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PAGE 1 MODIFICATION OF STANDARD LOAD LIFE EQUATION USED FOR DESIGNING ROLLING ELEMENT BEARINGS By NIKHIL DNYANESHWAR LONDHE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TH E DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2014 PAGE 2 2014 Nikhil Dnyaneshwar Londhe PAGE 3 To my loving parents w hose continuous support and encourage ment kept me motivated PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my parents for their support and encouragement during my graduate studies with thesis research. I would also like to thank all of my relatives and friends who have supported me throughout my education. Special thanks to my advisor, Prof. Nagaraj Arakere, for being a great mentor and friend throughout my graduate studies. His continued guidance helped me to explore different areas for thesis research and our countless fruitful discussions were very beneficial to my ongoing education. Another special thanks to Prof. Ghatu Subhash, for being my committee member and allowing me to avail facilities available in Center for Dynamic Response of Advanced Mate rials lab I am also grateful to Prof. Raphael Haftka, for offering the course on verification, validation, uncertainty quantification and reduction during my graduate studies. Through this course, I got a chance to learn powerful calibration techniques a nd apply those in my research. His insightful comments and suggestions regarding my research are greatly appreciated. I would also like to extend thanks to my lab mates Anup, Abir, Zhichao, Lihao and Bryan for their guidance and the wonderful time we spent together. The cheerful environment of our workplace always kept me motivated. I would also like to thank all the members of Mechanical & Aerospace Engineering Department, who ensured smooth running of our graduate program. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 1.1 Rolling Element Bearings ................................ ................................ .................. 12 1.2 Bearing Types ................................ ................................ ................................ ... 12 1.3 Bearing Materials ................................ ................................ .............................. 13 1.4 Current Study ................................ ................................ ................................ .... 14 2 SENSITIVITY OF BEARING FATIGUE LIFE TO THE VARIATIONS IN ELASTIC MODULUS OF RACEWAY MATERIAL ................................ .................. 17 2.1 Background ................................ ................................ ................................ ....... 17 2.2 Rolling Element Bearing Terminologies ................................ ............................ 19 2.3 Hertz Contact Stresses ................................ ................................ ..................... 22 2.4 Distribution of Radial Load inside Ball Bearing ................................ ................. 25 2.5 Basic Dynamic Capacity and Equivalent Radial L oad ................................ ....... 28 2.6 Stress Life relation ................................ ................................ ........................... 31 2.7 Numerical Analysis ................................ ................................ ........................... 32 2.7.1 Case A: Elastic Modulus of Raceway Material = 200 GPa ...................... 34 2.7.2 Case B: Elastic Modulus of Raceway Material = 180 GPa ...................... 36 2.8 Summary ................................ ................................ ................................ .......... 38 3 VALIDATION OF STANDARD LOAD LIFE EQUATION USED FOR DESIGNING ROLLING ELEMENT BEARINGS ................................ ...................... 42 3.1 Background ................................ ................................ ................................ ....... 42 3.2 Bearing Fatigue Life Dispersion ................................ ................................ ........ 43 3.3 Standard Load Life Equation ................................ ................................ ............ 45 3.4 Data Characteristics ................................ ................................ .......................... 47 3.5 Weibull Distribution Parameters ................................ ................................ ........ 51 3.6 Verification and Validation ................................ ................................ ................. 52 3.7 Statistical Calibration ................................ ................................ ......................... 53 3.8 Summary ................................ ................................ ................................ ........... 57 PAGE 6 6 LIST OF REFERENCES ................................ ................................ ............................... 68 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 69 PAGE 7 7 LIST OF TABLES Table page 2 1 Radial Load distribution inside bearing with elastic modulus of 200 GPa for raceway material ................................ ................................ ................................ 41 2 2 Radial Load distribution inside bearing with elastic modulus of 180 GPa for raceway material ................................ ................................ ................................ 41 3 1 Deep groove ball bearing (DGBB) geometry, load, speed and actual fatigue life data ................................ ................................ ................................ ............... 62 3 2 Angular contact ball bearing (ACBB) geometry, load, speed and actual fatigue life data ................................ ................................ ................................ ... 63 3 3 Cylindrical roller bearing (CRB) geometry, load, speed and actual fatigue life data ................................ ................................ ................................ .................... 64 3 4 Actual B earing life versus predicted life for DGBBs ................................ ............ 65 3 5 Actual Bearing l ife versus predicted l ife for ACBBs ................................ ......... 66 3 6 Actual Bearing life versus predicted life for CRBs ................................ .............. 66 3 7 Calibrated value of load life exponents ................................ ............................... 67 PAGE 8 8 LIST OF FIGUR ES Figure page 1 1 Typical cross section of rolling element bearings ................................ ............... 16 3 1 plot of data points for DGBBs and ACBBs ............................. 58 3 2 plot of data points for the CRBs ................................ .............. 58 3 3 Histogram of estimated load life exponents for ball bearings i.e. DGBBs and ACBBs ................................ ................................ ................................ ................ 59 3 4 Histogram of estimated load life exponents for cylindrical roller bear ings (CRBs) ................................ ................................ ................................ ................ 59 3 5 Empirical cumulative distribution of load life exponents for ball bearings ........... 60 3 6 Empirical cumulative distribution of load life exponents for cylindrical roller bearings ................................ ................................ ................................ .............. 60 3 7 plot of data points for DGBBs and ACBBs ............................ 61 3 8 plot of data points for CRBs ................................ ................... 61 PAGE 9 9 LIST OF ABBREVIATIONS ACBB AFB M A AISI ANSI CRB CVD DGBB ECDF ISO LP RCF SAE Angular contact ball b earing Ant i Friction Bearing Manufacturers Association American Iron and Steel I nstitute American National Standard Institute Cylindrical roller b earing Carbon Vacuum Degassed Deep groove ball b earing Empirical cumulative distribution f unction International O rganization of S tandards Lundberg and Palmgren Rolling contact f atigue Society of Automotive Engineers VAR VIMVAR Vacuum arc r e melt Vacuum induction m elt vacuum arc r e melt PAGE 10 10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for th e Degree of Master of Science MODIFICATION OF STANDARD LOAD LIFE EQUATION USED FOR DESIGNING ROLLING ELEMENT BEARINGS By Nikhil Dnyaneshwar Londhe May 2014 Chair: Nagaraj K. Arakere Major: Mechanical Engineering Since twentieth century, bearing manufacturers have sought to predict the fatigue endurance capabilities of rolli ng element bearings. The first generally accepted method to predict bearing fatigue life was published in 1940s by Lundberg and Palmgren (LP). The load life exponents used in the LP model were determined based on statistical analysis of the experimental da ta which was generated in 1940s. Hence this LP model tends to under predict bearing fatigue lives for modern bearing steels. As a part of thesis research, Elasticity and Statistical Analysis were performed to correct for bearing fatigue life under prediction. It has been reported that at subsurface depths gradient in material properties exists for case hardened steels. Study sho w that 10% decrease in elastic modulus results in 38.66% percent improvement in bearing fatigue life prediction. Therefore, t o account for this fact one more modification factor should be provided in standard load life equation. Based on the bearings endu rance data reported in the literature, validation analysis of basic LP formula was also performed to calibrate the values of load life exponent for rolling element bearings. Results indicate that to best represent the observed experimental data the load l ife exponent for ball bearings should be corrected