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Spiral Groove Bearing Multiphysics Modeling 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 min- ing 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 Sj¨oberg, 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] ! Eccentric velocity [rad/s] ρ Fluid density [kg/m3] σ Stress [Pa] τ Shear stress [Pa] υi Poission ratio " 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] 3 This page is intentionally left blank. 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].
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