply after z where it reaches maximum at b (at the step), then drops againafter the step in a non-linear manner. In a spiral groove bearing, there aretypically more than one of these pockets (grooves) in a given radial direction sothe pressure distribution should look like that in Figure 17 but repeated withthe same number of times as there is grooves while the magnitude of maximumpressure drops with every groove starting from the inner edge till reaching theouter edge of the bearing. However, this is for the case when grooves are shallowenough to allow for hydrodynamic pressure build-up due to wedge/restrictionaction.

38

3.11 Cavitation Model

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

f =

1, p - pa > 00, p - pa <-β

1 − 2(p−paβ )3 − 3(p−paβ )2, -β ≤ p− pa ≤ 0

(13)

where

ρ = ρ0f + α

1 + a(14)

η = η0f + α

1 + a(15)

where f is the degree of saturation of the fluid and it varies with the fluidpressure as seen in equation (12), β is the threshold pressure and α is a thresholdconstant that takes a value approaching zero. Then in the areas where pressureis 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 to9000 [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 notan easy task. Also, as far as my knowledge, similar simulations in literature tobe compared against are not present for the time being. Therefore it was neces-sary to validate the numerical models against some analytical solution that weare certain of. A modelling ladder was made starting from the most simple 2Danalytical solution of a Rayleigh step-bearing to make sure that the results wesee for every step in the ladder are sensible. The modelling ladder was dividedinto two parts,

1. Modelling ladder, part 1: In this part, the grooves are shallow enoughto allow for hydrodynamic pressure build-up due to restriction action andis shown in Figure 18. Pure rotational motion of the counter-surface isalso 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 deepas in Sandvik’s piston step geometry and is shown in Figure 19. The ec-centric 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-bearingand the spiral groove bearing according to their respective modelling laddersare shown and discussed.

4.1 Modelling ladder, part 1

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

4.1.1 Analytical - Rayleigh Step-bearing

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

Figure 20: Analytical pressure distribution.

42

4.1.2 Numerical - Rectangular Rayleigh Step-bearing Pad, DifferentWidth:Length Ratios

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

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

Width:length ratio Max. pressure [MPa]1:1 2.8

1.5:1 3.66:1 3.920:1 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 infinitelywide, the closer the maximum pressure value becomes of that of the analyticalmaximum pressure solution shown in Figure 20 and the closer the width becomesto the length, the smaller this maximum pressure value becomes due to sideleakage. This proves that the analytical solution does not suffice when we addanother dimension (width) and that the numerical solution of some form ofthe 2D Reynold’s equation is required for more accurate approximations of thepressure 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 thefurther away the pad is from the center of the bearing and the smaller the angleis between the left and right sides of the pad, the more rectangular it will looklike..The pressure distribution schematic is shown in Figure 22 and the maxi-mum pressure value is shown in Table 3.

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

Parameter ValueMax. pressure 2.78 [MPa]

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

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

44

4.1.4 Numerical - Cylindrical Pad, Different Circumferential Widths

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

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

Parameter Wide Typical NarrowMax. pressure [MPa] 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 threevariants.

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

45

4.1.5 Numerical - Straight Groove Bearing (Shallow Grooves)

In this subsection, the time-dependent results from the parameterized straightgroove bearing with shallow grooves are shown. The load sharing graph canbe seen in Figure 24 and the graphic representation of the hydrodynamic andasperity 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]- pressure in [Pa].

(b) Asperity pressure at 0.14 [s] - pres-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 sur-face, the pressure distribution can be seen to resemble that of the cylindricalRayleigh step-bearing pad shown in Figure 23 in that the maximum pressureoccurs on the step (groove-ridge boundary) due to the restriction/wedge actionbut it is most similar to the pressure distribution expected from a pocket bear-ing as shown in Figure 17 where the pressure is zero in the grooves but spikesup rapidly to reach a maximum at the groove-ridge boundary then falls downnon-linearly over the ridge, and thus the pressure distribution obtained in thissimulation can be deemed as sensible.