to 4.27 with 68.2% confidence bounds PAGE 11 11 as [3.15, 5.37], against current value of 3, and for roller bearings it should be corrected to 5.66 with 68.2% confidence bounds [4.42, 7.26] as against the current val ue of 3.33. Hence there is need to mod ify standard load life equation used for bearings design. PAGE 12 12 CHAPTER 1 INTRODUCTION 1.1 Rolling Element Bearings Rolling element bearings such as ball and roller bearing are one of the most widely used machine elements. They facilitate rotational and linear movement between two moving objects. Hence they are used in the complex mechanical systems such as internal combustion engines and jet engines to support main engine shafts. S ome of the oil and gas services industry components such as mud rotors, mud pumps and rotating tables of drilling rigs use bearings. Their primary objective is to reduce friction and bear contact stresses without undergoing any deformation. Rolling element bearings generally experience radial load, thrust load or combined radial and thrust load. In later cases, the combination of two loads along with bearing internal geometry features determine bearing contact angle. Typical cross section of the rolling ele ment bearing s is presented in F igure 1 1 1.2 Bearing Types R olling element bearing generally has three important components : inner & outer raceways and rolling elements. Depending upon the type of rolling element used bearing can be classified into four different categories: 1. Ball bearings which use spherical balls as rolling elements 2. Cylindrical roller bearings which use cylindrical ro llers 3. Tapered roller bearings which use tapered rollers 4. Spherical roller bearings which use spherical barrels between inner and outer raceway For majority of bearing applications, the inner and outer raceway groove curvature radii range from 51.5 to 53% of the rolling element diameter. Ball bearings PAGE 13 13 can be of deep groove or angular contact type. DGBBs are generally designed for radial load where as ACBBs are designed for combined radial and thrust load. Some applications use double rows of rolling elements to increase radial load carrying capacity of the bearing. The contact angle for angular contact ball bearings generally does not exceed Thrust ball bearings has contact angle of B earings having contact angle more than are classified as thrust bearings. They are suitable for high speed operations. Compared to ball bearings, roller bearings are designed to carry much larger supporting load. They are usually stiffer and provide better endurance compared to ball bearing of similar size. Manufactur ing of roller element bearing is much difficult compared to that of ball bearing. Tapered roller bearings can carry combinations of large radial and thrust load, but generally they are not used in high speed applications. Similarly spherical roller bearing s are used in heavy duty applications, but they have very high friction compared to cylindrical roller bearings and are not suitable for high speed operations. 1.3 Bearing Materials Through hardened chromium rich AISI 52100 steel is widely used in bearing industry. Generally it is hardened to 61 65 Rockwell C scale hardness. Some of the manufacturers also use case hardened steels such as AISI 3310, 4620, 8620 or VIMVAR M50 NiL for manufacturing rolling bearing components. These case hardened steels are gen erally very hard at surface. Some of the applications in aero space industry, which demand higher power transmission to weight ratio, use silicon nitride balls on steel raceway. These are called as hybrid bearings or ceramic bearings. Ceramic balls are lig hter than steel balls; hence the force on outer raceway is reduced. PAGE 14 14 This in turn reduces friction and rolling resistance. As well, ceramic balls are harder, smoother and have better thermal properties compared to steel balls. These factors lead to 10 tim es better fatigue life for ceramic bearings compared to steel bearings. Bearings generally fail because of formation of spall in rolling surfaces. The primary reason for this destruction of bearing component surfaces is rolling contact fatigue. Historical ly it has always been difficult to accurately formulate rolling contact fatigue. The tri axial state of stress and out of phase stress strain relationship makes endurance predictions really difficult. Therefore bearing industry extensively relies on empiri cal data of bearing fatigue lives. The current industrial standards used for middle of twentieth century. It uses endurance data of the bearings which were manufactured using steel and manufacturing practices available at that time. Over the period of past 70 years, there is been significant improvement in the quality of bearing steels and accuracy of manufacturing practices. These resulted into significant improvement in endu rance capabilities of bearing steels. Therefore the current industrial standards severely under predict bearing fatigue lives for modern steels. Hence based on latest endurance data there is need to modify existing standard load life equation used for bear ings design. 1.4 Current Study Chapters in the following sections contain detailed study aimed at modifying standard load life equations used for bearings design. As a part of thesis research, elasticity and statistical analysis were performed to correct for the bearing fatigue life under prediction. Recent Nano indentation tests indicate that for case harden steels there exists gradient in carbide volume fraction at sub surface depth. This gradient in PAGE 15 15 carbide volume fractions result in to gradient in mate rial properties, such as hardness and elastic modulus, at sub surface depth. Chapter 2 explains the improved endurance capabilities of the bearing steel due to this gradient in elastic modulus. Numerical analysis explains sensitivity of bearing fatigue lif e to the variations in elastic modulus of the raceway material. Chapter 3 contains detail procedure of the statistical analysis used to calibrate values of load life exponent for both ball and cylindrical roller bearings. Bearing endurance data reported in Harris and McCool (1) was used for validation of standard load life equation used for bearings design Calibration r esults are found to be significantly different from current industrial standards. Hence based on this analysis it is recommended that stan dard load life equation used for bearings design should be updated with calibrated values of load life exponent for the accurate prediction of fatigue lives of modern bearing steels. PAGE 16 16 Figure 1 1 Typical cross section of rolling element bearings PAGE 17 17 CHAPTER 2 SENSITIVITY OF BEARING FATIGUE LIFE TO THE VARIATIONS IN ELASTIC MODULUS OF RACEWAY MATERIAL 2.1 Background Historically it has been always difficult to accurately predict rolling contact fatigue lives for ball and roller e lement bearings. Because of the out of phase relation between stress & strain and complex internal stress fields & energy dissipation, rolling contact fatigue is significantly different from structural fatigue. Generally, fatigue is observed as formation o f spall either in rolling element or raceway surfaces. However it has been observed that at low to medium level of loads raceway surfaces fail first than rolling element surfaces. Crack initiation and propagation are the t wo phases which are observed to le ad to the formation of spall Crack initiation phase corresponds to the time from starting till micro structure crack commenc ement at sub surface layers. Crack Propagation corresponds to the time required by crack to grow from sub surface layers to surface layers. However, it is reported that majority of bearing life is consumed in crack initiation. Test samples show that many cracks are formed below the surface that d oes not propagate to the top layers Hence in 1940s, Weibull proposed that first crack g enerated in the subsurface layers must lead to the spall formation. Since then number of revolutions to the observance of first spall is considered as a sufficient accurate measurement for bearing fatigue life. In twentieth ce ntury Lundberg and Palmgren were the first researchers who successfully developed fundamental mathematical model for the characterization of rol ling contact fatigue. The entire rolling element bearing design standards which were developed in past century found there origin in basic L P theory. However over the period of past 60 years it has been observed that these standards tend to under predict PAGE 18 18 fatigue lives for the bearings manufactured from modern case hardened steels. The industrial standards provide methods of calculating ba sic dynamic capacity and fatigue life rating for bearings which are manufactured using conventional steel and traditional manufacturing practices. Currently in reality, due to significant improvement in bearing steel quality and accuracy of manufacturing pract ices, very high fatigue lives are observed for rolling element bearings. The major drawbacks in the basic LP model which lead to this gap between theory and reality can be listed as follows: 1. In the derivation of equations for Hertz stresses it was assumed that all the deformations occur in elastic range. 2. Effect of surface shear stress was not considered while calculating subsurface stresses. Only shear stresses generated by compressive normal stress were considered. 3. LP model is valid only when geometries are perfect. i. e. diametrical clearance between rolling elements and raceways is zero. 4. Effects of compressive residual stresses were not considered. 