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- http://digital.auraria.edu/AA00007242/00001
## Material Information- Title:
- Numerical analysis of lubricated contacts
- Creator:
- Mertz, Alexander M.
- Place of Publication:
- Denver, CO
- Publisher:
- University of Colorado Denver
- Publication Date:
- 2019
- Language:
- English
## Thesis/Dissertation Information- Degree:
- Doctorate ( Doctor of philosophy)
- Degree Grantor:
- University of Colorado Denver
- Degree Divisions:
- College of Engineering and Applied Sciences, CU Denver
- Degree Disciplines:
- Engineering and applied science
- Committee Chair:
- Welch, Samuel
- Committee Members:
- Rorrer, Ron
Ingber, Marc Graham, Alan Schreyer, Lynn
## Notes- Abstract:
- Ball bearing performance is predicated on the ball-race interface. Simple analytical models can predict the required torque and power lost if the interface is absent of lubricant. Lubricant is added to the interface to decrease bearing frictional power loss and bearing wear. Depending on the operating conditions of speed and load, hydrodynamic forces can fully separate the ball from the bearing interface. This is called full film lubrication. Although this is just one possible lubrication regime, it has the best performance in terms of friction and power lost. Engineers strive to operate bearings in this regime. In this work, the details of the lubricated interface are studied numerically with the finite element analysis (FEA) software COMSOL. A simplified version of the Navier-Stokes equations, the Reynolds equation, is used to model the fluid to reduce the problem size and avoid a highly nonlinear fluid structure interaction (FSI) problem. This is coupled with a full 3D elastic representation of the race and an equation that satisfies load continuity. The pressure in these interfaces can easily exceed 1GPa which has dramatic effects on the fluidâ€™s properties. Therefore, we model the fluid as Newtonian with temperature and pressure dependent viscosity and density. Multiple rheological models are compared. This method is then extended to include non-isothermal effects, surface roughness, and solid-solid contact in cases where the surface roughness is larger than the fluid film to study bearing performance under a variety of conditions. Elastic and plastic asperity contact models are considered. The result is an engineering tool that can aid in the design of bearing systems. Many results are specific to the system being studied, but where possible, results are represented as a function of nondimensional parameters. Propeller shaft bearings in marine and Naval applications operate on a similar set of principles as oil lubricated ball bearings. The geometry and lubricant differ significantly. However, the underlying physics are similar. Hydrodynamic forces can separate the shaft from the bearing depending on the operating conditions of speed and load, solid contact occurs if these requirements arenâ€™t met. The model developed for oil lubricated ball bearings is applied to water lubricated polymer lined journal bearings. The results of this model are presented non-dimensionally with analytic functions so that engineers can use them. A parameter study is presented consisting of over 500 unique solutions. Multiple second order effects are considered. Comparison to experimental data is presented.
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NUMERICAL ANALYSIS OF LUBRICATED CONTACTS
by ALEXANDER M. MERTZ B.S., New Mexico State University, 2010 M.S. University of Wisconsin Madison, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Engineering and Applied Science Program 2019 This thesis for the Doctor of Philosophy degree by Alexander M. Mertz has been approved for the Engineering and Applied Science Program by Samuel Welch, Chair Ron Rorrer, Advisor Marc Ingber Alan Graham Lynn Schreyer Date: August 3, 2019 11 Mertz, Alexander McMillen (PhD., Engineering and Applied Science) Numerical Analysis of Lubricated Contacts Thesis directed by Associate Professor Ronald Rorrer ABSTRACT Ball bearing performance is predicated on the ball-race interface. Simple analytical models can predict the required torque and power lost if the interface is absent of lubricant. Lubricant is added to the interface to decrease bearing frictional power loss and bearing wear. Depending on the operating conditions of speed and load, hydrodynamic forces can fully separate the ball from the bearing interface. This is called full him lubrication. Although this is just one possible lubrication regime, it has the best performance in terms of friction and power lost. Engineers strive to operate bearings in this regime. In this work, the details of the lubricated interface are studied numerically with the finite element analysis (FEA) software COMSOL. A simplified version of the Navier-Stokes equations, the Reynolds equation, is used to model the fluid to reduce the problem size and avoid a highly nonlinear fluid structure interaction (FSI) problem. This is coupled with a full 3D elastic representation of the race and an equation that satisfies load continuity. The pressure in these interfaces can easily exceed 1 GPa which has dramatic effects on the fluidâ€™s properties. Therefore, we model the fluid as Newtonian with temperature and pressure dependent viscosity and density. Multiple rheological models are compared. This method is then extended to include non-isothermal effects, surface roughness, and solid-solid contact in cases where the surface roughness is larger than the fluid him to study bearing performance under a variety of conditions. Elastic and plastic asperity contact models are considered. The result is an engineering tool that can aid in the design of bearing systems. Many results are specific to the system being studied, but where possible, results are represented as a function of nondimensional parameters. Propeller shaft bearings in marine and Naval applications operate on a similar set of principles as oil lubricated ball bearings. The geometry and lubricant differ significantly. However, the underlying physics are similar. Hydrodynamic forces can separate the shaft from the bearing depending on the operating conditions of speed and load, solid contact occurs if these requirements arenâ€™t met. The model developed for oil lubricated ball bearings is applied to water lubricated polymer lined journal bearings. The results of this model are presented non-dimensionally with analytic functions so that engineers can use them. A parameter study is presented consisting of over 500 unique solutions. Multiple second order effects are considered. Comparison to experimental data is presented. The form and content of this abstract are approved. I recommend its publication. Approved: Ronald Rorrer IV Thank you to all that have supported me throughout this process. v CONTENTS CHAPTER I INTRODUCTION.................................................. 1 II ELASTO-HYDRODYNAMIC LUBRICATION (EHL)........................ 10 2.1 Derivation of Reynoldâ€™s Equation........................... 10 2.2 Material Properties........................................ 17 2.3 Coupled EHL Model...........................................27 2.4 Weak Form and Stabilization.................................30 2.5 Mesh Convergence............................................32 2.6 Pressure Penalty Convergence................................36 2.7 Model Validation............................................37 2.8 EHL Results.................................................38 III MIXED-ELASTO-HYDRODYNAMIC LUBRICATION (MEHL)..................45 3.1 Elastic Asperity Deformation................................47 3.2 Plastic Asperity Deformation................................49 vi 3.3 Comparison of Plastic to Elastic Asperity Deformation 50 3.4 Implementation into EHL Model.................................52 3.5 Results of the Elasto-Hydrodnamic With Asperity The Plastic Contact Model.........................................................55 IV THERMAL-MIXED-EHL (TMEHL)........................................68 4.1 Model.........................................................69 4.2 TMEHL Results.................................................79 4.3 TMEHL Model Usefulness as Design Tool.........................86 V FULL BEARING SYSTEM MEHL.........................................88 5.1 Introduction..................................................89 5.2 Model.........................................................90 5.3 Dimensional Analysis..........................................99 5.4 Results and Discussion.......................................102 5.5 Conclusions..................................................129 BIBLIOGRAPHY..............................................................139 APPENDIX A Code For Fitting Viscosity And Density Models....................140 B Values Of Integrals Of The Type fn (H) = (s â€” H)n (f) (s) ds 147 VARIABLE LIST a Hertzian contact radius C-ii di) C\ Constants in multi mode exponential fluid load fraction model A Ashurst Hoover power law parameter B Ashurst Hoover power law parameter Bd Doolittle exponential parameter O o ~o o Characteristic lengths in the x, y, z directions respectively c Journal bearing radial clearance O^Icq" III o Contact parameter Cp Heat capacity at constant pressure d Hertzian indentation depth e error E Youngâ€™s modulus E* Combined modulus F Applied Force pr â€” logo: u Froude number 9 Gravitational constant G h jj = ctE1 Hamrock and Downson dimensionless material parameter h Film thickness ht Modified him thickness h0 Film thickness constant TT hR a2 Dimensionless him thickness Ht = 1 a2 Dimensionless modihed him thickness TT â€” h HsÂ«= v Solid contact gap thickness Ha Hardness k Thermal conductivity vm Kqr Bulk Modulus at reference state Kb r Derivative of bulk modulus with respect to pressure at reference state L = aE>( 2^)1/4 Moeâ€™s dimensionless speed parameter Euclidian vector norm M ~ F (oRoum\-3/4 Moeâ€™s dimensionless load parameter m Mass MX, My Mass flow rate per unit length in x, y directions respectively n Normal vector N Number of asperities per unit area Nc Number of individual asperity contacts for a given macro-contact P Fluid pressure Pc Solid contact pressure Ph Hertzian contact pressure P â€” P Ph Dimensionless fluid pressure Pc = â€” c Ph Dimensionless solid contact pressure Pent Cutoff pressure P&x-, P&y-, P&z Peclet number in the x, y and z direction respectively PID Relative Amount of artificial diffusion added Pf Cavitation penalty factor q= [qx,qv,qz\ Heat flux vector and components 9m Heat flux due to sliding asperity contact friction Qv Volumetric heating rate ^ Qvd Volumetric heating rate due to viscous dissipation Qc Volumetric heating rate due to work done on fluid r Radial coordinate in spherical system R Combined radius IX i?l, i?2 Sphere or cylinder radius Ra Mean surface roughness Rq = RrMS Root mean square surface roughness Re = hÂ°u.Po Volo Reynoldâ€™s number SRR ~ Um Slide to roll ratio Sq Combined surface roughness parameter t Temporal coordinate ^0 Characteristic time T = i Dimensionless time T Temperature AT = T - T0 Change in temperature Tavg = Jq Tdz Film averaged temperature ATavg = fQh (T - To) dz Film averaged temperature change To Reference Temperature Tr Reference temperature for rheological models Tg Glass transition temperature u = [u, V, w\ Displacements vector u with components u,v,w uo,vo, wo Characteristic velocities in x, y, z directions respectively ^a ? ^b x component of velocity of the lower (a) and upper (b) surfaces â€ž â€” Â«a+M6 tim â€” 2 Mean entrainment velocity rr â€” iToUm uHD - E,R Flamrock and Dowson dimensionless speed parameter V \^xi 'Vyi ^z\ Velocity vector and components u,v,w Dimensionless fluid velocities in x, y, z directions respectively V Specific volume Vo Specific volume at reference state ^molec Specific volume of a single molecule X WHD - E,R2 Hamrock and Dowson dimensionless load parameter x,y,z cartesian coordinates X, Y, Z Dimensionless Cartesian coordinates as Pressure viscosity coefficient for the Barus model O-Ch Pressure viscosity coefficient for the Cheng model (%-Roe Pressure viscosity coefficient for the Roelands model aT = -Jb; Thermal diffusivity P Asperity Radius Pk Temperature coefficient for bulk modulus P'k Temperature coefficient for bulk modulus derivative with respect to pressure Pch Temperature viscosity coefficient for the Cheng model I^Roe Temperature viscosity coefficient for the Roelands model Pdh Temperature coefficient for the Dowson Higgenson equation of state Pv - (t) fe)) Ashurst-Hoover scaling parameter Â£ = Ml i)o A Reynolds equation parameter e strain A Mesh size A4; A4/; A4s Total friction coefficient, friction due to viscous forces, and solid contact friction R Lame parameter P Density Po Reference density P=~ r P 0 Dimensionless density XI 4>(z) Probability distribution of asperity heights $ Viscous dissipation function Â£ = ho Ot-Tt 0 Dimensionless temporal parameter V Viscosity Vo Reference viscosity % Viscosity at glass transition temperature v = ^ 1 Vo Dimensionless viscosity -y 12umr)o R2 ~ a3Ph Reynolds equation parameter \ - f fPdQ A/ â€” F Fluid him load fraction a â€” II Podn Ac â€” p Asperity contact load fraction A = NE*/3isl Greenwood Williamson model stiffness parameter a Fluid and solid Cauchy stress tensor â€” h20po as - tovo Squeeze number o~i Standard deviation of asperity height distribution for surface i P Test function for Galerkin FEM UJ Rotational velocity Qf Lubrication domain a Solid domain (5 solid displacement e = i Dimensionless temperature A0 = T-TÂ° J- 0 Dimensionless temperature rise CHAPTER I INTRODUCTION Solid to solid contact with sliding is the main cause of wear in mechanical elements. While friction is essential to any mechanical system, it can cause excessive power loss and premature failure. Many mechanical systems are lubricated to combat this. Examples of lubrication stretch back as far as the 17th century BC, but it is only in the last 150 years or so that we have come to understand how lubrication works. In the late 1800â€™s, train axles were lubricated with heavy grease. This grease needed to be changed periodically which was done through a grease port. The grease ports were plugged with a cork to prevent leakage and contamination when not in use. It was observed that during normal operation of the bearing the grease port cork would be pushed out. The relative motion of the spinning axle on the stationary sleeve, combined with the immense weight of the train, generated a hydrodynamic pressure which pushed the cork out. Such observations led to laboratory experiments, the first of which was preformed by a Mr. Beauhamp Tower in 1883. His work measured the hydrodynamic pressure azimuthally around a brass sleeve with a spinning cylinder in it separated by a thin him of olive oil. He found a 625 PSI (42.5 ATM) peak pressure rise [64], It was hypothesized that for the fluid pressure to rise, there must be a him that completely separates the two surfaces. Inspired by this result, Osbourne Reynolds calculated a similar pressure rise by simplifying the Navier-Stokes equations [55]. For a continuum equation such as the Navier-Stokes equations to accurately predict the pressure, there must be a him sufficiently thick to be considered a continuum. This confirms the hypothesis of a significant fluid him separating 1 two surfaces supporting a significant load. This fluid film has been measured to nanometer level accuracy with modern equipment by Jubault [67]. As long as the fluid him is thicker thicker than the largest surface asperities wear and friction is minimized. Two surfaces separated by a fluid him carrying a normal load via hydrodynamic pressure is the essence of hydrodynamic lubrication. The hydrodynamic lubrication (HL) regime is dehned as when the huid him is much thicker than the elastic deformations due to the hydrodynamic pressure. Increasing the load further will elastically deform the two solids on the order of the him thickness or greater. This is called elasto-hydrodynamic lubrication, or EHL for short. If the load is further increased such that the him thickness is on the same order as the surface roughness, then solid contact occurs, this is known as mixed lubrication (ML), or mixed elasto-hydrodynamic lubrication (MEHL). Further decrease in him thickness almost completely removes the lubricant from the area of contact. There may be a few molecules thick of lubricant bonded to the solid surfaces. This is known as boundary lubrication. Plotting the hction coefficient, p, as a function of the Sommerfeld number,^ as shown in hgure 1.1, reveals the frictional characteristics of each lubrication regime. Where rj is the viscosity, oj is the rotational velocity, and p is the contact pressure. This is known as the Stribeck curve [61]. The Sommerfeld number, which is sometimes called the Hersey number, is proportional to the him thickness, therefore it is useful the think of the horizontal axis as him thickness. The wear rates increase dramatically at low him thicknesses due to increased solid solid contact. The dehnition of pressure in the Sommerfeld number is unclear. It may mean maximum huid pressure, or maximum Hertzian contact pressure (dehned in table 1.1), or the applied load divided by the bearing area depending on the author. The underlying meaning is consistent for each dehnition. Bearing geometries generally fall under two classihcations: conformal or non conformal (also known as counterformal) as shown in hgure 1.2. Conformal contacts spread the load out over a greater area reducing the peak pressure in the lubricant. Conversely non conformal contacts focus the load over a small area, greatly increasing the peak 2 V Figure 1.1. Stribeck curve: frictional coefficient as a function of the Sommerfeld number. pressure in the lubricant. The peak fluid pressure scales with the Hertzian contact pressure. Hertzian contact theory describes the contact pressures and deformations between two elastic contacting bodies in the absence of frictional forces. Point contacts occur when two bodies in contact at zero load contact only at one point. Examples are spheres contacting spheres and spheres contacting planes. Line contacts occur when two bodies in contact at zero load touch only on a single line. Examples are cylinders contacting cylinders and cylinders contacting planes. Table 1.1 defines the contact radii, a, contact pressures, ph, and indentation depths, cl, for line and point contacts, where E* is the combined modulus and R is the combined radii, v is Poissionâ€™s ratio, and F is the applied load. Note that these solutions are valid if i?2 = oo, and it becomes a half space. In this case, R = R\. Due to the fully coupled nature of the fluid-solid problem, and the nonlinear lubricant properties, analytic solutions do not exist. Authors have been interested in numerical solutions since the 1950s. The first attempt at a line contact was preformed by Grubin in 1949 [26]. In lieu of solving the coupled elasto-hydrodynamic problem he assumed the 3 Conformal Non-conformal Figure 1.2. Illustration of conformal (left) and non-conformal (right) contacts. Table 1.1: Definitions of contact radii, a, contact pressures, ph, and indentation depths, cl, combined modulus,if*, and combined radii, R, for line and point contacts. Point Line a (3 FR\li / 8RF V 4E* ) V wLE* Ph 3 F / FE* 'iwa2 V wLR d a2 a2 R 2 R E* i _ i-v'i , i-v'i E* E-\ E? 1 _ l-v'i , i-A E* E-\ E? R i - x + x ti ti\ Jrt2 ti ti\ ti2 elastic deformation was equal to the deformation encountered in the Hertzian [36] problem. Many early EHL solutions followed this blueprint of using the Hertzian elastic solution for line and point contacts [13, 68]. This provided a central him thickness solution that agreed well with experimental results. However, it failed to capture the pressure spike and exit contraction, as seen in the results in chapter II, that is exclusive to the coupled problem. The first coupled solution to the EHL problem was done by Hamroek and Dowson in 1976 [21, 22, 23], who later wrote an extensive textbook on fluid him lubrication [9]. Their numerical approach was to solve the Reynoldâ€™s equation via central finite difference iterating between the elastic and huicl problem using relaxation to ensure convergence. The steep pressure gradients that cause numerical instabilities were mitigated by combining the pressure and him thickness into a single variable, pH3/2. After solving for pH3/2, p is solved using the standard definition of him thickness and the solution of pH3/2. Major speedups came when Lubrecht [1, 2, 3] applied multilevel multi integration 4 (MLMI) and multi-grid techniques. MLMI allows a faster computation of the elastic deformation. Multi grid techniques are known to increase convergence rates by starting with a very coarse mesh and successively refining while smoothing out the largest wavelength errors first. The next jump in computational capability came in 2001 when Zhu and Hu [39] extended the classic formulation to include mixed lubrication using a unified formulation. This method allows for contacts to share loads between solid-solid contact and fluid him lubrication. Zhu and Hu did extensive heat transfer calculations, accounting for viscous dissipation, compressive heating, and solid-solid frictional heating. Coupling the EHL problem with the thermal problem requires additional degrees of freedom, making the convergence rate of the EHL problem more important. Other solid contact models implemented into EHL solvers include Kraker (2006), [18]. This method uses a modified definition of him thickness that conserves the mass of the lubricant trapped between the two rough surfaces. All solid contact is assumed to be plastic. This method is also a unihed formulation, but has only been used in water lubricated journal bearings to date. All the methods discussed previously were hnite difference method solutions that rely on relaxation techniques to stabilize the coupled system of equations. In 2008, Habchi [34] used a full system hnite element formulation with various stabilization techniques to solve classic EHL problems. Full system means all the equations are numerically solved simultaneously as a system of linear equations. By putting all equations into one system cross coupling terms are taken into account that allow a solution with no relaxation. This allows the full power of Newtonâ€™s method to be realized ensuring quadratic convergence. The best possible numerical model for these kinds of systems would have the stability and speed of the full system approach coupled with the multi regime capability of the Zhu and Hu or Kraker hnite difference formulations. Reynoldâ€™s equation is generally written for a compressible huid with a pressure and temperature dependent viscosity. It cannot handle phase change and will predict negative pressures at the diverging side of the contact. In reality the huid will cavitate at pressures 5 below the vapor pressure, and therefore large negative pressures cannot exist. There are multiple schemes to mitigate this unrealistic modeling result. The most simple is to truncate the negative pressures to the vapor pressure of the lubricant. More commonly used is the penalty method that penalizes negative pressures at the vapor pressure. Neither of these methods conserve the mass of the lubricant. This is only an issue in systems where the cavitation region is large compared to the contact area, or when cavitation pressures are over similar magnitude to fluid operating pressures. One sophisticated model that deals with this is the Jakobsson-Floberg-Olsson (JFO) model [28]. The JFO model solves two different PDEâ€™s, a modified Reynolds equation valid in the full him regime, and a PDE that describes the flow of vapors in the cavitation region. The interface between the two regions must be solved for each iteration, requiring special meshing strategies. Elrod [27] improved this model by introducing a modified Reynolds equation that is valid in both the full him region and the cavitation region which also satisfies conservation of mass. The downside of this solution is that the pressure difference between the fluid and vapor section is dependent on the bulk modulus of the fluid, which must be relaxed for convergence. This work involves developing numerical models that produce dimensionless results that are presented in a general form, with functional fits where possible, for engineers to use. This work covers two main topics: 1. Oil lubricated point contact 2. Water lubricated line contact Both problems involve a coupled fluid structure interaction problem. In chapter II, the assumptions required to derive the Reynolds equation from the Navier-Stokes equations are stated and justified. Many lubricant models are discussed and fitted to experimental data from the literature. The EHL model is fully explained and results are presented as well as 6 model validation. A mesh convergence study is presented. Dimensionless parameters from the literature are presented that uniquely identify EHL solutions. In chapter III, a statistical representation of surface roughness is presented, and multiple solid-solid contact models are implemented to extend the model into the mixed and boundary lubrication regime. These models can be used for the oil lubricated point contact and the water lubricated line contact problem. Two new dimensionless parameters are presented that account for the effects of solid-solid contact. A fitted model for the fraction of load supported by the fluid him as a function of dimensionless parameters is presented. Estimates of the friction coefficient of the rolling element bearing are also presented and discussed. In chapter 4, a thermal model is coupled with the MEHL model. Temperature rises come from three heating sources: viscous dissipation, frictional heating, and compressive heating. The temperature rise effects the fluidâ€™s material properties, in turn effecting the pressure and him thickness and the heating rates in the fluid. Dimensionless parameters are presented. Conditions that thermally degrade the lubricant are presented. The use of water as a lubricant is driven by environmental and defense concerns instead of tribological performance in marine propulsor shaft applications. Oil lubricated bearings require an oil seal where the shaft exits the ships hull which inevitably leaks. This leakage, which is considered normal operational consumption and acceptable practice, was estimated at between 130 to 244 million liters a year world wide in 2010 [24], For reference, the well known Exxon Valdez spilled 41.6 million liters of oil in 1989. This is an environmental concern, but also a defense concern as it is generally undesirable to have a detectible chemical leaking from warships. Water lubricated bearings were implemented because of these concerns. Oil-lubricated ball bearing contacts are considered in the chapters II, III, andIV. Sea water-lubricated journal bearings are studied in chapter V. The underlying physics of mixed-elasto-hydrodyanmic lubrication applies in both systems, therefore the models 7 developed in chapters II and III can be easily adapted to this new system. In chapter 5, the MEHL formulation is extended to solve on real world engineering surfaces so entire bearing assemblies may be modeled, instead of a small half space that is representative of a bearing race. Different bearing fixtures and their effects on lubrication performance are presented. The effect of fixture stiffness on lubrication performance for polymer lined water lubricated bearings is presented and compared to experimental data. A fitted model for the fraction of load supported by the fluid him as a function of dimensionless parameters is presented. Water lubricated bearings are prevalent in a wide variety of industrial and marine applications, and thus the prediction of bearing life has been a primary focus of engineering analysis. Operating in the full him hydrodynamic regime over the entire bearing, which prevents solid contact, increases bearing life. Of course, full him lubrication is not always possible given the size, load, and speed requirements of a bearing. The use of soft polymer or elastomer bearing liners extends the full him regime to lower speeds by increasing the conformity of the contact and therefore spreading the load out over a larger area. This is the basis of elastohydrodynamic lubrication. Current water lubrication models in the literature are quite sophisticated. A popular recent focus is modeling multiple lubrication regimes, spanning full him lubrication down to mixed and boundary lubrication [18, 37, 63]. This is usually done with a modified formulation of the Reynolds equation that is valid in both full him lubrication and mixed/boundary lubrication. Rarely are these models compared to experimental data. This may be due to the inherent difficulty in experimentally decoupling the reaction torque from the applied load. This may not seem like a hard problem, but coefficients of friction can be as low as 0.0001, meaning the torque can be 4 orders of magnitude lower than the applied load. To further complicate matters, the friction coefficient can be as high as 0.5, meaning the torque transducer also has to span four orders of magnitude. Another area of research has been the evaluation of transient effects, including stability 8 analysis of stave bearings. These models employ a transient formulation of the Reynolds equation that are used to determine the dynamic effects on the bearing in the hydrodynamic lubrication regime. In some cases, useful stiffness and damping coefficients were extracted for use in other larger scale dynamic models [52], The shape and size of the cavitation region is also of great interest as it affects the performance characteristics of the bearing. A variety of cavitation models that range in complexity have been proposed. Initial models simply truncated anything below the cavitation pressure to the cavitation pressure each iteration. A more sophisticated method penalizes pressures below the cavitation pressure in order to raise the pressure up to the cavitation pressure [51]. Neither of these methods conserves mass. To address this issue, the JFO [28] model was developed and although it conserves mass in the cavitation region, it requires solving one PDE in the cavitation region and another in the full him region. Also, the position of the interface between the two regimes must be solved during each iteration, which increases the complexity of the model. Elrod [27] extended the JFO model to work as a unified PDE in each regime. This work employs a penalty method for simplicity. The size and shape of the cavitation region should not effect our results too much due to the difference in peak fluid pressure and vapor pressure in our system. Peak pressures in water lubricated bearings are on the order of lMPa, the cavitation pressure for water is 3.169KPa at room temperature. We assume that a small change in pressure near the cavitation region does not significantly affect the elastic deformations we are focusing on. Most modeling efforts in the water lubricated soft bearing liner held that account for elasticity only account for the elastic bearing liner [44], A few studies have coupled a deformable shaft with a deformable bearing liner [35, 70, 71]. 9 CHAPTER II ELASTO-HYDRODYNAMIC LUBRICATION (EHL) The extreme aspect ratio of lubricating films, often 1000:1, makes traditional CFD meshes unreasonably large. Fluid him lubrication is therefore modeled with a set of major simplifications of the Navier Stokes equations, resulting in the Reynolds equation. The Reynolds equation is derived from the equations of motion and equation of continuity via an order of magnitude analysis in the following section. Pressures in point contact EF1L problems reach Gigapascal levels while him thickness are on the order of Nanometers to micrometers. Under these conditions the width and length of the him is on the order of 0.1 millimeters. Appropriate nondimensionalization must be applied to ensure numerical accuracy. The nondimensional stabilized and pressure penalized weak form of Reynolds equation are presented. 