4.2 Modelling Ladder - Part 2

In this subsection the solutions following the validation modelling ladder shownin Figure 19 are laid out. In this modelling ladder, the grooves are deep justlike in Sandvik’s piston step. Due to the depth of the grooves, a hydrodynamicpressure build-up due to the restriction action will not be present, but ratheronly due to a squeeze action over the ridges as explained previously. For eachvariant, the load sharing graph will be shown for 0.2 [s] of the loading cycleand also a graphic representation of the hydrodynamic pressure build-up due tosqueeze 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 parame-terized straight grooves and a rigid surface (ie. no coupling of Solid Mechanicsphysics). The load sharing graph can be seen in Figure 26 and the graphicrepresentation of the hydrodynamic and asperity pressures taken at 0.14 [s] areshown in Figure 27.

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

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

(b) Asperity pressure at 0.14 [s] - pres-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 groovebearing with a rigid surface (ie. no coupling of Solid Mechanics physics). Theload sharing graph can be seen in Figure 28 and the graphic representation ofthe hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure29.

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

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

(b) Asperity pressure at 0.14 [s] - pres-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 geom-etry with a rigid surface (ie. no coupling of Solid Mechanics physics). The loadsharing graph can be seen in Figure 30 and the graphic representation of thehydrodynamic 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-sure in [Pa].

(b) Asperity pressure at 0.14 [s] - pres-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 groovebearing with a deformed surface (ie. Solid Mechanics physics coupled). Theload sharing graph can be seen in Figure 32 and the graphic representation ofthe hydrodynamic and asperity pressures taken at 0.14 [s] are shown in Figure33.

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

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

(b) Asperity pressure at 0.14 [s] - pressurein [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 geom-etry with a deformed surface (ie. Solid Mechanics physics coupled). The loadsharing graph can be seen in Figure 34 and the graphic representation of thehydrodynamic 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] -pressure in [Pa].

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

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

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4.2.6 Validating the Implementation of Solid Mechanics Physics

In order to validate the implementation of the Solid Mechanics physics in themodels, a series of simulations were run for the parameterized spiral groovebearing with different degrees of elasticity of the bearing surface measured withthe equivalent Young’s modulus. The goal is to make sure that both the hydro-dynamic and asperity load in the load sharing graph would converge towardsthe rigid load sharing solution as the bearing surface is set to be more and morerigid. The convergence of hydrodynamic and asperity load shares towards therigid 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.

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5 Possible Design Changes

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

5.1 Shallower Grooves

If we were to compare, for example, the results for the Straight Groove Bearingin 4.1.5 where the grooves are shallow and in 4.2.1 where the grooves are exces-sively deep, it would be immediately clear that shallow grooves (in order of afew multiples the oil film thickness over the ridges) allow for full-film generationdue to the restriction action discussed previously and this leads to completemitigation of undesirable asperity contact. Such full-film generation is not pos-sible when the grooves are excessively deep as in the current design Sandvik’sbearing-set. The reason why the grooves were chosen to be excessively deep inthe first place is to allow for a larger volume of cold oil to flow in the grooves andcool 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 overthe ridges while keeping the depth of the grooves. However, if shallow grooveswere used, will the full-film generation of the lubricant reduce heating of theinterfaces to an extent that a large volume of cold oil flow in the grooves willnot be needed any more? This is an important question to investigate in fu-ture work where the thermal effects will be included in the model. Also, it isimportant to note that in cone crushers, the operating conditions in terms ofload and motion are extreme and may make impossible for a full-film of lubri-cant 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 thepiston-step using an adequate amount hydrostatic pressure, this can separatethe piston-step and step-washer surfaces from coming into contact. However, ifwe rely only on hydrostatic pressure and it ceases to exist momentarily due toa technical fault, the piston-step and step-washer surfaces will crash in a waythat can destroy the interface.

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5.3 Larger Surface Area of Ridges

If we were to choose to only enhance the squeeze action over the ridges, thenincreasing the surface area of the ridges on which squeeze action occurs whiledecreasing the surface area of the grooves can be a good choice. However, thetotal volume of oil that can flow in the grooves has to be kept constant in orderto not reduce the cooling capability in the bearing. The increase in surface areaof the ridges can be done by editing geometric parameters in the model in thefollowing manner:

1. Decreasing groove:ridge surface area ratio + increasing groove depth tokeep total groove volume constant.

2. Decreasing number of grooves + increasing groove depth to keep totalgroove volume constant.

And therefore, these two possibilities were tested and the results are shown in5.3.1 and 5.3.2.