5. All the properties of the material are assumed to remain constant throughout the 6. Detailed knowledge on mechanics of lubrication of concentrated contacts was not 7. Also temperature variation of material properties and hoop stresses which tend to reduce fatigue lives, were not given consideration. 8. Instead of orthogonal shear stress criterion, due to tri axial stress field, maximum sub surface shear stress and octahe dral shear stress can also be considered as a suitable failure criterion. Hence based on these issues there is need to modify existing load life equation used for bearings design to correct for under prediction of fatigue lives. To address one of such iss with respect to gradient in elastic modulus was performed for case hardened bearing steels. Following sections of this chapter contains standard ball bearing design problem PAGE 19 19 t aken from Harris ( 2 ) and results of the altercation of elastic properties of the bearing material on predicted fatigue lives. 2.2 Rolling Element Bearing Terminologies The various terminologies used in rolling element bearing analysis are as follows: Bore diameter of the bearing Outer most diameter of the bearing Inner raceway diameter Outer raceway diameter Pitch diameter of the bearing Rolling element diameter Diametrical clearance : Free angle of Misalignment are defined as: (2 1) (2 2) Osculation is defined as the ratio of radii of curvature of rolling element to that of the raceway in a direction perpendicular to the rolling direction. Mathematically it can be written as: (2 3) It determines the capacity of the bearing to carry the applied load. We can define f as: Means (2 4) PAGE 20 20 In no load state bearings have diametrical clearance as well as axial play. Once this axial play is removed, contact angle occurs between ball raceway contact and radial plane. This contact angle is called as free contact an gl e and it is estimated using clearance available under zero loading conditions. Let A be the distance between c enter of curvatures of the inner and outer raceway grooves and B is defined as the total curvature of the bearing. (2 5) Substituting E quation (2 4) in Equation (2 5) we get, (2 6) The free contact angle can be determined using Equation (2 7): (2 7) Bearings with diametrical clearance can float freely in axial direction under the condition of no load. Maximum axial movement of inner ring with respect to outer ring under the condition of no load is called as free end play. For single row ball beari ngs it can be calculated as, (2 8) It is observed that contact angles, end plays, maximum compressive Hertz stresses, deformations, load distribution integrals and fatigue lives are dependent on diametrical clearance. The maximum angular rotation between inner ring and outer ring of the bearings under no load conditions is called as free angle of misalignment. Its values are as small as fe w minutes and seconds. Harri s ( 2 ) gives the formulation for free angle of misalignment as foll ow, PAGE 21 21 (2 9) If bodies I and II are in rolling contact and their principal planes of radii of curvature are 1 and 2, then following nomenclature can be used: : Radius of curvature of body I in principal plane 1 : Radius of curvature of body I in principal plane 2 : Radius of curvature of body II in principal plane 1 : Radius of curvature of body II in principal plane 2 For a body with radius r, (2 10) Convex surfaces have positive curvature while concave surfaces have negative curvature. For analysis of contact between mating surfaces of revolution, following terminologies are used: Curvature Sum : (2 11) Curvature Difference: (2 12) For ball inner raceway contact Equation s (2 11) and (2 12) can be simplified further as (2 13) (2 14) Similarly, simplifying for ball outer raceway contact we ge t, PAGE 22 22 (2 15) (2 16) From Equation s (2 14) and (2 16) we can say, This means that elliptical area of greater ellipticity exists at inner raceway contact compared to outer r aceway contact. This will result into greater stress values at inner raceway compared to outer raceway. 2.3 Hertz Contact Stresses Due to small contact area, moderate loads induce large stresses on the surfaces of rolling elements and raceways. General ly in most of the rolling element bearing applications, the maximum compressive normal stresses are in the range of 1.4GPa to 3.5GPa. Sometimes for accelerated fatigue failure, laboratories perform bearings endurance tests at compressive normal stress up t o 5.5GPa. The effective area over which load is supported increases with depth at subsurface layers. This prevents high compressive Hertz stresses from permeating through entire raceway material. Therefore bulk failure of either raceway or rolling member is generally not a critical factor in bearing design. However rolling members are generally rigid machine elements, because of this Hertz stresses cause contact deformations which are of order of magnitude up to 0.025mm. Even this much destruction of rol ling surface is important factor in rolling bearing design. In 1881, Hertz proposed that when two bodies are in contact, instead of a point or line contact, a small contact area must be formed, causing the load to be distributed PAGE 23 23 over a surface. In determ ining these contact stresses, Hertz made following assumptions: 1. All deformations occur in elastic range and proportional limit of the material is not exceeded. 2. Loading is normal to the surface. Surface shear stress is not considered in determining subsu rface shear stresses. 3. The contact area dimensions are of low order of magnitude compared to radii of curvature of the bodies in contact. Hertz used the classical Elasticity approach by assuming stress function that satisfies compatibility equations and boundary conditions. He also assumed that shape of the deformed surface is that of an ellipsoid of revolution and the stress at geometrical center of elliptical contact area is given by, (2 17) where, Q is the normal load acting on the contact and a & b are the semi major axis and semi minor axis of the elliptical contact area. For point contact, Harris ( 2 ) gives following expressions for determining contact dimensions a & b: (2 18) (2 19) Similarly relative approach between two remote points on the contacting bodies can be given as, (2 20) Definition of terms used in E quations (2 18), (2 19) and (2 20) are as follows: (2 21) PAGE 24 24 (2 22) (2 23) (2 24) (2 25) (2 26) It should be noted that in E quations (2 18), (2 19) and (2 can be determined using Equation s (2 13) to (2 16). As pointed out earlier maximum compressive Hertz stress is always higher at inner raceway than at outer raceways. Hertz analysis only considers concentrated normal force applied on the contact. Jones, Thomas and Hoersch provided e quat ion s for principal stresses and occurring along Z axis at subsurface depths. Since the surface stress is maximum at Z axis, principal stresses were also fo und to be maximum along Z axis. aximum shear stress on z axis can be given as, (2 27) It should be noted that in the derivation of this e quation it is assumed that elements roll in the directio n of Y axis and raceway surface is along Z axis. The depth at which maximum shear stress occurs is approximately 0.467b for simple point contact and 0.786b for line contact. During bearing operation at ever y contact point the maximum shear stress on the z axis varies between 0 and For elements rolling along y direction, then for variation of y less than and greater than zero, the shear stress in th e sub surface PAGE 25 25 vary from negative to positive values, respectively. Hence the maximum variation of sh ear stress at any subsurface depth is 2 Lundberg and Palmgren named maximum value of this variation as maximum orthogonal shear stress. Since the maximum shear stress amplitude is always smaller than maximum orthogonal shear stress, it was assumed that maximum orthogonal shear stress is the significant criteria for fatigue failure of the surfaces in rolling contact. The subsurface depth at which maximum orthogonal shear stress occurs is approxima tely 0.49b. As well, it was observed that this stress generally occurs under the extremities of the contact ellipse with respect to the direction of motion. For case hardened bearing steels, carbides are considered to be weakness locations at which fatigu e failure is initiated. In general, the depth at which maximum orthogonal shear stress is observed is smaller than the depth at which maximum subsurface shear stress is observed. Hence for case hardening the depth at which maximum subsurface shear stress o bserved is used. 2.4 Distribution of Radial Load inside Ball B earing deformation and Q is the load acting on the con tact then according to Harris ( 2 ) following relation can be esta blished between them, (2 28) Where, K can be obtained by inverting Equation (2 20) and n = 1.5 for point contact & n = 1.11 for line contact, The total approach between two raceways can be obtained by adding individual approaches between rolling elements and each raceway. In mathematical form, PAGE 26 26 (2 29) Since both the contacts experience same amount of load Q, we can write (2 30) and, (2 31) For bearings experiencing only radial load, the radial deflection at any rolling element angular position is given by, (2 32) Where and is the diametrical clearance. Solving for maximum deflection and re substituting we get, (2 33) Where, (2 34) The effect of applied load is determined using angular extent of load zone which is defined as, (2 35) Zero clearance implies Equation (2 31) can be re written as, PAGE 27 27 (2 36) Combining Equation s (2 33) and (2 36) we get, (2 37) For bearing to be in static equilibrium, the sum of the vertical components of the rolling element loads must be equal to the applied radial load. Therefore using equilibrium conditions, we can see that (2 38) Substituting Equation (2 37) into Equation (2 38), we get (2 39) In the continuous integral form Equation (2 39) can be written as: (2 40) This expression can be reformulated as: (2 41) Where is defined as the load distribution integral which can be obtained by following expression, (2 42) Harris ( 2 ) gives different values of load distribu the y can be used for interpolat ion a t intermediate data points or Equation (2 42) can be solved in MATLAB. Solving Equation (2 (2 43) Substituting Equation (2 43) in Equation (2 41) we get, PAGE 28 28 (2 44) As a general practice for a given geometrical properties of the bearing under given operating conditions, above Equation s (2 34), (2 35), (2 42) and (2 44) are solved by tria l and error method. First value of are evaluated using Equation s (2 34), (2 42) and if Equation (2 is again repeated. 