2.1 Derivation of Reynoldâ€™s Equation To understand the origin of all the terms in Reynolds equation, we form a complete derivation here. Starting with the Navier-Stokes (NS) equation in Cartesian coordinates (See eqs. B6-1 - B6-3 in [43]). P dvx dt + V dvx x dx + V dvx V dy + V dvx z dz dp dx + r] d2vx dx2 d2vx dy2 d2vx dz2 P9x (2.1) P dVy dt + V dVy x dx + V ()Vy_ V dy + V dVy_ z dz dp dy + T] 92Vy dx2 92Vy dy2 92vy dz2 P9y (2.2) 10 , dvz X Qx (2.3) P dvz dt + V + V dvz V dy + V dvz z dz dp dz + r] d1 2vz dx2 dy2 d2vz d2z + P9z where p is the fluid density, vx,vy,vz are the fluid velocities in the x,y,z directions respectively, t is time, p is pressure, p is viscosity, gx,gy,gz are the gravitational constants in the x,y,z directions respectively. The equations of motion are nondimensionalised with the following dimensionless quantities: X x_ y lo 1 V_ bo z T = i z ho u v* y UO ^ w VO JL fj = V- P = hop Po ' Vo vo^oh Vz WO (2.4) where lo,bo,ho, uo,vo,wo are the characteristic lengths and velocities in the x,y,z directions respectively, t0 is the characteristic time, and p0 and p0 are the characteristic density and viscosity respectively. Substituting the dimensionless variables into the x component of NS yields: PPoUo .yomPE I fine h2 dx V Wo Multiplying by PÂ°y2 gives: '0 Uup dU lo dX Vvo dU bo dY Wwp dU ho dZ uo d2U I ^0 d2U I ^0 d2U l2 dX2 b2 dY2 h2 dZ2 ppogx (2.5) ( lo dU , t t dU_ , lo_vo.\/PLL i (o 4Â»o \\rdll A ytoÂ«o 9r ' dX ' bo uo 9Y ' ho uo 9Z J h2od2U Polo 1 dP h2u0po P 9X VqIq V hf}uopo P l2 dX2 h20 d2U b2 dY2 d2U dZ2 (2.6) Qx Noting that the pressure and viscous terms have the inverse modified Reynolds number for lubrication problems, Rex = feÂ°^Â°oPo and the gravitational term is the Froude number, Frx = ^yr, and defining the squeeze number as as = we multiply through by the modified Reynolds number. h-pPo dU | h2uppp j j Qlj h2vppo .. ijl . hpwppo iy dU tpr/p dr ' r/plp U dX ' r/pbp V dY ' r/p ' dZ 1 dP_ I 2 r^o dpj_ I by 8pu_ I dpj p dX ' p 112 dX2 "T" b2 dY2 "T" dZ2 bpUppo lpgx Void Â«o (2.7) 11 (2.8) ( Similarly, the y and x components of the NS equation are given by: hi d2U , hl a2U , d2U + RexFrx (â€ž dV \?s dr + RexU Â§Â£ + 1 dP I V ~h2 d2v hi d2V , p dY + J l2 dX2 â€œT bl dY2 "T" (â€ž dU W* dr + R&xU^y + ReyV If 1 dP pdZ + ? h20 d2W l2 dX2 + hi d2W , bl dY2 "T" d2V az2 'zVV dZJ + ReyFry (2.9) az2 (2.10) + RezFrz We can now preform an order of magnitude analysis (OOMA) to arrive at the Reynolds equation. Good estimates for the length scales and material properties for point contacts are introduced in table 2.1. Classical views on point contacts [9] and experimental results [67] show that the length scales, (l0,b0), scale with the Hertzian contact radii, and the fluid pressure, p, scales with the Hertzian contact pressure. Experimentally verified him thickness results are used for the last length scale, h0. Dowson and Higgins pressure density relationship and Roelands pressure viscosity relationship, defined in section 2.2, are used to calculate po and rjo at the Hertzian contact pressure for a given pressure viscosity coefficient, a Roe. Usually the coordinate system is oriented such that there is only motion in the x direction and therefore Vo = 0. Here we assume Vo = uo just to be rigorous. With no obvious choice for w0, we assume w0 = u0/10. The table below details the test case of a moderately loaded point contact, and the scales associated with it. The dimensionless parameters can be calculated with the numerical values listed in table 2.1: Rex = Rey (1.3x10 7m)2 (lm/s) (1100kg/m3) (220Pa â€¢ s) (1.6 x 10â€œ4m) ~ 6 x 10â€œ10 (2.11) 12 Table 2.1: Parameters used in OOMA R 0.0127[m] F 50 [N] E* 110[GPa] ^Roe 15 [1/GPa] Vo 220 [Pa s] Po 1100[kg/m3] a 1.6E-4[m] Ph 0.927[GPa] h0 1.3E-7[m] u0 1 [m/s] Vo 1 [m/s] Wo 0.1 [m/s] to Rez (1.3x10 7m) (O.lm/s) (1100kg/m3) (220Pa â€¢ s) ~ 7 x 10â€œ8 (2.12) hlpo = (1.3 x 10 7mf {1100kg/m3) _ 1Q_8 topo (220Pa â– s) (1.3 x 10-6s) ~ (2.13) ReTFr7 ReyFry 6x 10â€œlo^f = 6x 10 ui _10 (1.6 x 10â€œ4m) (9.81m/s2) (1 m/s)2 9x 10 -13 (2.14) Re7Fr7 = 7 x 10 -8 hog 7 x 10 W n _g(1.3xl0 7m) (9.81m/s2) (0.1 m/s)2 1 x 10 -12 (2.15) /lo h,Q (1.3 x 10 7m)2 lo b/, (1.6 x 10-4m)2 (2.16) 13 Substituting these values into Eqs. (2.8), (2.9), and (2.10) : (7 x 10-8dU 6 x 10-10 ([/f! + V$) + 7 x 10~8W^) 1 dP pdX 2 P 7 x 10â€œ7 d2U ax2 a2u dY2 a2u az2 + 9 x 10-13 (2.17) (7 x 10-Â®f+ 6x 10_lo[7 av ax + + 7 x 1 dP p dY 2 P 7 x 10â€œ7 a2v ax2 a2v dY2 d2V az2 + 9 x 10-13 (2.18) (7x 10-sf( + 6x IQâ€10 (Uff \rdW\ V dY ) 7 x 10-8tEf|) 1 dP p az 2 p 7 x 10â€œ7 a2w ax2 d2W dY2 d2W az2 + 1 x 10-12 (2.19) If we choose the scales wisely, each non-numerical term in Eqs. (2.17), (2.18), and (2.19) is on the order of one. Neglecting terms smaller than O (1): 1 dP 7] I"d2U ~p~dX+~p \dZP_ (2.20) 1 dP 7] I"d2V ~pdY+~p \dZP _ (2.21) 0 Id P p dZ 7] \d2W p YdZP (2.22) More generally stated, the Reynolds number terms can be neglected when the him thickness is much smaller than the width and length (or radius in the case of circular contacts), the squeeze term can be neglected for small fluid films and high viscosities, even with incredible short time scales, and the gravitational terms can be neglected due to the small length scales of the him and the O (l[m/sj) velocities. We also assume that there is no pressure gradient through the him, which implies that = 0. Therefore the dimensional equations of motion simplify to: 14 (2.23) dp d2vx dx ^ dz2 dp d2vy dy ^ dz2 which can be solved for velocity with the boundary conditions: (2.24) at z = 0, vx = ua vy = vz = 0 (2.25) at z = h, vx = Ub vy = 0 vz = wo where the velocity in the Z direction is the combination of squeeze flow and the X velocity times the slope of the surface: Solving for velocity: Wo dh ~dt + Uo dh dx (2.26) v X (2.27) * = Yr,% Z ~ ~h) <2'28> which are combinations of Pouiselle and Couette flow. The mass flow rates per unit length in the x and y directions, Mx and My are given by: fh Mx = / pvxdz Jo ph3 dp 12rj dx T phum (2.29) My ph3 dp 12rj dy (2.30) where um = Ua^Ub is the mean entrainment velocity. The mean entrainment velocity is the 15 mean velocity of the fluid from the perspective of stationary solids, it is the mean velocity of the fluid entrained into the contact. Note that these mass flow rates are per unit length, not per unit area. This is due to the reduction of the problem from three dimensional in x, y, and z to a two dimensional problem in x and y. To combine these terms we apply conservation of mass to a differential element neglecting flow in the z direction, where ^ is the change in mass per unit time. Alternatively, one could integrate the equation of continuity across the him thickness and apply Leibnizâ€™s formula to obtain the same result. dm_____ Mx \x Mx lai-i-cte My \y My ly+dy dt dx d y 2.31 dMx dMy dx dy (2.32) Combining Eqs. (2.29), (2.30), and (2.32), we arrive at Reynolds equation (note that um is constant in all spatial directions): 0 V + Um dph dx Using the nondimensional variables: (2.33) X = X a 5*1 e III III "3. III fj=^ 1 Vo p = _p_ Ph She hi ^ l2umV0R2 ~ a3Ph P - PH3 (2.34) where a is the Hertzian contact radius, h is the him thickness, p0 is the ambient density, p0 is the ambient viscosity, ph is the hertzian contact radius, A is an nondimensional parameter, and e is a nondimensional variable, we arrive at the dimensionless Reynolds equation: - V â€¢ (eVP) + 9{pH) dX (2.35) 16 2.2 Material Properties Lubricant Properties The extreme pressure environment encountered in lubrication problems drastically reduces the inter-molecular distances, locking up the fluid into what some authors call an amorphous solid [69]. The consequences of this behavior is increased lubricant density and viscosity. As seen in figure 2.1 the density can increase by up to 30%, and the viscosity can increase by several orders of magnitude under typical conditions. Very high pressure rheology (>lGPa) of oils is difficult and expensive to measure, requiring custom rheometers. Table 2.2 summaries rheological models that have been found to be useful for calculating the viscosity of lubricating oils as a function of temperature and pressure. The Barus model was the first attempt at modeling the viscosity rise of oils due to pressure rise. It was noticed early on that the viscosity rises exponentially and the slope of dln(~^vo) _ aB^ wqere aB the pressure viscosity coefficient, which has units of inverse pressure and is usually expressed as The Cheng model is an extension of this model to include temperature effects. Experiments show the slope of dln(~^vo) not constant at pressures exceeding 200 â€” 400 MPa. The Cheng and Barus models are useful despite these shortcomings because they are simple mathematically and have very few parameters making it easy to fit to data. The Roelands model is a significant improvement over the Barus and Cheng model. Its basis is still empirical, but it models the change in slope of dlYlM'no'> and temperature effects with only 4 parameters. Similar to the Barus and Cheng model, limdln(V>?o) =aRoe, however as p increases dlnCh7o) decreases. p_^ The last two models in table 2.2 model the change of viscosity due to the change in density, and require or have an implied equation of state. Bair (2012) [6] noted that the pressure and temperature dependent viscosity curves of several materials collapsed onto one master curve when plotted against a new thermodynamic scaling factor /3y = (^) (vJpT)) > where VmÂ°lec is the specific volume of a single molecule of lubricant normalized with the specific 17 volume of the macro lubricant at a temperature, T, and pressure, p, and 7 is a fitting parameter. Vmoiec can be calculated numerically by assuming each atom of a lubricant molecule is a sphere with a radius equal to the van der Waals radius of the atom. The authors of this model note that they were able to use low pressure data to extrapolate accurately to higher pressure data due to this scaling for propylene carbonate. The modified Williams-Landel-Ferry (WLF) model, [59], is based on the time temperature superposition principle, which means it is possible to collapse all curves onto one master curve by shifting each curve by the glass transition temperature at that pressure. Table 2.3 presents two equations of state that model the density (or specific volume) of lubricants as a function of pressure and temperature. Density changes approximately 30% over the pressure range of interest. This pales in comparison to the several orders of magnitude viscosity changes. Nevertheless, it is important to model density changes accurately because each term in the Reynolds equation is a mass flow rate, as seen in equations (2.32) and (2.33). The Dowson and Fligginson model is a three parameter empirical model that was developed in 1966 and has been used since the early days of computational tribology. This equation is reasonably accurate, however it only shifts the ^ (P) vertically as temperature changes, it can not change the shape of this curve. The Tait equation of state is a free volume model, relating the change in density to the temperature dependent bulk modulus, K. It is a seven parameter model that can fit a wide range of materials. A high pressure and specific volume data set for the commonly studied lubricant Squalene is presented in [5]. Squalene is regularly used in high pressure rheology because it is widely available in large quantities and inexpensive and representative of the pressure and temperature dependent rheology of paraffinic mineral oils. The viscosity data was taken in a temperature controlled high pressure falling body viscometer. A hydraulic system with a pressure intensiher is used to control the pressure in the cylindrical oil chamber. A needle 18 Table 2.2: Summary of Rheological models available for high pressure non-isothermal lubricants. Name Expression Parameters Requires EOS? Basis Barus [7] JL _ paBp Vo Po, olb No Empirical Cheng [13] Tj (ot-ChV+fich Tjj ) LjCh Vo ~ Vo j otcfi ftch i 'Ich No Empirical Roelands [57] JL - J(HVR)+9.67)F(p)G(T)] Vo F(p) = -l+[l+5.1xlO-9p]ZÂ° g(t) = (Ty_â€˜S) <7 Â«Poe ^0 â€” 5.1x10-9(ln(f]R)) q fiRoe(TR-138) Â°Â° â€” ln(r]R)+ 9.67 m At Roe $Roe Tr No Empirical Ashurst-Hoover Exponential Power Law [6] JL - JA^+B^} Vo pv = (t) (Vv(p) ) m A, B Q, Q 7 Vnolec Yes Scaling is Ashurst-Hoover, viscosity model is empirical Doolittle [19, 20] JL â€” qBd yo Vo m Bd Vo Yes Free Volume Modified WLF [59] ~C1 (Tâ€”Tg (p)'jF(p) LL â€” in C2+(T-Tg(p))F(p) Vg Tg (p) = Tg0 + Alin (1 + A2p) F (p) = 1 â€” Biln (1 + B2p) % Cl, c2 TgO Ai, A2 B, B2 Implied TTS/ Free Volume Table 2.3: Summary of equations of state for high pressure non-isothermal lubricants. Name Expression Parameters Basis Dowson & Higginson |9] Po [l + A6.7xio\ - Pdh (T - Tr) Po Pdh Tr Empirical Modified Tait Equation of State [62, 25] Vo 1 1+K'Jn f1 + Ko C1 +Ko)] K0 = K0Rexrp (-(3KT) K'o = Kmexp (~Pkt) = 1 + av (T - Tr) P = po (l?) Ko r, /3 k Kor, Pk Tr, av Po Free Volume is then dropped in the pressure chamber and its terminal velocity is measured via a linear variable differential transformer (LVDT). Viscosity measurements are taken from p = 0.1 â€” 1200 MPa and T = 20 â€” 100Â° C. The relative volume measurements are taken with a metal bellows piezometer, from p = 0.1 â€” 378 MPa and T = 40 and 100Â° C. Each rheological model is plotted against Bairâ€™s set of high pressure rheological data in figure 2.1. The parameter values as well as goodness of fit estimates are listed in table 2.4 for each model used. The parameter values for each rheological model and equation of state where obtained with a nonlinear regression done in Matlab using the command â€œhtnlmâ€. The Matlab code used to do this is attached in Appendix A. A very good initial guess is required since there are so many parameters. Matlab was unable to come up with a fit for the Ashurst-Hoover model, even when using an initial guess that fits the data very well. Therefore parameters are used from [6]. Similar difficulties were encountered in trying to fit the WLF model. This was remedied by fitting the function for Tg (p) to data before fitting the rest of the model. o Bair (2006), T=293K ---Ashurt-Hoover, T=293K ---Roelands, T=293K â– â€”Cheng, T=293K o Bair (2006), T=313K Ashurt-Hoover, T=313K Roelands, T=313K â€” Cheng, T=313K O Bair (2006), T=338K ---Ashurt-Hoover, T=338K ---Roelands, T=338K â€¢â€”Cheng, T=338K o Bair (2006), T=373K ---Ashurt-Hoover, T=373K ---Roelands, T=373K â€” Cheng, T=373K Figure 2.1. Rheological models for high pressure non-isothermal lubricants fit to experimental data for Squalene from [5]. It is very difficult to design a pressure vessel to withstand typical lubrication pressures in order to take direct viscosity measurements. These models are often extrapolated in the absence of very high pressure viscosity data. In figure 2.3 three viscosity models are 21 Table 2.4: Coefficient values and RMS error for each fit in figures 2.1 and 2.2. Model Parameter values R~2 adj R~2 Notes RMS Error Cheng fjch â€” 1-57 x 10 2 Pa- s, ach â€” 1.9 x 10 2 Gpa /3Ch = 2.38 x 103 K, lch = 9.74 .983 .983 0.593 Roelands Vr = 1.57 x 10-2 Pa â– s, aRoe = 17.5 ^ Prtoe = 3.65 x 10â€œ2 K, TR = 313.15 K 0.999 0.999 0.129 AH r]R = 2.4 x 10-8 Pa â– s A = 24.28Kq, C = 7.126 x 106 KQ q = 0.1051, Q = 1.981 7=3.92 Vmoiec = 0.8002 *Â£ - - Fit from [6]. Reported standard deviation of relative viscosity is 8.4%. - WLF % = 45.88 x 1012 Pa â€¢ s Ci = 17.98, C2 = 32.85Â° C Tg0 = -134.6Â° C A\ = 40.95Â° C, A2 = 8.01 x 10â€œ3 ^ B = 0.718, B2 = 8.92 x 10â€œ4 ^ 1 1 Fit doesnâ€™t pass the eyeball test 48.4 Tait K0R7.93GPa, f3K = -6.04 x 10â€œ3 Â± Km = 11.6, pK = -9.17 x 10-6 A Tr = 313.15 K, av = 8.26 x 10â€œ4 A PR = 858 S 1 1 .000519 DH pR = 852.3 ^ Pdh = 5-739 x 10â€œ4 A Tr = 313.15 K .979 .977 0.0081 Figure 2.2. Density models fit to data from [5] extrapolated to 5,000 MPa. The experimental data shows a very clear trend of reducing the pressure viscosity slope as pressure increases. The Barus and Cheng model have constant slopes, limiting their usefulness. At 5,000 MPa the there is a 10 order of magnitude difference between the highest prediction (Cheng) to the lowest (Roelands), and a 3 order of magnitude difference between the Roelands and Ashurst-Hoover model. Roelands and Ashurst-Hoover models are only separated by a factor of 4 at 2,000 MPa. This raises the question: what happens to these predictions if some amount of high pressure viscosity data is removed and fit is repeated? One would assume that rheological models with an accurate physical basis would be able to predict lubricant viscosity outside of the experimental envelope more accurately than an empirical model. Figures 2.4 and 2.5 show the Ashurst-Hoover and Roelands model extrapolated to 2, 000 MPa after cutting of varying amounts of high pressure data and refitting the model. Table 2.5 tabulates this data at 1,200 MPa and 2,000 MPa. The Ashurst-Hoover model predictions are reduced by a factor of 3.5 when cutting out pressure data above 800 MPa. The Roelands model is only reduced by a factor of 1.1 with the same data manipulation. Similarly, at 1,200 MPa, the Ashust-Hoover model is 23 Figure 2.3. Extrapolation of models in Figure 1.1 to 5 GPa. The Barus and Cheng models are the same by definition when fit to a single temperature. reduced by a factor of 1.4 and the Roelands model is reduced by a factor of 1.04. Surprisingly the Roelands model with no physical basis out performs the model based on a free volume approach. Both the Roelands model and the Ashurst-Hoover model provide a quality representation of viscosity up to 1, 200 MPa. In lieu of data at very high pressure, and assuming the lubricant does not undergo any major structural rearrangements the Roelands viscosity model provides a reasonable extrapolation for viscosity up to 2,000 MPa. Table 2.5: Roelands and Ashurst-Hoover model evaluations when omitting high pressure data from the fit. Pent Roelands rj (1,200 MPa) Roelands 77 (2,000 MPa) Ashurst-Hoover 77 (1,200 MPa) Ashurst-Hoover 77(2,000 MPa) oo 8, 632Pa â– s 3.94 x 106 Pa â– s 10,415 Pa-s 1.55 x 107 Pa â– s 1100 8,465 Pa-s 3.76 x 106 Pa â– s 9,880 Pa â– s 1.25 x 107 Pa â– s 1000 8,375 Pa-s 3.69 x 106 Pa â– s 9,790 Pa â– s 1.13 x 107 Pa â– s 900 8,184 Pa â– s 3.56 x 106 Pa â– s 8,690 Pa â– s 6.80 x 106 Pa â– s 800 8,287 Pa-s 3.60 x 106 Pa â– s 7, 223 Pa â– s 4.40 x 106 Pa â– s 24 Figure 2.4. Extrapolation of Anhutst-Hoover viscosity model, when using lower pressure data Figure 2.5. Extrapolation of Roelands viscosity model when using lower pressure data Bearing Materials Rolling element bearings are made of a rolling element which in this work is a ball, and two bearing surfaces for the ball to roll in between called races. This is depicted in figure 2.6. The inner race in figure 2.6 is shown cut in two parts. Each race has one concave curvature to conformally adapt to the ball. The inner race also has a convex curvature to 25 Inner Race half Outer Race Figure 2.6. Parts of a four point contact angular contact ball bearing. A 4-PÂ°int angular contact ball bearing by User:Silberwolf/CC-BY-2.5. wrap around the inner shaft, and the outer race has a second concave curvature to wrap around the ball path. The ball spacing is held together with a retaining clip that can also serve to hold grease pockets. Typical values of Youngâ€™s modulus and Poissonâ€™s ratio for hardened bearing steel are listed in table 2.6. Sapphire is included because it is commonly used in experimental setups that require an IR transparent bearing material, although it is far too brittle and expensive to be used in actual bearing applications. Rolling element bearings are designed to avoid bulk plastic deformation. When the applied load exceeds the design load, or in impact scenarios, the balls will indent the race and leave a mark. This is known as Brinelling. Bearing materials are commonly hardened to avoid Brinneling. Hardening processes usually include heating the material upwards of 1700Â° F then quenching, changing the structure of the steel. This structural change does not necessarily permeate through the entire body, therefore assuming a constant modulus for the entire elastic body is an approximation. The metallurgy of steel bearing materials is outside the scope of this work. Linear elasticity is assumed for the ball and race with a 26 constant modulus and Poissonâ€™s ratio. A much more complete discussion of bearing materials and hardening processes can be found in The Fundamentals of Fluid Film Lubrication [9]. Table 2.6: Linear elastic properties of commonly used bearing materials Material E (GPa) V Aluminum 62 0.33 Medium and high alloy Steel 200 0.30 Silicon Carbide 450 0.19 Sapphire 360 0.34 2.3 Coupled EHL Model In this section, we define an equivalent Youngâ€™s Modulus and Poissonâ€™s Ratio to reduce the two body solid problem to one solid body. The equivalent problem solves for the deformation of a half space, reducing the size of the problem by only requiring one solid domain. The sum of the ball and race deformations, hi and h2 respectively, is defined as a single equivalent deformation 6eq, as shown in figure 2.7: deg = hi + h2 (2.36) Atkin and Fox [4] provides an analytic solution for all three components of displacement for a half space loaded by a point force. Substituting the analytic solution into the previous equation and simplifying yields a relationship between the Lame material parameters of the ball and race and the one body Lame material parameters: Teg -y + 2 (1 - ueq) Ti â€” + 2 (1 - q) 1 Tl2 -y + 2(l â€”zz2) 12.37) where fij, /r2 and iif are the lame constants for the ball, race, and equivalent material 27 respectively, zq, zq, and veq are the Poissonâ€™s ratio for the ball, race, and equivalent material respectively, and r = ybr2 + y2 + z2. Any set of equivalent properties that satisfies the previous equation can be used to define an equivalent single body problem to replace the two body problem. Making the additional assumption that and the Meg Ml M 2 definition of the Lame constant, E = 2pL(l + u), we can solve the system of equations for Eeq and veq, which is displayed below: _ E\E2 (1 + zq)2 + Eâ€˜22Ei (1 + zq)2 _ EiU2 (1 + zq) + E2Vi (1 + zq) r Â£/â€” q and â€” 7_, , % 7_, / % ( [Al (1 + zq) + E-2 (1 + zq)] El (1 + u2) + E2 (1 + z/i) where E\, E2 and Eeq are the Youngâ€™s modulus for the ball, race, and equivalent material respectively. These equivalent material properties allow us to transform the two body problem to a single body problem as shown in figure 2.7, not to scale. Figure 2.7. Two body problem (left), and equivalent single body problem (right), not to scale. Note that the curvature of the rolling element is taken into account via the him thickness equation. The curvature of the race can be taken into account in multiple ways as will be discussed in chapter V. We can now define our domain and boundary conditions. The Reynolds equation will be solved on a two-dimensional boundary of our three-dimensional elastic domain. The fluid pressure will act as a boundary condition to the solid domain, and the solid deformation in the z direction effects the him thickness equation, leading to a fully coupled set of equations. The him thickness and huid pressure in the contact region is the solution to a coupled elasto-hydrodynamic system of equations as follows: 1. 2. 3. 4. 5. 6. 7. 8. Reynolds equation: â€”V â€¢ (eVP) + = 0, on i?/ Simplihed Cauchy momentum equation: V â€¢ a = 0, on Qs Hookeâ€™s Law: a = C: e, on Qs Dehnition of strain: e = (Vu)T + Vu , on i?. Load equation: ff pdQf = F, Globally Film thickness equation: h = + w + h0, Globally Any rheological model from table 2.2, on i?/ Any equation of state from table 2.3, on i?/ where the unknowns are huid pressure, P, solid stress, a, solid displacement/strain, u,e, the him thickness constant ho, Of and f2s are the huid and solid domains, respectively, and F is the applied load. The computational domain, shown in hgure 2.8, is a rectangular prism representing a half space. The size has been shown to be large enough to not impact the deformation in the lubrication domain [34], The boundary conditions for the solid problem: 1 2 1. Fixed: u = 0, on the surface denoted by BC1 2. Symmetry: Vu â€¢ n = 0, on the surface denoted by BC2 29 3. Applied boundary pressure: P on Qf All other surfaces of the solid domain are free to deform and have no applied loads. The highlighted subdomain is the lubricated surface where the Reynolds equation is solved, known as the lubrication domain which will be denoted by Qf. The remaining geometric features, such as the semicircle, are for mesh control and post-processing. Boundary conditions to Reynolds equation are shown in figure 2.8, and can be summarized as: 1. P = 0 on the Â±X and +Y edges of Qf 2. VP â€¢ n = 0 on the â€”Y edge of Qf Note that the edge of the lubrication domain that shares an edge with the solid symmetry boundary also has a symmetry boundary condition. The origin is located at the center of the semicircle on the lubrication domain as shown in figure 2.8 with the zâ€”axis coming out of the page. 2.4 Weak Form and Stabilization We derive the weak form of Reynolds equation in order to solve it with the finite element method. Multiplying Eq. (2.35) by a test function, ip, and integrating over the lubrication domain, i?j: - JipV- (eVP) dQf + J iP^^dQf = 0 Integrating by parts to reduce the order of the derivative on P: (2.39) - ip (eVP â€¢ n) |dnf eVip â– VPdi7/ + 9{pH) dX d Qf = 0 (2.40) 30 60a Figure 2.8. Domain and boundary conditions for the EHL problem. The three-dimensional block is the solid domain, Qs, the highlighted two-dimensional domain is the fluid domain, Qf. The domain sizes are scaled by the Hertzian contact radius, a. where n is the outward pointing vector normal to the boundary. Integrating by parts on the last term: - Vâ€™ (sVP â€¢ n) \dnf + f eVVâ€™ â€¢ VPdi?/ + 4â€™pHnx \dnf - f pH^dfif = 0 (2.41 The test function, -0, is defined to be zero on the boundaries eliminating the boundary terms. J sViâ€™p â– VPdQf - J pH^rdQf = 0 (2.42) We arrive at the weak form of Reynolds equation. The Reynolds equation predicts that converging surfaces will produce a positive pressure gradient, while diverging surfaces will produce negative pressures. If the diverging surfaces are steep enough, the liquid pressure will drop below the vapor pressure and the liquid will cavitate. As discussed earlier in 31 chapter I, there are several cavitation models to handle this ranging in complexity and accuracy. Since operating pressures in highly loaded point contacts are on the order of 1 GPa, and typical vapor pressures for mineral oils are on the order of 0.1 kPa, a simple penalty method will suffice. The weak form augmented with the pressure penalty method is as follows: r dp eVpp â€¢ VPdi?/ â€” / Tp/min(P, 0)pp = 0 12.43) Pf is a very large number and, therefore when the pressure is negative the last term is several orders of magnitude larger than the others. Hence, Eq. (2.43) simplifies to pp ~ 0 at any node where the pressure is negative, and forces negative pressures to be on the order rsyp J Pf of the negative pressure is presented in section 2.6. of vPd% f PH ex aQf' Qn ejje(q of Pf on the solution and the magnitude At very large loads the viscosity can increase several orders of magnitude and the him thickness can be very small, and hence, e 1. Standard Galerkin finite element schemes fails in convection dominated systems, such as the Reynolds equation when e 1. Therefore it is prudent to include stream wise upwind Petrov Galerkin/Galerkin least squares (SUPG/GLS) and isotropic diffusion (ID) stabilization schemes from [34], SUPG/GLS stabilization methods are consistent, while ID is inconstant. Even though [34] reported success using this method, it was not able to be replicated in this work. Appendix ?? demonstrates a successful implementation of SUPG/GLS for a linear convection, diffusion, source problem. The usefulness of this code is compromised in some of the parameter space without stabilization as will be seen in section 2.8. Also shown in section is the effect of ID on accuracy. 2.5 Mesh Convergence The numerical solution to the standard Galerkin finite element representation of Reynolds equation fails at high load, as will be shown later in section 2.8. Therefore care 32 must be taken to ensure the terms in the Reynolds equation do not produce unnecessary shows the Reynolds equation depends on the gradient of the cube of the z component of problem. This is problematic as the fluid domain shares boundary elements with the solid domain. Therefore, a unique approach to mesh convergence must be taken given the coarse elements can be used for the solid problem, especially far away the contact zone. The extreme pressure gradients require a very fine mesh in the Hertzian contact area. The discrepancy in mesh size in the contact zone can be corrected by using higher order elements for the fluid problem than the solid problem. Therefore the meshing strategy is as follows: 1. Using Hertzian contact as a test problem: decrease mesh size near and far from the contact zone until we get good agreement with the Hertzian analytic solution. Repeat with linear, quadratic, and cubic elements. 2. With an appropriate solid mesh, refine the fluid mesh until the d2 norm of the fluid him pressure converges to an appropriate level. 3. Check that Hertz error, defined below, is smaller than 1 x 10-3. Solid Problem Convergence The effects of mesh size and order near and far from the contact zone with the Hertzian contact test problem are studied. Figure 2.9 shows an example mesh for the Hertzian contact problem. There are two mesh zones, one near the contact zone and the perturbations. Expanding the V ( term to include the definition of him thickness the solid displacement, w: V quadratic solid elements to ensure this term is continuous. The mesh requirements for the solid problem do not necessarily coincide with the huid multi-physical nature of the problem. Previous authors [34] have discovered that very 33 remainder of the geometry. The surface deformation in the lubrication zone is the only coupling variable from the solid problem. The numerically calculated indentation depth, dcamsou at the center point of the contact zone will be compared to the analytically calculated Hertzian indentation depth, duertz- The error in the numerical simulation can then be defined as &Hertz â€” dHertz &C omsol dH ertz which will be referred to as the Hertz error. Figure 2.9. Mesh before and after convergence study, a) Before, isometric view b) after, isometric view c) before, lubrication zone d) after, lubrication zone. The near zone is refined from A/a = 6toA/a = 0.333 with linear, quadratic, and cubic elements. The far mesh zone is represented with a coarse (figure 2.9a) and a refined mesh (2.9b). The Hertz error is plotted as a function of the mesh size in figure 2.10 (left), the Hertz error is plotted as a function of the number of degrees of freedom is shown in figure 34 2.10 (right). Figure 2.10. H and P convergence for the Hertzian contact problem. Hertz error as a function of mesh size for multiple cases (left), and Hertz error as a function of number of degrees of freedom (right) for the same cases. Figure 2.10 (right) shows that the quadratic refined mesh yields sub 1% error at under 15,000 degrees of freedom. This point corresponds to a A/a = 1.0. The fluid problem will be solved on the surface of this mesh, therefore we must now refine the surface mesh until the fluid problem converges. Fluid Problem Convergence It was immediately clear that A/a = 1.0 was far too large for the fluid solution. The highest order fluid elements (7th order) were chosen to maximize the converged mesh size. Table 2.7 shows the relative d2 norm error, equation 2.44, for pressure evaluated at 22 points within one Hertzian radius to compare meshes. Column two of this table should be interpreted as the relative error, e, between the mesh size stated and a mesh of half that size. 35 2 (2.44) 22 e = N(U-A/2) i= 1 Table 2.7: d2 norm error, calculated with Eq. (2.44), for several mesh sizes Mesh Size (A/a) Â£2 Norm Relative Error DOF 1 2 2.222 x 10-2 16,264 1 4 9.760 x 10â€œ4 51,197 1 8 2.220 x 10â€œ5 93,663 1 16 1.901 x 10â€œ6 195,909 1 32 3.348 x 10â€œ7 440,669 1 64 - 1,237,592 Newton iterations are performed until the d2 norm of the relative residual reaches ~ 1 x 10-5. Newton iterations are terminated when the relative residual between iterations is less than 1 x 10-6. Using a mesh size of A provides accuracy on the order of the error estimate, but it is very costly. A mesh size of | provides sufficient accuracy at less than half the number of degrees of freedom, which solves roughly 4 times faster. A mesh size of | on the lubrication surface with 7th order elements, quadratic elements of size 1 or smaller in the near zone and the refined mesh in the far zone are used in this work unless other wise specified. This mesh can be seen in 2.9b and 2.9d. The Hertz error is equal to 0.00082 when using this mesh. 2.6 Pressure Penalty Convergence In this section we study the effect of the pressure penalty factor, pf as shown in Eq. (2.43) on the minimum pressure in the lubrication domain. Table 2.8 shows the minimum pressure for values of pf ranging from 1 x 104 to 1 x 107. The pressures never reach zero, but the minimum pressure in the lubrication domain 36 Table 2.8: Minimum film pressure as for several values of the penalty factor. Pf min (P) 1 X 104 -1.13 X io- -2 1 X 105 -1.14 X io- -3 1 X 106 -2.20 X io- -4 1 X 107 -3.50 X io- -5 decreases with an increasing penalty factor. The minimum pressure is insignificant with a penalty factor greater than 1 x 10-6. Penalty factors greater than 1 x 10' can cause convergence issues. A penalty factor of 1 x 106 is used in this work. 2.7 Model Validation There are no relevant analytic solutions to validate this code due to the highly nonlinear and coupled nature of this problem. Therefore we validate our model with results from other numerical models. Figure 2.11 (left) compares dimensionless pressure and him thickness to a similar finite element model as described in [51]. This data was obtained by digitally analyzing the figure from the paper which has some error associated with it. Figure 2.11 (right) compares the dimensionless pressure and him thickness in the X and Y directions through the center of the contact to a finite difference code written by Zheng Shi of Seagate. Figure 2.11. Comparison of current model to a comparable model from the literature [51] (left). Comparison of current model to a finite difference model [60] (right). Comparing this code to others with the same assumptions does not tell us much about the 37 validity of our assumptions, or give us insight into the physics, but it does increase confidence that we have a valid solution to a well posed problem with no bugs or errors. 