5.3.1 Decreasing groove:ridge surface area ratio

Herein, 5 different variants of the parameterized spiral groove bearing are mod-elled, with different groove depths and groove:ridge surface area ratios. Thegeometric parameters of the different variants are shown in Table 5 where G:Ris 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 asperityload shares for all variants are shown in Figure 37. The results were truncatedat 0.18 [s] because some of the simulations faced trouble in convergence beyondthis point in time.

Variants G:R GD [m] Total groove volume [m3]1 1:2 5.7E-4 1E-42 1:3 6E-3 1E-43 1:4 8E-3 1E-44 1:5 1E-2 1E-45 1:6 1.7E-2 1E-4

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

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(a) Hydrodynamic load share.

(b) Asperity load share.

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

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5.3.2 Decreasing number of grooves

Herein, 5 different variants of the parameterized spiral groove bearing are mod-elled, with different groove depths and number of grooves. The geometric pa-rameters of the different variants are shown in Table 6 where GN is the numberof grooves and GD is the groove depth and are set such that the total groovevolume is kept constant.

Variants GN G:R GD [m] Total groove volume [m3]1 7 1/3 6E-3 1E-42 6 0.28 7E-3 1E-43 5 0.235 8.4E-3 1E-44 4 0.19 8.8E-3 1E-45 3 0.14 1.4E-2 1E-4

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

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(a) Hydrodynamic load share.

’

(b) Asperity load share.

Figure 38: Load sharing for different groove numbers.

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6 Discussion

In this section, the results shown in the previous section as well as the insightgained 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 sim-ulation results are logical such that every simulation result behaves as expectedin light of the simulations that comes before it in the ladder. However, the log-ical links between successive simulations may not be clear by simply looking atthe 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 atest rig would be very useful, although the operating conditions that Sandvik’sbearing-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 Figure34 and that of the parameterized spiral groove bearing in Figure 30, it is notablethat the squeeze action in Sandvik’s bearing is more prominent and thus asperitycontact 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 (moreasperity contact), is that as the ridges deform, they face maximum deformationat the center of the ridge and the step washer (opposite surface) moves towardsthe ridge with an amount that is proportional to the deformation. It was no-ticed that as the step washer moves towards the deformed ridges, its surface getsreally close to the sides of the ridge, and so higher asperity pressure is noticedon the sides of each ridge. This can be clear when looking at Figure 33 (b) andFigure 35 (b).

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6.4 Design Changes

The design changes proposed in Section 5 may well and truly be useful in miti-gating asperity friction on the interfaces in Sandvik’s bearing, however a numberof factors are not considered in the simulations and should be considered beforeapplying such changes. These factors include:

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

2. The distance between grooves in the current design of the piston-step wastailored to allow the whole surface of the piston-step to be exposed to thelubricating 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 afull-film of lubricant due to restriction action under the extreme conditionsposed on it in the cone crusher should be investigated experimentally, asit is believed that it may be impossible for such a full-film of lubricantto be generated, in which case the operation of the bearing may ceasedetrimentally.

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

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7 Conclusions

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

When looking at the load sharing graphs obtained from both models in Section4, which describe how the applied thrust load is partitioned between hydro-dynamic and asperity loads in the lubricated interfaces of the bearing, it canbe concluded that Sandvik’s bearing runs under severe mixed-lubrication con-ditions. This is believed to be, as explained previously, due to the excessivedepth of the grooves which make it impossible for a full-film of lubricant to begenerated due to a restriction action and therefore, the only method by whicha lubricant film is generated in the interfaces of the bearing-set is the squeezeaction. The pressure sustained in the lubricant film due to squeeze action is notsufficient to fully support the thrust load that the bearing-set is subject to andso asperities come into contact.

In Section 5, several possible design changes which can either enhance thesqueeze 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 thesedesign changes are mentioned. Overall, the objectives of the thesis have beenmet, although improvements have to be made in order to have models thatproduce very realistic and reliable results. These improvements to be made arediscussed in the following section.

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8 Future Work

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

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Appendix

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

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

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