2.5 Basic Dynamic Capacity and Equivalent Radial Load Fatig ue life of ball bearing experiencing point contact is given by Equation (2 45) (Harris ( 2 )) (2 45) where, is the basic dynamic capacity of the bearing which is defined as the load that rolling element contact will support for one million revolutions of th e bearings. Lundberg and Palmgren ( 3 ) mathematically defined it as : (2 46) Here, upper sign corresponds to inner raceway contact and lower sign corresponds to outer raceway contact, R is the crowning radius of the rolling element, D is the rolling element diameter, r represents inner and outer raceway radii profile and Z is numbe r of rolling elements per row. For ball bearings Equation (2 46) can be simplified as, (2 47) Generally failure occurs at inner raceway, because maximum compressive hertz stress is usually higher at inner raceway contact than at outer raceway contact. PAGE 29 29 During rolling bearing operation, at a single point of time many contacts exists on the circum ference of inner raceway and each point on the raceways experiences cyclic load. Therefore, due to statistical nature of fatigue loading, complete load cycle must be considered for prediction of fatigue lives of the raceways. Lundberg and Palmgren ( 3 ) dete rmined that cubic mean load can be used for point contact loading. Hence for a ring rotating with respect to applied load equivalent radial load can be given as, (2 48) Equation (2 48) can be rewritten in continuous format as: (2 49) Therefore for rotating raceway Equation (2 45) can be represented as: (2 50) From Weibull slope distribution curve we know probability of survival of any given contact point on the non rotating raceway is given by, (2 51) U sing the product law of probability, the probability of failure of the ring can be expressed as the product of the probability of failure of the individual contacts. Hence using this fact and the empirical relation developed by L undberg and P almgren ( 3 ) : we get following relationship : (2 52) where, represents equivalent radial load experienced by non rotating raceway, (2 53) Equation (2 53) can also be written in discrete numerical format as PAGE 30 30 (2 54) Therefore, fatigue life of non rotating raceway can be given as, (2 55) Again using the product law of probability we can say that probability of survival of entire bearing is the product of the probabilities of survival of the individual raceways. The probability of survival of the rotating or inner raceway can be given as, (2 56) For non rotating raceway, (2 57) For entire bearing, (2 58) Solving for the case of from Equation s (2 56), (2 57) and (2 58) we get, (2 59) Based on the statistical analysis done in 1955, U.S. National Bureau of Standards recommended that e = 10/9 for point contact. Substituting this is in Equation (2 59) we get, (2 60) Equation (2 60) provides fundamental approach for calculation of fatigue lives of ball bearings which experience radial load on point contact. However, Equation (2 60) can also be used for evaluating fatigue lives of the bearings under combined loading PAGE 31 31 conditions. It should be noted that for using Equation (2 60) normal load at each element location must be determined using load distribution integral. It is obse rved that rolling elements such as balls or rollers never fail at lower or medium range of loads. It is reasoned that ball changes rotational axis readily. Hence, the entire ball surface is subjected to stress, spreading the stress cycles over greater volu me. This reduces the probability of ball failure before the raceway failure. Therefore bal l failure was not considered in the development of LP fatigue life model. 2.6 Stress Life relation For point contacts, by combining Equation s (2 17), (2 18) and (2 19) we can with con ( Zaretsky, et al. (4)), (2 61) Similarly using Equation (2 (2 62) Combining Equation s (2 61) and (2 62) we get, (2 63) As we can see from Equation (2 63) due to vary high value of stress life exponent of 9, bearing fatigue life is highly sensitive to maximum compressive hertz stress experienced by the raceways. Equation (2 63) can be further simplified to relate the bearing fatigue lives under two different states of stresses as: (2 64) PAGE 32 32 Using Equation (2 64) properties of state 2 can be evaluated based on the properties of the state 1. Similar expression can be derived for geometries in line contact: (2 65) It should be noted that in the derivation of Equation (2 65) conservative estimate of load life exponent is used. 2.7 Numerical Analysis Using the example s provided in Harris (2) numerical analysis was performed to measure the effect of gradient in elastic modulus on bearing fatigue lives. Details of this numerical stu dy are provided in this section Consider the case of 209 single row radial deep groove ball bearing. Inner raceway diameter = 52.291 mm Outer r aceway diameter = 77.706 mm Rolling element diameter = 12.7 mm Z: No. of balls = 9 : Inner groove radius = 6.6 mm : Outer groove radius = 6.6 mm : Radial load = 8900 N : Elastic modulus of raceway material (steel) = 200GPa : Elastic modulus of rolling element material (steel) = 200GPa PAGE 33 33 From Equation s (2 1) and (2 2) we can determine bearing pitch diameter an d diametrical clearance as: Using Equation (2 4) we can determine the osculation values for the inner and outer raceways as: Also, imply that free contact angle for the bearing Solving Equation (2 8) we get free end play for bearing as: From Equation (2 9) free angle of misalignment can be calculated as: We can see that free angle of misalignment is very small. For inner rac eway ball contact, curvature sum and curvature difference were calculated using Equation s (2 13) and (2 14) respectively: Similarly for outer raceway ball contact, curvature sum and curvature difference can be calculate d as: PAGE 34 34 We can see that curvature difference value is higher at inner raceway compared to outer racew ay. This indicates that ellipticity ratio is higher at inner raceway contact area compared to outer raceway contact area Because of this peak compressive hertz stresses are always higher at inner raceway than at outer raceway. After determining the geometrical properties of the bearing, contact stresses and deformations were determined. 2.7.1 Case A: Elastic Modulus of Raceway M aterial = 200 GPa For ball inner raceway contact Equation s (2 21), (2 22), (2 23), (2 24), (2 25) and (2 26) were simultaneously solved in MATLAB and the results obtained were as follows: Similarly for ball outer raceway contact we get, After solving Equation (2 20) we get following expre ssion: (2 66) Comparing Equation (2 31) with Equation (2 66), we can say tha t, (2 67) Solving Equation (2 67) for both inner and outer raceway contacts we get, Using Equation (2 30) we can determine, PAGE 35 35 Using the standard tables available for Load distribution integ ral in Harris ( 2 ) Equation s (2 34) and (2 44) were solved by trial & error approach and results were found to converge to These values were used to solve Equation (2 41) to determine the maximum load experienced by the rolling element raceway contact: Angular spacing between the rolling elements degrees Therefore, load experienced by each rolling element were obtained from Equation (2 37) and the values are provided in Table 2.1. Next step in the bearing analysis involved calculation of maximum compressive hertz stresses experienced by each raceway. For this Inner and outer raceway contact dimensions were determined using Equation s (2 18) and (2 19) as: Peak compressive hertz stress experienced by inner and outer raceways can be calculated using Equation (2 17) as: Using Equation (2 47) basic dynamic cap acity was determined for inner and outer raceway as: Cubic mean equivalent radial load experienced by inner and outer raceways was calculated using Equation s (2 48) and (2 54) as: PAGE 36 36 Based upon dynamic capacity and equivalent radial load we can calculate fatigue lives for both inner and outer raceway using Equation s (2 50) and (2 55) respectively: Equation (2 60) can be used to obtain combined fatigue life of the entire bearing as: 2.7.2 Case B: Elastic Modulus of Raceway M aterial = 180 GPa Now let us assume that elastic modulus of the bearing raceway material is decreased by 10% from 200GPa to 180GPa, due to gradient in the hardness of the case hardened steels at sub surface depths. Also it was assumed that gradient in and elastic modulus of the ball remains constant. After solving Equation (2 20) we get following expression : (2 68) Comparing Equation (2 31) with Equation (2 68), we can say that (2 69) Solving Equation (2 69) for both inner and outer raceway contacts we get, Using Equation (2 30) we can determine, Similar to Case