2.8 EHL Results This section presents results for the EHL model. A solution consists of a fluid him pressure profile, the separation distance between the ball and race, and the combined deformations of the ball and race. In EHL the following six parameters determine the him thickness and pressure profiles: F, E', R, um, and a. The Buckingham-n theorem states the number of dimensionless groups to describe the solution is equal to the number of parameters that govern the solution minus the number of dimensions, which is three (mass, length, time). For the EHL problem there are six parameters and three dimensions leaving three dimensionless parameters. Hamrock and Dowson [23] found these parameters to be the load, material, and speed parameter respectively as shown in eq. 2.45. Whd = lRR2 Ghd = aE' Uhd = (2-45) Another method of dimensional analysis, the optimum similarity analysis first used by Moes, [47], which uses a computer program to analyze the equations and parameters in a system to calculate the optimum dimensionless parameters yields results that are a combination of the Hamrock and Dowson parameters as shown in eq. 2.46. M ee WHD (2UHD)~3/i = ^ (2â„¢f )â€œ3/4 ^ ^ L EE Ghd (2Uhd)1,â€˜ = aE' (2lff)l/i Any combination of parameters on the right hand side of Eq. (2.46) that yield the same M and L values will have the same solution, assuming the rheological model and equation of state is held constant. Figure (2.12) details the areas on the M-L plane where we obtain a converged result. Green shows where a smooth converged result was found. Yellow represents a rough converged result, as shown in figure 2.14. Anywhere in the red, the 38 o o o o oooo o o o oooo o o o oo Smooth Converged Result Rough Converged Result o No Convergence 25 3 Log (M) Figure 2.12. Convergence area of the code. Green = smooth converged result Yellow = perturbed converged result, Red = No convergence. convection term dominates and the standard Galerkin formulation fails as described by Habchi [34], Many solutions were obtained at different M and L values by choosing a F and a small a obtaining a solution, and slowly increasing a until the solver fails to converge. Figure 2.13 shows the dimensionless pressure profile P (X) at Y = 0 for several values of M and L. Solutions are easily obtainable at low M and L, as either M or L increases, a better initial condition must be specified for convergence. Better initial conditions are obtained by taking smaller steps in a and obtaining intermediate solutions. Steps as small as 0.0001a'mQ,iT were taken. For values of M and L greater than 10,000 and 13, respectively, the solution becomes unstable and large oscillations in pressure occur. These solutions still converge, even with extreme pressure oscillations as seen in figure 2.14, but their accuracy and usefulness are questionable. Solutions at low M show a non zero pressure gradient at the entrance to the lubrication domain. This indicates that the size of the numerical lubrication domain is having an effect on the solution. This problem is ignored as solutions at low M are not of 39 interest to engineers. These solutions are very low load, and tend to have thick fluid films that do not always satisfy the Reynolds equation assumptions. Furthermore, very thick films tend to have very low wear rates which is ideal in practice, but uninteresting computationally. Therefore we do not increase the size of the lubrication domain, and hence the total size of the problem, to obtain more accurate solutions for a parameter space that is not of interest and that does not always satisfy our assumptions. Figure 2.13. Pressure profiles for multiple vales of M and L. Effect of Isotropic Diffusion Convergence becomes problematic as the pressure oscillations grow. Isotropic diffusion (ID) was implemented to reduce the pressure oscillations. ID is an inconsistent 40 Figure 2.14. Example solution with extreme pressure oscillations. stabilization method the effect of its magnitude must be studied by increasing the PID parameter until the solution is significantly effected. Figure 2.15 shows the effect of isotropic diffusion on the pressure profile. The M and L values were chosen because this solution is already smooth, and it has a large and steep pressure spike. This allows us to see the maximum effect of isotropic diffusion on the accuracy. The majority of the pressure profile is unaffected, but the pressure spike is spread out and its magnitude is decreased. The minimum him thickness increased from 0.6489 at PID=0 to 0.6515 at PID=1, which is a 0.4 increase. This is accompanied by a 5.1 decrease in the height of the pressure spike. Overall the effect of ID is quite small. It is not recommended to go above PID=1. Figure 2.16 shows the effect of isotropic diffusion on an unstable solution. Pressure Spike and Exit Contraction Classic EHL solutions demonstrate a pressure spike and reduction in him thickness at the exit of the contact. The reason for this spike and contraction is rarely discussed, but tersely explained as a result of the pressure viscosity coefficient, a. This section aims at a more in depth explanation. The effect of the pressure viscosity coefficient is examined first in hgure 1. The pressure viscosity coefficient has a pronounced effect on him thickness. Increasing 41 Figure 2.15. Effect of Isotropic Diffusion stabilization on the pressure profile, expanded view of the pressure spike inset. Figure 2.16. Pressure profiles of the unstabilized (left) and stabilized (right, PID=1) for an EHL contact M=8064 and L = 9.3. the viscosity several orders of magnitude in the contact region provides significantly more lift. Surprisingly, the case with no pressure viscosity coefficient shows a exit contraction but no pressure spike. The pressure spike must not be caused by the exit contraction if the pressure spike is absent in the presence of the exit contraction. The exit contraction seems to be caused by the difference between the fluid pressure and the equivalent Hertzian contact pressure, as show below(hgure under construction). This is related to the no cavitation condition. 42 â– Pa=l[l/GPa] â– H a=l[l/GPa] â– P ct=5[l/GPa] â– H a=5[l/GPa] -Pu=10[l/GPa] H a=10[l/GPa] â– Pa=15[l/GPa] â– H u=15[l/GPa] A Figure 2.17. Pressure and film thickness for several values of pressure viscosity coefficient. Note the pressure spike for larger pressure viscosity coefficients and the exit contraction in him thickness. The pressure viscosity coefficient does indeed cause the pressure spike, but how? To answer this we must examine what is happening at the entrance and exit of the contact. At the entrance the motion of the ball and race is drawing fluid in, as the fluid is squeezed into the gap the pressure and therefore viscosity and density rise. The pressure driven flow counter acts the flow due to the movement of the races. It becomes harder and harder to flow into the contact region until the fluid reaches the center. After passing the center of the contact the pressure driven flow and the flow due to the movement of the ball and race are in the same direction. The pressure is decreasing rapidly causing the viscosity and density to fall rapidly. EHL Model Usefulness as Design Tool The results from this type of model have been used to calculate the minimum him thickness for rolling element bearings since the 60â€™s. This model does a good job determining whether or not there is solid contact for a given geometry, lubricant, and load. As is, this model does not predict the amount of solid contact or itâ€™s location. This model also fails to account for temperature effects, and assumes the bearing is a half space. 43 Chapter III expands this model to include a statistical model for solid solid contact. Chapter IV expands the model from chapter III to include thermal effects. Chapter V uses the model from chapter III to study a real world bearing geometry. 44 CHAPTER III MIXED-ELASTO-HYDRODYNAMIC LUBRICATION (MEHL) Asperity contact occurs when the him thickness is on the order of the asperity size. When asperities are in contact, force is transmitted between the two surfaces by direct solid to solid contact instead of hydrodynamic forces. Many theorize that there is a monolayer of lubricant trapped between the contacting surfaces that modify friction and reduce wear. In this chapter we review elastic and plastic asperity contact models and implement them into the EHL solver. This allows the model to make accurate predictions through every lubrication regime described in chapter I. This is also a critical step in calculating temperature rises in lubricated contacts as frictional heating is a substantial component to the overall heat generation, as will be seen in chapter IV. Direct computation of asperity contacts for full scale bearings is extremely computationally expensive due to the number of asperity contacts within the lubricated contact. The number of possible asperity contacts can be estimated as the number of asperities per unit area, also known as the asperity density, N, times half the Hertzian contact area, l/27ra2 which accounts for the plane of symmetry discussed in chapter II. Table 3.1 shows contact radii and the estimated number of contacts, Nc, for a 1 cm radius steel ball contacting a plate with a combined modulus of 100 GPa for several different applied loads, F. A typical value for asperity density, N, for highly polished bearing components is 4 according to spectral and deterministic measurements reported in [40]. Typically, at least 10 surface elements per asperity are required for convergence in the contact problem. The final mesh presented in figure 2.9 had 1320 elements representing the 45 Table 3.1: Contact radius, a, as a function of applied load, F, number of elements required for the given asperity density, N. F[N] a= (1^)Â® (hm) Nc 0.1 19.5 2400 1 42.2 11,000 10 91 240,000 100 195 1,100,000 lubrication surface. The mesh density would have to be doubled to get a rough estimate of even the easiest case, F = 0.1 N. The mesh would quickly get out of hand, requiring two hundred times the standard mesh density for a typical case of F = 10 N. A statistical representation of surface roughness is therefore preferred to explicitly modeling each asperity. The following sections outline approaches that model asperity contact statistically, using differing assumptions about how the asperities deform. Common assumptions of the following models: 1. Asperity height distribution of each surface is Gaussian 2. Two rough surfaces can be represented by one rough surface with a composite surface roughness and one smooth surface. The composite surface roughness is equal to Sq = \Ai"Taf, where o\ and 02 are the standard deviations of the Gaussian asperity height distributions for the two rough surfaces. 3. The asperity deformation is either fully elastic or fully plastic. The probability of encountering an asperity of size z away from the mean surface is: ^<2)=cSPexp(^) <3' Where a is standard deviation of the normally distributed (f>(z). Given a population of n 46 measurements of distance from the mean surface, Zi, of a rough normally distributed surface, the definition of standard deviation is: a \ n Y & -y 13.2) i= 1 where the mean of this distribution, fi, is zero by definition. This perfectly coincides with the commonly used definition of surface roughness, Rq, also known as Rrms- Rq â€” Rrms \ 1 n Y 13.3) i= 1 therefore if it is assumed that a surface has a normally distributed asperity height we can use the commonly reported Rq = Rrms roughness parameters in the following models. Rq is related to the more commonly used Ra surface roughness parameter by Rq = /2Ra, where Râ€ž is defined as: 1 n 13.4) i= 1 Surface characterization studies have shown that many common engineering surfaces have normally distributed roughnesses [31, 8]. 3.1 Elastic Asperity Deformation The Greenwood Williamson model, [30], is the original rough contact model. It assumes that each asperity can be approximated by a spherical cap of radius /3, the contact between each asperity and the smooth surface is Hertzian, and that asperity deformation is elastic. Rearranging the Hertzian point contact expressions found in table 1.1 in terms of the indentation depth, d, the contact area and the load are: 47 The probability of asperity of distance s = z/Sq from the nominal dimensionless him thickness of HSq = h/Sq contacting the hat surface is written as: where (z) is the probability distribution of asperity heights, assumed to be Gaussian. Assuming the smooth surface contacts the highest point of the asperity, which is reasonable if the slope of the smooth surface near the asperity is small, the indentation depth will be z â€” h and the resulting contact area and load is: dehning each expression on a differential area where the separation between nominal surfaces is h, and the asperity density per unit area is N: We now have a set of expressions that can be integrated to provide a continuous function of contact area and contact force for elastically deforming asperity contact as a function of nominal surface separation. Assuming the contact area and pressure are constant in each differential element, and dehning a dimensionless asperity height, Eq. (3.9) can be written as: for all z > u. The expected value for each of these expressions is: A, = ttI3S, (s - Hs,)
F< = iE'piS, (s - Hs,)i 4>(s) ds
Ot'j' h2 |

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PAGE 1 NUMERICALANALYSISOFLUBRICATEDCONTACTS by ALEXANDERM.MERTZ B.S.,NewMexicoStateUniversity,2010 M.S.UniversityofWisconsinMadison,2012 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy EngineeringandAppliedScienceProgram 2019 PAGE 2 ThisthesisfortheDoctorofPhilosophydegreeby AlexanderM.Mertz hasbeenapprovedforthe EngineeringandAppliedScienceProgram by SamuelWelch,Chair RonRorrer,Advisor MarcIngber AlanGraham LynnSchreyer Date:August3,2019 ii PAGE 3 Mertz,AlexanderMcMillenPhD.,EngineeringandAppliedScience NumericalAnalysisofLubricatedContacts ThesisdirectedbyAssociateProfessorRonaldRorrer ABSTRACT Ballbearingperformanceispredicatedontheball-raceinterface.Simple analyticalmodelscanpredicttherequiredtorqueandpowerlostiftheinterfaceisabsent oflubricant.Lubricantisaddedtotheinterfacetodecreasebearingfrictionalpowerloss andbearingwear.Dependingontheoperatingconditionsofspeedandload,hydrodynamic forcescanfullyseparatetheballfromthebearinginterface.Thisiscalledfulllm lubrication.Althoughthisisjustonepossiblelubricationregime,ithasthebest performanceintermsoffrictionandpowerlost.Engineersstrivetooperatebearingsinthis regime.Inthiswork,thedetailsofthelubricatedinterfacearestudiednumericallywiththe niteelementanalysisFEAsoftwareCOMSOL.AsimpliedversionoftheNavier-Stokes equations,theReynoldsequation,isusedtomodeltheuidtoreducetheproblemsizeand avoidahighlynonlinearuidstructureinteractionFSIproblem.Thisiscoupledwitha full3Delasticrepresentationoftheraceandanequationthatsatisesloadcontinuity.The pressureintheseinterfacescaneasilyexceed 1 GPa whichhasdramaticeectsonthe uid'sproperties.Therefore,wemodeltheuidasNewtonianwithtemperatureand pressuredependentviscosityanddensity.Multiplerheologicalmodelsarecompared.This methodisthenextendedtoincludenon-isothermaleects,surfaceroughness,and solid-solidcontactincaseswherethesurfaceroughnessislargerthantheuidlmto studybearingperformanceunderavarietyofconditions.Elasticandplasticasperity iii PAGE 4 contactmodelsareconsidered.Theresultisanengineeringtoolthatcanaidinthedesign ofbearingsystems.Manyresultsarespecictothesystembeingstudied,butwhere possible,resultsarerepresentedasafunctionofnondimensionalparameters. PropellershaftbearingsinmarineandNavalapplicationsoperateonasimilar setofprinciplesasoillubricatedballbearings.Thegeometryandlubricantdier signicantly.However,theunderlyingphysicsaresimilar.Hydrodynamicforcescan separatetheshaftfromthebearingdependingontheoperatingconditionsofspeedand load,solidcontactoccursiftheserequirementsaren'tmet.Themodeldevelopedforoil lubricatedballbearingsisappliedtowaterlubricatedpolymerlinedjournalbearings.The resultsofthismodelarepresentednon-dimensionallywithanalyticfunctionssothat engineerscanusethem.Aparameterstudyispresentedconsistingofover500unique solutions.Multiplesecondordereectsareconsidered.Comparisontoexperimentaldatais presented. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:RonaldRorrer iv PAGE 5 Thankyoutoallthathavesupportedmethroughoutthisprocess. v PAGE 6 CONTENTS CHAPTER IINTRODUCTION.........................1 IIELASTO-HYDRODYNAMICLUBRICATIONEHL.........10 2.1DerivationofReynold'sEquation.................10 2.2MaterialProperties.......................17 2.3CoupledEHLModel......................27 2.4WeakFormandStabilization...................30 2.5MeshConvergence.......................32 2.6PressurePenaltyConvergence..................36 2.7ModelValidation........................37 2.8EHLResults..........................38 IIIMIXED-ELASTO-HYDRODYNAMICLUBRICATIONMEHL.....45 3.1ElasticAsperityDeformation...................47 3.2PlasticAsperityDeformation...................49 vi PAGE 7 3.3ComparisonofPlastictoElasticAsperityDeformation........50 3.4ImplementationintoEHLModel.................52 3.5ResultsoftheElasto-HydrodnamicWithAsperityThePlasticContact Model.............................55 IVTHERMAL-MIXED-EHLTMEHL.................68 4.1Model.............................69 4.2TMEHLResults........................79 4.3TMEHLModelUsefulnessasDesignTool.............86 VFULLBEARINGSYSTEMMEHL.................88 5.1Introduction..........................89 5.2Model.............................90 5.3DimensionalAnalysis......................99 5.4ResultsandDiscussion.....................102 5.5Conclusions..........................129 BIBLIOGRAPHY.............................139 APPENDIX ACodeForFittingViscosityAndDensityModels............140 BValuesOfIntegralsOfTheType f n H = R 1 H s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H n s ds ......147 vii PAGE 8 VARIABLELIST a Hertziancontactradius a i , b i , c i , d i , e i Constantsinmultimodeexponentialuidloadfractionmodel A AshurstHooverpowerlawparameter B AshurstHooverpowerlawparameter B D Doolittleexponentialparameter l 0 ;b 0 ;h 0 Characteristiclengthsinthe x;y;z directionsrespectively c Journalbearingradialclearance C R S q Contactparameter ^ C p Heatcapacityatconstantpressure d Hertzianindentationdepth e error E Young'smodulus E Combinedmodulus F AppliedForce Fr l 0 g x u Froudenumber g Gravitationalconstant G HD E 0 HamrockandDownsondimensionlessmaterialparameter h Filmthickness h t Modiedlmthickness h 0 Filmthicknessconstant H hR a 2 Dimensionlesslmthickness H t h t R a 2 Dimensionlessmodiedlmthickness H S q h S q Solidcontactgapthickness H a Hardness k Thermalconductivity viii PAGE 9 K 0 R BulkModulusatreferencestate K 0 0 R Derivativeofbulkmoduluswithrespecttopressureatreferencestate L E 0 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 0 u m E 0 R 1 = 4 Moe'sdimensionlessspeedparameter ` 2 Euclidianvectornorm M F E 0 R 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(2 0 u m E 0 R )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = 4 Moe'sdimensionlessloadparameter m Mass M x ;M y Massowrateperunitlengthin x;y directionsrespectively n Normalvector N Numberofasperitiesperunitarea N C Numberofindividualasperitycontactsforagivenmacro-contact p Fluidpressure p c Solidcontactpressure p h Hertziancontactpressure P p p h Dimensionlessuidpressure P c p c p h Dimensionlesssolidcontactpressure P cut Cutopressure Pe x ;Pe y ;Pe z Pcletnumberinthe x;y and z directionrespectively PID RelativeAmountofarticialdiusionadded p f Cavitationpenaltyfactor q =[ q x ;q y ;q z ] Heatuxvectorandcomponents q Heatuxduetoslidingasperitycontactfriction Q V Volumetricheatingrate w m 3 Q VD Volumetricheatingrateduetoviscousdissipation Q C Volumetricheatingrateduetoworkdoneonuid r Radialcoordinateinsphericalsystem R Combinedradius ix PAGE 10 R 1 ;R 2 Sphereorcylinderradius R a Meansurfaceroughness R q = R RMS Rootmeansquaresurfaceroughness Re h 2 0 u 0 0 l 0 Reynold'snumber SRR u b )]TJ/F10 6.9738 Tf 6.227 0 Td [(u a u m Slidetorollratio S q Combinedsurfaceroughnessparameter t Temporalcoordinate t 0 Characteristictime t t 0 Dimensionlesstime T Temperature T = T )]TJ/F11 9.9626 Tf 9.962 0 Td [(T 0 Changeintemperature T avg = R h 0 Tdz Filmaveragedtemperature T avg = R h 0 T )]TJ/F11 9.9626 Tf 9.963 0 Td [(T 0 dz Filmaveragedtemperaturechange T 0 ReferenceTemperature T R Referencetemperatureforrheologicalmodels T g Glasstransitiontemperature u =[ u;v;w ] Displacementsvector u withcomponents u;v;w u 0 ;v 0 ;w 0 Characteristicvelocitiesin x;y;z directionsrespectively u a ;u b xcomponentofvelocityoftheloweraandupperbsurfaces u m u a + u b 2 Meanentrainmentvelocity U HD 0 u m E 0 R HamrockandDowsondimensionlessspeedparameter v =[ v x ;v y ;v z ] Velocityvectorandcomponents U;V;W Dimensionlessuidvelocitiesin x;y;z directionsrespectively V Specicvolume V 0 Specicvolumeatreferencestate V molec Specicvolumeofasinglemolecule x PAGE 11 W HD F E 0 R 2 HamrockandDowsondimensionlessloadparameter x;y;z cartesiancoordinates X;Y;Z DimensionlessCartesiancoordinates B PressureviscositycoecientfortheBarusmodel Ch PressureviscositycoecientfortheChengmodel Roe PressureviscositycoecientfortheRoelandsmodel T = k C p Thermaldiusivity AsperityRadius K Temperaturecoecientforbulkmodulus 0 K Temperaturecoecientforbulkmodulus derivativewithrespecttopressure Ch TemperatureviscositycoecientfortheChengmodel Roe TemperatureviscositycoecientfortheRoelandsmodel DH TemperaturecoecientfortheDowsonHiggensonequationofstate V )]TJ/F7 6.9738 Tf 6.665 -4.147 Td [(1 T V molec V p;T Ashurst-Hooverscalingparameter " H 3 0 Reynoldsequationparameter strain D Meshsize , f , s Totalfrictioncoecient,frictiondueto viscousforces,andsolidcontactfriction L Lamparameter Density 0 Referencedensity 0 Dimensionlessdensity xi PAGE 12 z Probabilitydistributionofasperityheights Viscousdissipationfunction h 2 0 T t 0 Dimensionlesstemporalparameter Viscosity 0 Referenceviscosity g Viscosityatglasstransitiontemperature 0 Dimensionlessviscosity 12 u m 0 R 2 a 3 p h Reynoldsequationparameter f RR pd F Fluidlmloadfraction c RR p c d F Asperitycontactloadfraction NE 1 2 S 3 2 q GreenwoodWilliamsonmodelstinessparameter FluidandsolidCauchystresstensor s h 2 0 0 t 0 0 Squeezenumber i Standarddeviationofasperityheightdistributionforsurface i TestfunctionforGalerkinFEM ! Rotationalvelocity f Lubricationdomain s Soliddomain soliddisplacement = T T 0 Dimensionlesstemperature = T )]TJ/F10 6.9738 Tf 6.226 0 Td [(T 0 T 0 Dimensionlesstemperaturerise xii PAGE 13 CHAPTERI INTRODUCTION Solidtosolidcontactwithslidingisthemaincauseofwearinmechanicalelements. Whilefrictionisessentialtoanymechanicalsystem,itcancauseexcessivepowerlossand prematurefailure.Manymechanicalsystemsarelubricatedtocombatthis.Examplesof lubricationstretchbackasfarasthe17thcenturyBC,butitisonlyinthelast150years orsothatwehavecometounderstandhowlubricationworks. Inthelate1800's,trainaxleswerelubricatedwithheavygrease.Thisgreaseneeded tobechangedperiodicallywhichwasdonethroughagreaseport.Thegreaseportswere pluggedwithacorktopreventleakageandcontaminationwhennotinuse.Itwasobserved thatduringnormaloperationofthebearingthegreaseportcorkwouldbepushedout.The relativemotionofthespinningaxleonthestationarysleeve,combinedwiththeimmense weightofthetrain,generatedahydrodynamicpressurewhichpushedthecorkout.Such observationsledtolaboratoryexperiments,therstofwhichwaspreformedbyaMr. BeauhampTowerin1883.Hisworkmeasuredthehydrodynamicpressureazimuthally aroundabrasssleevewithaspinningcylinderinitseparatedbyathinlmofoliveoil.He founda625PSI.5ATMpeakpressurerise[64].Itwashypothesizedthatfortheuid pressuretorise,theremustbealmthatcompletelyseparatesthetwosurfaces.Inspired bythisresult,OsbourneReynoldscalculatedasimilarpressurerisebysimplifyingthe Navier-Stokesequations[55].ForacontinuumequationsuchastheNavier-Stokes equationstoaccuratelypredictthepressure,theremustbealmsucientlythicktobe consideredacontinuum.Thisconrmsthehypothesisofasignicantuidlmseparating 1 PAGE 14 twosurfacessupportingasignicantload.Thisuidlmhasbeenmeasuredtonanometer levelaccuracywithmodernequipmentbyJubault[67].Aslongastheuidlmisthicker thickerthanthelargestsurfaceasperitieswearandfrictionisminimized. Twosurfacesseparatedbyauidlmcarryinganormalloadviahydrodynamic pressureistheessenceofhydrodynamiclubrication.ThehydrodynamiclubricationHL regimeisdenedaswhentheuidlmismuchthickerthantheelasticdeformationsdue tothehydrodynamicpressure.Increasingtheloadfurtherwillelasticallydeformthetwo solidsontheorderofthelmthicknessorgreater.Thisiscalledelasto-hydrodynamic lubrication,orEHLforshort.Iftheloadisfurtherincreasedsuchthatthelmthicknessis onthesameorderasthesurfaceroughness,thensolidcontactoccurs,thisisknownas mixedlubricationML,ormixedelasto-hydrodynamiclubricationMEHL.Further decreaseinlmthicknessalmostcompletelyremovesthelubricantfromtheareaofcontact. Theremaybeafewmoleculesthickoflubricantbondedtothesolidsurfaces.Thisisknown asboundarylubrication.Plottingthectioncoecient, ,asafunctionoftheSommerfeld number, ! p asshowningure1.1,revealsthefrictionalcharacteristicsofeachlubrication regime.Where istheviscosity, ! istherotationalvelocity,and p isthecontactpressure. ThisisknownastheStribeckcurve[61].TheSommerfeldnumber,whichissometimes calledtheHerseynumber,isproportionaltothelmthickness,thereforeitisusefulthe thinkofthehorizontalaxisaslmthickness.Thewearratesincreasedramaticallyatlow lmthicknessesduetoincreasedsolidsolidcontact.Thedenitionofpressureinthe Sommerfeldnumberisunclear.Itmaymeanmaximumuidpressure,ormaximum Hertziancontactpressuredenedintable1.1,ortheappliedloaddividedbythebearing areadependingontheauthor.Theunderlyingmeaningisconsistentforeachdenition. Bearinggeometriesgenerallyfallundertwoclassications:conformalornon conformalalsoknownascounterformalasshowningure1.2.Conformalcontactsspread theloadoutoveragreaterareareducingthepeakpressureinthelubricant.Conversely nonconformalcontactsfocustheloadoverasmallarea,greatlyincreasingthepeak 2 PAGE 15 Figure1.1. Stribeckcurve:frictionalcoecientasafunctionoftheSommerfeldnumber. pressureinthelubricant.ThepeakuidpressurescaleswiththeHertziancontactpressure. Hertziancontacttheorydescribesthecontactpressuresanddeformationsbetweentwo elasticcontactingbodiesintheabsenceoffrictionalforces.Pointcontactsoccurwhentwo bodiesincontactatzeroloadcontactonlyatonepoint.Examplesarespherescontacting spheresandspherescontactingplanes.Linecontactsoccurwhentwobodiesincontactat zeroloadtouchonlyonasingleline.Examplesarecylinderscontactingcylindersand cylinderscontactingplanes.Table1.1denesthecontactradii, a ,contactpressures, p h , andindentationdepths, d ,forlineandpointcontacts,where E isthecombinedmodulus and R isthecombinedradii, isPoission'sratio,and F istheappliedload.Notethat thesesolutionsarevalidif R 2 = 1 ,anditbecomesahalfspace.Inthiscase, R = R 1 : Duetothefullycouplednatureoftheuid-solidproblem,andthenonlinearlubricant properties,analyticsolutionsdonotexist.Authorshavebeeninterestedinnumerical solutionssincethe1950s.TherstattemptatalinecontactwaspreformedbyGrubinin 1949[26].Inlieuofsolvingthecoupledelasto-hydrodynamicproblemheassumedthe 3 PAGE 16 Figure1.2. Illustrationofconformalleftandnon-conformalrightcontacts. Table1.1: Denitionsofcontactradii, a ,contactpressures, p h ,andindentationdepths, d , combinedmodulus, E ,andcombinedradii, R ,forlineandpointcontacts. Point Line a )]TJ/F19 7.9701 Tf 6.675 -4.976 Td [(3 FR 4 E 1 3 q 8 RF LE p h 3 F 2 a 2 q FE LR d a 2 R a 2 2 R E 1 E = 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 1 E 1 + 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [( 2 2 E 2 1 E = 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 1 E 1 + 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 2 E 2 R 1 R = 1 R 1 + 1 R 2 1 R = 1 R 1 + 1 R 2 elasticdeformationwasequaltothedeformationencounteredintheHertzian[36]problem. ManyearlyEHLsolutionsfollowedthisblueprintofusingtheHertzianelasticsolutionfor lineandpointcontacts[13,68].Thisprovidedacentrallmthicknesssolutionthatagreed wellwithexperimentalresults.However,itfailedtocapturethepressurespikeandexit contraction , asseenintheresultsinchapterII,thatisexclusivetothecoupledproblem. TherstcoupledsolutiontotheEHLproblemwasdonebyHamrockandDowsonin 1976[21,22,23],wholaterwroteanextensivetextbookonuidlmlubrication[9].Their numericalapproachwastosolvetheReynold'sequationviacentralnitedierence iteratingbetweentheelasticanduidproblemusingrelaxationtoensureconvergence.The steeppressuregradientsthatcausenumericalinstabilitiesweremitigatedbycombiningthe pressureandlmthicknessintoasinglevariable, pH 3 = 2 .Aftersolvingfor pH 3 = 2 , p issolved usingthestandarddenitionoflmthicknessandthesolutionof pH 3 = 2 . MajorspeedupscamewhenLubrecht[1,2,3]appliedmultilevelmultiintegration 4 PAGE 17 MLMIandmulti-gridtechniques.MLMIallowsafastercomputationoftheelastic deformation.Multigridtechniquesareknowntoincreaseconvergenceratesbystarting withaverycoarsemeshandsuccessivelyreningwhilesmoothingoutthelargest wavelengtherrorsrst.Thenextjumpincomputationalcapabilitycamein2001whenZhu andHu[39]extendedtheclassicformulationtoincludemixedlubricationusingaunied formulation.Thismethodallowsforcontactstoshareloadsbetweensolid-solidcontactand uidlmlubrication.ZhuandHudidextensiveheattransfercalculations,accountingfor viscousdissipation,compressiveheating,andsolid-solidfrictionalheating.Couplingthe EHLproblemwiththethermalproblemrequiresadditionaldegreesoffreedom,makingthe convergencerateoftheEHLproblemmoreimportant.Othersolidcontactmodels implementedintoEHLsolversincludeKraker,[18].Thismethodusesamodied denitionoflmthicknessthatconservesthemassofthelubricanttrappedbetweenthe tworoughsurfaces.Allsolidcontactisassumedtobeplastic.Thismethodisalsoaunied formulation,buthasonlybeenusedinwaterlubricatedjournalbearingstodate. Allthemethodsdiscussedpreviouslywerenitedierencemethodsolutionsthatrely onrelaxationtechniquestostabilizethecoupledsystemofequations.In2008,Habchi[34] usedafullsystemniteelementformulationwithvariousstabilizationtechniquestosolve classicEHLproblems.Fullsystemmeansalltheequationsarenumericallysolved simultaneouslyasasystemoflinearequations.Byputtingallequationsintoonesystem crosscouplingtermsaretakenintoaccountthatallowasolutionwithnorelaxation.This allowsthefullpowerofNewton'smethodtoberealizedensuringquadraticconvergence. Thebestpossiblenumericalmodelforthesekindsofsystemswouldhavethestabilityand speedofthefullsystemapproachcoupledwiththemultiregimecapabilityoftheZhuand HuorKrakernitedierenceformulations. Reynold'sequationisgenerallywrittenforacompressibleuidwithapressureand temperaturedependentviscosity.Itcannothandlephasechangeandwillpredictnegative pressuresatthedivergingsideofthecontact.Inrealitytheuidwillcavitateatpressures 5 PAGE 18 belowthevaporpressure,andthereforelargenegativepressurescannotexist.Thereare multipleschemestomitigatethisunrealisticmodelingresult.Themostsimpleisto truncatethenegativepressurestothevaporpressureofthelubricant.Morecommonly usedisthepenaltymethodthatpenalizesnegativepressuresatthevaporpressure.Neither ofthesemethodsconservethemassofthelubricant.Thisisonlyanissueinsystemswhere thecavitationregionislargecomparedtothecontactarea,orwhencavitationpressures areoversimilarmagnitudetouidoperatingpressures.Onesophisticatedmodelthatdeals withthisistheJakobsson-Floberg-OlssonJFOmodel[28].TheJFOmodelsolvestwo dierentPDE's,amodiedReynoldsequationvalidinthefulllmregime,andaPDE thatdescribestheowofvaporsinthecavitationregion.Theinterfacebetweenthetwo regionsmustbesolvedforeachiteration,requiringspecialmeshingstrategies.Elrod[27] improvedthismodelbyintroducingamodiedReynoldsequationthatisvalidinboththe fulllmregionandthecavitationregionwhichalsosatisesconservationofmass.The downsideofthissolutionisthatthepressuredierencebetweentheuidandvaporsection isdependentonthebulkmodulusoftheuid,whichmustberelaxedforconvergence. Thisworkinvolvesdevelopingnumericalmodelsthatproducedimensionlessresults thatarepresentedinageneralform,withfunctionaltswherepossible,forengineersto use.Thisworkcoverstwomaintopics: 1.Oillubricatedpointcontact 2.Waterlubricatedlinecontact Bothproblemsinvolveacoupleduidstructureinteractionproblem.InchapterII,the assumptionsrequiredtoderivetheReynoldsequationfromtheNavier-Stokesequationsare statedandjustied.Manylubricantmodelsarediscussedandttedtoexperimentaldata fromtheliterature.TheEHLmodelisfullyexplainedandresultsarepresentedaswellas 6 PAGE 19 modelvalidation.Ameshconvergencestudyispresented.Dimensionlessparametersfrom theliteraturearepresentedthatuniquelyidentifyEHLsolutions. InchapterIII,astatisticalrepresentationofsurfaceroughnessispresented,andmultiple solid-solidcontactmodelsareimplementedtoextendthemodelintothemixedand boundarylubricationregime.Thesemodelscanbeusedfortheoillubricatedpointcontact andthewaterlubricatedlinecontactproblem.Twonewdimensionlessparametersare presentedthataccountfortheeectsofsolid-solidcontact.Attedmodelforthefraction ofloadsupportedbytheuidlmasafunctionofdimensionlessparametersispresented. Estimatesofthefrictioncoecientoftherollingelementbearingarealsopresentedand discussed. Inchapter4,athermalmodeliscoupledwiththeMEHLmodel.Temperaturerisescome fromthreeheatingsources:viscousdissipation,frictionalheating,andcompressiveheating. Thetemperatureriseeectstheuid'smaterialproperties,inturneectingthepressure andlmthicknessandtheheatingratesintheuid.Dimensionlessparametersare presented.Conditionsthatthermallydegradethelubricantarepresented. Theuseofwaterasalubricantisdrivenbyenvironmentalanddefenseconcernsinsteadof tribologicalperformanceinmarinepropulsorshaftapplications.Oillubricatedbearings requireanoilsealwheretheshaftexitstheshipshullwhichinevitablyleaks.Thisleakage, whichisconsiderednormaloperationalconsumptionandacceptablepractice,was estimatedatbetween130to244millionlitersayearworldwidein2010[24].Forreference, thewellknownExxonValdezspilled41.6millionlitersofoilin1989.Thisisan environmentalconcern,butalsoadefenseconcernasitisgenerallyundesirabletohavea detectiblechemicalleakingfromwarships.Waterlubricatedbearingswereimplemented becauseoftheseconcerns. Oil-lubricatedballbearingcontactsareconsideredinthechaptersII,III,andIV.Sea water-lubricatedjournalbearingsarestudiedinchapterV.Theunderlyingphysicsof mixed-elasto-hydrodyanmiclubricationappliesinbothsystems,thereforethemodels 7 PAGE 20 developedinchaptersIIandIIIcanbeeasilyadaptedtothisnewsystem. Inchapter5,theMEHLformulationisextendedtosolveonrealworldengineering surfacessoentirebearingassembliesmaybemodeled,insteadofasmallhalfspacethatis representativeofabearingrace.Dierentbearingxturesandtheireectsonlubrication performancearepresented.Theeectofxturestinessonlubricationperformancefor polymerlinedwaterlubricatedbearingsispresentedandcomparedtoexperimentaldata. Attedmodelforthefractionofloadsupportedbytheuidlmasafunctionof dimensionlessparametersispresented. Waterlubricatedbearingsareprevalentinawidevarietyofindustrialandmarine applications,andthusthepredictionofbearinglifehasbeenaprimaryfocusofengineering analysis.Operatinginthefulllmhydrodynamicregimeovertheentirebearing,which preventssolidcontact,increasesbearinglife.Ofcourse,fulllmlubricationisnotalways possiblegiventhesize,load,andspeedrequirementsofabearing.Theuseofsoftpolymer orelastomerbearinglinersextendsthefulllmregimetolowerspeedsbyincreasingthe conformityofthecontactandthereforespreadingtheloadoutoveralargerarea.Thisis thebasisofelastohydrodynamiclubrication. Currentwaterlubricationmodelsintheliteraturearequitesophisticated.Apopularrecent focusismodelingmultiplelubricationregimes,spanningfulllmlubricationdownto mixedandboundarylubrication[18,37,63].Thisisusuallydonewithamodied formulationoftheReynoldsequationthatisvalidinbothfulllmlubricationand mixed/boundarylubrication.Rarelyarethesemodelscomparedtoexperimentaldata.This maybeduetotheinherentdicultyinexperimentallydecouplingthereactiontorquefrom theappliedload.Thismaynotseemlikeahardproblem,butcoecientsoffrictioncanbe aslowas 0 : 0001 ,meaningthetorquecanbe4ordersofmagnitudelowerthantheapplied load.Tofurthercomplicatematters,thefrictioncoecientcanbeashighas0.5,meaning thetorquetransduceralsohastospanfourordersofmagnitude. Anotherareaofresearchhasbeentheevaluationoftransienteects,includingstability 8 PAGE 21 analysisofstavebearings.ThesemodelsemployatransientformulationoftheReynolds equationthatareusedtodeterminethedynamiceectsonthebearinginthe hydrodynamiclubricationregime.Insomecases,usefulstinessanddampingcoecients wereextractedforuseinotherlargerscaledynamicmodels[52]. Theshapeandsizeofthecavitationregionisalsoofgreatinterestasitaectsthe performancecharacteristicsofthebearing.Avarietyofcavitationmodelsthatrangein complexityhavebeenproposed.Initialmodelssimplytruncatedanythingbelowthe cavitationpressuretothecavitationpressureeachiteration.Amoresophisticatedmethod penalizespressuresbelowthecavitationpressureinordertoraisethepressureuptothe cavitationpressure[51].Neitherofthesemethodsconservesmass.Toaddressthisissue,the JFO[28]modelwasdevelopedandalthoughitconservesmassinthecavitationregion,it requiressolvingonePDEinthecavitationregionandanotherinthefulllmregion.Also, thepositionoftheinterfacebetweenthetworegimesmustbesolvedduringeachiteration, whichincreasesthecomplexityofthemodel.Elrod[27]extendedtheJFOmodeltowork asauniedPDEineachregime.Thisworkemploysapenaltymethodforsimplicity.The sizeandshapeofthecavitationregionshouldnoteectourresultstoomuchduetothe dierenceinpeakuidpressureandvaporpressureinoursystem.Peakpressuresinwater lubricatedbearingsareontheorderof 1 MPa ,thecavitationpressureforwateris 3 : 169 KPa atroomtemperature.Weassumethatasmallchangeinpressurenearthe cavitationregiondoesnotsignicantlyaecttheelasticdeformationswearefocusingon. Mostmodelingeortsinthewaterlubricatedsoftbearinglinereldthataccountfor elasticityonlyaccountfortheelasticbearingliner[44].Afewstudieshavecoupleda deformableshaftwithadeformablebearingliner[35,70,71]. 9 PAGE 22 CHAPTERII ELASTO-HYDRODYNAMICLUBRICATIONEHL Theextremeaspectratiooflubricatinglms,often1000:1,makestraditionalCFD meshesunreasonablylarge.Fluidlmlubricationisthereforemodeledwithasetofmajor simplicationsoftheNavierStokesequations,resultingintheReynoldsequation.The Reynoldsequationisderivedfromtheequationsofmotionandequationofcontinuityvia anorderofmagnitudeanalysisinthefollowingsection.PressuresinpointcontactEHL problemsreachGigapascallevelswhilelmthicknessareontheorderofNanometersto micrometers.Undertheseconditionsthewidthandlengthofthelmisontheorderof0.1 millimeters.Appropriatenondimensionalizationmustbeappliedtoensurenumerical accuracy.ThenondimensionalstabilizedandpressurepenalizedweakformofReynolds equationarepresented. 2.1DerivationofReynold'sEquation TounderstandtheoriginofallthetermsinReynoldsequation,weformacomplete derivationhere.StartingwiththeNavier-StokesNSequationinCartesiancoordinates Seeeqs.B6-1-B6-3in[43]. @v x @t + v x @v x @x + v y @v x @y + v z @v x @z = )]TJ/F22 7.9701 Tf 10.746 5.256 Td [(@p @x + h @ 2 v x @x 2 + @ 2 v x @y 2 + @ 2 v x @z 2 i + g x .1 @v y @t + v x @v y @x + v y @v y @y + v z @v y @z = )]TJ/F22 7.9701 Tf 10.598 5.256 Td [(@p @y + h @ 2 v y @x 2 + @ 2 v y @y 2 + @ 2 v y @z 2 i + g y .2 10 PAGE 23 @v z @t + v x @v z @x + v y @v z @y + v z @v z @z = )]TJ/F22 7.9701 Tf 10.5 5.256 Td [(@p @z + h @ 2 v z @x 2 + @ 2 v z @y 2 + @ 2 v z @ 2 z i + g z .3 where istheuiddensity, v x ;v y ;v z aretheuidvelocitiesinthe x;y;z directions respectively, t istime, p ispressure, isviscosity, g x ;g y ;g z arethegravitationalconstants inthe x;y;z directionsrespectively.Theequationsofmotionarenondimensionalisedwith thefollowingdimensionlessquantities: X x l 0 Y y b 0 Z z h 0 U v x u 0 V v y v 0 W v z w 0 t t 0 0 0 P h 2 0 p 0 u 0 l 0 .4 where l 0 ;b 0 ;h 0 ;u 0 ;v 0 ;w 0 arethecharacteristiclengthsandvelocitiesinthe x;y;z directions respectively, t 0 isthecharacteristictime,and 0 and 0 arethecharacteristicdensityand viscosityrespectively.Substitutingthedimensionlessvariablesintothe x componentofNS yields: 0 u 0 1 t 0 @U @ + Uu 0 l 0 @U @X + Vv 0 b 0 @U @Y + Ww 0 h 0 @U @Z = )]TJ/F22 7.9701 Tf 10.494 5.256 Td [( 0 u 0 h 2 0 @P @X + 0 h u 0 l 2 0 @ 2 U @X 2 + u 0 b 2 0 @ 2 U @Y 2 + u 0 h 2 0 @ 2 U @Z 2 i + 0 g x .5 Multiplyingby l 0 0 u 2 0 gives: l 0 t 0 u 0 @U @ + U @U @X + l 0 b 0 v 0 u 0 V @U @Y + l 0 h 0 w 0 u 0 W @U @Z = )]TJ/F22 7.9701 Tf 16.241 5.256 Td [( 0 l 0 h 2 0 u 0 0 1 @P @X + 0 l 0 h 2 0 u 0 0 h h 2 0 l 2 0 @ 2 U @X 2 + h 2 0 b 2 0 @ 2 U @Y 2 + @ 2 U @Z 2 i + l 0 g x u 2 0 .6 NotingthatthepressureandviscoustermshavetheinversemodiedReynoldsnumberfor lubricationproblems, Re x h 2 0 u 0 0 0 l 0 andthegravitationaltermistheFroudenumber, Fr x l 0 g x u 2 0 ,anddeningthesqueezenumberas s h 2 0 0 t 0 0 ,wemultiplythroughbythe modiedReynoldsnumber. h 2 0 0 t 0 0 @U @ + h 2 0 u 0 0 0 l 0 U @U @X + h 2 0 v 0 0 0 b 0 V @U @Y + h 0 w 0 0 0 W @U @Z = )]TJ/F19 7.9701 Tf 10.545 4.707 Td [(1 @P @X + h h 2 0 l 2 0 @ 2 U @X 2 + h 2 0 b 2 0 @ 2 U @Y 2 + @ 2 U @Z 2 i + h 2 0 u 0 0 0 l 0 l 0 g x u 2 0 .7 11 PAGE 24 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( s @U @ + Re x U @U @X + Re y V @U @Y + Re z W @U @Z = )]TJ/F19 7.9701 Tf 10.545 4.707 Td [(1 @P @X + h h 2 0 l 2 0 @ 2 U @X 2 + h 2 0 b 2 0 @ 2 U @Y 2 + @ 2 U @Z 2 i + Re x Fr x .8 Similarly,theyandxcomponentsoftheNSequationaregivenby: )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [( s @V @ + Re x U @V @X + Re y V @V @Y + Re z W @V @Z = )]TJ/F19 7.9701 Tf 10.545 4.707 Td [(1 @P @Y + h h 2 0 l 2 0 @ 2 V @X 2 + h 2 0 b 2 0 @ 2 V @Y 2 + @ 2 V @Z 2 i + Re y Fr y .9 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( s @U @ + Re x U @W @X + Re y V @W @Y + Re z W @W @Z = )]TJ/F19 7.9701 Tf 10.545 4.708 Td [(1 @P @Z + h h 2 0 l 2 0 @ 2 W @X 2 + h 2 0 b 2 0 @ 2 W @Y 2 + @ 2 W @Z 2 i + Re z Fr z .10 WecannowpreformanorderofmagnitudeanalysisOOMAtoarriveatthe Reynoldsequation.Goodestimatesforthelengthscalesandmaterialpropertiesforpoint contactsareintroducedintable2.1.Classicalviewsonpointcontacts[9]andexperimental results[67]showthatthelengthscales, l 0 ;b 0 ,scalewiththeHertziancontactradii,and theuidpressure, p ,scaleswiththeHertziancontactpressure.Experimentallyveriedlm thicknessresultsareusedforthelastlengthscale, h 0 : DowsonandHigginspressuredensity relationshipandRoelandspressureviscosityrelationship,denedinsection2.2,areusedto calculate 0 and 0 attheHertziancontactpressureforagivenpressureviscosity coecient, Roe .Usuallythecoordinatesystemisorientedsuchthatthereisonlymotion inthe x directionandtherefore v 0 =0 .Hereweassume v 0 = u 0 justtoberigorous.With noobviouschoicefor w 0 ; weassume w 0 = u 0 = 10 .Thetablebelowdetailsthetestcaseofa moderatelyloadedpointcontact,andthescalesassociatedwithit. Thedimensionlessparameterscanbecalculatedwiththenumericalvalueslistedin table2.1: Re x = Re y = : 3 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 m 2 m=s kg=m 3 Pa s : 6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 m u 6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(10 .11 12 PAGE 25 Table2.1: ParametersusedinOOMA R 0.0127[m] F 50[N] E 110[GPa] Roe 15[1/GPa] 0 220[Pas] 0 1100[kg/m 3 ] a 1.6E-4[m] p h 0.927[GPa] h 0 1.3E-7[m] u 0 1[m/s] v 0 1[m/s] w 0 0.1[m/s] t 0 h 0 w 0 =1 : 3[ s ] Re z = : 3 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 m : 1 m=s kg=m 3 Pa s u 7 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 .12 s = h 2 0 0 t 0 0 = : 3 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 m 2 kg=m 3 Pa s : 3 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 s u 7 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 .13 Re x Fr x = Re y Fr y =6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(10 ag u 2 0 =6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(10 : 6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 m : 81 m=s 2 m=s 2 u 9 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(13 .14 Re z Fr z =7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(8 h 0 g w 2 0 =7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(8 : 3 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 m : 81 m=s 2 : 1 m=s 2 =1 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(12 .15 h 2 0 l 2 0 = h 2 0 b 2 0 = : 3 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 m 2 : 6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 m 2 =7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 .16 13 PAGE 26 SubstitutingthesevaluesintoEqs.2.8,2.9,and2.10: )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(8 @U @ +6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(10 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(U @U @X + V @U @Y +7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(8 W @U @Z = )]TJ/F19 7.9701 Tf 10.546 4.707 Td [(1 @P @X + h 7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 @ 2 U @X 2 + @ 2 U @Y 2 + @ 2 U @Z 2 i +9 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(13 .17 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(7 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 @V @ +6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(10 U )]TJ/F22 7.9701 Tf 7.062 -4.977 Td [(@V @X + V @V @Y +7 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 W @V @Z = )]TJ/F19 7.9701 Tf 10.546 4.707 Td [(1 @P @Y + h 7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 @ 2 V @X 2 + @ 2 V @Y 2 + @ 2 V @Z 2 i +9 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(13 .18 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(7 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 @U @ +6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(10 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(U @W @X + V @W @Y +7 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(8 W @W @Z = )]TJ/F19 7.9701 Tf 10.545 4.707 Td [(1 @P @Z + h 7 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 @ 2 W @X 2 + @ 2 W @Y 2 + @ 2 W @Z 2 i +1 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(12 .19 Ifwechoosethescaleswisely,eachnon-numericalterminEqs.2.17,2.18,and 2.19isontheorderofone.Neglectingtermssmallerthan O : 0= )]TJ/F15 11.9552 Tf 10.587 8.087 Td [(1 @P @X + @ 2 U @Z 2 .20 0= )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @P @Y + @ 2 V @Z 2 .21 0= )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @P @Z + @ 2 W @Z 2 .22 Moregenerallystated,theReynoldsnumbertermscanbeneglectedwhenthelm thicknessismuchsmallerthanthewidthandlengthorradiusinthecaseofcircular contacts,thesqueezetermcanbeneglectedforsmalluidlmsandhighviscosities,even withincredibleshorttimescales,andthegravitationaltermscanbeneglectedduetothe smalllengthscalesofthelmandthe O [ m=s ] velocities.Wealsoassumethatthereis nopressuregradientthroughthelm,whichimpliesthat @ 2 W @Z 2 =0 .Thereforethe dimensionalequationsofmotionsimplifyto: 14 PAGE 27 @p @x = @ 2 v x @z 2 .23 @p @y = @ 2 v y @z 2 .24 whichcanbesolvedforvelocitywiththeboundaryconditions: at z =0 ;v x = u a v y = v z =0 at z = h;v x = u b v y =0 v z = w 0 .25 wherethevelocityinthe Z directionisthecombinationofsqueezeowandthe X velocity timestheslopeofthesurface: w 0 = @h @t + u 0 @h @x .26 Solvingforvelocity: v x = 1 2 @p @x )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(z 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(zh + z h u b )]TJ/F21 11.9552 Tf 11.955 0 Td [(u a + u a .27 v y = 1 2 @p @y )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(z 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(zh .28 whicharecombinationsofPouiselleandCouetteow.Themassowratesperunitlength inthe x and y directions, M x and M y aregivenby: M x = Z h 0 v x d z = )]TJ/F21 11.9552 Tf 10.695 8.088 Td [(h 3 12 @p @x + hu m .29 M y = Z h 0 v y d z = )]TJ/F21 11.9552 Tf 10.695 8.088 Td [(h 3 12 @p @y .30 where u m = u a + u b 2 isthemeanentrainmentvelocity.Themeanentrainmentvelocityisthe 15 PAGE 28 meanvelocityoftheuidfromtheperspectiveofstationarysolids,itisthemeanvelocity oftheuidentrainedintothecontact.Notethatthesemassowratesareperunitlength, notperunitarea.Thisisduetothereductionoftheproblemfromthreedimensionalin x;y; and z toatwodimensionalproblemin x and y .Tocombinethesetermsweapply conservationofmasstoadierentialelementneglectingowinthe z direction,where @m @t is thechangeinmassperunittime.Alternatively,onecouldintegratetheequationof continuityacrossthelmthicknessandapplyLeibniz'sformulatoobtainthesameresult. @m @t = M x j x )]TJ/F21 11.9552 Tf 9.298 0 Td [(M x j x + dx d x + M y j y )]TJ/F21 11.9552 Tf 9.298 0 Td [(M y j y + dy d y .31 0= @M x @x + @M y @y .32 CombiningEqs.2.29,2.30,and2.32,wearriveatReynoldsequationnotethat u m is constantinallspatialdirections: 0= r h 3 12 r p + u m @h @x .33 Usingthenondimensionalvariables: X x a Y y a Z za h 2 0 0 P p p h H hR a 2 12 u m 0 R 2 a 3 p h " H 3 .34 where a istheHertziancontactradius, h isthelmthickness, 0 istheambientdensity, 0 istheambientviscosity, p h isthehertziancontactradius, isannondimensional parameter,and " isanondimensionalvariable,wearriveatthedimensionlessReynolds equation: )-222(r " r P + @ H @X =0 .35 16 PAGE 29 2.2MaterialProperties LubricantProperties Theextremepressureenvironmentencounteredinlubricationproblemsdrastically reducestheinter-moleculardistances,lockinguptheuidintowhatsomeauthorscallan amorphoussolid[69].Theconsequencesofthisbehaviorisincreasedlubricantdensityand viscosity.Asseeningure2.1thedensitycanincreasebyupto30%,andtheviscositycan increasebyseveralordersofmagnitudeundertypicalconditions.Veryhighpressure rheology>1GPaofoilsisdicultandexpensivetomeasure,requiringcustom rheometers.Table2.2summariesrheologicalmodelsthathavebeenfoundtobeusefulfor calculatingtheviscosityoflubricatingoilsasafunctionoftemperatureandpressure. TheBarusmodelwastherstattemptatmodelingtheviscosityriseofoilsduetopressure rise.Itwasnoticedearlyonthattheviscosityrisesexponentiallyandtheslopeof d ln = 0 dp = B ,where B isthepressureviscositycoecient,whichhasunitsofinverse pressureandisusuallyexpressedas 1 GPa .TheChengmodelisanextensionofthismodelto includetemperatureeects.Experimentsshowtheslopeof d ln = 0 dp isnotconstantat pressuresexceeding 200 )]TJ/F15 11.9552 Tf 11.955 0 Td [(400 MPa .TheChengandBarusmodelsareusefuldespitethese shortcomingsbecausetheyaresimplemathematicallyandhaveveryfewparameters makingiteasytottodata.TheRoelandsmodelisasignicantimprovementoverthe BarusandChengmodel.Itsbasisisstillempirical,butitmodelsthechangeinslopeof d ln = 0 dp andtemperatureeectswithonly4parameters.SimilartotheBarusandCheng model, lim p ! 0 d ln = 0 dp = Roe ,howeveras p increases d ln = 0 dp decreases. Thelasttwomodelsintable2.2modelthechangeofviscosityduetothechangeindensity, andrequireorhaveanimpliedequationofstate.Bair[6]notedthatthepressure andtemperaturedependentviscositycurvesofseveralmaterialscollapsedontoonemaster curvewhenplottedagainstanewthermodynamicscalingfactor V = )]TJ/F19 7.9701 Tf 7.611 -4.976 Td [(1 T V molec V p;T ,where V molec V isthespecicvolumeofasinglemoleculeoflubricantnormalizedwiththespecic 17 PAGE 30 volumeofthemacrolubricantatatemperature, T ,andpressure, p ,and isatting parameter. V molec canbecalculatednumericallybyassumingeachatomofalubricant moleculeisaspherewitharadiusequaltothevanderWaalsradiusoftheatom.The authorsofthismodelnotethattheywereabletouselowpressuredatatoextrapolate accuratelytohigherpressuredataduetothisscalingforpropylenecarbonate. ThemodiedWilliams-Landel-FerryWLFmodel,[59],isbasedonthetime temperaturesuperpositionprinciple,whichmeansitispossibletocollapseallcurvesonto onemastercurvebyshiftingeachcurvebytheglasstransitiontemperatureatthat pressure. Table2.3presentstwoequationsofstatethatmodelthedensityorspecicvolume oflubricantsasafunctionofpressureandtemperature.Densitychangesapproximately 30%overthepressurerangeofinterest.Thispalesincomparisontotheseveralordersof magnitudeviscositychanges.Nevertheless,itisimportanttomodeldensitychanges accuratelybecauseeachtermintheReynoldsequationisamassowrate,asseenin equations2.32and2.33. TheDowsonandHigginsonmodelisathreeparameterempiricalmodelthatwas developedin1966andhasbeenusedsincetheearlydaysofcomputationaltribology.This equationisreasonablyaccurate,howeveritonlyshiftsthe V V 0 P verticallyastemperature changes,itcannotchangetheshapeofthiscurve.TheTaitequationofstateisafree volumemodel,relatingthechangeindensitytothetemperaturedependentbulkmodulus, K .Itisasevenparametermodelthatcantawiderangeofmaterials. Ahighpressureandspecicvolumedatasetforthecommonlystudiedlubricant Squaleneispresentedin[5].Squaleneisregularlyusedinhighpressurerheologybecauseit iswidelyavailableinlargequantitiesandinexpensiveandrepresentativeofthepressure andtemperaturedependentrheologyofparanicmineraloils.Theviscositydatawastaken inatemperaturecontrolledhighpressurefallingbodyviscometer.Ahydraulicsystemwith apressureintensierisusedtocontrolthepressureinthecylindricaloilchamber.Aneedle 18 PAGE 31 Table2.2: SummaryofRheologicalmodelsavailableforhighpressurenon-isothermallubricants. Name Expression Parameters RequiresEOS? Basis Barus [7] 0 = e B p 0 ; B No Empirical Cheng [13] 0 = e Ch p + Ch 1 T )]TJ/F20 5.9776 Tf 11.157 3.258 Td [(1 T R + Ch p T 0 ; ch ch ; ch No Empirical Roelands [57] 0 = e [ln R +9 : 67 F p G T ] F p = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1+ [ 1+5 : 1 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 p ] Z 0 G T = T )]TJ/F19 7.9701 Tf 6.586 0 Td [(138 T R )]TJ/F19 7.9701 Tf 6.586 0 Td [(138 )]TJ/F22 7.9701 Tf 6.587 0 Td [(S 0 Z 0 = Roe 5 : 1 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 ln R S 0 = Roe T R )]TJ/F19 7.9701 Tf 6.587 0 Td [(138 ln R +9 : 67 0 Roe Roe T R No Empirical Ashurst-Hoover ExponentialPowerLaw [6] 0 = e [ A q V + B Q V ] V = )]TJ/F19 7.9701 Tf 7.132 -4.541 Td [(1 T V molec V P 0 A;B q;Q V molec Yes ScalingisAshurst-Hoover, viscositymodelisempirical Doolittle [19,20] 0 = e B D V V 0 0 B D V 0 Yes FreeVolume ModiedWLF [59] g =10 )]TJ/F23 5.9776 Tf 5.756 0 Td [(C 1 T )]TJ/F23 5.9776 Tf 5.756 0 Td [(T g p F p C 2 + T )]TJ/F23 5.9776 Tf 5.756 0 Td [(T g p F p T g p = T g 0 + A 1 ln + A 2 p F p =1 )]TJ/F11 10.9091 Tf 10.909 0 Td [(B 1 ln + B 2 p g C 1 ;C 2 T g 0 A 1 ;A 2 B;B 2 Implied TTS/FreeVolume 19 PAGE 32 Table2.3: Summaryofequationsofstateforhighpressurenon-isothermallubricants. Name Expression Parameters Basis Dowson&Higginson [9] 0 h 1+ 0 : 6 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 p 1+1 : 7 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 p )]TJ/F21 11.9552 Tf 11.955 0 Td [( DH T )]TJ/F21 11.9552 Tf 11.955 0 Td [(T R i 0 DH T R Empirical ModiedTaitEquationofState [62,25] V V 0 =1 )]TJ/F19 7.9701 Tf 22.091 4.707 Td [(1 1+ K 0 0 ln h 1+ p K 0 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ K 0 0 i K 0 = K 0 R exp )]TJ/F21 11.9552 Tf 9.298 0 Td [( K T K 0 0 = K 0 0 R exp )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [()]TJ/F21 11.9552 Tf 9.299 0 Td [( 0 K T V 0 V R =1+ V T )]TJ/F21 11.9552 Tf 11.955 0 Td [(T R = 0 )]TJ/F22 7.9701 Tf 6.675 -4.638 Td [(V R V K 0 R ; K K 0 0 R ; 0 K T R ; V 0 FreeVolume 20 PAGE 33 isthendroppedinthepressurechamberanditsterminalvelocityismeasuredviaalinear variabledierentialtransformerLVDT.Viscositymeasurementsaretakenfrom p =0 : 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1200 MPa and T =20 )]TJ/F15 11.9552 Tf 11.956 0 Td [(100 C .Therelativevolumemeasurementsaretaken withametalbellowspiezometer,from p =0 : 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(378 MPa and T =40 and 100 C . EachrheologicalmodelisplottedagainstBair'ssetofhighpressurerheologicaldatain gure2.1.Theparametervaluesaswellasgoodnessoftestimatesarelistedintable2.4 foreachmodelused.Theparametervaluesforeachrheologicalmodelandequationofstate whereobtainedwithanonlinearregressiondoneinMatlabusingthecommandtnlm. TheMatlabcodeusedtodothisisattachedinAppendixA.Averygoodinitialguessis requiredsincetherearesomanyparameters.Matlabwasunabletocomeupwithatfor theAshurst-Hoovermodel,evenwhenusinganinitialguessthattsthedataverywell. Thereforeparametersareusedfrom[6].Similardicultieswereencounteredintryingtot theWLFmodel.Thiswasremediedbyttingthefunctionfor T g p todatabeforetting therestofthemodel. Figure2.1. Rheologicalmodelsforhighpressurenon-isothermallubricantsttoexperimentaldataforSqualenefrom[5]. Itisverydiculttodesignapressurevesseltowithstandtypicallubricationpressuresin ordertotakedirectviscositymeasurements.Thesemodelsareoftenextrapolatedinthe absenceofveryhighpressureviscositydata.Ingure2.3threeviscositymodelsare 21 PAGE 34 Table2.4: CoecientvaluesandRMSerrorforeachtingures2.1and2.2. Model Parametervalues R^2 adjR^2 Notes RMSError Cheng Ch =1 : 57 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 Pa s; Ch = )]TJ/F8 10.9091 Tf 8.484 0 Td [(1 : 9 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 1 GPa Ch =2 : 38 10 3 K; Ch =9 : 74 .983 .983 0.593 Roelands R =1 : 57 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 Pa s; Roe =17 : 5 1 GPa Roe =3 : 65 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 K;T R =313 : 15 K 0.999 0.999 0.129 AH R =2 : 4 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 Pa s A =24 : 28 K q ;C =7 : 126 10 6 K Q q =0 : 1051 ;Q =1 : 981 =3.92 V molec =0 : 8002 cm 3 g Fitfrom[6]. Reportedstandard deviationof relativeviscosityis 8 : 4% . WLF g =45 : 88 10 12 Pa s C 1 =17 : 98 ;C 2 =32 : 85 C T g 0 = )]TJ/F8 10.9091 Tf 8.485 0 Td [(134 : 6 C A 1 =40 : 95 C;A 2 =8 : 01 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 1 MPa B =0 : 718 ;B 2 =8 : 92 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 1 MPa 1 1 Fitdoesn'tpass theeyeballtest 48.4 Tait K 0 R 7 : 93 GPa; K = )]TJ/F8 10.9091 Tf 8.485 0 Td [(6 : 04 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 1 K K 0 0 R =11 : 6 ; 0 K = )]TJ/F8 10.9091 Tf 8.485 0 Td [(9 : 17 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 1 K T R =313 : 15 K; V =8 : 26 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 1 K R =858 kg m 3 1 1 .000519 DH R =852 : 3 kg m 3 DH =5 : 739 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 1 K T R =313 : 15 K .979 .977 0.0081 22 PAGE 35 Figure2.2. Densitymodelsttodatafrom[5] extrapolatedto 5 ; 000 MPa .Theexperimentaldatashowsaverycleartrendofreducing thepressureviscosityslopeaspressureincreases.TheBarusandChengmodelhave constantslopes,limitingtheirusefulness.At 5 ; 000 MPa thethereisa10orderof magnitudedierencebetweenthehighestpredictionChengtothelowestRoelands,and a3orderofmagnitudedierencebetweentheRoelandsandAshurst-Hoovermodel. RoelandsandAshurst-Hoovermodelsareonlyseparatedbyafactorof 4 at 2 ; 000 MPa . Thisraisesthequestion:whathappenstothesepredictionsifsomeamountofhigh pressureviscositydataisremovedandtisrepeated? Onewouldassumethatrheologicalmodelswithanaccuratephysicalbasiswouldbeableto predictlubricantviscosityoutsideoftheexperimentalenvelopemoreaccuratelythanan empiricalmodel.Figures2.4and2.5showtheAshurst-HooverandRoelandsmodel extrapolatedto 2 ; 000 MPa aftercuttingofvaryingamountsofhighpressuredataand rettingthemodel.Table2.5tabulatesthisdataat 1 ; 200 MPa and 2 ; 000 MPa . TheAshurst-Hoovermodelpredictionsarereducedbyafactorof 3 : 5 whencutting outpressuredataabove 800 MPa .TheRoelandsmodelisonlyreducedbyafactorof1.1 withthesamedatamanipulation.Similarly,at 1 ; 200 MPa ,theAshust-Hoovermodelis 23 PAGE 36 Figure2.3. ExtrapolationofmodelsinFigure1.1to5GPa.TheBarusandChengmodels arethesamebydenitionwhenttoasingletemperature. reducedbyafactorof1.4andtheRoelandsmodelisreducedbyafactorof1.04. SurprisinglytheRoelandsmodelwithnophysicalbasisoutperformsthemodelbasedona freevolumeapproach.BoththeRoelandsmodelandtheAshurst-Hoovermodelprovidea qualityrepresentationofviscosityupto 1 ; 200 MPa .Inlieuofdataatveryhighpressure, andassumingthelubricantdoesnotundergoanymajorstructuralrearrangementsthe Roelandsviscositymodelprovidesareasonableextrapolationforviscosityupto 2 ; 000 MPa . Table2.5: RoelandsandAshurst-Hoovermodelevaluationswhenomittinghighpressure datafromthet. P cut Roelands ; 200 MPa Roelands ; 000 MPa Ashurst-Hoover ; 200 MPa Ashurst-Hoover ; 000 MPa 1 8 ; 632 Pa s 3 : 94 10 6 Pa s 10 ; 415 Pa s 1 : 55 10 7 Pa s 1100 8 ; 465 Pa s 3 : 76 10 6 Pa s 9 ; 880 Pa s 1 : 25 10 7 Pa s 1000 8 ; 375 Pa s 3 : 69 10 6 Pa s 9 ; 790 Pa s 1 : 13 10 7 Pa s 900 8 ; 184 Pa s 3 : 56 10 6 Pa s 8 ; 690 Pa s 6 : 80 10 6 Pa s 800 8 ; 287 Pa s 3 : 60 10 6 Pa s 7 ; 223 Pa s 4 : 40 10 6 Pa s 24 PAGE 37 Figure2.4. ExtrapolationofAnhutst-Hooverviscositymodel,whenusinglowerpressure data Figure2.5. ExtrapolationofRoelandsviscositymodelwhenusinglowerpressuredata BearingMaterials Rollingelementbearingsaremadeofarollingelementwhichinthisworkisaball, andtwobearingsurfacesfortheballtorollinbetweencalledraces.Thisisdepictedin gure2.6.Theinnerraceingure2.6isshowncutintwoparts.Eachracehasoneconcave curvaturetoconformallyadapttotheball.Theinnerracealsohasaconvexcurvatureto 25 PAGE 38 Figure2.6. Partsofafourpointcontactangularcontactballbearing. A4-pointangular contactballbearing byUser:Silberwolf/CC-BY-2.5. wraparoundtheinnershaft,andtheouterracehasasecondconcavecurvaturetowrap aroundtheballpath.Theballspacingisheldtogetherwitharetainingclipthatcanalso servetoholdgreasepockets. TypicalvaluesofYoung'smodulusandPoisson'sratioforhardenedbearingsteelare listedintable2.6.Sapphireisincludedbecauseitiscommonlyusedinexperimentalsetups thatrequireanIRtransparentbearingmaterial,althoughitisfartoobrittleandexpensive tobeusedinactualbearingapplications. Rollingelementbearingsaredesignedtoavoidbulkplasticdeformation.Whenthe appliedloadexceedsthedesignload,orinimpactscenarios,theballswillindenttherace andleaveamark.ThisisknownasBrinelling.Bearingmaterialsarecommonlyhardened toavoidBrinneling.Hardeningprocessesusuallyincludeheatingthematerialupwardsof 1700 F thenquenching,changingthestructureofthesteel.Thisstructuralchangedoes notnecessarilypermeatethroughtheentirebody,thereforeassumingaconstantmodulus fortheentireelasticbodyisanapproximation.Themetallurgyofsteelbearingmaterialsis outsidethescopeofthiswork.Linearelasticityisassumedfortheballandracewitha 26 PAGE 39 constantmodulusandPoisson'sratio.Amuchmorecompletediscussionofbearing materialsandhardeningprocessescanbefoundin TheFundamentalsofFluidFilm Lubrication [9]. Table2.6: Linearelasticpropertiesofcommonlyusedbearingmaterials Material E GPa n Aluminum 62 0 : 33 MediumandhighalloySteel 200 0 : 30 SiliconCarbide 450 0 : 19 Sapphire 360 0 : 34 2.3CoupledEHLModel Inthissection,wedeneanequivalentYoung'sModulusandPoisson'sRatioto reducethetwobodysolidproblemtoonesolidbody.Theequivalentproblemsolvesforthe deformationofahalfspace,reducingthesizeoftheproblembyonlyrequiringonesolid domain. Thesumoftheballandracedeformations, 1 and 2 respectively,isdenedasa singleequivalentdeformation eq ,asshowningure2.7: eq = 1 + 2 .36 AtkinandFox[4]providesananalyticsolutionforallthreecomponentsofdisplacement forahalfspaceloadedbyapointforce.Substitutingtheanalyticsolutionintotheprevious equationandsimplifyingyieldsarelationshipbetweentheLammaterialparametersofthe ballandraceandtheonebodyLammaterialparameters: 1 L eq z 2 r 2 +2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( eq = 1 L 1 z 2 r 2 +2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 + 1 L 2 z 2 r 2 +2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 .37 where L 1 , L 2 and L eq arethelamconstantsfortheball,race,andequivalentmaterial 27 PAGE 40 respectively, 1 , 2 ,and eq arethePoisson'sratiofortheball,race,andequivalent materialrespectively,and r = p x 2 + y 2 + z 2 .Anysetofequivalentpropertiesthatsatises thepreviousequationcanbeusedtodeneanequivalentsinglebodyproblemtoreplace thetwobodyproblem.Makingtheadditionalassumptionthat 1 L eq = 1 L 1 + 1 L 2 andthe denitionoftheLamconstant, E =2 L + ,wecansolvethesystemofequationsfor E eq and eq ,whichisdisplayedbelow: E eq = E 2 1 E 2 + 1 2 + E 2 2 E 1 + 2 2 [ E 1 + 2 + E 2 + 1 ] 2 and eq = E 1 2 + 2 + E 2 1 + 1 E 1 + 2 + E 2 + 1 .38 where E 1 , E 2 and E eq aretheYoung'smodulusfortheball,race,andequivalent materialrespectively.Theseequivalentmaterialpropertiesallowustotransformthetwo bodyproblemtoasinglebodyproblemasshowningure2.7,nottoscale. Figure2.7. Twobodyproblemleft,andequivalentsinglebodyproblemright,notto scale. Notethatthecurvatureoftherollingelementistakenintoaccountviathelm thicknessequation.Thecurvatureoftheracecanbetakenintoaccountinmultiplewaysas willbediscussedinchapterV. Wecannowdeneourdomainandboundaryconditions.TheReynoldsequationwill besolvedonatwo-dimensionalboundaryofourthree-dimensionalelasticdomain.The uidpressurewillactasaboundaryconditiontothesoliddomain,andthesolid 28 PAGE 41 deformationinthe z directioneectsthelmthicknessequation,leadingtoafullycoupled setofequations. Thelmthicknessanduidpressureinthecontactregionisthesolutiontoacoupled elasto-hydrodynamicsystemofequationsasfollows: 1.Reynoldsequation: r " r P + @ H @X =0 ,on f 2.SimpliedCauchymomentumequation: r =0 ,on s 3.Hooke'sLaw: = C : ,on s 4.Denitionofstrain: = 1 2 h r u T + r u i ,on s 5.Loadequation: RR p d f = F ,Globally 6.Filmthicknessequation: h = x 2 2 R x + y 2 2 R y + w + h 0 ,Globally 7.Anyrheologicalmodelfromtable2.2,on f 8.Anyequationofstatefromtable2.3,on f wheretheunknownsareuidpressure, P ,solidstress, ,soliddisplacement/strain, u;" ,the lmthicknessconstant h 0 , f and s aretheuidandsoliddomains,respectively,and F istheappliedload.Thecomputationaldomain,showningure2.8,isarectangularprism representingahalfspace.Thesizehasbeenshowntobelargeenoughtonotimpactthe deformationinthelubricationdomain[34].Theboundaryconditionsforthesolidproblem: 1.Fixed: u =0 ,onthesurfacedenotedbyBC1 2.Symmetry: r u n =0 ,onthesurfacedenotedbyBC2 29 PAGE 42 3.Appliedboundarypressure: P on f Allothersurfacesofthesoliddomainarefreetodeformandhavenoappliedloads.The highlightedsubdomainisthelubricatedsurfacewheretheReynoldsequationissolved, knownasthelubricationdomainwhichwillbedenotedby f .Theremaininggeometric features,suchasthesemicircle,areformeshcontrolandpost-processing.Boundary conditionstoReynoldsequationareshowningure2.8,andcanbesummarizedas: 1. P =0 onthe X and + Y edgesof f 2. r P n =0 onthe )]TJ/F21 11.9552 Tf 9.299 0 Td [(Y edgeof f Notethattheedgeofthelubricationdomainthatsharesanedgewiththesolidsymmetry boundaryalsohasasymmetryboundarycondition.Theoriginislocatedatthecenterof thesemicircleonthelubricationdomainasshowningure2.8withthe z )]TJ/F17 11.9552 Tf 9.299 0 Td [(axiscomingout ofthepage. 