A Equation s (2 34) and (2 44) were solved by trial & error approach and results were found to converge to PAGE 37 37 These values were used to sol ve Equation (2 41) to determine the maximum load experienced by the rolling element raceway contact: Angular spacing between the rolling elements degrees Therefore, load experienced by each rolling element were obtai ned from Equation (2 37) and the values are provided in Table 2.2. Comparing values in Table 2.1 and 2.2 we can see that the normal element loads acting at each rolling element locations are increased in magnitude except at This is owing to the fact that angular extent of load zone is slightly increased from 86.52 deg. in case A to 86.624 deg. in case B. Next step in the bearing analysis is calculation of maximum compressive hertz stresses experienced by each raceway. For this Inner and outer race way contact dimensions were determined using Equation s (2 18) and (2 19) as: Peak compressive hertz stress experienced by inner and outer raceways can be calculated using Equation (2 17) as: (4) approach and Equation (2 64), we can relate bearing fatigue lives with peak hertz stresses experience d by the inner raceway as, (2 70) Solving Equation (2 70) we get fatigue life of the inner raceways of the bearings in Case B as, PAGE 38 38 Simila rly fatigue life of the outer raceway of the bearings in Case B was found to be, From Equation (2 60) and above two results we can calculate, combined fatigue life of the bearing as: Percent change is bearing fatigue life from Case A to Case B can be calculated as: Therefore, if we decrease elastic modulus of the raceway material by 10% then, the combined bearing fatigue life i s improved by 38.66 % from Case A to Case B. This analysis underlines the fact that bearing fatigue lives are highly sensitive to the gradient in elastic modulus and because of this gradient there is need to provide modification factors in standard life rating equation for case hardened bearing steels. 2.8 Summary In modern manufacturing practices, to avoid wear and tear of surface layers, metal objects are case hardened. Case hardening allows fabrication of metals with hard surface layers and soft cores beneath them. It is generally performed on steels with low carbon content and it involves infusing carbon atoms into the case depths. Since it is very difficult to machine hardened surfaces, case hardening is done when the part has been formed into its fin al shape. Case hardened steels are particularly used in the applications in which metal parts are continuously subjected to deformation stresses. The soft core beneath the metal surface helps in absorbing stresses without cracking. PAGE 39 39 Hardness of the material is an engineering property and it is related to its yield strength, whereas, elastic modulus is an intrinsic material property which is related to the atomic bonding. Indentation hardness is the measure of materials ability to resist total deformation. Fo r most of the ceramics, contribution of elastic and plastic deformation to the total deformation is same. Hence, recent Nano indentation te sts data reported by Klecka et al. (5) indicate that hardness is directly dependent on elastic modulus. Also it is been reported that for case hardened bearing steels, there exists gradient in carbide volume fraction at sub surface depths. These results into gradient in material properties such as hardness and elastic modulus at sub surface depths. As a part of thesis research, aim of this analysis was to relate gradient in elastic modulus to the improved fatigue capabilities of the rolling element bearings. For elasticity analysis, sample bearing design problem was used from Harris ( 2 ) The sensitivity of bearing fati gue life to the variations of elastic modulus of the raceway material was studied. It was observed that if elastic modulus of the case hardened bearing steel raceway is decreased by 10% then L10 life of the bearing will be increased by 38.66%. It can be re asoned that decrease in elastic modulus and hardness result into decreased resistance to material deformation. Hence the effective area over which load is distributed increases at sub surface depths very rapidly. It was found that contact area dimensions f or point contact increases by 1.752% along semi major axis and by 1.77% along the semi minor axis. This in turn decreases the peak compressive hertz stress value by 3.57 % and hence we see the observed improvement in bearing fatigue life. PAGE 40 40 From this analysis we can say that bearing fatigue lives are highly sensitive to the variations in the elastic modulus of the bearing steel material. Hence it is reasonable to assume that gradient in elastic modulus of bearing steel is one of the contrib uting factors to the improvement in actual fatigue lives for the rolling element bearings. Therefore, life modification factors must be provided in standard life rating equation for accurate prediction of fatigue lives of case hardened bearing steels. PAGE 41 41 Table 2 1 Radial Load distribution inside bearing with elastic modulus of 200 GPa for raceway material Load distribution inside ball bearing (N) 0 1 4527.88 40 0.766 2845.39 40 0.766 2845.39 80 0.1737 65.451 80 0.1737 65.451 120 0.5 0 120 0.5 0 160 0.9397 0 160 0.9397 0 Table 2 2 Radial Load distribution inside bearing with elastic modulus of 180 GPa for raceway material Load distribution inside ball bearing (N) 0 1 4521.44 40 0.766 2847.031 40 0.766 2847.031 80 0.1737 71.46 80 0.1737 71.46 120 0.5 0 120 0.5 0 160 0.9397 0 160 0.9397 0 PAGE 42 42 CHAPTER 3 VALIDATION OF STANDARD LOAD LIFE EQUATION USED FOR DESIGNING ROLLING ELEMENT BEARINGS 3.1 Background Since twentieth century bearing manufacturers and users have sought to predict the fatigue endurance capabilities of rolling element bearings. The first generally accepted method to predict bearing fatigue life was published in 1940s by Lundberg and Palmgr en (LP). This method was adopted as a basis for all the standards developed in past century. The load life exponents used in the standard LP equation were determined by the statistical analysis of the experimental data which was generated in 1940s. Over th e period of past 70 years there is significant improvement in the quality of bearing steels. This resulted into increased fatigue lives of rolling components of the bearings and the current load life equation used for bearings design tends to under predict bearings life. Hence there is a need to correct standard life rating equation used for bearings design. It is well known fact that even if rolling element bearings in service are properly lubricated, properly aligned, and properly loaded, raceway an d rolling element contact surfaces tend to damage because of material fatigue. Because of the probabilistic nature of the fatigue of surfaces in rolling contact, rolling bearing theory postulated that no rotating bearing can give unlimited service hours. T he stresses repeatedly experienced by these rolling contact surfaces are extremely high as compared to other stresses acting on engineering structures. Hence the probability that rolling contact fatigue will have infinite endurance is close to zero. Roll ing contact fatigue is considered as a chipping off of metallic particles from the surface layers of raceways and/ or rolling elements. It is found that this flaking PAGE 43 43 usually commences as a crack from sub surface layers and propagate to the surface forming a pit or spall in the lo ad carrying surface. Lundberg and Palmgren ( 3 ) were the first researcher s They also suggeste d that fatigue cracking originates at weak points below the surface of the material. These weak points include microscopic slag inclusions and metallurgical dislocations that can be detected only by laboratory methods. V arious experimental studies so far h ave confirmed LP theory that fai lure initiates at weak points (Harris (2)) Hence, changing the homogeneity, metallurgical structure, and chemical composition of steel significantly affects the endurance capabilities of the bearing. 3.2 Bearing Fatigue Life Dispersion Bearing subsurface material is subjected to a large number of rolling contact fatigue (RCF) stress cycles (~10 10 ) with complex tri axial stress state and changing planes of maximum shear stress during a loading cycle. Bearing raceway surfac es subjected to RCF experience highly localized cyclic micro plastic loading, leading to localized material degradation nucleation and propagation of subsurface cracks, and the detachment of material (spall) fr om the main body of the bearing Lundber g an d Palmgren (3) applied the Weibull statistical strength theory to the stressed volume under Hertzian elastic contact to obtain the probability of survival of the volume susceptible to subsurface initiated fatigue cracks. The LP crack initiation based life prediction worked expected L 10 life of about 2,000 hours. However, these theories do not work well for Vacuum Arc Re melt (VIM VAR) bearing steels, which have fewer and smaller non metallic inclusions. Since RCF effects are PAGE 44 44 very localized the spatial aspects of the microstructure (primary inclusions oxides, and secondary inclusions carbides and nitrides) and their attributes (vol ume fraction, size, morphology and distribution) play an important role in bearing fatigue life dispersion. Even when nominally identical sets of bearings are te sted under nominally identical operating conditions such as load, speed, lubrication, and environmental conditions bearings fail according to a dispersion that varies over a wide range of values. Because of this life dispersion, bearing life is typically expressed by Life, define d as the number of cycles at which 90% of the bearings survive under a rated load and speed The volume of RCF affected material V is estimated to be V ( 3 1) where a is the semi major axis of the contact ellipse, is the depth where orthogonal shear stress is maximum, and is circumference of the raceway at depth The probability of survival of bearing raceway subjected to the maximum orthogonal shear stress number of revolutions is given by (Lundberg and Palmgren (3)) ( 3 2) Here c e and h are empirical constants and u is number of stress cycles per revolution. Equation ( 3 2) can further be simplified for a specific bearing under a particular load as : (3 3) where A is the life which 63.21% of the bearings in a sample will successfully pass. Further simplification leads to, PAGE 45 45 ( 3 4) Equation ( 3 4) represents Weibull distribution for bearing fatigue lives. The exponent e is Weibull slope and it is a measure of bearing fatigue life dispersion. Lower value of e indicates greater dispersion of fa tigue life. It is clear from Equation ( 3 4) that vs p lot will be a straight line fit. 3.3 Standard Load Life Equation Currently ball and roller bearing RCF life predi ction is based on the ANSI Sta ndard 9:1990(6) life rating equation given as : ( 3 5) Harris (2) and Zaretsky (4) discuss the systematic procedure of obtain ing standard life rating Equation ( 3 5) from basic LP Equation ( 3 2). Constants and are life modification factors, which are determined by the operating conditions of the bearings. The basic dynamic capacity of bearing, C is the rated load that rolling element raceway contact wi ll successfully withstand for one million revolutions with reliability level of 100 s. is the equivalent radial load experienced by the bearing and in the standard method of life calculation is given by ( 3 6) where, X and Y are radial and axial load factors, and and are the applied axial and radial loads, respectively. Load factor values are dependent on the nominal contact angle of the bearings. PAGE 46 46 Both the ANSI 9, 11:1990 (6, 7) and ISO 281:199 1 (8) standards have recommended the values of life adjustment factors to be used for bearings design. It was observed that life adjustment factors are independent of load life exponent, except for the case when inner ring raceways ex perience tensile hoop stresses due to interference fits. In LP Equation ( 3 2), c and h depend upon exponent s e and p, which are determined empirically. Evaluating the endurance test data of approximately 1500 be arings, Lundberg and Palmgren ( 3 ) determined that for point contact (ball bearings) e = 10/9 c = 31/3 and h =7/3. Harris ( 2 ) showed that load life exponent can be expressed as, ( 3 7) For line contact (roller bearings) Lundberg and Palmgren (9) reported that ( 3 8) and e = 9/8 From Equation s ( 3 7) and ( 3 8), L undberg and Palmgren ( 3 9 ) concluded that load life exponent p = 3 for point contact and p = 4 for line contact. However, Lundberg and Palmgren (9) conservatively chose to use p = 10/3 for roller bearings. Exponents c and h were found to be constant for both ball and roller bearings, indicating that they are material constants. The load life exponent 3 for ball bearings and 4 for roller bearin gs was determined empirically based on the endurance tests data generated in 1950s. These values pertain to rolling bearings of specific design, properly manufactured from good quality steel, and are based on work by Lundberg and Palmgren ( 3 9) cond ucted during that time The load rating and life calculation formulas developed in the middle of PAGE 47 47 twentieth century is representative of the manufacturing practices, materials, and lubricating methods available at that time. Recent endurance data for bearings man ufactured from VIMVAR steels has indicated that bearings fail at much higher fatigue lives. The current life rating Equation ( 3 5) tends to under predict bearing life for the majority of applications. Based on the latest endurance tests data available the current load life exponent values for bearings must be updated to bridge the existing gap between theory and practice. Zaretsky et al (4) recommends that the values of load life exponent should be changed to 4 and 5 for ball and roller bearings, respectively. To check the accuracy of existing and recommended load life exponents, validation analysis of the standard load life equation was pe rformed. The data presented in Harris and McCool (1) was used to calibrate the value of load life exponent so that observed bearing lives can be best represented on the logarithmic plot of ratio of predicted and actual fatigue lives. 3.4 Data Char acteristics In 1993, under a study sponsored by the United States Navy, ball and roller bearing fatigue endurance data were collected from various bearing manufacturers, power transmission applications and laboratory tests of bearing endurance. Endurance d ata of bearings was obtained from 10 separate sources. Of the 62 data sets reported in Harris and McCool ( 1 ) 47 data sets were found satisfactory for validation analysis. Out of the 15 data sets which were not considered for this analysis, 11 data sets re ported 0 or 1 failures. Hence no or Weibull estimates were available for these data sets. The remaining four data sets were found to over predict bearing lives from the LP model and hence were rejected for this study. PAGE 48 48 The collected endurance data contains fatigue lives for deep groove ball bearings (DGBB s ), angular contact ball bearings (ACBB s ), and cylindrical roller bearings (CRBs). The DGBBs and CRBs were operating under pure radial load. Some of the ACBBs were subjected to pure axial loading an d the rest were subjected to combined radial and axial loads in which axial load is dominant. Some of the ACBBs data was also obtained from tests in which load and speed acted according to an estimated time variant duty cycle. In such cases, equivalent cub ic mean loa d specified by LP is utilized (Harris and McCool (1)) (3 9) In Equation ( 3 9), is the number of revolutions experienced under load and k is the total number of different op erating conditions. N is the total number of revolutions experienced during one load speed cycle. For each experiment m ean speed can be found as (3 10) Tables 3 1, 3 2 and 3 3 represents geometry features of each bearing used along with the operating conditions maintained in each experiment for DGBBs, ACBBs and CRBs respectively. The bearing components: inner & outer raceways, balls and rollers were manufactured from through hardening steels such as CVD 52 100, VIMVAR M50, VAR M50 and case hardened steels such as carburi zed SAE 8620 and VIMVAR M50 NiL.VV represents VIMVAR steel and V represents VAR steel PAGE 49 49 From the data we can see that four types of lubricants were used : mineral oil, Mil L 7808, Mil L 23699 and ester type grease. As well, geometry features like pitch diameter (dm) and ratio values for each bearing tested are also specified. These values were obtained from Equation s ( 3 11) and ( 3 12) respectively. (3 11) where and are inner and outer raceway diameter respectively. ( 3 12) where D is the rolling element diameter and is the bearing contact angle. The basic dynamic capacity rating for each bearing is provided based on CVD 52100 steel. Dimensionless ratio /C indicate the loads experienced by each bearing. is the equivalen t radial load for DGBBs and CRBs and equivalent thrust load for ACBBs. Mean shaft speed, sample data set size and number of failed bearings are also provided for each operation. Last column in tables 3 1, 3 2 and 3 3 provides unbiased maximum likelihood es timates of the lives in hours by Harris and McCool ( 1 ) Data lines 9, 11, 28, 32, 35, 42, 44, 48, 57, 59 and 60 correspond to 0 or 1 failures. Hence there life and Weibull Slope estimates were not reported. Tables 3 4, 3 5 and 3 6 contain media n unbiased maximum likelihood estimates of the shape parameter for the Weibull distribution of DGBBs, ACBBs and CRBs. It should be noted that data line entry in Tables 3 4, 3 5 and 3 6 corresponds to the data line entry in Table 3 1, 3 2 and 3 3. From the Weibull slope estimates for each data set, we can see that the quality of endurance data is mixed. Higher values of the Weibull slope indicate narrow dispersion and smaller values indicate wide dispersion of fatigue life values of the PAGE 50 50 bearings in the given data set. Harris ( 2 ) reported that for commonly used bearing steels, values of e are observed in the range 1.1 to 1.5 and for modern, ultraclean, vacuum re melted steels, they are in the range of 0.8 to 1.0. Harris and McCool ( 1 ) mentioned that some data is from bearing testing, in which operating conditions were carefully monitored. Some data are from poorly monitored tests of small samples of bearings with few failures. Some data are from testing of large samples of bearings wit h few failures. Some of the data were obtained from tests with less controlled time variant load and speed duty cycle, leading to large uncertainties in the observations. In some cases, failures of the bearings were not examined as per the laboratory stand ards. These led to substantial scatter in both lives and Weibull slopes. Tables 3 4, 3 5 and 3 6, also show the peak Hertzian compressive stresses experienced by bearings in each o f the operations. Harris ( 2 ) provides the systematic procedure for ca lculating dimensions and maximum compressive stresses experienced by the contact areas in both point contact and line contact. For an elliptical contact area, the normal stress at point (x,y) within the contact area is given by, (3 13) Where, Q is the normal load acting on the contact, a & b are semi major and minor axes of the contact area respectively. For ideal line contact, i.e. when lengths of the two bodies in contact are equal, the compressi ve stress distribution over the surface can be given as (3 14) PAGE 51 51 where b is the semi width of the contact surface. lubrication film thickness also provided for each bearing operation and is defined as: (3 15) where h is the minimum lubricant film thickness between the rolling elements and raceways, and is the composite rms roughness of the and McCool ( 1 ) recommends that when =1, = 1; for sufficient lubrication between rolling contacts 1 ; and for poor lubrication < 1. Using the standard load life Equation ( 3 5), theoretically predicted values for lives were calculated for each operating bearing. The LP life factors used in this study were obtained from ANSI 9, 11:1990 (6, 7) and ISO 281:1991 (8) standards and they are multiplicative. Tables 3 4, 3 5 and 3 6 represent the ratio of theoreticall y predicted fatigue lives, from conventional LP model, to the experimental fatigue lives for DGBBs, ACBBs and CRBs respectively. It is clear that majority of the L(LP)/L(act) ratios are close to 0, indicating that current theoretical LP model severely und er predicts bearing fatigue lives. To correct for this under prediction, validation analysis was performed to statistically calibrate the load life exponent values for both ball bearings and cylindrical roller bearings. 