2.4WeakFormandStabilization WederivetheweakformofReynoldsequationinordertosolveitwiththenite elementmethod.MultiplyingEq.2.35byatestfunction, ,andintegratingoverthe lubricationdomain, f : )]TJ/F27 11.9552 Tf 11.955 16.272 Td [(Z r " r P d f + Z @ H @X d f =0 .39 Integratingbypartstoreducetheorderofthederivativeon P : )]TJ/F21 11.9552 Tf 11.955 0 Td [( " r P n j @ f + Z " r r P d f + Z @ H @X d f =0 .40 30 PAGE 43 Figure2.8. DomainandboundaryconditionsfortheEHLproblem.Thethree-dimensional blockisthesoliddomain, s ,thehighlightedtwo-dimensionaldomainistheuiddomain, f .ThedomainsizesarescaledbytheHertziancontactradius, a . where n istheoutwardpointingvectornormaltotheboundary.Integratingbypartson thelastterm: )]TJ/F21 11.9552 Tf 11.955 0 Td [( " r P n j @ f + Z " r r P d f + Hn x j @ f )]TJ/F27 11.9552 Tf 11.291 16.272 Td [(Z H @ @X d f =0 .41 Thetestfunction, ,isdenedtobezeroontheboundarieseliminatingtheboundary terms. Z " r P r P d f )]TJ/F27 11.9552 Tf 11.955 16.272 Td [(Z H @ P @X d f =0 .42 WearriveattheweakformofReynoldsequation.TheReynoldsequationpredictsthat convergingsurfaceswillproduceapositivepressuregradient,whiledivergingsurfaceswill producenegativepressures.Ifthedivergingsurfacesaresteepenough,theliquidpressure willdropbelowthevaporpressureandtheliquidwillcavitate.Asdiscussedearlierin 31 PAGE 44 chapterI,thereareseveralcavitationmodelstohandlethisrangingincomplexityand accuracy.Sinceoperatingpressuresinhighlyloadedpointcontactsareontheorderof1 GPa,andtypicalvaporpressuresformineraloilsareontheorderof0.1kPa,asimple penaltymethodwillsuce.Theweakformaugmentedwiththepressurepenaltymethodis asfollows: Z " r P r P d f )]TJ/F27 11.9552 Tf 11.955 16.272 Td [(Z H @ P @X d f + p f min P; 0 P =0 .43 p f isaverylargenumberand,thereforewhenthepressureisnegativethelasttermis severalordersofmagnitudelargerthantheothers.Hence,Eq.2.43simpliesto P u 0 atanynodewherethepressureisnegative,andforcesnegativepressurestobeontheorder of R " r P r P d f )]TJ/F28 7.9701 Tf 6.587 6.42 Td [(R H @ P @X d f p f .Astudyontheeectof p f onthesolutionandthemagnitude ofthenegativepressureispresentedinsection2.6. Atverylargeloadstheviscositycanincreaseseveralordersofmagnitudeandthelm thicknesscanbeverysmall,andhence, " n 1 .StandardGalerkinniteelementschemes failsinconvectiondominatedsystems,suchastheReynoldsequationwhen " n 1 . ThereforeitisprudenttoincludestreamwiseupwindPetrovGalerkin/Galerkinleast squaresSUPG/GLSandisotropicdiusionIDstabilizationschemesfrom[34]. SUPG/GLSstabilizationmethodsareconsistent,whileIDisinconstant.Eventhough[34] reportedsuccessusingthismethod,itwasnotabletobereplicatedinthiswork.Appendix ?? demonstratesasuccessfulimplementationofSUPG/GLSforalinearconvection, diusion,sourceproblem.Theusefulnessofthiscodeiscompromisedinsomeofthe parameterspacewithoutstabilizationaswillbeseeninsection2.8.Alsoshowninsection istheeectofIDonaccuracy. 2.5MeshConvergence ThenumericalsolutiontothestandardGalerkinniteelementrepresentationof Reynoldsequationfailsathighload,aswillbeshownlaterinsection2.8.Thereforecare 32 PAGE 45 mustbetakentoensurethetermsintheReynoldsequationdonotproduceunnecessary perturbations.Expandingthe r h 3 12 r p termtoincludethedenitionoflmthickness showstheReynoldsequationdependsonthegradientofthecubeofthe z componentof thesoliddisplacement, w : r 0 @ x 2 + y 2 2 R + w + h 0 3 12 r p 1 A .Wemustthereforeuseatleast quadraticsolidelementstoensurethistermiscontinuous. Themeshrequirementsforthesolidproblemdonotnecessarilycoincidewiththeuid problem.Thisisproblematicastheuiddomainsharesboundaryelementswiththesolid domain.Therefore,auniqueapproachtomeshconvergencemustbetakengiventhe multi-physicalnatureoftheproblem.Previousauthors[34]havediscoveredthatvery coarseelementscanbeusedforthesolidproblem,especiallyfarawaythecontactzone. TheextremepressuregradientsrequireaverynemeshintheHertziancontactarea.The discrepancyinmeshsizeinthecontactzonecanbecorrectedbyusinghigherorder elementsfortheuidproblemthanthesolidproblem.Thereforethemeshingstrategyisas follows: 1.UsingHertziancontactasatestproblem:decreasemeshsizenearandfarfromthe contactzoneuntilwegetgoodagreementwiththeHertziananalyticsolution.Repeat withlinear,quadratic,andcubicelements. 2.Withanappropriatesolidmesh,renetheuidmeshuntilthe ` 2 normoftheuid lmpressureconvergestoanappropriatelevel. 3.CheckthatHertzerror,denedbelow,issmallerthan 1 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 . SolidProblemConvergence Theeectsofmeshsizeandordernearandfarfromthecontactzonewiththe Hertziancontacttestproblemarestudied.Figure2.9showsanexamplemeshforthe Hertziancontactproblem.Therearetwomeshzones,onenearthecontactzoneandthe 33 PAGE 46 remainderofthegeometry. Thesurfacedeformationinthelubricationzoneistheonlycouplingvariablefromthe solidproblem.Thenumericallycalculatedindentationdepth, d Comsol ; atthecenterpointof thecontactzonewillbecomparedtotheanalyticallycalculatedHertzianindentation depth, d Hertz .Theerrorinthenumericalsimulationcanthenbedenedas e Hertz d Hertz )]TJ/F22 7.9701 Tf 6.586 0 Td [(d Comsol d Hertz ,whichwillbereferredtoastheHertzerror. Figure2.9. Meshbeforeandafterconvergencestudy.aBefore,isometricviewbafter, isometricviewcbefore,lubricationzonedafter,lubricationzone. Thenearzoneisrenedfrom =a =6 to =a =0 : 333 withlinear,quadratic,andcubic elements.Thefarmeshzoneisrepresentedwithacoarsegure2.9aandarenedmesh 2.9b.TheHertzerrorisplottedasafunctionofthemeshsizeingure2.10left,the Hertzerrorisplottedasafunctionofthenumberofdegreesoffreedomisshowningure 34 PAGE 47 2.10right. Figure2.10. HandPconvergencefortheHertziancontactproblem.Hertzerrorasa functionofmeshsizeformultiplecasesleft,andHertzerrorasafunctionofnumberof degreesoffreedomrightforthesamecases. Figure2.10rightshowsthatthequadraticrenedmeshyieldssub1%errorat under15,000degreesoffreedom.Thispointcorrespondstoa D /a =1 : 0 .Theuidproblem willbesolvedonthesurfaceofthismesh,thereforewemustnowrenethesurfacemesh untiltheuidproblemconverges. FluidProblemConvergence Itwasimmediatelyclearthat D /a =1 : 0 wasfartoolargefortheuidsolution.The highestorderuidelementsthorderwerechosentomaximizetheconvergedmeshsize. Table2.7showstherelative ` 2 normerror,equation2.44,forpressureevaluatedat22 pointswithinoneHertzianradiustocomparemeshes.Columntwoofthistableshouldbe interpretedastherelativeerror, e ,betweenthemeshsizestatedandameshofhalfthat size. 35 PAGE 48 e = 22 X i =1 P 4 i )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 4 = 2 i 2 .44 Table2.7: ` 2 normerror,calculatedwithEq.2.44,forseveralmeshsizes MeshSize D /a ` 2 NormRelativeError DOF 1 2 2 : 222 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 16,264 1 4 9 : 760 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 51,197 1 8 2 : 220 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 93,663 1 16 1 : 901 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 195,909 1 32 3 : 348 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 440,669 1 64 1,237,592 Newtoniterationsareperformeduntilthe ` 2 normoftherelativeresidualreaches 1 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 : Newtoniterationsareterminatedwhentherelativeresidualbetweeniterations islessthan 1 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 .Usingameshsizeof 1 16 providesaccuracyontheorderoftheerror estimate,butitisverycostly.Ameshsizeof 1 8 providessucientaccuracyatlessthanhalf thenumberofdegreesoffreedom,whichsolvesroughly4timesfaster.Ameshsizeof 1 8 on thelubricationsurfacewith7thorderelements,quadraticelementsofsize 1 orsmallerin thenearzoneandtherenedmeshinthefarzoneareusedinthisworkunlessotherwise specied.Thismeshcanbeseenin2.9band2.9d.TheHertzerrorisequalto 0 : 00082 when usingthismesh. 2.6PressurePenaltyConvergence Inthissectionwestudytheeectofthepressurepenaltyfactor, p f asshowninEq. 2.43ontheminimumpressureinthelubricationdomain.Table2.8showstheminimum pressureforvaluesof p f rangingfrom 1 10 4 to 1 10 7 . Thepressuresneverreachzero,buttheminimumpressureinthelubricationdomain 36 PAGE 49 Table2.8: Minimumlmpressureasforseveralvaluesofthepenaltyfactor. p f min P 1 10 4 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 13 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 1 10 5 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 14 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 1 10 6 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 20 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 1 10 7 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 50 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 decreaseswithanincreasingpenaltyfactor.Theminimumpressureisinsignicantwitha penaltyfactorgreaterthan 1 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 .Penaltyfactorsgreaterthan 1 10 7 cancause convergenceissues.Apenaltyfactorof 1 10 6 isusedinthiswork. 2.7ModelValidation Therearenorelevantanalyticsolutionstovalidatethiscodeduetothehighly nonlinearandcouplednatureofthisproblem.Thereforewevalidateourmodelwithresults fromothernumericalmodels.Figure2.11leftcomparesdimensionlesspressureandlm thicknesstoasimilarniteelementmodelasdescribedin[51].Thisdatawasobtainedby digitallyanalyzingthegurefromthepaperwhichhassomeerrorassociatedwithit. Figure2.11rightcomparesthedimensionlesspressureandlmthicknessinthe X and Y directionsthroughthecenterofthecontacttoanitedierencecodewrittenbyZhengShi ofSeagate. Figure2.11. Comparisonofcurrentmodeltoacomparablemodelfromtheliterature[51] left.Comparisonofcurrentmodeltoanitedierencemodel[60]right. Comparingthiscodetootherswiththesameassumptionsdoesnottellusmuchaboutthe 37 PAGE 50 validityofourassumptions,orgiveusinsightintothephysics,butitdoesincrease condencethatwehaveavalidsolutiontoawellposedproblemwithnobugsorerrors. 2.8EHLResults ThissectionpresentsresultsfortheEHLmodel.Asolutionconsistsofauidlm pressureprole,theseparationdistancebetweentheballandrace,andthecombined deformationsoftheballandrace.InEHLthefollowingsixparametersdeterminethelm thicknessandpressureproles: F , E 0 , R , 0 , u m ,and .TheBuckinghamp theoremstates thenumberofdimensionlessgroupstodescribethesolutionisequaltothenumberof parametersthatgovernthesolutionminusthenumberofdimensions,whichisthreemass, length,time.FortheEHLproblemtherearesixparametersandthreedimensionsleaving threedimensionlessparameters.HamrockandDowson[23]foundtheseparameterstobe theload,material,andspeedparameterrespectivelyasshownineq.2.45. W HD F E 0 R 2 G HD E 0 U HD 0 u m E 0 R .45 Anothermethodofdimensionalanalysis,theoptimumsimilarityanalysisrstusedby Moes,[47],whichusesacomputerprogramtoanalyzetheequationsandparametersina systemtocalculatetheoptimumdimensionlessparametersyieldsresultsthatarea combinationoftheHamrockandDowsonparametersasshownineq.2.46. M W HD U HD )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 = 4 = F E 0 R 2 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(2 0 u m E 0 R )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 = 4 L G HD U HD 1 = 4 = E 0 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(2 0 u m E 0 R 1 = 4 .46 AnycombinationofparametersontherighthandsideofEq.2.46thatyieldthesame M and L valueswillhavethesamesolution,assumingtherheologicalmodelandequationof stateisheldconstant.Figure2.12detailstheareasonthe M L planewhereweobtaina convergedresult.Greenshowswhereasmoothconvergedresultwasfound.Yellow representsaroughconvergedresult,asshowningure2.14.Anywhereinthered,the 38 PAGE 51 Figure2.12. Convergenceareaofthecode.Green=smoothconvergedresultYellow= perturbedconvergedresult,Red=Noconvergence. convectiontermdominatesandthestandardGalerkinformulationfailsasdescribedby Habchi[34]. Manysolutionswereobtainedatdierent M and L valuesbychoosinga F anda small obtainingasolution,andslowlyincreasing untilthesolverfailstoconverge. Figure2.13showsthedimensionlesspressureprole P X at Y =0 forseveralvaluesof M and L . Solutionsareeasilyobtainableatlow M and L ,aseither M or L increases,abetter initialconditionmustbespeciedforconvergence.Betterinitialconditionsareobtainedby takingsmallerstepsin andobtainingintermediatesolutions.Stepsassmallas 0 : 0001 max weretaken.Forvaluesof M and L greaterthan10,000and13,respectively,the solutionbecomesunstableandlargeoscillationsinpressureoccur.Thesesolutionsstill converge,evenwithextremepressureoscillationsasseeningure2.14,buttheiraccuracy andusefulnessarequestionable. Solutionsatlow M showanonzeropressuregradientattheentrancetothe lubricationdomain.Thisindicatesthatthesizeofthenumericallubricationdomainis havinganeectonthesolution.Thisproblemisignoredassolutionsatlow M arenotof 39 PAGE 52 interesttoengineers.Thesesolutionsareverylowload,andtendtohavethickuidlms thatdonotalwayssatisfytheReynoldsequationassumptions.Furthermore,verythick lmstendtohaveverylowwearrateswhichisidealinpractice,butuninteresting computationally.Thereforewedonotincreasethesizeofthelubricationdomain,and hencethetotalsizeoftheproblem,toobtainmoreaccuratesolutionsforaparameter spacethatisnotofinterestandthatdoesnotalwayssatisfyourassumptions. Figure2.13. PressureprolesformultiplevalesofMandL. EectofIsotropicDiusion Convergencebecomesproblematicasthepressureoscillationsgrow.Isotropic diusionIDwasimplementedtoreducethepressureoscillations.IDisaninconsistent 40 PAGE 53 Figure2.14. Examplesolutionwithextremepressureoscillations. stabilizationmethodtheeectofitsmagnitudemustbestudiedbyincreasingthePID parameteruntilthesolutionissignicantlyeected.Figure2.15showstheeectof isotropicdiusiononthepressureprole.The M and L valueswerechosenbecausethis solutionisalreadysmooth,andithasalargeandsteeppressurespike.Thisallowsusto seethemaximumeectofisotropicdiusionontheaccuracy.Themajorityofthepressure proleisunaected,butthepressurespikeisspreadoutanditsmagnitudeisdecreased. Theminimumlmthicknessincreasedfrom0.6489atPID=0to0.6515atPID=1,whichis a 0 : 4 increase.Thisisaccompaniedbya 5 : 1 decreaseintheheightofthepressurespike. OveralltheeectofIDisquitesmall.ItisnotrecommendedtogoabovePID=1. Figure2.16showstheeectofisotropicdiusiononanunstablesolution. PressureSpikeandExitContraction ClassicEHLsolutionsdemonstrateapressurespikeandreductioninlmthicknessat theexitofthecontact.Thereasonforthisspikeandcontractionisrarelydiscussed,but terselyexplainedasaresultofthepressureviscositycoecient, .Thissectionaimsata moreindepthexplanation.Theeectofthepressureviscositycoecientisexaminedrst ingure1. Thepressureviscositycoecienthasapronouncedeectonlmthickness.Increasing 41 PAGE 54 Figure2.15. EectofIsotropicDiusionstabilizationonthepressureprole,expanded viewofthepressurespikeinset. Figure2.16. Pressureprolesoftheunstabilizedleftandstabilizedright,PID=1for anEHLcontact M =8064and L =9 : 3 . theviscosityseveralordersofmagnitudeinthecontactregionprovidessignicantlymore lift.Surprisingly,thecasewithnopressureviscositycoecientshowsaexitcontractionbut nopressurespike.Thepressurespikemustnotbecausedbytheexitcontractionifthe pressurespikeisabsentinthepresenceoftheexitcontraction.Theexitcontractionseems tobecausedbythedierencebetweentheuidpressureandtheequivalentHertzian contactpressure,asshowbelowgureunderconstruction.Thisisrelatedtotheno cavitationcondition. 42 PAGE 55 Figure2.17. Pressureandlmthicknessforseveralvaluesofpressureviscositycoecient. Notethepressurespikeforlargerpressureviscositycoecientsandtheexitcontractionin lmthickness. Thepressureviscositycoecientdoesindeedcausethepressurespike,buthow?To answerthiswemustexaminewhatishappeningattheentranceandexitofthecontact.At theentrancethemotionoftheballandraceisdrawinguidin,astheuidissqueezedinto thegapthepressureandthereforeviscosityanddensityrise.Thepressuredrivenow counteractstheowduetothemovementoftheraces.Itbecomesharderandharderto owintothecontactregionuntiltheuidreachesthecenter.Afterpassingthecenterof thecontactthepressuredrivenowandtheowduetothemovementoftheballandrace areinthesamedirection.Thepressureisdecreasingrapidlycausingtheviscosityand densitytofallrapidly. EHLModelUsefulnessasDesignTool Theresultsfromthistypeofmodelhavebeenusedtocalculatetheminimumlm thicknessforrollingelementbearingssincethe60's.Thismodeldoesagoodjob determiningwhetherornotthereissolidcontactforagivengeometry,lubricant,andload. Asis,thismodeldoesnotpredicttheamountofsolidcontactorit'slocation.Thismodel alsofailstoaccountfortemperatureeects,andassumesthebearingisahalfspace. 43 PAGE 56 ChapterIIIexpandsthismodeltoincludeastatisticalmodelforsolidsolidcontact. ChapterIVexpandsthemodelfromchapterIIItoincludethermaleects.ChapterVuses themodelfromchapterIIItostudyarealworldbearinggeometry. 44 PAGE 57 CHAPTERIII MIXED-ELASTO-HYDRODYNAMICLUBRICATIONMEHL Asperitycontactoccurswhenthelmthicknessisontheorderoftheasperitysize.When asperitiesareincontact,forceistransmittedbetweenthetwosurfacesbydirectsolidto solidcontactinsteadofhydrodynamicforces.Manytheorizethatthereisamonolayerof lubricanttrappedbetweenthecontactingsurfacesthatmodifyfrictionandreducewear.In thischapterwereviewelasticandplasticasperitycontactmodelsandimplementtheminto theEHLsolver.Thisallowsthemodeltomakeaccuratepredictionsthroughevery lubricationregimedescribedinchapterI.Thisisalsoacriticalstepincalculating temperaturerisesinlubricatedcontactsasfrictionalheatingisasubstantialcomponentto theoverallheatgeneration,aswillbeseeninchapterIV. Directcomputationofasperitycontactsforfullscalebearingsisextremelycomputationally expensiveduetothenumberofasperitycontactswithinthelubricatedcontact.The numberofpossibleasperitycontactscanbeestimatedasthenumberofasperitiesperunit area,alsoknownastheasperitydensity, N ,timeshalftheHertziancontactarea, 1 = 2 a 2 whichaccountsfortheplaneofsymmetrydiscussedinchapterII.Table3.1showscontact radiiandtheestimatednumberofcontacts, N C ,fora 1 cm radiussteelballcontactinga platewithacombinedmodulusof 100 GPa forseveraldierentappliedloads, F .Atypical valueforasperitydensity, N ,forhighlypolishedbearingcomponentsis 4 1 m 2 accordingto spectralanddeterministicmeasurementsreportedin[40]. Typically,atleast10surfaceelementsperasperityarerequiredforconvergenceinthe contactproblem.Thenalmeshpresentedingure2.9had1320elementsrepresentingthe 45 PAGE 58 Table3.1: Contactradius, a ,asafunctionofappliedload, F ,numberofelementsrequired forthegivenasperitydensity, N . F [ N ] a = )]TJ/F19 7.9701 Tf 6.675 -4.976 Td [(3 FR 4 Ee 1 3 m N C 0 : 1 19 : 5 2400 1 42 : 2 11,000 10 91 240,000 100 195 1,100,000 lubricationsurface.Themeshdensitywouldhavetobedoubledtogetaroughestimateof eventheeasiestcase, F =0 : 1 N .Themeshwouldquicklygetoutofhand,requiringtwo hundredtimesthestandardmeshdensityforatypicalcaseof F =10 N .Astatistical representationofsurfaceroughnessisthereforepreferredtoexplicitlymodelingeach asperity.Thefollowingsectionsoutlineapproachesthatmodelasperitycontact statistically,usingdieringassumptionsabouthowtheasperitiesdeform. Commonassumptionsofthefollowingmodels: 1.AsperityheightdistributionofeachsurfaceisGaussian 2.Tworoughsurfacescanberepresentedbyoneroughsurfacewithacompositesurface roughnessandonesmoothsurface.Thecompositesurfaceroughnessisequalto S q = p 2 1 + 2 2 ,where 1 and 2 arethestandarddeviationsoftheGaussianasperity heightdistributionsforthetworoughsurfaces. 3.Theasperitydeformationiseitherfullyelasticorfullyplastic. Theprobabilityofencounteringanasperityofsize z awayfromthemeansurfaceis: z = 1 p 2 2 exp )]TJ/F21 11.9552 Tf 13.977 8.088 Td [(z 2 2 2 .1 Where isstandarddeviationofthenormallydistributed z .Givenapopulationof n 46 PAGE 59 measurementsofdistancefromthemeansurface, z i ,ofaroughnormallydistributed surface,thedenitionofstandarddeviationis: = v u u t 1 n n X i =1 z i )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 .2 wherethemeanofthisdistribution, ,iszerobydenition.Thisperfectlycoincideswith thecommonlyuseddenitionofsurfaceroughness, R q ,alsoknownas R RMS : R q = R RMS = v u u t 1 n n X i =1 z i 2 .3 thereforeifitisassumedthatasurfacehasanormallydistributedasperityheightwecan usethecommonlyreported R q = R RMS roughnessparametersinthefollowingmodels. R q isrelatedtothemorecommonlyused R a surfaceroughnessparameterby R q = p = 2 R a ; where R a isdenedas: R a = 1 n n X i =1 j z i j 2 .4 Surfacecharacterizationstudieshaveshownthatmanycommonengineeringsurfaces havenormallydistributedroughnesses[31,8]. 3.1ElasticAsperityDeformation TheGreenwoodWilliamsonmodel,[30],istheoriginalroughcontactmodel.Itassumes thateachasperitycanbeapproximatedbyasphericalcapofradius ,thecontactbetween eachasperityandthesmoothsurfaceisHertzian,andthatasperitydeformationiselastic. RearrangingtheHertzianpointcontactexpressionsfoundintable1.1intermsofthe indentationdepth, d ,thecontactareaandtheloadare: 47 PAGE 60 a i = dF i = 4 3 E 1 2 d 3 2 i .5 Theprobabilityofasperityofdistance s = z=S q fromthenominaldimensionlesslm thicknessof H S q = h=S q contactingtheatsurfaceiswrittenas: prob )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s>H S q = Z 1 H S q s d s .6 where z istheprobabilitydistributionofasperityheights,assumedtobeGaussian. Assumingthesmoothsurfacecontactsthehighestpointoftheasperity,whichisreasonable iftheslopeofthesmoothsurfaceneartheasperityissmall,theindentationdepthwillbe z )]TJ/F21 11.9552 Tf 11.955 0 Td [(h andtheresultingcontactareaandloadis: A i = s i )]TJ/F21 11.9552 Tf 11.955 0 Td [(H F i = 2 3 E 1 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s i )]TJ/F21 11.9552 Tf 11.955 0 Td [(H S q 3 2 .7 forall z>u .Theexpectedvalueforeachoftheseexpressionsis: A i = S q R 1 H S q )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s )]TJ/F21 11.9552 Tf 11.956 0 Td [(H S q s d s F i = 2 3 E 1 2 S q R 1 H S q )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H S q 3 2 s d s .8 deningeachexpressiononadierentialareawheretheseparationbetweennominal surfacesis h ,andtheasperitydensityperunitareaisN: dA = NS q da R 1 H S q )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H S q s d s dF = 2 3 NE 1 2 S q da R 1 H S q )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H S q 3 2 s d s .9 Wenowhaveasetofexpressionsthatcanbeintegratedtoprovideacontinuousfunction ofcontactareaandcontactforceforelasticallydeformingasperitycontactasafunctionof nominalsurfaceseparation.Assumingthecontactareaandpressureareconstantineach dierentialelement,anddeningadimensionlessasperityheight,Eq.3.9canbewritten as: 48 PAGE 61 a = NS q f 1 p c = 2 3 f 3 2 .10 Where NE 1 2 S 3 2 q ,theintegralsoftheform f n )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(H S q = R 1 H S q )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H S q n s ds must beevaluatednumericallywhen n isanoninteger.Selectvaluesof f 0 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(H S q , f 1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(H S q , f 3 = 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(H S q aswellasrelevantanalyticalsolutionsandMATLABcodethatperforms numericalintegrationforthenonintegervaluesof n aregiveninAppendixB. Equation3.10determinesthecontactpressureandthecontactareaasafunctionofhow closethenominalsurfacesare.Thecombinedsurfaceroughnessparameter, S q ; determines howclosethesurfaceshavetobebeforesolidcontactoccurs.Theparameter determines themagnitudeofcontactpressure.Typicalvaluesare N =5 10 6 1 mm 2 ;E =200 GPa , S q =0 : 01 m ,and =1000 S q .Usingthesevalues,thecontactpressureat H S q =0 is p c =0 : 89 GPa . Notethatthenominalsurfacedistance H canbenegativeiftheasperitiesare signicantlydeformed.NegativelmthicknessposeaproblemtosolutionstotheEHL problem,thisisdiscussedandremediedinsection3.4. 3.2PlasticAsperityDeformation BowdenandTaylor[11]derivedcontactmodelassumingeachasperitywasinpurelyplastic deformation.Startingagainwiththeprobabilityofadimensionlessasperityofheight s = z=S q contactingatadimensionlessseparationdistanceof H = h=S q ,andnotingthat thisisalsotheratioofcontactareatototalarea, a c : prob )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s>H S q = a c = Z 1 H S q s ds .11 whichanintegralofthetype f 0 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(H S q ,thesolutionof f 0 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(H S q isdiscussedinAppendixB andispresentedhere: 49 PAGE 62 f 0 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(H S q = a c = 1 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(erf H S q p 2 .12 Ifallasperitiesareundergoingpurelyplasticdeformation,thecontactpressureisby denitionthehardnessofthesofterofthetwomaterials,denotedas H a .Thecontact pressureisthen: p c = H a 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(erf H S q p 2 .13 Thisisatwoparametermodel,thecombinedsurfaceroughness, S q ,andthesofterofthe twomaterialshardness, H S q .Thesurfaceroughnessparameterdictateshowclosethe surfaceshavetobebeforesolidcontactoccurs.Thecontactpressureisthehardnessofthe softermaterialbydenitionofplasticdeformation.Thecontactpressureinanelementis thedimensionlesscontactareamultipliedbythehardness.Thelimitingcaseathigh load/lowspeedis a c =1 and p c = H a .Atypicalvalueforthehardnessofrollingelement bearingsis H a =6 : 8 GPa .Itshouldbenotedthatthecontactpressureinthismodelonly dependsonmaterialpropertiesofonematerial,whereasintheGreenwoodWilliamson modelthecontactpressuredependsonbothcontactingmaterialselasticproperties.Inthe plasticdeformationmodel,noassumptionismadeabouttheasperitiesshapeandthe contactareaofasingleasperityisconstantasitdeforms.Thisisincontrasttotheelastic modelthatassumeseachasperityhasahemisphericalcap,wherethecontactareaofa singleasperityincreasesasthecontactingsurfacesgetcloser. 3.3ComparisonofPlastictoElasticAsperityDeformation Figure3.1showsthecontactpressureasafunctionofnominalapproachdistanceforthe plasticandelasticasperitycontactmodels.Thereisnoobviouschoicefor sincethe parameterisacombinationofmaterialpropertiesandsurfacetopography,whichcanvary greatlydependingonapplication.Arangeoftypicalvaluesfor isplottedingure3.1. 50 PAGE 63 Sincethecontactpressurecannotexceedthehardnessofthesoftermaterialtheelastic contactpressureistruncatedatthehardness,whichinthiscaseis 6 : 8 GPa . Figure3.1. Elasticandplasticasperitycontactpressureasafunctionofnominalapproach distancefor =1 GPa , =25 GPa , =60 GPa Figure3.2showsthedimensionlesscontactpressure, P c p c p h ,forboththeelasticand plasticcontactmodelsatvariousloadsasafunctionofthe x coordinatenormalizedwith theHertziancontactradius.Bothcontactmodelsexhibitlowerpeakpressuresandlarger contactareasthantheHertzsolution.Thiseectdiminishesastheloadincreases.The contactradiusandcontactpressureapproachesHertzianwithincreasingload,asmoreand moreasperitiesaredrawnintothecontact.Thisistrueforbothcontactmodels.Atloads above 1 N thetwocontactmodelsarenearlyindistinguishable.Thepeakpressuresdiers by3.4%at F =0 : 01 N ,whichisaverylowloadforMEHLcontacts. Thetruncatedelasticdeformationmodelgivesamorecompletedescriptionofthe contact,butrequiresarigorousdescriptionofthesurfacetopographyandthematerial propertiesnearthesurfacetocomputethefourparametersrequired.Theplasticcontact modelontheotherhandismuchsimpler,butrequiresonlytwoparametersthatare commonlyreportedforengineeringsurfaces.Giventhatthetwomodelsyieldsimilar 51 PAGE 64 Figure3.2. DimensionlesscontactpressurefortheelasticsolidandplasticdashedasperitycontactmodelsforvariousloadscomparedtotheHertziancontactpressureprole. Thecontactmodelparametersare =2 : 36 GPa and H a =6 : 8 GPa fortheelasticand plasticmodelrespectively. results,andintheabsenceofdetailedexperimentaldatathatwouldallowusto dierentiatebetweenthesetwomodels,weusetheplasticasperitycontactmodeldueto it'ssimplicity.Thisgreatlyreducesthecomplexityoftheupcomingdimensionalanalysis. 