3.5 Weibull Distribution Parameters The Weibull distribution is a continuous probability distribution and its probability densit y function is represented as (Papoulis and Pillai (10)) : (3 16) PAGE 52 52 scale parameter of the distributio n respectively. If x represents Equation ( 3 16) gives distribution for which the failure rate is proportional to a power of time. The cumulative distribution function for Equation ( 3 16) is (Papoulis and Pillai (10)): (3 17) where F represents the cumulative probability of x For x 0, Equation ( 3 17) can be rearranged as (3 18) Substituting in Equation ( 3 18) we get, (3 19) Comparing Equation s ( 3 19) and ( 3 4), we can conclude that the shape parameter: ( 3 20) and the scale parameter: (3 21) Equation ( 3 4) can be solved for each bearing data set. Values of Weibull slope e and parameter A were determined using given data for each data set. Equations ( 3 20) and ( 3 21) were used to determine shape and scale parameters for each data set. These Weibull distribution parameters were used to represent corresponding data sets in Statistical Calibration. 3.6 Verification and Validation Issues surrounding verification and validation include validity of the theoretical model and uncertainty quantification. As we can see from Table 3 4, 3 5 and 3 6 fatigue PAGE 53 53 lives predicted by current theoretical LP model differ significantly from field data. Figure 3 1 and 3 2 represents accuracy of the LP model. Figure 3 1 represents values for ball bearings (DGBBs and ACBBs) in the given data set. As shown in the figure, if the LP model accurately predicts fatigue life data we should get a straight line fit at =0 for al l the data sets. Similar to Figure 3 1, Figure 3 2 represents the values for rolle r element bearings (CRBs) in the given data set. From Figure 3 1 and 3 2 we can see that there is significant under prediction of fatigue lives from the LP model. The LP based life models with current load life exponents generally result in wide variation in predicted vs. observed life and thus are not reliable as a design tool for the new generations of bearings and bearing steels. This can result in the use of over sized bearings and increased weight penalty for the required load conditions, a serious con cern in aerospace applications. To address this issue and update the load life exponents using the available and more recent experimental data, validation and calibration analysis was performed for the standard life rating Equation ( 3 5) and the results ob tained are explained in following sections. 3.7 Statistical Calibration Since bearing preloading conditions are not specified, calibration study cannot be performed for life modification factors. However, based on the observed fatigue lives and operating load ing conditions validation study can be performed to calibrate the values of load life exponent for both ball and roller element bearings. For each individual data s et, standard life rating Equation ( 3 5 ) can be rearranged as: (3 22) PAGE 54 54 For 90% reliability and load life exponent of 3 for ball bearings, Equation ( 3 22) can be written as (3 23) Equations ( 3 22) and ( 3 23) can be combined to rewrite generalized relation between with L(LP) we get, (3 24) Based on given loading conditions and observed fatigue lives, Equation ( 3 24) s: (3 25) Similarly for roller bearings, load life exponent can be derived as, (3 26) To improve the agreement with experimental data it is necessary to calibrate the in LP model. Before beginning calibration analysis there is need to quantify the uncertainty in each data set. After the studying the given data, it was observed that there is epistemic uncertainty corresponding to limited number of data points in each data set. Because of the finite number of tests we have errors in the distributions of fatigue lives. This epistemic uncertainty can be reduced by method s imilar to Bootstrap Sampling technique. Based on the estimated Weibull distribution parameters for each data set, 5000 n virtual data samples of the fatigue lives were created corresponding to each experimental data set. It should be noted that here n PAGE 55 55 re presents the number of bearings tested in the experimental data set and the number 5000 was decided based on the fact that results were found to converge to a unique value at this number. As well, it resulted in considerable saving of computational time. F or this 5000 n virtual samples of fatigue lives, 10 th percentile that is values were determined for each virtual data set of n samples. These 5000 estimates of lives were considered to represent the uncertainty in conducting that individual exp eriment. Corresponding to 5000 estimates of life, using Equation ( 3 25), 5000 values of load life exponent were determined for each experimental data set. Figure 3 3 and 3 4 represents histograms of these estimated load life exponents for ball and ro ller bearings respectively. Figure 3 3 represents 5000 X 36 estimates of the load life exponen t for DGBBs and ACBBs and Figure 3 4 represent s 5000 X 11 estimates of the load life exponent for CRBs. Figure s 3 5 and 3 6 represents corresponding empiric al cumulative distribu tions for histograms in F igure s 3 3 and 3 4 respectively. Figure 3 3 and 3 4 represents on histograms presented in F igure 3 3 and 3 4, to determine value of load life exponent two different criterions can be used: 1. Maximum likelihood estimate 2. Median estimate such that ECDF = 0.5 From F igure 3 3 and 3 4 we can see that maximum likelihood estimate of load life exponent for ball and cyli ndrical roller bearings are 4.2 and 4.8 respectively. However, when graph was plotted for cylindrical roller bearings it was found that number of data points were not symmetric about line. Generally, after PAGE 56 56 the calibration, it expected that experimental data points fall symmetric about this line. Hence median value of posterior was used to estimate the calibrated value of load life exponent for both ball and cylindrical roller bearings. Table 3 7 shows the ca librated value of load life exponents for both point contact and line contac t. For estimation, ECDF = 0.5 criterion was used to ensure that after calibration data points are symmetric about y= line on the logarithmic plot of ratios of predicted to actual fatigue lives. From T able 3 7, we can see that for ball bearings, calibrated value of load life exponent is 4.27 with 68.2% confidence bounds as [3.15, 5.37]. This value is much higher than currently used industrial standard of 3. We can also see that the posterior distribution of p is very wide, owing to the fact that experimental fatigue lives are widely scattered. Similarly for cylindrical roller bearings the calibrated value of load life exponent was found to be 5.66 with 68.2% confidence bounds as [4.4 2, 7.26]. This value is also much higher compared to current industrial standard of 10/3. Using the calibrated value of load life exponents, the ratios of predicted to actual fatigue lives were recalculated and are presented in Tables 3 4, 3 5 and 3 6. The y were plotted on logarithmic graphs similar to Figure 3 1 and 3 2 to validate the results. Fig ure 3 7 represents plot for ball bearin gs i.e. DGBBs and ACBBs and Figure 3 8 represents corresponding plot for cylindrical roller beari ngs. We can see from the plot that number of data points are symmetric about y=0 axis, as expected from the output of calibration. In this statistical calibration analysis it was assumed that LP model represents reality without any discrepancy. Hence tech niques available for statistical calibration without any discrepancy were used in this study. PAGE 57 57 3.8 Summary After analyzing the extensive rolling bearing fatigue data gathered by United States Na vy in 1993 (Harris and McCool (1)) it was found that the widely used LP method and the derivat ive standard life rating equation tend to significantly underestimate bearing fatigue lives. Therefore, due to this under prediction bearing users tend to design oversized bearings and mechanisms than are necessary Thi s is critical issue for aerospace engine components in which generally higher values of power transmission to weight ratios are demanded. This underlines the need for a more realistic, rolling bearing life prediction model To