3.4ImplementationintoEHLModel Theimplementationofeithersolidcontactmodelspreviouslydiscussedrequires: 1.Arepresentationofroughnessinthelmthicknessequation,6 2.Anadditionaltermintheloadequationtoaccountfortheloadcarriedbyasperity contact,5 3.Modiedboundaryloadon s toaccountforthesolidcontactpressure,2 52 PAGE 65 Thesethreemodicationsarediscussedinthissection,andthemodiedformulationsis presented. FilmThicknessEquation Thedenitionoflmthicknessrequiresfurtherexaminationforroughcontact.Theexplicit andnominallmthicknessateachpointinthelmis: h explicit = F x;y + w x;y + h 0 + R x;y h nominal = F x;y + w x;y + h 0 .14 where F x;y isafunctionrepresentingthecurvatureofthecontact, w x;y istheelastic deformationinthedirectionofthelmthickness, h 0 istheapproachdistanceofthe objects,and R x;y isafunctionrepresentingthedeviationofthesurfaceaboutthemean duetosurfaceroughness.Sincewearenotexplicitlymodelingthedeformationofthe asperities w willnotcompensatefor R whenthelmthicknessisontheorderofthe asperityheight.Thelmthicknesswillbenegativeinthatcase,whichisanonphysical result.Onemightexpectthatthisisremediedbytrackingthedistancebetweentheat surfaceandthemeanoftheroughsurface,howeverthesameproblemoccursasshownin gure3.3.Thereforeamodieddenitionoflmthicknessthatisalwaysgreaterthanor equaltozeroisrequired. ChengweiandLinquing[15]didexactlythis.Aneweectivelmthickness, h t ,isdened asthevolumeoflubricantdividedbythetotalsurfacearea.Thisconservesthetotal volumeoflubricantasthecontactareaincreases.Italsocannotbenegativebydenition asneitherthevolumeoflubricantnorthesurfaceareacanbenegative.Foraninnitesimal approach dh ,thecorrespondingchangein dh t canbewrittenas: dh t = )]TJ/F21 11.9552 Tf 11.956 0 Td [(a c d h .15 53 PAGE 66 Figure3.3. Explicit,nominal,andmodiedlmthicknessesforaroughcontactwithdecreasing h 0 . where a c isthefractionofareaexperiencingsolid-solidcontactand 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(a c isthewettedor lubricatedareafraction.RearrangingandassumingaGaussiandistributionofasperitysize: dh t dh =1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(a c = Z h 1 p 2 S 2 q exp )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(s 2 2 d s .16 Integratingtwice: h t S q = 1 2 h S q 1+ erf 1 p 2 h S q + 1 p 2 exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 h S q 2 ! .17 Plottingthedimensionlessmodiedlmthickness, h t =S q ,asafunctionofthedimesionless explicitlmthickness, h=S q ,showsthatforlmsthickerthan h=S q > 3 , h t asymptotically approaches h ,andfor h=S q < )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 , h t asymptoticallyapproacheszero.Thedimensionless modiedlmthickness, H t h t R a 2 ,canbedirectlysubstitutedintoReynoldsequation,eq 2.35.Therewillbenochangeforthicklms,andforthinlmsexperiencingsolidcontact 54 PAGE 67 theequationwillrepresenttheaverageowratesoverareascontainingmanyasperities. Figure3.4. Modiedlmthicknessnormalizedwiththecombinedsurfaceroughnessparameter, h t =S q ,asafunctionofnominallmthicknessnormalizedwiththecombinedsurface roughnessparameter, h=S q . 3.5ResultsoftheElasto-HydrodnamicWithAsperityThePlasticContact Model InthissectionwepresentresultsfortheMEHLproblemwiththeplasticasperitycontact model.Withthenewphysicsmodelcoupledin,thestandardEHLdimensionless parametersarenolongersucient.Anewdimensionlessanalysisisperformed,uncovering twodimensionlessparametersthatwhencoupledwiththestandardMoe'sorHamrockand DowsonEHLparameters,completelydescribeaMEHLsolution.Thenalresultofthis chapteristheabilitytocalculatetheloadcarriedbytheuidandsolidpressures 55 PAGE 68 respectivelyasafunctionoftheaforementioneddimensionlessparameters.Anexpression fortheminimumandcentrallmthicknessasafunctionofthedimensionlessparameters couldalsobecalculated,butitwouldbeoflimiteduseasthedenitionforminimumlm thicknessforroughcontactsisnotexplicit. Alsowiththismodel,aterseestimateofrollingfrictioncanbecalculated.Thetotal rollingfrictionisacombinationofthefollowingeects: 1.Coulombfrictionontherolling/slidingcontactduetosolidcontact,ignoring micro-slip. 2.Viscousdragontherollingelement 3.Pushingapooloflubricantinfrontoftherollingelement andaretrivialtocalculateinourmodel,howeverrequiresanassumptionof howthickthelubricantisontheballandrace.Additionally,theReynoldsequation assumptionsbreakdownastheoillmgetsthickfarfromthecontactcenter.Ourmodel doesnotaccountfor,howevermanybearingsaredesignedtominimizebylimiting theamountofoilinthebearingduringproduction. MEHLResultsforSeveralParameters Lifto Figure3.6showstheuidandsolidcontactpressurecontoursformeanentrainment velocitiesof u m =1 m=s downto u m =1 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 m=s .Fluidentersfromtheleftandis ejectedtotheright.As u m decreasestheloadcarriedbytheuidlmdecreasesandthe loadcarriedbyasperitycontactincreases.Asperitycontactloadisslightlyconcentratedat theexitcontractioninintermediatecaseswheresometheloadissplitbetweentheuid 56 PAGE 69 u m S q Figure3.5. Transitionfromfulllmlubricationtopleft, u m =1 m=s ,toboundary lubricationbottomright, u m =10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 m=s .Insettablegive M and L valuesforeachsubplot. S q =10 nm , H a =6 : 8 GPa .Dimensionlesssolidcontactpressure, P c ,inred.Dimensionless uidpressure, P ,inblue. 57 PAGE 70 lmandasperitycontact.Thiscanbeobservedincases u m =0 : 1 m=s through u m =0 : 001 m=s ofgure3.6.Thesizeofthehighpressureuidlmdecreasesastheload thelmissupportingdecreases,thismovestheexitcontractionintothecenterofthe contact.Recallthatthelocationsandcauseofthepressurespikeandexitcontractionwere discussedinsubsection. SolidandFluidLoadFractions Figure3.5showstheuidandsolidcontactpressureprolesformanycaseswithdierent u m andcompositesurfaceroughnesses, S q .Atlow u m ,thelmthicknessismuchsmaller thanthecombinedasperityheight, S q ; andtheentiretyoftheloadiscarriedbyasperity contact,ascanbeseeninthebottomrowofgure3.5.Athigh u m thelmthicknessis muchlargerthan S q andtheentiretyoftheloadiscarriedbytheuidlm,ascanbeseen intheupperrowofgure3.5.Infacttheloadratiocarriedbytheuidandsolidpressure, f and c respectively,iscalculatedwithEq.3.18 f = RR pd F c = RR p c d F .18 andplottedasafunctionof u m ingure3.7.Thetransitionbetweenboundaryandfull lmlubricationhappensatlowerspeedsforsmall S q ,andhigherspeedsforlarge S q . DimensionalAnalysis Eachrowofgure3.5hasidentical M and L parameters,howeverthepressureprolesare notequaloncethemeanentrainmentvelocityislowenoughforsolidsolidcontacttooccur. Itisnotsurprisingthattheintroductionofasolidcontactmodelwithtwonewphysical parameters S q ;H a shouldrequireamodieddimensionlessvariableset.InMEHLthe 58 PAGE 71 Figure3.6. Lubricantpressuretopandsolidcontactpressurebottomforseveraldierent valuesof u m . S q =10 nm forallcases. followingeightparametersdeterminethelmthicknessandpressureproles: F , E 0 , R , 0 , u m , , S q ,and H a .TheBuckinghamp theoremstatesthenumberofdimensionlessgroups todescribethesolutionisequaltothenumberofparametersthatgovernthesolution minusthenumberofdimensions.FortheMEHLproblemthereareeightparametersand 59 PAGE 72 Figure3.7. Solid, s ,anduid, f ,loadfractionasafunctionofthemeanentrainment velocity, u m ,forseveralvaluesofasperitysize, S q =[5 nm; 10 nm; 20 nm ] . threespatialdimensionsleavingvedimensionlessparameters.Wecanstillrecoverthe threeHamrockandDowsonparameters,recallequation2.45fromsection2.8,andthe followingtwoadditionalparameters: G = E 0 H a C = S q R .19 Thedimensionlessparameter G relatesthecontactpressuretothethemodulusofthe combinedsurface,whichinasenseistheratioofstinessoftheplasticallydeforming asperitiestotheelasticallydeformingsubstrate.Wecallthisthesecondmaterial parameter,therstbeing G HD .ThedimensionlessparameterCistheratiooftheasperity sizetotheballradius.Recallthatforbothcontactmodelsthesolidcontactpressurewasa functionofthedimensionlessexpression h=S q .Thisseemslikeanobviouschoicefor C ,but wedonotknow h apriori .Valuesforminimumandcentrallmthicknesscanbeobtained fromthemodeldescribedinchapterII.Accuratepowerlawtsoftheform: H = c 1 U c 2 HD G c 3 HD W c 4 HD .20 60 PAGE 73 Figure3.8. Solid, s ,anduid, f ,loadfractionasathedimensionlesssolidcontact parameter, C forseveralvaluesof M . havebecalculatedby[9]forcertainrangesof U HD ;G HD ; and W HD ,butthelmthickness doesnotfollowapowerlawoverlargesrangesof U HD ;G HD ; and W HD .Itwouldtherefore bediculttouseanexpressionlikeEq.3.20todescribe H wellenoughtodenea dimensionlessparameter.Wethereforeusetheaforementioneddenitionof C = S q R . Figures3.8and3.9show f and s asfunctionof C forseveralvaluesof M and L for atypicalballbearingmaterial, G =16 : 5 .Eachlineongures3.8and3.9ismadeupof approximatelyninetysevenuniquesolutions.Thesixcurvescombinedrepresent579 uniquesolutionsthatspan 377 PAGE 74 Figure3.9. Solid, s ,anduid, f ,loadfractionasathedimensionlesssolidcontact parameter, C forseveralvaluesof L . SolidandFluidLoadFractionExpressions ClassicEHLresultsallowanengineertodetermineminimumandcentrallmthicknessas afunctionofthedimensionlessparametersU,G,WorMandL.Thisisusefultoestimate thelubricationregime,butitdoesnotprovideanyinformationpastthat,suchasthesolid anduidloadfractions.Inthissectionwepresent f asafunctionof M , L ,and C viaa multimodeexponentialoftheform: f = n X i =1 a i exp )]TJ/F24 11.9552 Tf 5.479 -9.683 Td [()]TJ/F21 11.9552 Tf 9.299 0 Td [(b i M c i L d i C e i .21 where a i , b i , c i , d i ,and e i arettingconstantsthatwillbedeterminedbyanonlinear regressionttothenumericalresultsfor f presentedingures3.8and3.9viathe'tnlm' functioninMatlab.Thesolidloadfractioncanbecalculatedwiththisexpressionas c =1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( f .Fittingcoecientswerecalculatedfor n =[1 ; 2 ; 3 ; 4 ; 5 ; 6] ,coecientsand ` 2 62 PAGE 75 Figure3.10. Fluidloadfractionsnumericaldatacirclesaswellaststothisdatausing Eq.3.21linesasafunctionofthedimensionlessparameter C forseveralvaluesof L . Onlyeverythirdpointofthenumericaldataisplottedforvisibility. n =6 . normerrorestimatescanbefoundintable3.2.Errordecreaseswitheverysuccessivemode addedwithdiminishingreturnsfor n> 6 .Eq.3.21contains 30 parameterfor n =6 .There isnoconcernforoverttingasthereareatmost 30 parameter n =6 and 579 unique numericalsolutions. FrictionCoecientEstimate Thetotalfrictionalforcesonaballbearinginthemixedlubricationregimeisequaltothe sumoftheuidfrictionplusthesolidcontactfriction.Theuidfrictioncoecient, f ,is theuidfrictionalforcedividedbytheappliedload.Theuidfrictionalforceisequalto theintegratedshearstressonthebearingsurface: f = RR @v x @z d F .22 63 PAGE 76 Table3.2: Coecients a i , b i , c i , d i ,and e i andL2normerrorestimatesfortsofEq.3.21tothe579numericaldatapoints presentedingures3.8and3.9. n a i b i c i d i e i ` 2 1 a 1 =1 : 2051 b 1 =3 : 3352 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(6 c 1 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 : 5588 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 1 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 : 9460 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 1 =1 : 1531 ` 2 =0 : 58 2 a 1 =4 : 8654 10 )]TJ/F20 5.9776 Tf 5.757 0 Td [(1 a 2 =5 : 8240 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 b 1 =7 : 6956 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 b 2 =3 : 1928 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 c 1 =2 : 1078 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 2 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 : 4445 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 1 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 5005 d 2 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 : 4703 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 1 =2 : 0954 e 2 =1 : 5796 ` 2 =0 : 25 3 a 1 =3 : 1800 10 )]TJ/F20 5.9776 Tf 5.757 0 Td [(1 a 2 =5 : 1257 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 a 3 =2 : 2613 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 b 1 =1 : 7447 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 b 2 =3 : 3459 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(9 b 3 =2 : 4947 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(10 c 1 =1 : 4693 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(2 c 2 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 : 0431 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 3 =5 : 3196 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 1 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 : 0874 d 2 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 : 4727 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 3 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 : 9824 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 1 =1 : 8866 e 2 =1 : 5516 e 3 =3 : 0510 ` 2 =0 : 20 4 a 1 =1 : 3635 10 )]TJ/F20 5.9776 Tf 5.757 0 Td [(1 a 2 =2 : 9395 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 a 3 =3 : 6286 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 a 4 =2 : 2790 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 b 1 =1 : 7556 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(8 b 2 =6 : 5131 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(14 b 3 =1 : 2171 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(11 b 4 =1 : 8698 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(14 c 1 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 : 5263 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 2 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(9 : 7850 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 3 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 : 5597 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 4 =3 : 1778 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 1 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 8839 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 2 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 2306 d 3 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 5513 d 4 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 : 5463 e 1 =1 : 5180 e 2 =2 : 3774 e 3 =2 : 5082 e 4 =3 : 6370 ` 2 =0 : 13 5 a 1 =1 : 1471 a 2 =8 : 7483 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(2 a 3 =3 : 5297 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 a 4 =1 : 3423 a 5 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 2397 b 1 =8 : 7822 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(3 b 2 =1 : 2774 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(21 b 3 =1 : 7560 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(10 b 4 =2 : 0884 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(3 b 5 =9 : 3497 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(6 c 1 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 : 5307 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 2 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 : 0060 c 3 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 : 6222 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 4 =2 : 6533 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 5 =9 : 4150 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(2 d 1 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 5558 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(2 d 2 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 6212 d 3 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 2840 d 4 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 0131 d 5 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 1547 e 1 =5 : 4020 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 2 =3 : 4009 e 3 =1 : 7104 e 4 =8 : 2667 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 5 =1 : 4440 ` 2 =0 : 030 6 a 1 =1 : 1842 a 2 =8 : 9173 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(2 a 3 =3 : 6010 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 a 4 =1 : 4762 a 5 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 : 8828 a 6 =1 : 5146 b 1 =1 : 1169 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 b 2 =4 : 7250 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(21 b 3 =3 : 4842 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(10 b 4 =3 : 6421 10 )]TJ/F20 5.9776 Tf 5.757 0 Td [(3 b 5 =5 : 2225 10 )]TJ/F20 5.9776 Tf 5.757 0 Td [(6 b 6 =8 : 1345 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(7 c 1 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 : 6372 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 2 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 : 0134 c 3 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 : 2776 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 4 =2 : 3762 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 c 5 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 : 0798 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(2 c 6 = )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 : 0365 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 1 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 : 3467 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(3 d 2 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 4858 d 3 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 2090 d 4 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(9 : 1703 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 d 5 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 1075 d 6 = )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : 1957 e 1 =5 : 1267 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 2 =3 : 2891 e 3 =1 : 6772 e 4 =7 : 6133 10 )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 e 5 =1 : 4723 e 6 =1 : 6401 ` 2 =0 : 021 64 PAGE 77 Figure3.11. Fluidloadfractionsnumericaldatacirclesaswellaststothisdatausing Eq.3.21linesasafunctionofthedimensionlessparameter C forseveralvaluesof M . Onlyeverythirdpointofthenumericaldataisplottedforvisibility. n =6 . recall v x isthesumofPoiseuilleandCouetteowasdenedinchapterI,substituting v x andsimplifying: f = 1 F ZZ h 2 @p @x + S RR u m h d .23 Wheretheslidetorollratioisadimensionlessquantitydenedas S RR u b )]TJ/F22 7.9701 Tf 6.586 0 Td [(u a u m .Theslide torollratioisboundedbetween )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 S RR 1 .Purerollingoccurswhen S RR =0 ,and pureslipoccurswhen S RR =1 or S RR = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 .Thesignsigniesthedirectionthecontactis slippingin, + or )]TJ/F21 11.9552 Tf 9.299 0 Td [(x . Similarly,thesolidcontactfrictioncoecient, s ,isdenedasthefrictionalforceof solidcontactdividedbytheappliedload.ThefrictionalforceisassumedtobeCoulomb: c = a RR p c d F .24 65 PAGE 78 Where a istheasperitycontactfrictioncoecient.Andthecombinedsystemfrictional coecientis s = f + c .Figure3.12showsthetotal,uidandsolidfrictioncoecients forseveralvaluesof SRR .Thetotalfrictioncoecientisdominatedbysolidcontactatlow speed,asuidvelocityincreasesthecontactfullyseparatesandonlytheuidfriction matters.Highervaluesoftheslidetorollratiodramaticallyincreasetheuidfriction forces.Theviscosityinthecontactisextremelyhigh,thereforeevenverysmallslipspeeds generatetremendousviscousdrag.Thisslipwillalsogeneratelargeaviscousdissipation heatingtermaswillbeseeninchapterIV. EHLModelUsefulnessasDesignTool Inthischapteramulti-regimelubricationtoolwaspresented.Itiscapableofpredicting uidandsolidcontactpressuresasafunctionoffourdimensionlessparameters: M , L , C , and )]TJ/F17 11.9552 Tf 7.314 0 Td [(.Accuratecurvetsforthesolidanduidloadfractionswerepresentedasafunction of M , L ,and C foratypicalvalueof )]TJ/F17 11.9552 Tf 11.215 0 Td [(asseeninindustry.Thistoolcanalsobeusedto predicttotalfrictioncoecientandpower-loss.Powerlossisinthemainlyintheformof heat,whichwillbethetopicofthenextchapter.Thischaptercontainsmorethan600 solutionstoa152,551degreeoffreedommodel.Eachsolutiontakesapproximately15 minutestorun.Wherepossible,theruntimeisreducedbyusingapreviouslyrunsolution asaninitialcondition. Acknowledgment ThankyoutotheSeagatecoorperationforsponsoringthiswork. 66 PAGE 79 Figure3.12. Solid,uid,andcombinedfrictioncoecientasafunctionofmeanentrainment velocity u m forseveralvaluesofslidetorollratio SRR . 67 PAGE 80 CHAPTERIV THERMAL-MIXED-EHLTMEHL Lubricantpropertiesareextremelytemperaturesensitive,asseeningures2.1and2.2 fromchapterIIwhichshowthetemperatureandpressuredependentviscosityanddensity ofsqualene.ThelmthicknessisverysensitivetolubricantviscosityasseeninchapterII. Therefore,determiningthetemperatureofthelubricantisimportantforthepredictionof quantitieswithinthelubricationregime.Inthischapterweextendourmodeltoinclude thermaleects. Thepurposeofcalculatingthetemperatureriseistwofold.One,theaccuratepredictionof lmthicknessispredicatedonanaccuratedescriptionofthelubricantproperties,which canonlybedonetakingtemperatureriseintoaccount.Two,carbonbasedlubricants undergochemicalreactionssuchasoxidationandevaporationofthelowmolecularweight speciesathightemperature,bothofwhichresultinahigherviscosityofthebaseoiland thegrease.Itiscommoninpracticetoseeevidenceofburnandcharredlubricantsinworn outbearings.Theincreasedinviscosityreducestheoil'sabilitytoreowintotheballpath resultinginastarvedcontactwithincreasedsolid-solidcontact,temperaturerise,and runawaylubricantdegradation[58].Theaccuratepredictionofthisdestructivecycleis paramounttothedesignoflonglifebearings. Figure4.1depictslubricantenteringthecontactareaatanambientinlettemperature undergoingheatingandbeingejected.Heatisgeneratedintheuidandcarriedawayby forcedconvectionandconductionintheuidandsolid.Theconvectioninthesolid domainscomesfromthesolidbodymovementduetotherollingandslidingmotions, 68 PAGE 81 whichisalsothedrivingforcebehindtheuidvelocity. Thedicultiesinmeshingtheuiddomainin3DwerediscussedinchapterII.There,we usedanorderofmagnitudeanalysisOOMAtoreducetheequationsofmotionsothatan analyticalsolutiontotheuidvelocitycouldbesolvedfor.Thisalloweda2Dmeshto representa3Duidvolume.Wefacethesameprobleminthethermalproblem.Herewe cannotreducetheuiddomainfrom3Dto2Dunlessweneglectconvectiveheattransferin theuid,whichleadstotheoverpredictionoftemperature. TheresultoftheOOMAshowsthattheenergyequationfortheuidtransienttermison thesameorderastheconvectiveterms.Bothtermsaresmallcomparedtotheconductive terminthezdirectionforthinlms.Theliteraturehasoverwhelminglydecidedthatthe convectivetermsandthezdiusivetermareimportantwhilethetransienttermisnot [39,33].Includingthetransienttermcomplicatestheanalysis,thereforeweonlystudythe steadystatebehaviorinsection4.2.Theresultofneglectingthetransienttermistheover predictionoftemperatureriseincaseswithsmallheatingterms,andnearlynoeectwhen theheatingtermsareverylarge.Insection4.2thetemperatureriseiscalculatedasa functionofthedimensionlessparameters M , L ,and C forconstantvaluesofthe dimensionlessmaterialparameters G : Thevaluesoftheparametersareheldatconstant valuesrelatedtooillubricatedsteelonsteelcontactsinordertoreducethesizeofthe sensitivitystudy. 4.1Model Inthissectiontheassumptionsandequationsforthetemperaturecalculationare presented.Startingwiththefullheatequationandpresentandorderofmagnitudeanalysis fortheuidandsoliddomainsandintroducenondimensionalvariables.Theweakformis thenpresentedalongwithboundaryconditions.Thenalsubsectiondiscusseshow 69 PAGE 82 Figure4.1. Diagramofheatinginthecontactarea temperatureiscoupledwiththeMEHLproblem. 4.1.1EectofTemperatureonLubricant Fitsofvariousrheologicalmodelstoexperimentaldatawerediscussedinsection2.2.Here wetakeacloserlookattheeectoftemperatureonoilrheology.Figure4.2showsthe viscosityasfunctionoftemperatureforseveralpressures.At 0 : 1 MPa theviscosity decreasesbyafactorofapproximately 2 : 3 per 20 K increaseintemperature.At 1200 MPa theviscositydecreasesbyafactorof 25 fora 20 K temperaturerise. Thedecreaseinviscosityreducestheviscousdissipationterm,inturnloweringthe temperaturerise.Thedecreaseinviscosityalsodecreaseslmthickness,whichincreases theviscousdissipationtermbyincreasingtheshearrate, @v x @z and @v y @z .Thedecreaseinlm thicknessalsolowerstemperaturerisebydecreasingthelengthoftheconductivepathfor heattoleavethecontact.Theproductofthesecompetingeectscanonlybedetermined withafullycoupledthermaluidstructureinteractioncode. Thereductionofviscosityathightemperaturealsobluntstheinherentnumerical 70 PAGE 83 Figure4.2. Viscosityasafunctionoftemperatureforselectedpressuresrangingfrom 0 : 1 MPa to 1200 MPa .Experimentaldatafrom[5]areshownwithcircles,tstothisdata withtheRoelandsmodelareshownwithsolidlines.TheconstantsusedintheRoelands modelwerediscussedinsection2.2andpresentedintable2.4. 71 PAGE 84 instabilitydiscussedinsection2.8,whichallowsustostudyhighervaluesofpressure viscositycoecientbeforetheGalerkinformulationfails. 4.1.2GoverningEquations Theequationofenergyforageneralstateofmatterwithvelocity v =[ v x ;v y ;v z ] iswritten as: ^ C p @T @t + v x @T @x + v y @T @y + v z @T @z = h @ @x )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(k @T @x + @ @y k @T @y + @ @z )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(k @T @z i )]TJ/F27 11.9552 Tf 11.291 9.684 Td [()]TJ/F22 7.9701 Tf 7.56 -4.428 Td [(@ ln @ ln T p Dp Dt )]TJ/F15 11.9552 Tf 11.955 0 Td [( : r v .1 from[43].Wheretheoperator D denotesthesubstantialderivative.Thelasttwotermson therighthandsiderepresenttheheatingdotocompressionofthematerialandheating duetoviciousdissipation.Whichisdenedasthecombinedvolumetricheatingterm, Q V = )]TJ/F27 11.9552 Tf 11.291 9.684 Td [()]TJ/F22 7.9701 Tf 7.56 -4.428 Td [(@ ln @ ln T p Dp Dt )]TJ/F15 11.9552 Tf 11.955 0 Td [( : r v .Thegoalofthisnumericalmodelistodeterminehow importantthecompressiveandviscousdissipationheatingtermsare.Thesetermsare diculttoestimatesincetheyarederivativesWeneglectthesetermstosimplifythe OOMAsinceitisdiculttoestimatepressure,velocitygradients,andviscositywithouta solutiontoacoupledTMEHLmodel,butwillincludethesourcetermslateronintheFEA analysis.Theheatsourcesarediscussedfurtherinsection4.1.3.Noassumptionabout stateismadehere,theseequationsarevalidforuidandsolid. Deningthefollowingdimensionlessvariables: X x l 0 Y y b 0 Z z h 0 U v x u 0 V v y v 0 W v z w 0 t t 0 0 0 T T 0 .2 Substitutingandrearranging,andmultiplyingby h 2 0 = T 0 k : h 2 0 T t 0 @ @ + l 0 u 0 T h 2 0 l 2 0 U @ @X + b 0 v 0 T h 2 0 b 2 0 V @ @Y + h 0 w 0 T W @ @Z = h h 2 0 l 2 0 @ 2 @X 2 + h 2 0 b 2 0 @ 2 @Y 2 + @ 2 @Z 2 i .3 wherethethermaldiusivityisdenedas T = k 0 ^ C p andthesubscript T isusedto dierentiatefromthepressurenumbers,andthePcletnumbersinthe x;y and z 72 PAGE 85 directionsdenedas Pe x l 0 u 0 T , Pe y b 0 v 0 T ,and Pe z h 0 w 0 T respectively,andthe dimensionlesstemporalparameter, h 2 0 T t 0 : @ @ + Pe x U h 2 0 l 2 0 @ @X + Pe y V h 2 0 b 2 0 @ @Y + Pe z W @ @Z = h h 2 0 l 2 0 @ 2 @X 2 + h 2 0 b 2 0 @ 2 @Y 2 + @ 2 @Z 2 i .4 wearriveatthedimensionlessheatequation.Theassumedscalesforthethermalproblem arelistedintable4.1.Twosetsofscalesareassumedsincethematerialpropertiesand convectivevelocitiesfortheuidandsoliddomaindier.Therelevanttimescaleassumed hereis t 0 = l 0 u 0 ,whichistheaverageresidencetimeoftheuidinthecenteroftheHertzian contactarea,whichistheaverageamountoftimethattheuidissubjectedtoheating. Usingthisdenitionof t 0 ,andcombiningthePcletnumberswiththeirrespectiveaspect ratios,Eq.4.4canbewritten: h 2 0 u 0 T l 0 @ @ + h 2 0 u 0 T l 0 U @ @X + h 2 0 v 0 T b 0 V @ @Y + h 0 w 0 T W @ @Z = h h 2 0 l 2 0 @ 2 @X 2 + h 2 0 b 2 0 @ 2 @Y 2 + @ 2 @Z 2 i .5 Notethattheprefactoronthetransienttermandthe x convectiontermareidentical,and alsoequaltotheprefactoronthe y convectiontermif v 0 b 0 = u 0 l 0 ,whichisacommon condition.Thismeansthattheconvectivetermsarebydenitionofthescalesassociated withtheTEHLproblemareasimportantasthetransientterms.Thereforeanyanalysis thatassumessteadystatewhileusingconvectivetermsisneglectingtransienteectsand thereforeoverestimatingtemperaturerise.Thisisacommonassumption[33,39].The steadystateassumptionisonepossibleexplanationforthediscrepancyingures2and3 of[33],wheretheresultsofthenumericalTEHLmodelismoresensitivetotemperature risethantheexperiments. Fillingthesescaleswithnumericalvaluesyieldsfortheuid: )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 : 3 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 @ @ +1 : 13 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 U @ @X +1 : 13 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 V @ @Y +1 : 4 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 W @ @Z = h 6 : 6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 @ 2 @X 2 +6 : 6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 @ 2 @Y 2 + @ 2 @Z 2 i .6 Itiscleartoseethatforthisintermediatevalueoflmthicknessthetransientand 73 PAGE 86 Table4.1: ParametersusedinOOMAfortheuiddomainleftandsoliddomainsright. Note T 0 isnotrequiredasitcancelsoutduringtheprocessofnondimensionalization. Fluid Solid Parameter Value Parameter Value R 0 : 0127 m R 0 : 0127 m F 50 N F 50 N E 110 GPa E 110 GPa Roe 15 1 GPa Roe 15 1 GPa 0 220 Pa s 0 220 Pa s 0 818 kg=m 3 0 818 kg=m 3 a 1 : 6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 m a 1 : 6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 m p h 0 : 927 GPa p h 0 : 927 GPa h 0 1 : 3 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 [ m ] h 0 1 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 m u 0 1 m=s u 0 1 m=s v 0 1 m=s v 0 0 w 0 1 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 m=s w 0 0 t 0 l 0 u 0 =1 : 6 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 s t 0 l 0 u 0 =1 : 6 E )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 s T 9 : 35 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 m 2 s T 1 : 25 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 m 2 s T 0 300 K T 0 300 K 74 PAGE 87 convectivetermsaremuchsmallerthan O ,howeverforareasonablylargelmthickness thatis10timesthatoftheassumedlmthickness,thesetermswillbeonorderof O : 1 , anditwouldbeunwisetoneglectthem.Sinceweareusinganiteelementrepresentation oftheenergyequation,itistrivialtoincludetheconvectiveterms.Weneglectthe transienttermhereforsimplicity,butitseectsshouldbestudiedinfuturework. Droppingalltermssmallerthan O ,excludingtheconvectiveterms,leaves h 2 0 u 0 T l 0 U @ @X + h 2 0 v 0 T b 0 V @ @Y = @ 2 @Z 2 whichwaswrittenas: h 2 0 u 0 T l 0 U @ @X + h 2 0 v 0 T b 0 V @ @Y = @ 2 @Z 2 + h 2 0 kT 0 Q V .7 beforeapplyingtheproductrule.Andforthesoliddomain: 0 @ @ +500 U h 2 0 l 2 0 @ @X +0 V h 2 0 b 2 0 @ @Y +0 W @ @Z = h 3 : 9 @ 2 @X 2 +3 : 9 @ 2 @Y 2 + @ 2 @Z 2 i .8 Hereweseeconvectioninthe y and z directionsarezerobydenition,thetransientterm issmall,andallothertermsare O orlarger.Droppingalltermssmallerthan O and includingthevolumetricheatsourceterm: Pe x U h 2 0 l 2 0 @ @X = h 2 0 l 2 0 @ 2 @X 2 + h 2 0 b 2 0 @ 2 @Y 2 + @ 2 @Z 2 + h 2 0 kT 0 Q V .9 Equations4.7and4.9arethedimensionlessheatequationsfortheuidandsolid domainrespectively.Thegeometrytheseequationsaresolvedonareshowningure4.3. Thedomainisrectangularwiththreelayers.Thebottomlayerisasolidandrepresentsthe race,themiddleisuidandrepresentsthelubricant,andtheupperlayerissolidand representstheball.Theuiddomaincanberepresentedasarectanglesincetheitis non-dimensionalizedwiththelmthicknessinthe z direction.Thetemperatureis symmetricaboutthe x )]TJ/F21 11.9552 Tf 11.955 0 Td [(z planesoonlyhalfoftheproblemismodeled.Thecylindrical objectshowningure4.3isformeshrenementintheheatgenerationregion. 75 PAGE 88 Figure4.3. Geometryandboundaryconditionsforthethermalproblem 4.1.3HeatSources Heatgenerationcomesfromthreesources: 1.Viscousdissipationofenergywithinthelubricant, Q VD = )]TJ/F15 11.9552 Tf 11.291 0 Td [( : r v 2.Workdonecompressingtheuid, Q C = )]TJ/F27 11.9552 Tf 11.291 9.684 Td [()]TJ/F22 7.9701 Tf 7.56 -4.428 Td [(@ ln @ ln T p Dp Dt 3.Frictionalheatingoftheslidingasperity-asperitycontact, q = a p c u slide Thersttwoareaddedtotheheatequationasavolumetricheatsource, Q V .The frictionalheatingtermhasunitsof w m 3 andismodeledasaheatuxboundarycondition betweentheuidandsoliddomain.Tocalculatetheviscousdissipationtermwemust knowthevelocitycomponentswithintheuid.DuringthederivationofReynoldsequation, thevelocitywasassumedtobeonlyinthe x and y directionsaswrittenbelow: u = 1 2 @p @x )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(z 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(zh + z h u h )]TJ/F21 11.9552 Tf 11.955 0 Td [(u 0 + u 0 .10 76 PAGE 89 v = 1 2 @p @y )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(z 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(zh .11 Theviscousdissipationtermcanthenbecalculatedas[43]: Q VD = : r v =2 )]TJ/F22 7.9701 Tf 6.675 -4.976 Td [(@u @x 2 + @v @y 2 + )]TJ/F22 7.9701 Tf 6.675 -4.976 Td [(@w @z 2 + h @v @x + @u @y i 2 + h @w @y + @v @z i 2 + @u @z + @w @x 2 )]TJ/F19 7.9701 Tf 13.151 4.707 Td [(2 3 h @u @x + @v @y + @w @z i 2 .12 Simplifying,given u = u x;y;z , v = v x;y;z , w =0 : Q VD =2 )]TJ/F22 7.9701 Tf 6.675 -4.977 Td [(@u @x 2 + @v @y 2 + h @v @x + @u @y i 2 + @v @z 2 + )]TJ/F22 7.9701 Tf 6.675 -4.976 Td [(@u @z 2 )]TJ/F19 7.9701 Tf 13.151 4.708 Td [(2 3 h @u @x + @v @y i 2 .13 Thebiggestcontributorhereis @u @z when SRR 6 =0 .Thevelocitiesaresolvedforonthe samemeshasthetemperatures,andthegradientsaretakennumerically.Thecompressive heatingtermisshownbelow. Q C = )]TJ/F27 11.9552 Tf 11.291 16.856 Td [( @ ln @ ln T p Dp Dt .14 expandingthesubstantialderivative: Q C = )]TJ/F27 11.9552 Tf 11.291 16.857 Td [( @ ln @ ln T p @p @t + @p @x + @p @y + @p @z .15 where @p @t =0 duetotheassumedsteadystatemodel,and @p @z =0 duetotheReynolds equationassumptions.Thepressuregradientsin x and y arecomputedfromthederivatives oftheshapefunction.The )]TJ/F22 7.9701 Tf 7.56 -4.428 Td [(@ ln @ ln T p termisre-writtenas T )]TJ/F22 7.9701 Tf 7.56 -4.428 Td [(@ @T p andtakenfromthe mathematicaldenitionoftheequationofstate. Thefrictionalheatingtermisafunctionoftheslidingvelocity, u slide = u b )]TJ/F21 11.9552 Tf 11.956 0 Td [(u a ,while theReynoldsequationdependsonthemeanentrainmentspeed, u m = u a + u b 2 .Recallthat u a and u b arethevelocitiesoftheupperballandlowerracesurfacesrespectively.The slidingvelocityalsogeneratesalargecontributiontotheviscousdissipationfunctiondue tothe @u @z term.Acommondimensionlesswaytorepresenttherelativeamountofslidingin acontactisthe slidetorollratio , SRR ,denedas: SRR = u slip u m = u 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(u 2 1 2 u 1 + u 2 .16 Thecombinationofslidetorollratiowiththemeanentrainmentvelocitycompletely 77 PAGE 90 describesthekinematicsofthecontact. 