address this issue, validati on analysis of the standard load life equation used for bearings design was performed in this study. It was assumed that LP model represents reality without any discrepancy. The current values of the load life exponent for ball and roller bearings are dete rmined based on the statistical study done in 1955 and there is need to calibrate these values, based on latest experimental data. The epistemic uncertainties in the reported experimental data were reduced by generating 5000 virtual samples of fatigue live s for each data set. Using statistical calibration techniques it was found that load life exponent values should be corrected for ball bearings from 3 to 4.27 with 68.2% confidence bounds as [3.15, 5.37] and for cylindrical bearings it should be corrected from 10/3 to 5.66 with 68.2% confidence bounds as [4.42, 7.26]. Based on calibrated model, the logarithmic plot of ratios of predicted to actual fatigue life values show that data points are equally distributed across y=0 line for DGBBs, ACBBs and CRBs. H ence, it is recommended that calibrated value of load life exponent should be used for bearings design in order to optimize bearing size for given load application. PAGE 58 58 Figure 3 1 plot of data points for DGBBs and ACBBs Fig ure 3 2 plot of data points for the CRBs PAGE 59 59 Fig ure 3 3 Histogram of estimated load life exponents for ball bearings i.e. DGBBs and ACBBs Figure 3 4 Histogram of estimated load life exponents for cylindrical roller bearing i.e. CRBs PAGE 60 60 Figure 3 5 Empirical cumulative distribution of load life exponents for ball bearings Figure 3 6 Empirical cumulative distribution of load life exponents for cy lindrical roller bearings PAGE 61 61 Figure 3 7 plot of data points for DGBBs and ACBBs Figure 3 8 plot of data points for CRBs PAGE 62 62 Table 3 1 Deep groove ball bearing (DGBB) geometry, load, speed and actual fatigue life data Data Line. Material Lubricant dm (mm) C(N) /C Speed (rpm) Sample Size No. of failures L10 life (hr) 1 52100 mineral oil 72.5 0.241 52800 0.357 1500 40 22 527 2 52100 Mil L 7808 72.5 0.241 52800 0.357 6000 11 7 147 3 52100 mineral oil 43.5 0.255 21200 0.379 8000 6 3 21 4 52100 mineral oil 46 0.207 19500 0.354 2000 33 4 3503 5 52100 Mil L 23699 260.35 0.067 96800 0.919 263 67 9 1644 6 52100 mineral oil 72.5 0.241 52800 0.357 6000 37 7 1956 7 52100 mineral oil 43.5 0.255 21200 0.212 8000 28 3 654 8 52100 mineral oil 72.5 0.241 52800 0.53 6000 37 23 1723 9 52100 mineral oil 72.5 0.241 52800 0.357 6000 10 52100 mineral oil 72.5 0.241 52800 0.357 1500 41 2 8856 11 52100 mineral oil 43.5 0.255 21200 0.379 8000 12 52100 mineral oil 72.5 0.241 52800 0.357 6000 79 23 807 13 52100 mineral oil 72.5 0.241 52800 0.53 6000 40 11 513 14 52100 mineral oil 46 0.207 19500 0.354 2000 60 21 433 15 52100 mineral oil 43.5 0.255 21200 0.379 8000 30 12 115 16 52100 mineral oil 43.5 0.255 21200 0.379 8000 30 8 257 17 52100 mineral oil 43.5 0.255 21200 0.379 8000 103 43 1813 18 52100 mineral oil 43.5 0.255 21200 0.379 8000 29 29 19 19 8620 car mineral oil 43.5 0.255 21200 0.379 8000 29 8 240 20 8620 car mineral oil 43.5 0.255 21200 0.379 8000 29 12 290 21 8620 car mineral oil 43.5 0.255 21200 0.379 8000 57 57 296 22 VV M50 Mil L 7808 72.5 0.241 52800 0.357 3200 30 7 1723 23 VV M50 Mil L 7808 72.5 0.241 52800 0.357 3500 28 3 2678 24 VV M50 Mil L 7808 72.5 0.241 52800 0.357 3200 37 6 1133 25 VV M50 Mil L 7808 72.5 0.241 52800 0.357 3200 40 33 203 26 M50NiL mineral oil 72.5 0.241 52800 0.357 3500 40 6 1202 27 M50NiL mineral oil 72.5 0.241 52800 0.357 3500 40 5 1744 28 V M50 Mil L 23699 260.35 0.067 96800 0.919 263 29 V M50 Mil L 23699 72.5 0.12 20500 0.366 21200 18 11 58.3 PAGE 63 63 Table 3 2 Angular contact ball bearing (ACBB) geometry, load, speed and actual fatigue life data Data Line. Material Lubricant dm (mm) C(N) /C Speed (rpm) Sample Size No. of failures L10 life (hr) 30 52100 grease 165 0.108 121000 0.993 263 8 5 15 31 VV M50 Mil L 23699 388.47 0.092 100000 0.544 3130 33 2 385900 32 VV M50 Mil L 23699 244.6 0.103 56800 1.645 10320 33 VV M50 Mil L 23699 388.47 0.092 100000 0.544 3130 20 2 1292 34 VV M50 Mil L 23699 388.47 0.092 100000 0.544 3130 362 7 103900 35 VV M50 Mil L 23699 154.99 0.122 99500 0.224 12000 36 VV M50 Mil L 23699 223.95 0.112 54300 1.393 12400 634 2 3264000 37 VV M50 Mil L 23699 223.95 0.112 54300 0.349 9800 64 2 20260 38 VV M50 Mil L 23699 2.97 0.117 56300 1.344 12400 1199 3 17200 39 VV M50 Mil L 23699 154.99 0.122 99500 0.224 12000 20 3 504 40 VV M50 Mil L 23699 79.38 0.114 32200 0.159 19000 30 3 3010 41 M50NiL Mil L 23699 154.99 0.122 99500 0.224 25000 12 2 41 42 V M50 grease 165 0.108 121000 0.993 263 43 V M50 Mil L 23699 65 0.15 34300 0.359 5500 17 5 344.2 44 V M50 Mil L 23699 72.5 0.213 64300 0.148 5500 45 V M50 Mil L 23699 72.5 0.213 64300 0.244 5500 10 2 1503 46 V M50 Mil L 23699 72.5 0.176 56400 0.592 9700 10 5 67 47 V M50 Mil L 23699 72.5 0.176 56400 0.592 9700 10 4 70 48 V M50 Mil L 23699 72.5 0.213 64300 0.408 350 PAGE 64 64 Table 3 3 C ylindrical roller bearing (CRB) geometry, load, speed and actual fatigue life data Data Line Material Lubricant dm (mm) C (N) /C Speed (rpm) Sample Size No. failures L10 life (hr) 49 52100 Mil L 23699 141.1 0.142 1150000 0.091 1050 6 4 14 50 52100 Mil L 23699 57.5 0.193 47100 0.35 19400 6 6 46 51 52100 Mil L 23699 87.5 0.194 123000 0.38 7540 6 4 63 52 52100 Mil L 23699 65 0.215 76100 0.322 19400 7 4 4 53 52100 Mil L 23699 95 0.189 138000 0.387 7540 6 3 8 54 52100 Mil L 23699 77.5 0.116 50000 0.441 2550 6 3 41 55 VV M50 Mil L 23699 29.06 0.277 22700 0.073 27600 5321 13 14290 56 V M50 Mil L 23699 72.49 0.207 95600 0.325 9700 6 6 83 57 V M50 Mil L 23699 65 0.215 76100 0.322 19400 58 V M50 Mil L 23699 141.1 0.142 205000 0.51 1050 6 2 22 59 V M50 Mil L 23699 77.5 0.116 50000 0.441 2550 60 V M50 Mil L 23699 95 0.189 138000 0.387 7540 61 V M50 Mil L 23699 57.5 0.193 47100 0.35 19400 6 4 25 62 V M50 Mil L 23699 87.5 0.194 123000 0.38 7540 19 2 895 PAGE 65 65 Table 3 4 Actual B earing life versus predicted life for DGBBs Data Line. Weibull slope (e) (MPa) L (LP)/L (act) L(cal)/ L(act) 1 2.22 3880 14.6 0.3 1.24 3 2.22 4030 0.3 0.1 0.39 4 0.65 3410 15.0 0.4 1.47 5 0.65 3120 9.1 0.6 0.71 6 0.89 3410 3.3 0.7 2.45 7 1.33 3720 8.8 0.3 2.26 8 2.65 3410 10.0 0.3 0.59 10 0.70 3410 3.3 0.1 0.54 12 1.21 2970 5.0 0.7 2.54 14 0.93 3720 8.8 0.5 1.76 15 0.70 3720 8.8 0.4 1.47 16 0.72 3720 8.8 0.2 0.66 17 1.36 2970 5.0 0.3 1.05 19 0.69 3720 8.8 0.2 0.63 20 1.20 3720 8.8 0.2 0.52 21 0.95 3720 8.7 0.2 0.66 22 2.29 3410 1.1 0.2 0.76 23 1.06 3410 1.0 0.1 0.38 24 0.72 3550 1.1 0.2 0.79 26 0.68 3410 1.4 0.7 2.46 27 1.23 3410 1.4 0.2 0.74 29 2.29 3120 1.1 0.3 1.04 PAGE 66 66 Table 3 5 Actual Bearing l ife versus predicted life for ACBBs Data Line. Weibull slope (e) (MPa) L (LP)/L (act) L(cal)/L (act) 30 1.91 2700 0.3 0.3 0.30 31 0.20 1280 3.6 0.1 0.17 33 1.10 1970 5.1 0.4 0.96 34 0.95 1280 3.6 0.3 0.63 36 0.67 1280 5.6 0.0 0.01 37 0.80 1250 2.1 0.5 1.82 38 1.15 1690 2.2 0.01 0.01 39 0.81 1970 2.1 0.9 6.01 40 1.08 2230 3.0 0.2 1.62 41 0.14 3660 4.2 0.1 0.87 43 0.84 2010 1.0 0.3 1.12 45 1.14 2070 1.5 0.2 1.34 46 0.84 2340 3.2 0.9 1.75 47 0.69 2340 1.8 1.0 1.87 Table 3 6 Actual Bearing life versus predicted life for CRBs Data Line Weibull Slope (MPa) L(LP)/ L(act) L(cal)/ L(act) 49 0.75 2290 0.82 0.10 25.76 50 3.63 2460 0.75 0.06 0.73 51 1.23 2350 1.20 0.04 0.37 52 0.74 2390 0.97 0.64 8.97 53 1.90 2340 0.38 0.21 1.92 54 1.08 2230 1.48 0.15 1.03 55 1.00 1220 0.80 0.03 12.51 56 2.43 2460 0.75 0.05 0.64 58 0.48 2390 0.97 0.15 0.74 61 0.67 2350 1.20 0.13 1.52 62 0.21 2150 1.32 0.01 0.11 PAGE 67 67 Table 3 7 Calibrated value of load life exponents Bearing Type Calibrated value of 'p' 68.2 % confidence bounds Ball Bearings 4.27 [3.15, 5.37] Cylindrical Roller Bearings 5.66 [4.42, 7.26] PAGE 68 68 LIST OF REFERENCES (1) Harris, T. A., and McCool, J. I. Journal of Tribology 118 pp 297 310 (2) Harris, T. A. ( 1991 ) Rolling Bearing Analysis A Wiley Inter science Publication, John Wiley a nd Sons Inc., New York ( 3 ) Lundberg, G. and Palmgren, A. (1947) Acta. Polytech nica Mechanical 1 (3 7 ) ( 4 ) Zaretsky, E. V Vleck, B. L. and Hendricks R. C. (2005) 2005 213061 ( 5 ) Klecka, M. A., Subhash, G. and Arakere, N. K. ( 2013 ) Property Relationships in M50 NiL and P675 Case Tribo logy Transactions 00 : pp 1 14 (6) American National Standards Institute, American National S tandard (ANSI/AFBMA) Std. 9 (1990), Load Ratings and Fatigue Life for Ball Bearings (7) American National Standards Institute, American National St andard (AN SI/AFBMA) Std. 11 (1990) Load Ratings and Fatigue Life for Roller Bearings. (8) International Standards Organization, International Standard ISO 281 (1991) Rolling Bearings Dynamic Lo ad Ratings and Rating Life (9) Lundberg, G. and Palmgren, A. (1952) Acta Polytech nica Mech anical 2 ( 4, 96 ) (10) Papoulis, A. and Pillai, S. U. (2002), Probability, Random Variables and Stochastic Processes, McGraw Hill, 4 th E dition PAGE 69 69 BIOGRAPHICAL SKETCH Nikhil Londhe was born in Maharashtra State of India. He obtained degree in mechanical e ngineering from Birla Institute of Technology and Science, Pilani He jo ined University of Florida, in f all 2012 to pu rsue Master of Science degree in the Department of Mechanical and Aerospace Engi neering with specialization in solid mechanics, design and m anufacturing. From August 2012 he is pursuing his m equation us ed for designing rolling element bearings. His research work under the guidance of Dr. N. K. Arakere is aimed at elasticity and statistical analysis to correct for bearing fatigue life under prediction from standard load life model. Upon graduation he inte nds to pursue professional career in a erospace industry. 