4.1.4Discretizationandboundaryconditions Theenergyequationsfortheuidandsoliddomains,equations4.7and4.9respectively,are solvedinCOMSOLwiththecoecientformpartialdierentialPDEequationtoolbox. ThistoolboxsolvesthefollowinggeneralformPDEwithastandardGalerkinformulation. e a @ 2 T @t + d a @T @t + r )]TJ/F21 11.9552 Tf 9.299 0 Td [(c r T )]TJ/F21 11.9552 Tf 11.955 0 Td [(T + + r T + aT = f .17 Wherethecoecients e a ;d a ;;;a arezeroforbothdomains. c =1 , = h Pe x U h 2 0 l 2 0 ;Pe y V h 2 0 b 2 0 i , f = h 2 0 kT 0 Q V fortheuiddomain,andsimilarly c =1 ; = h Pe x U h 2 0 l 2 0 ; 0 i ;f = h 2 0 kT 0 Q V forthesoliddomain.RecallthatfortheReynolds equationtheweakformwasspecied.Thiswasdonetogivegreatercontrolin implementingstabilizationtechniqueswhicharenotrequired,exceptincasesofveryhigh entrainmentvelocity. Themeshusedforthismodelisshowningure4.4.Theupperdomainrepresentsthe ballandthelowerdomainrepresentstherace.Themiddledomainrepresentstheuidlm withameshrenementinthecontactarea.Anx-ysliceofthisdomainhasthesame dimensionsasthelubricationdomainfortheuidproblem.Fluidpressuresandlm thicknessaremappedontothismeshusingabuiltinCOMSOLfunction.Theuidmeshin thethermaldomaincloselyresemblesthemeshdensityfortheuidproblem,althoughitis notaperfectmatch. 4.1.5CouplingwithMEHLproblem Theeectoftemperatureonviscosityanddensitywerediscussedinsections2.2and 4.1.1.Thisisafullycoupledmodel,changesintemperatureeecttheviscosityand densityusedinthecoupleduid-solidproblem,andthepressuresandvelocitiesinturn eecttheheatequationthroughtheheatsourcesandconvectiveterms. Furthermore,temperaturevariationsareexpectedinthelmthicknessdirection z 78 PAGE 91 Figure4.4. Meshusedforthermaldomain.Meshisrenedincontactarea. coordinate.RecallthattheassumptionsmadeinchapterIIincludeconstantuid densityandviscosityinthelmthicknessdirection.Peiran[53]reformulatedReynolds equationtoallowfordensityandviscosityvariationsinthelmthicknessdirectiondueto temperatureandshearthinningeects.Thisformulationisimplemented. 4.2TMEHLResults InthissectionTMEHLmodelresultsarepresentedforvariousdimensionlessoperating parameters M , L , C , SRR forconstantvaluesof T and a .Thedominateheating mechanismandthemaximumtemperaturerisedependsontheseparameters.Twomain casesarepresented,therstisathicklmthathasnosolidcontact.Viscousdissipation andcompressiveheatingeectsareinvestigatedinthiscase. M , SRR ,and L areshownto havesubstantialeectsontherateofviscousdissipation.Thesecondcasehasasperitieson theorderofthelmthicknessresultinginmixedlubricationcontactexhibitingasperity contactheating. 79 PAGE 92 Table4.2: Parametersusedfortheviscousdissipationstudy Parameter Value F 23 N R 0 : 0127 m E 224 GPa u m 15 m s T;fluid 1 : 37 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 m 2 s T;solid 1 : 25 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 m 2 s S q 10 nm c 0 : 3 H 6 : 8 GPa 4.2.1ThickFilmSolution Resultsofthethermalmodelonthelmpressureandlmthicknesscomparedtoan isothermalsolutionareshowningure4.5fortheoperatingparameterslistedintable4.2, theRoelandsrheologicalparametersandtheDowsonandHiggensonequationofstate parameterslistedintable2.4.Themaximumtemperatureriseinthiscaseis 69 : 4 C . Reportednon-dimensionally 0 = T 0 )]TJ/F22 7.9701 Tf 6.586 0 Td [(T 0 T 0 ,where T 0 =300 K .Theincreasedtemperature lowerstheuidviscosity,whichinturnlowersthelmthickness,asseenbythesolidand dashedblacklinesingure4.5.Furthermore,thepressurespikeandexitcontractionare shiftedtotheleftandexaggerated. Mid-planetemperatureriseisshowningure4.6.Themid-planeoftheuidlmisdened as Z =0 : 5 .Thatmaximumtemperatureriseoccursneartheexitcontraction,seenin gure4.6at X =0 : 25 and Y =0 .Themaximumheatingrateinthiscaseoccurswherethe pressurespikeis,duetothelargeincreaseinviscosity. Thetotalheatingratein watts isplottedasafunctionofslidetorollratio SRR ,along withthecontributionofthetopthreetermsinequation4.13ingure4.7.Therateof 80 PAGE 93 Figure4.5. Filmpressure,thicknessandaveragetemperatureriseplottedalongthecenterlineofthecontact.MEHLmodelshownwithdashes,TMEHLmodelshownwithsolidlines. Filmissucientlythickinbothcasestopreventsolidcontact,therefore p c =0 .Maximum temperatureriseis 69 : 4 C . viscousdissipationisdominatedbythe @u @z termacrossallslidetorollratios. Thelmaveragedtemperaturerise,denedas T avg = R h 0 T )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 0 dz ,isausefulwayto visualizetemperaturerisebecauseitcanbeplottedona2Ddomain,revealinghowto temperaturechangesthroughthecontact.Maximumlmtemperatureriseisshownin gure4.8asafunctionofslidetorollratioSRRforseveralvaluesof : IncreasingSRR increasestheslipvelocity,whichissquaredintheviscousdissipationterm.Onewould expectthemaximumlmtemperaturetoincreasequadratically,butthetemperaturerise alsodecreasesuidviscositywhichinturndecreaseslmthickness,reducingtheviscous dissipationterm.Theresultofthenonlinearsystemisanapparentlinearincreasein maximumtemperatureasafunctionofslidetorollratio. Todeterminetherelativecontributiontothetotaltemperatureriseofthecompressive heating,wesettheviscousdissipationtermtozeroarticiallyandrerunthesame 81 PAGE 94 Figure4.6. Filmaveragedtemperatureriseonthelubricationdomain. Figure4.7. Integratedtotalpowerlossduetoviscousdissipationasafunctionofslideto rollratioSRR,includingselectedcomponentsoftheviscousdissipationterm. 82 PAGE 95 Figure4.8. T avg asafunctionofslidetorollratioforseveralvaluesofpressureviscosity coecient, . parameters.Thetemperatureriseforthiscaseisshowningure4.9.Themaximum temperatureriseforthiscaseislessthan 0 : 1 C ,risingininlettothecontactwherethe lubricantisbeingcompressed,andfallingbelowambientattheexitofthecontract.The uidleavesslightlycoolerthanitentered.Thisworksmuchlikeanairconditioner,where thecoolantiscompressed,cooledbytheambienttemperaturesurroundings,and uncompressedtoalowerthanambienttemperature.Thepressureandlmthicknessare indistinguishablefromtheisothermalcase. 4.2.2Thinlmresults ThissectionpresentsTMEHLresultswiththinlmsinthemixedlubricationregime.In thisregime,asperitycontactsupportsthemajorityoftheappliedload.Theuidlmis 83 PAGE 96 Figure4.9. Mid-planetemperaturerisewithzeroviscousdissipation. partiallyruptured,whichpreventslargepressurebuildups.Withoutlargepressure gradientstheviscousdissipationandcompressiveheatingtermsaresmall.Themaximum bulktemperatureriseinthiscaseis 8 : 06 C .Theincreasedtemperaturelowerstheuid viscosity,whichinturnlowersthelmthickness,asseenbythesolidanddashedblack linesingure4.10.Thedecreaseinlmthicknessduetotemperatureriseincreasesthe solidcontactpressure,andinturndecreasestheuidpressure. Themaximumlmaveragedtemperatureriseinthedomainisforthethinlmcase isshowningure4.11asafunctionofslidetorollratio.Whilethemaximum bulk temperatureriseinthiscaseis 8 : 06 C ,thatabsolutemaximumtemperaturerisedueto asperitycontactheatingcanbemuchhigherconsideringtheheatgeneratedbysliding frictionisnotstatisticallysmearedlikeinourmodel.RecallthatinchapterIIIcontact modelswereconsideredthatallowedustotreatthesolidcontactpressureasaspatially 84 PAGE 97 Table4.3: Parametersusedfortheviscousdissipationstudy Parameter Value F 23 N R 0 : 0127 m E 224 GPa u m 15 m s T;fluid 1 : 37 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 m 2 s T;solid 1 : 25 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 m 2 s S q 10 m c 0 : 3 H 6 : 8 GPa Figure4.10. Filmpressure,thicknessandaveragetemperatureriseplottedalongthecenterlineofthecontact.MEHLmodelshownwithdashes,TMEHLmodelshownwithsolid lines.Maximumtemperatureriseis 8 C . 85 PAGE 98 Figure4.11. T avg asafunctionofslidetorollratioforseveralvaluesofpressureviscosity coecient, . averagedvariable,insteadofadiscontinuousvariable.Whilethisworksreasonablywellfor determiningthesolidcontactstiness,itvastlyunderestimatesthemaximumtemperature rise.Todeterminetheactualmaximumtemperaturerise,adiscretemodelofanasperity wouldneedtobeconsidered.Theactualmaximumtemperatureriseforslidingcontactsis generallyontheorderof 100 C [11]. 4.3TMEHLModelUsefulnessasDesignTool Thischaptercontains25solutionstoa338,664degreeoffreedommodel.Eachsolution takesontheorderof1hourtorun.Themodelcanbeusedtoaccuratelydeterminethe lubricationregimeforagivensetofoperatingconditions.Additionally,thismodeloutputs maximumtemperatureriseandthesolidcontactloadfraction.Inthecaseoffrictional heating,thiscodeoutputsthebulktemperaturerise,butdoesnotaccuratelymodelthe 86 PAGE 99 maximumtemperaturesatthecontactinterfacewhichcanbeordersofmagnitudelarger. Otherapproachestothisproblemrelyonanexplicitmodelofsurfaceroughness, requiringmanymoredegreesoffreedomandincreasingsolutiontime.Thesemodelsalso relyonsuccessiveoverrelaxationforrobustnesswhichrequireshundredsofiterationsto converge.Thismodelusesthefullsystemapproach,solvingtheuid,solid,thermal,and loadequationinonesystemofequations.Thistakesintoaccountcouplingtermsthat providestability.ThereforenorelaxationisrequiredandthefullpowerofNewtonsmethod canbetakenadvantageof.Ittakesontheorderof10iterationstoconverge. Acknowledgment ThankyoutotheSeagatecoorperationforsponsoringthiswork. 87 PAGE 100 CHAPTERV FULLBEARINGSYSTEMMEHL AsseeninchapterIII,thereasonoilissuchagreatlubricantisthepressuredependent viscosity.Thelmbecomesstierasthelmpressureincreases.Typicalwaterpressuresin waterlubricatedbearingsareontheorderof 1 MPa .Seawaterhasaveryweakrheological pressuredependence,astheviscositychangesbylessthan1%betweenatmospheric pressureand 1 MPa [54].Thereforeasoftbearinglinerisusedtoincreasetheload capacityofwaterlubricatedbearingsbyincreasingcontactconformalityandspreadingthe loadoutoveralargerarea.Typicalbearinglinershaveamodulusontheorderof 400 MPa ,roughly 500 timeslowerthanthemodulusofsteel,whichisatypicaloil lubricatedcontactingelement.Thespecicgeometryofthisbearingwillbefullydescribed inthefollowingsection. Thegoalofthisworkistocreateamodelthatisexperimentallyvalidatedtostudywater lubricatedbearingsspanningtheregimesbetweenfulllmandboundarylubrication.With theproposedmodeltheeectofmultiplesecondordereects,aswellasprobefundamental questionsinlubricationisstudied.Themodelcanalsobeusedasadesigntoolfornew bearingsystems. Thisworkdiersfrompreviouswaterlubricatedbearingmodelsinthatitsolvesthe Reynold'sequationandtheassociatedsoliddeformationsonaphysicallyrelevantgeometry, insteadofonahalfspaceorunrolledbearinglikecurrentmodelsintheliterature[18,44]. Thismethodallowsfortheapplicationofrealworldboundaryconditionsandstudyingthe eectsofhousingbending,whichhasnotbeendonebeforeonwaterlubricatedsystems. 88 PAGE 101 Furthermore,thisworkisexperimentallyvalidated.SimilarlytochapterIII,tsto dimensionlessloadratiosarereportedasafunctionofrelevantdimensionlessparameters. Oneofthelargestissueswithmultiphysicsmodelsofthistypeisguaranteeing convergence.Thisistypicallyaddressedbysegregatingandtheniteratingbetweenthe solidanduidmodelswithnumericalrelaxation.Thiscantakehundredstothousandsof iterationsforasinglepoint.Thefullsystemapproach,proposedby[34],isimplemented here.Thefullsystemapproachwasimplementedinallpreviouschaptersaswell.Theidea behindthefullsystemapproachistosolveallphysicssimultaneouslyinonesystemof equations.Bydoingso,crosscouplingtermsareutilizedthatstabilizethesolution.Forthe modelinthischapter,itisfoundthatthefullsystemapproachworksverywellinthefull lmregime,butgenerallyfailsinthemixedandboundaryregimes.Howeverapartial systemapproachworkswelliftheeccentricityoftheshaftisspecied.Anovelsolutionis implementedwherethestinesscoecientsofthebearingareusedtocalculatethethe equilibriumpositionoftheshaft. 5.1Introduction ExperimentsinsupportofthisworkwereperformedatXdotEngineeringandAnalysisof Charlottesville,Virginia.Theybuiltanddesignedatestrigcapableofmeasuringthe reactiontorqueonthebearinghousingasafunctionofshaftspeedandappliedload.The housingwasdesignedtobesucientlystitominimizeitsdeection.Theshaftissolid andsupportedveryneartheendsofthehousinginordertomaximizeitsbendingstiness. Theeectsofhousingbendingisdiscussedinsection5.4. Thethreecomponentsofapolymerlinedbearingconsideredherearetheshaft,linerand bearinghousing,asshowninFigure5.1,withtheparameterslistedintable5.1.Theliner andthehousingmakeupthebearingassembly.Inthisstudywewillexaminegeometry wherethebearingassemblyisassumedexibleandtheshaftisconsideredtoberigid.This workshows,forthersttime,thatthehousingdeformation,specicallybendingand 89 PAGE 102 Table5.1: Parametersusedinthisstudythatrepresentthetestrig. Parameter Quantity Description r s 38 : 1 mm Shaftradius T L 25 : 4 mm Linerthickness T H 6 : 35 mm Housingthickness L S 406 : 4 mm Shaftlength L H 304 : 8 mm Bearinglength L unsup 50 : 8 mm Unsupportedshaftlength c 0 : 1016 mm Bearingradialclearance 1 mPa s Lubricantviscosity F 6422 : 5 N Appliedload E H 70 GPa Housingmodulus H 0 : 45 HousingPoisson'sratio E L 200 GPa Linermodulus L 0 : 3 LinerPoisson'sratio ovalization,mayplayanimportantroleindeterminingliftospeed. 5.2Model Thissectiondescribestheequations,boundaryconditions,mesh,andsolutionprocedure usedforthismodelforthewater-lubricatedjournalbearingproblem.Themodelis comparedandcontrastedtomodelsdiscussedinpreviouschapters.Adimensionalanalysis ispresented,andeightdimensionlessvariablesarepresentedthatcontroltheuidandsolid loadfractions,andthesystemcoecientoffriction. 5.2.1GoverningEquations Amixedelastohydrodynamicmodelthatincludedsolidcontactforoillubricatedball bearingswasproposedinchaptersIIandIII.Asimilarmodelforwater-lubricatedjournal bearingsisproposedinthissection,withdierentlubricantandgeometry.Modelingwater doesnotrequirethecomplicatedrheologicalandstatemodelsdiscussedinchapterI.The 90 PAGE 103 Figure5.1. Crosssectionoftheundeformedshafthousingassemblyleft,sideviewof shaftandhousingassemblyright.Theshaftisshowninlightgrey,thepolymerlinerin purple,andthehousingindarkgrey.Thebluegaprepresentstheradialbearingclearance, C. changeingeometrydoes,however,requiremodication.Onegoalofthismodelisto determinetheeectofmacrohousingdeformationsonthelubricationperformance,which requiresaccuraterepresentationofthegeometryofthebearinglinerandhousing.A commonsimplicationintheliteratureistounrollthebearingandsolvethecoupled uid-solidproblemonaatplane.InthisworkwerequireasolutiontoReynoldsequation onacylindricalsurface.WestartwiththemodieddimensionalReynoldsequationin cylindricalcoordinatesfrom[9]: 1 R 2 @ @ h 3 t G @p @ + @ @z z h 3 t G z @p @z )]TJ/F21 11.9552 Tf 11.956 0 Td [( s ! 2 @ h @ =0 .1 where ; z aretheowfactorsinthe and z forthePoiseuilleterms, s istheow factorfortheCouetteowterm, G and G z arecorrectionfactorsforturbulence.Flow factorsaccountforthedierenceinowrateduetoroughsurfacesbeingassumedsmooth. Theyarecommonlyassumedtobeunityforsimplicity,althoughtherehavebeenmany numericalstudies[48,49,65,32,45]thatcalculatethesefactorswitha3Drepresentation ofthecontactareaincludingsurfaceroughness.Thesemodelsaretoodetailedtoscaleup toanentirebearing,buttheeectcanbecapturedintheowfactorterms. 91 PAGE 104 Theturbulencecorrectioncoecients, G and G z ,isrequiredforbearingsoperating athighspeed.Theturbulencecorrectionfactorsreplacethe 12 inthedenominatorofthe ReynoldsequationwithafunctionoftheReynoldsnumberthatis 12 atlowspeed,and increasesslightlyathigherspeed,eectivelyraisingtheviscosityoftheuid.Again,these modelsareverydetailedandarediculttoscaleuptoanentirebearing,buttheeects canbecapturedwiththecoecients[66,50,38,17]. Inthiswork,theowfactorsareassumedtobeunity,andtheturbulentowfactors areassumedtobe 12 forsimplicity.Themodularnatureofthiscodeallowsforthesetobe easilymodiedinthefutureifdesired.Preliminaryworkshowsthattheeectsoftheseare smallforthebearingconsideredinthischapter. Thedimensionalnominallmthicknessequationforthissystemis: h explicit = c )]TJ/F21 11.9552 Tf 11.955 0 Td [(e x cos )]TJ/F21 11.9552 Tf 11.955 0 Td [(e y sin + ucos + vsin .2 where c istheradialclearanceseegure5.1, e x = u shaft c and e y = v shaft c arethe dimensionlesscoordinatesofthecenteroftheshaftseegure5.2,and isthestandard cylindricalangularcoordinateseegure5.2.Theexplicitlmthicknessismodiedand nondimensionalizedwiththesamefunctionasinchapterIII,seeequation3.17.Andthe correspondingcontactpressureisfromtheplasticasperitydeformationmodelinchapter IIIequation3.13. Figure5.2showsadrawingofacutawayofajournalbearingwiththehousing,liner, lubricant,andshaftshown.Alsoshownaretheexamplepressuresresultingfromuidow andsolidcontact.Atequilibrium,theseloadsmustbeequaltotheloadvector, [0 ;F ] and thesumoftheforcesin x and y directionsare: P F x = RR [ p + p c n x + r + a p c n y ] d W =0 P F y = RR [ p + p c n y + r + a p c n x ] d W = F .3 where r istheshearstressduetouidowand a istheasperity-asperitycontactfriction 92 PAGE 105 Figure5.2. Cutawayofjournalbearingshowingcoordinatesystem,deningtheeccentricities,andshowingthenormalforcesactingontheshaftduetotheuidowandsolidcontact. Notshownaretheshearstressesfromtheuidoworthesolidcontact. 93 PAGE 106 coecient.Notethatsincethesolutionforpressureactsnormaltoacurvedsurfaceithas x and y components.Similarly,theshearstressactstangentialtothesurfaceandalsohas x and y components.Thenumericalschemeincludestwowaysofsolvingforeccentricities thatsolvetheloadequationsateachspeed.Theywillbediscussedinsection5.2.4. ThedeformationsarecalculatedinasimilarmannerasdiscussedinchapterII,with possibleboundaryconditionsshowningure5.4andwillbediscussedlater.Thesolid meshnowhastwodomains,representingthecompliantlinerandthestihousing.Itis assumedthatthereisnoslipattheinterface.Inrealworldapplicationsthelinerinstalled withaninterferencetorbondedtothehousingwithanepoxy.Bothmethodsminimize interfacialslip.Futureworkshouldbedonetostudytheeectofslipatthisinterface. 5.2.2BoundaryConditions Inthissectiontheboundaryconditionsfortheuidandthesolidproblemarepresentedin ageneralway,aspartofthisworkisstudyingtheeectoftheboundaryconditionsonthe lubricationperformance.Dirichletboundaryconditionsforuidpressureareimplemented whereverthelubricantencountersanopenreservoirofuid,weatheritbewaterorair.To modeltheexperimentalsetupconsideredinthiswork,thepressureisspeciedas atmosphericontheboundarieshighlightedingure5.3. Fourdierentboundaryconditionsonthehousingareconsidered,simulatingvarious supportstructuresofthehousingasshowningure5.4.CaseArepresentsasupport structurethatisinnitelyrigidandattachesalongthelengthandcircumferenceofthe bearinghousing.Thisisanidealcaseandisusedasareference.ForcasesB,C,andD,we assumethatthesupportstructuresarerigidandthatthebearinghousingisassumedrigid atthelineorareaofcontact.CaseBrepresentsabearinggeometrywheretheendofthe bearinghasaangeboltedtoarigidxture.CasesCandDrepresentayokerigidly attachedtohalfofthebearinginthecenter.CaseConlyallowscompressionoftheliner 94 PAGE 107 Figure5.3. LocationsofDirichletboundaryconditionsforReynoldsequation 95 PAGE 108 Figure5.4. Computationaldomainforfourxtures,zerodisplacementsolidboundary conditionsareappliedonthehighlightedsurfaces/boundaries.Theinnerannularregionis thepolymerbearingliner,andtheouterannularregionisthealuminumhousing. directlyabovethexture,whereascaseDallowsforcompressionofthelinerand ovalizationunderthexture.AllresultsfromcasesB,C,andD,arecomparedwithcase A,arigidliner,housing,andshaft,forcomparisonandanalyses.Allresultsoutsideof subsection5.4.3usetheboundaryconditionsshownincaseA. 5.2.3Mesh Inthissectionthemeshingstrategyforthischapterisdescribed.Therearetwomeshes usedinthischapter,oneforthebasemodel,andaseparatemeshthatisusedforthe modelincorporatingbendingeectsforsection5.4.3.Bothmesheshavethesamesurface 96 PAGE 109 meshthatcontainsalltheReynoldsequationandsolidcontactelements.Thebending eectmodelmeshincludesanextraannulardomainforthebearinghousing,madeof aluminum,wheresolidboundaryconditionsareapplied.Bothmeshesareshowningure 5.5,mesh 1 ontheleftandmesh 2 ontheright.Mostoftheresultsinthischapteruse mesh 1 ,thatdoesnotincludethehousingdomain,andtheboundaryconditionsshownin caseAofgure5.4areappliedtotheoutsideofthebearingliner. Mesh 1 has 110 ; 892 DOF,mesh 2 has 258 ; 276 DOF.solutiontimesformesh 1 areon theorderof 12 hoursfora 100 pointstribeckcurve.Solutionstimesformesh 2 areonthe orderof 36 hours.Becauseoftheincreasedsolutiontimerequiredtoaccountforhousing bending,mostoftheworkinthischapterfocusesonrigidhousings.Theeectofexible housingsandxturelocationisconsideredinsection5.4.3.Itisfoundtobeasecond-order eectfortheexperimentalsetupconsideredinthischapter.Realworldbearingssystems thatmountthebearingxtureoutsideofthehullhavemuchmoreexiblehousingsthan studiedhere. Themaindriverformeshdensityisthesolidcontactelements.Iftheyarelargerthan L 20 forthecenteror L 175 fortheedgeinthezdirection,thestinesscannotbecalculated accuratelyandthesolverwillnotconverge.Theuidgradientsforthisproblemismuch smallerthaninpreviouschapterssincethepressuresaremuchsmallerMPavsGPaand thelengthscalesaremuchlargermmvs m m.Themappedsurfacemeshwasrenedboth radiallyandlengthwiseinthecontactareauntilbotheccentricitycomponentswere convergedtovesignicantgures.Itwasfoundthatameshsizeof R 50 isthe circumferentialdirectionwassucientinthecontactarea. 5.2.4SolutionProcedure Fullsystemsolutionofthismodelfailsinthetransitionbetweenfulllmlubricationand mixedlubrication.Thisislikelyrelatedtothestinessmatrixlosingsymmetryduringthe 97 PAGE 110 Figure5.5. Meshesusedforthischapter,rigidhousingleftandexiblehousingright. Detailofedgerenementshowninbottom. transitionregion.Moreworkshouldbedonetounderstandwhythismethodfails,asthe methodisveryecientandconvenient.Apossibleworkaroundwouldbetoextract informationaboutthestinessfromtheJacobiananduseittotondtheshaft equilibriumposition.Abruteforceversionofthismethodhasbeenimplementedthatis veryrobustbutontheorderofvetimesslowerthanthefullsystemsolution.Aowchart ofthenumericalimplementationisshowningure5.6. Aprovenmethodtoworkthroughthisissueistosolvetheproblemtransientlywith articialdampinginthesolidelements[18].Thismethodisstable,butrequires6iterations ontheloadequations,totaling150iterationsonthecoupleduid-structureinteraction problem.Themethoddescribedinthissectionusuallyconvergesin3iterationsontheload equations,requiringatotalof30iterationsonthecoupleduid-structureinteraction problem. TheinitialconditionisfoundbystartingathighspeedwithaGaussianuidpressureeld andaxedeccentricity.Theeccentricityisiterativelyincreasedtoaneducatedguessvalue thatnearlysolvestheloadequations.Thisinitialconditioncanbeusedforthefully 98 PAGE 111 Figure5.6. Solutionprocedureforthewaterlubricatedjournalbearingproblem. coupledsolverorthedecoupledsolver.Inthefullycoupledsolver,theuidpressure,solid displacement,andshafteccentricitiesaresolvedforsimultaneously.Eachsolutionisused astheinitialguessforthenextslowerspeeduntilthereissignicantsolidcontactandthe fullycoupledmethodfails.Atspeedslowerthanthefullycoupledsolvercanreach,the shafteccentricitiesaresegregatedfromtherestofthemodel.Theloadequationsarethen solvedbyrstsolvingforthelinearizedstinessofthesolidcontactanduidlmabout thecurrenteccentricity.Usingthisstinessaestimateoftheequilibriumpositionis calculated.Iterationsareperformeduntiltheeuclidiannormofthedimensionlessload vectorisbelowthespeciedthreshold,usually 1 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 . 5.3DimensionalAnalysis Therehasbeenextensiveworkonndingappropriatenon-dimensionalnumbersforoil lubricatedpointcontacts,asseeninchapterIIwhichwasextendedinchaptersIIIandIV. Inthissectionweconsidertheanalogousdimensionlessgroupsforthecurrentiso-viscous MEHLproblemforalinecontact.Thedimensionalparametersinvolvedinthisstudyare brokendownbycategoryintable5.2. Missingfromthislististheradialclearanceofthebearing,denedas c = R o )]TJ/F21 11.9552 Tf 11.955 0 Td [(R i , 99 PAGE 112 Table5.2: DimensionalparametersforthelinecontactMEHLproblem Category Parameter Description MLT Unit Geometric L Bearinglength L D Bearinginnerdiameter L R 0 Contactconformaility L T Linerthickness L LinerMechanicalProperties E 0 CombinedModulus M LT 2 H Hardness M LT 2 Rheological Viscosity M LT Topological S q Combinedsurfaceroughness L Operational ! Shaftspeed 1 T F Appliedload ML T where R o and R i aretheradiusofthelinerandshaftrespectively.Thecombinedcontact radius, 1 R 0 = 1 R i )]TJ/F19 7.9701 Tf 16.302 4.708 Td [(1 R o ,isanotherwaytorepresentthesameinformation.Theuseof R 0 over c ispreferredbecauseofitsuseinHertziancontactmechanicsandpointcontactMEHL dimensionlessgroups. Table5.2liststentotalparametersyieldingsevendimensionlessgroupsinthree dimensions.Notlistedintable5.2istheapparentasperityfrictioncoecient, a ,which diersfromthecalculatedormeasuredcoecientoffrictionduetouidfrictionallosses andgeometricnon-linearity.Listedin5.3aretheseven groupsfromthedimensional analysisalongwiththedimensionlessasperitycontactfrictioncoecient.Notethatthe twoHamrockandDowsonparameterscanbecombinedtocreateagroupanalogoustothe Sommerfeldnumber u m R 0 F .Therstthreeareaspectratiosassociatedwiththevarious lengthscalesofthebearing.Thenweseethefamiliar C and G groupsfromchapterIII,and thefamiliarHamrockandDowsonspeedandloadparametersfromchapterII. Recallthe y componentoftheloadequationfromeq.5.3: X F y = ZZ [ p + p c n y + r + a p c n x ] d W = F .4 Groupingtheuidandsolidtermsandnon-dimesnionalizing: 100 PAGE 113 Table5.3: Non-dimensionalparametersforthelinecontactMEHLproblem Non-dimesnionalGroup Description TypicalValues L D Ratioofbearinglengthtodiameter.Typically4formarineapplications 4 R 0 D Ratioofbearingconformailitytodiameter.Relatedto c D . 0 : 05 T D Ratiooflinerthicknesstobearingsize 0 : 08 C = S q R 0 Ratioofcombinedcontactradiustocombinedasperitysize 3 : 85 10 6 G = E 0 H Ratiooflinercombinedmodulustolinerhardness 8 : 8 U HD = u m R 0 E 0 HamrockandDowsonspeedparameter 6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(10 W HD = F ER 0 2 HamrockandDowsonloadparameter 5 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 a Asperitycontactfrictioncoecient 0 : 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 5 101 PAGE 114 1 F ZZ [ p c n y + a p c + p n y + r n x ] d W =1 .5 Wecannowdenethesolidanduidloadratiosas c = 1 F RR p n y + r n x d W and f = 1 F RR p n y + r n x d W respectivelyandthedimensionlessloadequation becomes: c + f =1 .6 thisissimilartothedenitionsof c and f fromchapterIII,buttheyincludethe shearstressesduetotheattitudeangleoftheshaft.Fitstonumericaldataofthesolid contactanduidloadratios, s , f ,arepresentedinsection5.4asafunctionof C , G ; U HD , W HD ,and a forxedvaluesof, L D , R 0 D and T D similarlytochapterIII. 5.4ResultsandDiscussion Resultsforasetoftypicalparameters,asdiscussedintable5.5,arepresentedinthis section.Acompletesolutionatanygivenspeedisdescribedbytheuidpressure, p ,the solidcontactpressure, p c ,thesoliddeformationsoftheliner, u;v;w ,andtheshaft eccentricitycomponents, e x ;e y .Filledcontourplotsoftheuidleftandsolidcontact middlepressureasafunctionofshaftspeedareshowningure5.7.Linerdeformations, p u 2 + v 2 + w 2 ,areondeformedsurfacesingure5.7right.Notethatthepressurescales areconsistentforeachspeedanddeformationscalesareconsistentthroughthewhole gure.AsseeninchapterIII,theuidpressuregeneratesenoughlifttocompletely separatethesurfacesatshaftspeedsabove N =550 rpm .Astheshaftspeeddecreases below N =100 rpm ,theshaftandlinerbegintotouchsignicantlyasseeningure5.7.At N =1 rpm themagnitudeoftheuidpressureisthreetimessmallerthanthesolidcontact pressure.At N =0 : 1 rpm theuidpressureis100timessmallerthanthesolidcontact pressure.ThetransitionfromEHLtoMLisaccompaniedbyadramaticshiftinshaft 102 PAGE 115 positionasshownintheshaftlocusplotingure5.8.Theshaftlocusplotshowsthe y componentofeccentricityasafunctionofthe x componentofeccentricityformultipleshaft speeds.Thisisthepaththeshafttakesthroughthetransitionfromfulllmtoboundary lubrication.Duringfulllmlubrication,theuidpressurebuildsupontheleadingedgeof thecontact,pushingtheshafttotheright.Atthespeeddecreases,asmallergapisrequired tobuildsucienthydrodynamicpressure,andtheshaftmovesdownandtotheleft,as seeningure5.8between N =1000 rpm and N =100 rpm .Thereissignicantsolid contactbelow N =100 rpm .Thedragofthesolidcontactcausestheshafttoclimbupthe leftsideofthebearing,ascanbeseeningure5.8between N =100 rpm and N =0 : 1 rpm . Nowwelookatintegratedquantitiesofthesolutiondescribedaboveingures5.7 and5.8.Thesystemcoecientoffriction, ,ascalculatedby: = 1 F ZZ fluid + solid d W .7 asafunctionofspeedfor12dierentcasesareshowningures5.9,5.10,5.11, and5.12.Thebluecurveisthebasecaseshowningures5.7and5.8,asdescribedintable 5.5andisconsistentineachgure.Figure5.9showstheeectofcombinedsurface roughnessontheStribeck.Eachcurvefollowsthesametrendinfulllmlubricationand transitionsintoMEHLwhenthelmthicknessreachesroughly 3 S q .Figure5.10showsthe eectofthehardnessontheStribeckcurve.Hardermaterialscarryahighernormalload foragivencontactarea,thereforethethetransitionfromfulllmtoMEHLhappensat higherspeed.Figure5.11showstheeectofbearingradialclearanceontheStribeckcurve. Smallerclearancestightertsgeneratemorehydrodynamiclift,shiftingthethestribeck curvetotheleft.Thiseectissimilartoreducingcombinedasperitysize.Figure5.12shows theeectofasperitycontactfrictioncoecientontheStribeckcurve.Thelimitof atlow speedisproportionalto a ,althoughtheyarenotequalduetotheattitudeangleofthe bearingseegure5.8.Figure5.13showstheeectofappliedloadontheStribeckcurve. Higherloadshavealowercoecientoffrictioninfulllmlubricationduetoincreased 103 PAGE 116 Figure5.7. Fluidpressureleftandsolidcontactpressuremiddleformultipleshaft speeds.Pressurescalesareconsistentforeachspeed.Soliddeformationsrightonadeformed surfacedeformationsarescaledup400x.Deformationscaleisconsistentthroughoutgure. 104 PAGE 117 Figure5.8. Shaftlocusplotforatypicalsolution,parametersdescribedin5.5.Shaftspeeds correspondingtothecontourplotsingure5.7arelabeled. contactconformaility.Althoughthecoecientoffrictionislower,thetorquerequirementis stillhigher.HigherloadstransitionintheMEHLathigherspeedsthanlowerloads. 5.4.1SolidandFluidLoadRatiosasaFunctionofDimensionlessParameters SimilartochapterIIItstothesolidloadfractionsarepresentedtorepresentthe 528 solutionspresentedingures5.9,5.10,5.12,and5.13.Thettingcoecientsin equation5.8arepresentedintable5.4for n =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 .Theminimumerrorisobtained with n =5 and n =6 , ` 2 =0 : 148 . Thetotalnumberofnumericaldatapointsusedinthetsis 528 .Eachmodehas 7 ttingparameters,for n =6 therearetotal 42 ttingparameters,thusthereisnoconcern forovertting.UnlikethetsinchapterIII,theerrordoesnotdecreasewitheach successivemode. c =1 )]TJ/F21 11.9552 Tf 11.956 0 Td [( f = n X i =1 a i exp )]TJ/F24 11.9552 Tf 5.479 -9.684 Td [()]TJ/F21 11.9552 Tf 9.299 0 Td [(b i U c i W d i C e i G fi g i .8 105 PAGE 118 Figure5.9. Systemfrictioncoecientasafunctionofshaftspeedformultiplevaluesof combinedsurfaceroughness, S q . Figure5.10. Systemfrictioncoecientasafunctionofshaftspeedformultiplevaluesof linerhardness, H . 106 PAGE 119 Figure5.11. Systemfrictioncoecientasafunctionofshaftspeedformultiplevaluesof radialclearance, c . Figure5.12. Systemfrictioncoecientasafunctionofshaftspeedformultiplevaluesof asperitycontactfrictioncoecient, a . 107 PAGE 120 Table5.4: Coecients a i , b i , c i , d i ,and e i andL2normerrorestimatesfortsofEq.5.8tothe528numericaldatapoints presentedingures3.8and3.9. n Values ` 2 1 ` 2 =0 : 596 2 ` 2 =0 : 175 3 ` 2 =0 : 153 4 ` 2 =0 : 149 5 ` 2 =0 : 148 6 ` 2 =0 : 148 108 PAGE 121 Figure5.13. Systemfrictioncoecientasafunctionofshaftspeedformultiplevaluesof appliedload, F . 5.4.2ExperimentalValidation ThissectionbrieydescribesacustomtestrigbuildandoperatedbyXdotEngineering andAnalysis.Aselectionofdatacollectedbythetestrigispresented,processed,and comparedtothenumericalmodeldescribedinthischapter.Themodelisdeterminedtot theexperimentaldatareasonablywell.Shortcomingsofthemodelandtheexperimentare discussed,andrecommendationsforfutureworkarepresented. ExperimentalTestDevice Inthissectionprovidesabriefdescriptionofthecustomexperimentaltestdevicebuiltby XdotEngineeringandAnalysis.Themaingoalofthetestdeviceistomeasurethereaction torqueonthebearingimposedbyaspinningshaftwithanappliednormalload.Adiagram ofthetestrigisshowningure5.15.Theshaftisspunbyadualdrivemotorand supportedbytworollingelementbearings.Thetestbearingandhousingismountedonthe shaft,andtheloadisappliedtothehousingbyasupportarm.Thetorquetransmitted 109 PAGE 122 Figure5.14. Solidloadfractionasafunctionof U HD forseveralvaluesof C , W HD , G ,and a .Numericaldataisrepresentedwithsolidlines,tsofeq.5.8with n =6 seetable5.4 shownwithcircles.Dimensionaldescriptionsofthesecurvescanbeseeningures5.9, 5.10,5.12,and5.13.Euclidiannormofthettingerroris ` 2 =0 : 148 . 110 PAGE 123 Figure5.15. Freebodydiagramoftestrig throughfromtheshaftthroughthetestbearingismeasuredonaloadcellconnectedtothe loadapplicationarm.Apictureofthetestrigisshowningure5.16. Thewiderangeofoperatingtorquesandspeedsrequiredmakethedesignofthisrig complicated.Themotormustbecapableofmaintainingaconstantspeedat 0 : 1 rpm and at 1000 rpm withvaryingtorquerequirements.Ahighandlowspeedmotorwerecoupled togethertoaccomplishthis.Thetorquesensormustbeaccurateatverylowandveryhigh torques,posinganotherchallenge. RawData Atypicalrunwillhavemultiplesinusoidalsweepsfrom 0 : 1 rpm to 600 rpm andbackto 0 : 1 rpm .Rawdataforaselectedrunisshowningure5.17.Thetopleftofgure5.17 showstheshaftspeedasafunctionoftime,thetoprightshowsthemeasuredreaction torqueasafunctionoftime,andthebottomleftshowsthemeasuredreactiontorqueasa functionofshaftspeed.AbroadoutlineofaStribeckcurvecanbeseeninthebottomleft ofgure5.17.Inthissectionweanalyzetherawtorqueasafunctionofspeeddata,lter outanytransientorstartupeects,andaveragemultiplecurvestoobtainanaverage 111 PAGE 124 Figure5.16. Pictureoftestdevice,fromErikSwansonatXdotEngineeringandAnalysis. Stribeckcurve.Wealsolookforhysteresis,ordierencesbetweenthetorquedatawhile rampingupinspeedversusrampingdowninspeed. Therststepinourdataprocessingistoseparatetherampupdatafromtheramp downdataasshowningure5.18.Thetorquedataisseparatedinto'rampup,'showin redifthedatapointwasmeasuredwhileshaftspeedwasincreasing,or'rampdown,' showinblackifthetorquewasmeasuredasspeedwasdecreasing.Thenabinaveraged torqueandabinaveragedshaftspeediscalculatedforrampupandrampdown,asdened ingure5.19,for100log-spacedbins.Therampupandrampdownbinaveragedtorques asafunctionofbinaveragedshaftspeedisshowningure5.19.Figure5.19showsminimal dierencesintorquebetweenrampingupandrampingdown.Withoutasystematic uncertaintyanalysisofthetestrig,wecannotcondentlyapplyerrorbarstothisdata. Wethereforeignorehysteresisandaveragetherampupdataandtherampdowndata togetherinthesamemannerasshowningure5.19. 112 PAGE 125 Figure5.17. Rawdatafromthetestrig.Shaftspeedasafunctionoftimetopleft, reactiontorqueasafunctionoftimetopright,andreactiontorqueasafunctionofshaft speedbottomleft.Testwasperformedwiththegeometryandoperatingconditionslisted intable5.1. Figure5.18. Seperatingtherampupredandrampdownbluespeedandtorqueraw data. 113 PAGE 126 Figure5.19. Rawdatafromthetestrig.Shaftspeedasafunctionoftimetopleft, reactiontorqueasafunctionoftimetopright,andreactiontorqueasafunctionofshaft speedbottomleft. Results Thebinaveragedexperimentaldatadiscussedpreviouslyisplottedagainstthenumerical modeldescribedinthischapteringure5.20.Thevaluesoftheparametersusedinthe modelandtheirjusticationislistedintable5.5.Allparametersaremeasuredwhere possible.Theexceptionsarethewaterviscosity,linercombinedmodulus,asperitycontact frictioncoecient,andthenonzerotorqueoset.Atextbookvalueisusedforwater viscosity[43].ThemanufactureofthebearinglinerprovidesvaluesforYoung'smodulus andPoison'sratio,whichareusedtocalculatedthecombinedmodulus, E 0 . Theasperitycontactfrictioncoecientdependsonthedetailsofthecontactpair, including:mechanicalproperties,surfacetopography,thecontactgeometrycurvedorat, therelativeslidingspeed,andthepresenceoflubricantsorothercontaminants.Onemay beabletomeasureitusinganatomicforcemicroscope,butalloftheabovevariableswould havetobeaccountedfor.Inlieuofthiscomplicatedexperiment,weassume a =0 : 45 so themodelandtheexperimentagreeintheboundarylubricationregime.Thebearing manufacturelistssystemfrictioncoecientsrangingfrom 0 : 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(0 : 35 dependingonthe presenceoflubricantsandtheslidingspeed,butwithnomentionofhowitwasmeasured. 114 PAGE 127 Thenonzerotorqueoset,T ,isadicultparametertoaccountforasitdepends onthespecicbuild.WecancalculateanapproximateT byttingalinetothehigh speedportionoftheexperimentaldataandtakingT tobetheyintercept.Thisprovides anestimatebecauseonewouldexpectthetorquetobezeroatzeroshaftspeed. Theagreementbetweentheexperimentalandnumericaldataisshownforthree dierentappliedloadsingure5.20.Thereisgoodagreementneartheliftopointandthe boundarylubricationregime.Theagreementinthefulllmlubricationregimeisadequate, butthereisanobviousslopedierencethatmaybeduetoturbulenteectsathighshaft speed.ThereisalsosomediscrepancyinthetransitionfromMEHLtoBLrpm.Thismay beduetomicroEHLeects,oraresultofanassumedconstantvalueofasperitycoecient offrictionforallspeeds,whichisdiscussedfurtherinsection5.4.5. 5.4.3Bending Inbearingmodelsthataccountforbending,thehousingisconsideredtobeinnitelyrigid andtheshaftisconsideredtobeexible.Inthisworkweconsidertheconverse.Thisisin ordertodeterminetheeectxturelocationonlubricationperformance.Small deformationsofthehousingmayseeminsignicant,butconsideringthatthedierence betweenfulllmlubricationandasperitycontactcanbelessthanamicron,evensmall deformationscanbeimportant. Figure5.21ashowstheuidpressuredistributiononthedeformedbearinglinerfor thesupportedalonglengthcaseAat.Thesoftpolymerlinerhasspreadtheload circumferentially,increasingtheeectiveloadcarryingareacomparedtoarigidliner.The maximumpressureislessandtheminimumlmthickness,uniformacrossthelengthofthe bearing,ismorethantherigidlinercasenotshown.ThisisatypicalresultforanEHL modelandisrepresentativeofwhatisseenintheliterature. Figure5.21b-dshowtheuidpressuredistributionsforthexed-at-ends, xed-at-center-top,andxed-at-center-bottomcases,respectively,atthesameoperating 115 PAGE 128 Figure5.20. Experimentalvalidationat F =6404 N top, F =9606 N middle,and F =12 ; 808 N bottom. 116 PAGE 129 Table5.5: DimensionalparametersforthelinecontactMEHLproblem Category Parameter Description Value Justication Geometric L Bearinglength 12 in Asmachined D Bearinginnerdiameter 3 in Asmachined R 0 Contactconformaility 151 : 5 in Calculatedfrommeasured shaftandliner T Linerthickness 0 : 25 in Asmachined Liner Mechanical Properties E 0 CombinedModulus 86 MPa Frommanufacturer Thordoninc H Hardness 10 MPa ConvertedfromshoreD durometerand Rheological Viscosity 0 : 001 Pa s Textbookforfreshwater Topological S q Combinedsurface roughness 0 : 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m Measuredwithstylus prolometerafterbreakin Operational ! Shaftspeed 0 : 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(600 rpm Halleectsensor F Appliedload 6404 N; 9606 N; 12 ; 808 N Straingaugeloadcell Tribological a Asperitycontactfriction coecient 0.45 ttoexperimentaldata TestRig Correction T Nonzerotorqueoset 0 : 432 in lbf ttonumericalmodel 117 PAGE 130 Figure5.21. Fluidpressuredistributionsplottedondeformedinnersurfaceofthepolymer linerforeachcasedescribedingure5.4at N =1000 rpm .Notethepressurescaleondis signicantlydierentthana-c. 118 PAGE 131 Figure5.22. FluidPressuredistributionsplottedondeformedinnersurfaceofthepolymer linerforeachcorrespondingcaseingure5.4at N =100 rpm . 119 PAGE 132 andloadingconditionsasinFigure5.21aonthedeformedbearingliner.Figure5.21bshows thatatthecenterofthebearing,theuidpressureforcesthehousingtobend,increasing thelmthicknessandreducingthepressure.Thisconcentratestheloadneartheendsof thebearingwheretheboundaryconditionsarexedsolidandzerouidpressure.Theopen endofthebearingforcesthepressurestodropsignicantlyneartheendofthebearing, resultinginzeropressureandconsequentlynobearinglinerdeformation,diminishingthe EHLeectandcausingalmthicknesses3timessmallerthanthexedalonglengthcase. Figure5.21cshowsthepressuredistributionforthexed-at-center-topcaseat 1000 rpm .Theloadisconcentratedinthecentercomparedtothecasewherethehousingis heldalongtheentireperiphery,duetotheendsofthebearingassemblybendingawayfrom theshaft.Figure5.21dshowsthepressuredistributionforthexed-in-center-bottomcase. Theloadishighlyconcentratedatthecenterofthebearing,althoughtheloadisspread outinthecircumferentialdirection. Figure5.22showsresultsforthesameboundaryconditionsat 100 rpm .Theshapeof thepressuredistributionatthislowerspeedisverysimilartothatathighspeed,but slightlymoreconcentratedaswouldbeexpectedduetotheloadbeingsupportedovera smallerarea.Asisexpected,thelmthicknessdecreasesatlowerspeeds. Figure5.23showstheareaofthebearingsurfacewherethelmthicknessisbelow 2 m forseveraldierentspeeds,assumingthisiswheresolidcontactwouldoccur.The shapeofthesolidcontactareaisuniqueforeachsetofboundaryconditions.ForCaseA, theasperitycontactwilloccurattheexitcontractionoftheEHLcontact.CaseBshows solidcontactneartheuidboundaryconditionsattheentranceandexit,whichisalsovery nearthesolidboundaryconditions.CaseCexhibitssolidcontactinlocationssimilarlyto caseA,butmorecentralizedduetothexed-at-centerboundarycondition.CaseDshows solidcontactatthecenterofthebearingdirectlyabovethebearingsupport.Thisis unsurprisingbecausetheendsoftheshaftbendawayfromthexedboundarycondition, increasingthelmthicknessawayfromthecenterandlocalizingtheloadatthecenter. 120 PAGE 133 Figure5.23. Contoursshowingareaswherethelmthicknessislessthan 2 m atdierent shaftspeedsforeachconguration.Theseareaswillexperiencesolidcontact. 121 PAGE 134 Figure5.24. Crosssectionofthedeformedlinerandhousinginthecenteroftheassembly forxedincenter,topleftandxedincenter,bottomright.500xdeformationscale. N =1000 rpm . Figure10showsthattheamountofsolidcontactchangesasyoumovedownthelengthof thebearing,andthatitisdependentonthelocationofthesupportstructureofthe bearinghousing. Figure5.24showsthedeformedshapeofthepolymerbearinglinerandthehousing forcasesCleftandDright,bothscaledwitharelativemaximumdisplacementof 0 : 1 . SinceCaseCissupportedonthetopandloadedatthebottom,itstretchesoutthe housingintoanegglikeshape.Thisincreasestheconformityofthecontactbydecreasing theeectiveclearance,whichiswhytheminimumlmthicknessisthehighestofallthe casesstudied.Holdingthebearinghousingatthebottomdoesnotgeneratethiseect becausethehousingissupporteddirectlyundertheload.OnecouldalsothinkofcasesC andDashavingthesameboundaryconditionsbutloadedinoppositedirections,sothat onemayconcludethattheperformanceofthisbearingdependsontheloadingdirection. 5.4.4Weareects Thebearingsstudiedinthissectionshowobvioussignsofwearaftertesting.Apictureof theworninareaisshowningure5.25.Thepolishedregionspansthelengthofthe bearingandabout30*circumferentially.Thisiswheretheshaftandlinersawsignicant solidcontactforanextendedperiodoftime.Itishypothesizedherethatwearhastwo 122 PAGE 135 eectsonthesurfaceofthebearingliner: 1.Polishing 2.Bulkmaterialremoval Thepolishingeectcanbedetectedwiththenakedeyeandbytouch.Alsoprolometer measurementsshowthatthebearingsurfacehasan R a =10 m asmachined,and R a =0 : 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m afteratypicalroundoftesting.Thereareprobablymoredetailedeects tothesurfacetopographythathaveyettomemeasured,butcircumferentialgroovescan beseen.Itisassumedthatthechangeinsurfacetopographyoftheshaftisnegligibledue tothelargedierenceinhardnessbetweenthepolymerlinerandthesteelshaft.Asseenin gure5.9,reducingthesurfaceroughnessshiftstheStribeckcurvetotheleft. Bulkmaterialremovaleectstheshapeofthebearingsurface.Mostbearing simulationcodesassumeaperfectlycylindricalsurface.Thisisnotthecaseforbearings thathavebeeninserviceforalongtime.Thetypicallifetimeofawaterlubricatedbearing is30years.InthissectionweuseamanipulatedformofReyeshypothesisfrom[29],eq 5.9,toestimatethechangeinbearinglinerthickness,anduseourmodeltocalculatethe resultingchangeinlubricationperformance: w d = K R! D t p c H .9 where Q isthetotalwearvolume, K isadimensionlessconstant, F istheappliedload, R! t istheslidinglength,and H isthehardnessofthesofterofthetwocontacting materials.Writingequation5.9inaformthatcanbesolvedtransientlyforweardepth: w i +1 d = w i d + K R! D t p c H .10 123 PAGE 136 Where w d isthecumulativeweardepthineachsurfaceelement.Equation5.11canbe non-dimensionalizedbydivingbothsidesoftheequationbythebearingradialclearance, c : W i +1 d = W i d + K R! D t c p c H .11 where W i d isthedimensionlessweardepthattimestep i .Thedimensionlessparameteron theright-handsideoftheequation, K R! D t c representsthedimensionlessrateofwear,and p c H dictateswhereinthebearingthewearoccurs. Tofullycouplethewearproblemtothecombineduidstructure+loadequation problemdescribedinsection5.2,theweardepthmustbetakenintoaccountintheexplicit lmthicknessequation: h explicit = c )]TJ/F21 11.9552 Tf 11.955 0 Td [(e x cos )]TJ/F21 11.9552 Tf 11.955 0 Td [(e y sin + ucos + vsin + w d ;z .12 where h explicit coupleswiththemodiedlmthicknessequation, h t ,andtheReynolds equationsintheusualwayasdescribedinsection5.2.Sincetherestofthemodelis alreadyquitelarge,itisundesirabletosolveittransiently.Aquasistaticmodelfor p;p c ;e x ;e y ; and u;v;w iscoupledwithatransientmodelfortheweardepth. Thewearconstant, K ,maydependonseveraloperatingparametersandthedetailof thesurfacetopology.Withoutareliablemeasurementofthedimensionlesswearconstant thismodelcannotpredicttherateofmaterialremoval.Inthiswork,itisassumedthatthe materialremovalrateismuchslowerthananytransientbearingeectscausedbywear. Figure5.26showstheweardepthonthebearingafteranintegratedwearparameter of K R! D t c .whichroughlycorrespondstoxhoursatyrpmwithawearconstantofz. Figure5.27showstheeectofthewearproleshowningure5.26ontheStribeck curve.Themodelsareindistinguishableinthefulllmlubricationregime.Thewornin modelliftsoataslightlyslowerspeedthantheunwornbearing.Theworninbearing 124 PAGE 137 Figure5.25. PictureofworninbearingfromXdotEngineeringandAnalysis. remainsatalowercoecientoffrictionthroughoutthetransitionintoboundary lubrication. Figure5.28showsthedierenceinshaftlocusbetweenthewornandunwornmodel forthewearproledescribedingure5.26.Similarly,togure5.27,themodelslargely agreeinthefulllmregime.Theworninmodelgoesdeeperintothebearingsmaller e y thantheunwornbearing,whichisunsurprisingsincethereis 1 m lessmaterialthere.This trendcontinuesuntilthetwomodelconvergeagainintheboundarylubricationregime. Itwashypothesizedthatwearineectscouldclosetheremaininggapbetweenthe experimentalandnumericalresultspresentedingure5.20.Basedontheresultsingure, itdoesnotappearthatwearininuencestheshapeoftheStribeckcurveinthe 10 rpm rangewherethereispooragreementbetweenthenumericalandexperimentalresults.More workneedstobedonetobuildamorerobustwearmodelthatcancalculatewearproles onthesameorderasseeninexperiments. 125 PAGE 138 Figure5.26. Wearprole, W d [ m ] ,after t = xhours . K = x: Figure5.27. ComparisonofwornandunwornStribeckcurve. 126 PAGE 139 Figure5.28. Shaftlocusshowingthepathoftheshaftforboththewornandunworncase. Figure5.29. Weardepthprolesasafunctionofcircumferentialangleforthecenterand edgeofthebearing. 127 PAGE 140 5.4.5Asperitycontactfrictioncoecientasafunctionofspeed Thetransitionbetweenstaticanddynamicfrictioncoecientsarepoorlyunderstood. Classicfrictiontheorystatesthatthereisadierencebetweenstaticanddynamicfriction coecient.Thereisnoprovenlawgoverninghowacontacttransitionsfromstaticto dynamic.Itisbelievedtobegovernedbyasperitiesconformingtoeachother,increasing therealcontactarea.Somestudieshaveshownatimedependenceonrealcontactarea, evenformetals.Sincethebearinglinerundoubtablyhasaviscousresponseaswellasan elasticresponse,it'sreasonabletoconsiderafrictioncoecientthatisafunctionof contacttime,whichisafunctionofshaftspeed.Inthissectionanempiricalmodelis proposedfortheasperitycontactfrictioncoecientasafunctionofrelativeslidingspeed thattransitionsthefrictioncoecientfromstatictodynamic.Thegeneralformofthe modelisinspiritedbythedisagreementbetweenthenumericalandexperimentalmodel seeningure5.20,whereplottingtheratioof experimental numerical showsthequalityofagreement, andhintsatthefunctionalityof a N . Theproposedempiricalmodelfor a N is: a = 8 > > > > > < > > > > > : static N< 1 aN b 1 N 100 dynamic N> 100 .13 whichisapowerlawmodelwithhighandlowlimits,plottedasafunctionofspeedin gure5.30for a =1 : 35 and b = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 9 .Thismodelsharplytransitionsfrompowerlaw behaviortoit'shighandlowlimits.Thediscontinuousderivativedoesnotappeartocause anyinstabilitynumerically,andsincethereisnoexperimentaljusticationforthismodel, smoothtransitionsarenotconsidered.Thestribeckcurvewiththeasperitycontactfriction coecientmodelinequation5.13isshowningure5.30. Theagreementisnotperfect,butthetrendismuchimproved.Theresultsshownin 128 PAGE 141 Figure5.30. Shaftlocusshowingthepathoftheshaftforboththewornandunworncase. 5.30haveinspiredfrictioncoecientexperimentsatUCDenver.Thetestrigiscurrently underconstruction. 5.5Conclusions Thischapterpresentedanexperimentallyvalidatedmulti-regimelubricationmodelfor waterlubricatedjournalbearings.Theagreementbetweentheexperimentalresultsandthe modelisreasonablygoodovertherangeofspeedsconsidered,seegure5.20.The disagreementinthehighspeeddatacanlikelybeattributedtoinertialeects,whichthis modelneglects.Thedisagreementinthetransitiontoboundarylubricationcanbe explainedbyatransitionintheasperitycontactcoecientoffrictionasthespeedlowers andthelubricantissqueezedoutfromin-betweentheasperities.Anexperimental apparatusisbeingbuilttotestthisatUCDenvertotestthishypothesis. Thisnewmodelcoversalllubricationregimeswithasimpliedalgorithmthat requiresanorderofmagnitudelessiterationsforconvergencethanpreviousapproaches. Previousnumericalapproachesformulti-regimelubricationrequiresomesortofrelaxation 129 PAGE 142 toconverge.Mostnitedierenceschemesusesuccessiveover-relation,whichisan iterativesolutiontechniquethatheavilyweightspreviousiterationsresultsthataremixed inwiththenewsolution[39].Itisverystable,butquiteslowsincethetechniquerelieson solutionstopreviousiterations.Previousniteelementapproachestomulti-regime lubricationrelyonpseudo-timesteppingalgorithmswitharticialdampingtoensure convergencetothenon-linearsystem[18].Theniteelementmethodproposedhereusesa steadystatefullsystemapproachwithnorelaxationtocalculateuidandsolidcontact stinesscoecientsatxedeccentricity.Thestinesscoecientsareusedtoestimatethe equilibriumshaftpositionatthenextiteration. Thisnewmodelhighlightstheimportanceofxturelocationinwaterlubricated bearingsforthersttime.Fixturelocationcanbeusedtoimprovelubricationperformance byallowingthebearingtobendawayfromtheshaftinlocationswherethereissolid contact.Thisdecreasesthesolidcontactloadfraction,andthereforedecreasesthewear rate,seesection5.4.3. Themainpurposeofpropulsershaftbearingistosupporttheweightofthepropulser andthepropulsershaft.Thesebearingsareperiodicallycheckedforwear,andreplaced whenaspeciedweardepthismet.Thesebearingsareexpectedtohavelifetimesof10-30+ years.Bearingoperatinginfull-lmlubricationhavepracticallyzerowearrates.Toaid engineersinthedesignofthesebearingsystems,thesolidloadfractions, c ; arepresented withatasafunctionoffourrelevantdimensionlessparametersthatrepresentthe material,geometric,andoperatingparametersofthebearing,seesection5.4.1fordetails. When c =0 ,thebearingisoperatinginfulllmlubricationregimeandthewearratesare practicallyzero.Withthetsof c presentedinthischapter,anyengineercandetermine thelevelofsolidcontactpresentinabearingwithoutacomplicatedniteelementmodel. Thischaptercontainsover3,000solutionstoamodelwith110,000degreesof freedom.Eachsolutiontakesapproximately10minutestosolve.Forthe400solutionwith housingbending,themodelcontains258,000degreesoffreedom. 130 PAGE 143 Acknowledgment ThankyoutotheOceofNavalResearchforsponsoringthiswork. 131 PAGE 144 BIBLIOGRAPHY [1]BrandtA.andLubrechtA.A.Multigrid,analternativemethodforcalculatinglm thicknessandpressureprolesinelastohydrodynamicallylubricatedlinecontacts. ASMEJournalofTribology ,108:551,1986. [2]BrandtA.andLubrechtA.A.Multilevelmatrixmultiplicationandfastsolutionof integralequations. JournalofComputationalPhysics ,90:348,1990. [3]LubrechtA.A.Thenumericalsolutionoftheelastohydrodynamicallylubricatedline andpointcontactproblemusingmultigridtechniques. 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IndustrialLubricationandTribology ,68:416,2016. 139 PAGE 152 APPENDIXI CodeForFittingViscosityAndDensityModels TheMATLABcodeinthissectiontakesaninputleofviscosityanddensityasafunction ofpressureandtemperatureandtsthefollowingmodels,describedindetailintables2.2 and2.3,usingnonlinearregression: 1.Cheng 2.Roelands 3.ModiedWLF 4.Ashurst-Hoover 5.TaitEquationofstate 6.DowsonandHiggenson clc clear closeall %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%Readindata %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fn_in='sql_bair_2006_tbl4_table.csv'; [npmu,nprho,mu,rho]=mat_read_tablefn_in; 140 PAGE 153 %%%%changepressureunitstoMPa mu:,1=mu:,1*1E-6; rho:,1=rho:,1*1E-6; %mu2=munpmu+1:npmu+npmu,: %mu=mu2; %npmu=14; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%FitCheng %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TR_cheng=munpmu+1,2; muR_cheng=munpmu+1,3; tbl=tablemu:,1,mu:,2,logmu:,3/muR_cheng; cheng_fun=@b,xb.*x:,1+b*./x:,2-1/TR_cheng+b.*x:,1./x:,2; b=[1E-9;1;0.1]; mdl=tnlmtbl,cheng_fun,b; beta_cheng=mdl.Coecients.Estimate; mu_cheng=muR_cheng*expcheng_funbeta_cheng,mu:,1:2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%FitRoelands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TR_roe=munpmu+1,2; muR_roe=munpmu+1,3; tbl=tablemu:,1,mu:,2,logmu:,3/muR_roe; roe_fun=@b,x... logmuR_roe+9.67*... -1++5.1E-3.*x:,1... .^b/.1E-3*logmuR_roe+9.67.*... 141 PAGE 154 x:,2-138./TR_roe-138... .^-b*TR_roe-138/logmuR_roe+9.67... ; b=[.001;0.6]; mdl=tnlmtbl,roe_fun,b; beta_roe=mdl.Coecients.Estimate; mu_roe=muR_roe*exproe_funbeta_roe,mu:,1:2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%FitTaitEOS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TR_tait=munpmu+1,2; tbl=tablerho:,1,rho:,2,rho:,3; b2scale=1E3; tait_fun=@b,x -1/+b*log+x:,1./b*b2scale*exp-b*x:,2*+b.*... +b*x:,2-TR_tait; %k0k00bkav b=[11.74;1E-3;-.0063;0.0001]; mdl=tnlmtbl,tait_fun,b; beta_tait=mdl.Coecients.Estimate; vvR_tait=tait_funbeta_tait,rho:,1:2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%FitAshurst-Hoover %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TR_AH=TR_tait; %muR_AH=munpmu+1,3; %%%%CalculateBetaV 142 PAGE 155 vinfR=0.833;%cm^3/g Vmolec=0.8002;%cm^3/g R=0.6683; gamma=3.92; VR=vinfR/R; V_AH=VR*tait_funbeta_tait,mu:,1:2;%calcspecicvolumeviaTaitmodel betaV_data=./mu:,2.*Vmolec./V_AH:,1.^gamma; b4scale=1E6; b1scale=1E-8; %muRBqCQ b=[2.4;24.28;0.1051;7.126;1.981]; tbl=tablebetaV_data,logmu:,3; AHbeta_fun=@b,xlogb*b1scale+b*x.^b+b*b4scale*x.^b; %AHbeta_fun=@b,xlogb*b6scale+ b.*./x:,2.*Vmolec./x:,1.^b.^b+ b*b3scale*./x:,2.*Vmolec./x:,1.^b.^b; options=statset'MaxIter',10; beta_AH=nlintbetaV_data,logmu:,3,AHbeta_fun,b,options; %beta_AH=mdl.Coecients.Estimate; %beta_AH=b;%tnlmdoesn'tdoagreatjobhere...JustuseBair'scoecientsforthe extrapolation mu_AH=expAHbeta_funb,betaV_data; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%Plotthedataandts %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% color_dot={'bo','go','ro','co','mo','yo','ko','wo'}; color_line={'b','g','r','c','m','y','k','w'}; 143 PAGE 156 color_dline={'b-.','g-.','r-.','c-.','m-.','y-.','k-.','w-.'}; color_dd={'b:','g:','r:','c:','m:','y:','k:','w:'}; hFig1=subplot,2,1; holdo counter=1; fori=1:lengthnpmu semilogymucounter:counter+npmui-1,1,mucounter:counter+ npmui-1,3,color_dot{i}... ,mucounter:counter+npmui-1,1,mu_chengcounter:counter+ npmui-1,color_dd{i}... ,mucounter:counter+npmui-1,1,mu_roecounter:counter+ npmui-1,color_dline{i}... ,mucounter:counter+npmui-1,1,mu_AHcounter:counter+npmui-1,color_line{i}... ; counter=counter+npmui; holdon end % legend['T=293.15';'T=313.15';'T=338.15';'T=373.15'],'Location','NorthWest' ylabel'LogViscosity[mPa*s]' xlabel'Pressure[MPa]' title'Viscosity-Pressure-Temperature[C]Relationship' axissquare counter=1; hFig2=subplot,2,2; fori=1:lengthnprho plotrhocounter:counter+nprhoi-1,1,rhocounter:counter+ 144 PAGE 157 nprhoi-1,3,color_dot{i}... ,rhocounter:counter+nprhoi-1,1,vvR_taitcounter:counter+ nprhoi-1,color_line{i}... counter=counter+nprhoi; holdon end legend['T=293.15';'T=313.15'],'Location','SouthWest' ylabel'RelativeVolume[1]' xlabel'Pressure[MPa]' title'RelativeVolume-Pressure-Temperature[C]Relationship' axissquare %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%EXTRAPOLATERECKLESSLY!!!! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% np=100; p=1.3*logspace1,9,np'./1E6; T=zerosnp,1; T:=TR_cheng; %%%%cheng mu_cheng=muR_cheng*expcheng_funbeta_cheng,[p,T]; %%%%Roelands mu_roe=muR_roe*exproe_funbeta_roe,[p,T]; %%%%Tait vvR_tait=tait_funbeta_tait,[p,T]; V_tait=VR*vvR_tait; %plotp,vvR_tait 145 PAGE 158 %%%%AH mu_AH=expAHbeta_funb,[V_tait,T]; betaV=./T.*Vmolec./V_tait:,1.^b; gure semilogyp,mu_cheng,p,mu_roe,p,mu_AH,'-o',munpmu+1:npmu+... npmu,1,munpmu+1:npmu+npmu,3,'o' legend'Cheng','Roelands','AH','Bair','Location','NorthWest' %axis[0200001E10] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%Bair006plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %gure %betavplot=sort[betaV_data,mu_AH],1,'ascend'; %semilogy./betavplot:,1,betavplot:,2,1./betaV_data,mu:,3,'o' Forfuturework,Iwouldrecommendexploringtheopensourcestatisticsand regressiontoolboxdevelopedbySandiaNationalLabs,Dakota.Moreinformationcanbe foundhere:https://dakota.sandia.gov/.Iwasmadeawareofthispackageafterwritingthe MATLABcode.IbelieveDakotaismorepowerfulandversatilethanMatlab'sNLMFIT function.IheardofthispackageattheannualSTLEmeetingduringatalkcorrelating moleculardimesnionsoflubricantstomacrorheologicalparameters. 146 PAGE 159 APPENDIXII ValuesOfIntegralsOfTheType f n H = R 1 H s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H n s ds Thefollowingtableprovidesusefulvaluestotheintegralsofthetype f n H = R 1 H s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H n s ds ,takenfrom[46]. Despitesomeauthorsclaims,thereisananalyticsolutionto f 0 : Itisincludedbelow aswellas f )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 H = R H s )]TJ/F21 11.9552 Tf 11.955 0 Td [(H n s ds forcompleteness,wherethesuperscript-1 denotesthattheintegrationiscomingfromtheoppositedirection. f 0 H = 1 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(erf H p 2 .1 f )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 H = 1 2 1+ erf H p 2 .2 Itshouldalsobenotedthatitispossibletogetananalyticsolutiontoany f n where n isanintegerbyintegrationbyparts.Sincetheonlycriticallyimportantintegralsofthis typeare n =0 and n =3 = 2 forMEHLproblemswedonotdiscussthesefurther.Instead, MATLABcodeforthenumericalintegrationofthesefunctionsisprovidedbelow. 147 PAGE 160 Table2.1: Valuesof f 0 H ;f 1 H ,and f 3 = 2 H takenfrom[46]. H f 0 f 1 f 3 = 2 H f 0 f 1 f 3 = 2 0 0.5000 0.3989 0.42999 1.8 0 : 3583 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 1428 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 1149 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0.1 0.46202 0.3509 0.3715 1.9 0 : 2872 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 1105 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 8773 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.2 0.4207 0.3069 0.3191 2.0 0 : 2275 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 8490 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 6646 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.3 0.3821 0.2668 0.2725 2.1 0 : 1786 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 6468 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 4995 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.4 0.3446 0.2304 0.2313 2.2 0 : 1390 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 4887 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 3724 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.5 0.3085 0.1987 0.1951 2.3 0 : 1072 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 3662 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 2754 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.6 0.2743 0.1687 0.1636 2.4 0 : 8198 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 2720 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 2020 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.7 0.2420 0.1429 0.1363 2.5 0 : 6210 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 2004 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 1469 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.8 0.2119 0.1202 0.1127 2.6 0 : 4661 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 1464 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 1060 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0.9 0.1841 0.1004 0 : 9267 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 2.7 0 : 3467 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 1060 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 0 : 7587 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 1.0 0.1587 0 : 8332 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 7567 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 2.8 0 : 2555 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 7611 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 0 : 5380 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 1.1 0.1357 0 : 6862 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 6132 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 2.9 0 : 1866 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 5417 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 0 : 3784 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 1.2 0.1151 0 : 5610 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 4935 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3.0 0 : 1350 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 0 : 3822 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 0 : 2639 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 1.3 0 : 9680 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 4553 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 3944 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3.2 0 : 6871 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 0 : 1852 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 0 : 1251 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 1.4 0 : 8076 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 3667 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 3129 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3.4 0 : 3369 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 0 : 8666 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 0 : 5724 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 1.5 0 : 6681 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 2930 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 2463 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3.6 0 : 1591 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 0 : 3911 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 0 : 2529 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 1.6 0 : 5480 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 2324 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 1925 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3.8 0 : 7235 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 0 : 1702 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 0 : 1079 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 1.7 0 : 4457 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 : 1829 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 0 : 1493 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 4.0 0 : 3167 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 0 : 7145 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 0 : 4438 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 148 PAGE 161 function[Fn]=fnn,low,high,N %fncalcluatesthefunctionofthetype: %Fnh=ints-h^n*phisdsfromhtoinf %wherephiisagaussiandistribution %nisthetypeofofintegraltobecalculated %lowisthelowerboundofthefunction %highistheupperboundofthefunction %Nisthenumberofpointsthefunctionisevaluatedat fun=@n,h,s1/sqrt*pi*s-h.^n.*exp-s.^2/2; h=linspacelow,high+high-low/N*2,N+2; fori=1:lengthh-2 y=funn,hi,hi:lengthh; t1=thi:lengthh',y','cubicinterp'; Fni=integratet1,high,hi; end end 149 |