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Page i Acknowledgement Page ii Table of Contents Page iii Page iv Page v Abstract Page vi Page vii Chapter 1. High speed, high power milling Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Chapter 2. Literature search Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Chapter 3. Experimental equipment Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Chapter 4. Thermal analysis Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Chapter 5. Experimental results and discussion Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Page 133 Page 134 Page 135 Page 136 Page 137 Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 Page 145 Page 146 Page 147 Page 148 Page 149 Page 150 Page 151 Page 152 Page 153 Page 154 Page 155 Page 156 Chapter 6. Bearing loads Page 157 Page 158 Page 159 Page 160 Page 161 Page 162 Page 163 Page 164 Page 165 Page 166 Page 167 Page 168 Page 169 Page 170 Page 171 Page 172 Page 173 Page 174 Page 175 Page 176 Page 177 Page 178 Page 179 Page 180 Page 181 Page 182 Page 183 Page 184 Page 185 Page 186 Page 187 Page 188 Page 189 Page 190 Page 191 Page 192 Chapter 7. Conclusions Page 193 Page 194 Page 195 Page 196 Page 197 Page 198 Page 199 Page 200 Page 201 Page 202 Appendix. Computer program listings Page 203 Page 204 Page 205 Page 206 Page 207 Bibliography Page 208 Page 209 Page 210 Page 211 Biographical sketch Page 212 Page 213 Page 214 
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DESIGN OF HIGH SPEED, HIGH POWER SPINDLES BASED ON ROLLER BEARINGS By ISMAEL A. HERNANDEZROSARIO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1989 ACKNOWLEDGMENTS The author wants to extends his sincere gratitude to Dr. Jiri Tlusty, Dr. Scott Smith and H. S. Chen. The deepest of all gratitude goes to my loving wife Laura. This research was funded under National Science Foundation grant # MEA8401442 Unmanned Machining, High Speed Milling. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................... ii ABSTRACT ............. .................. .......... vi CHAPTER 1. HIGH SPEED, HIGH POWER MILLING Introduction .................................. 1 Development of High Speed Milling .............. 2 High Speed, High Power Machining ............... 4 Goals and Scope ................................ 6 2. LITERATURE SEARCH Analytical Developments ........................ 8 High Speed Bearings: Experimental Results ..... 38 3.EXPERIMENTAL EQUIPMENT High Speed, High Power Milling Machine ........ 43 Test Spindles ........................ ....45 Oil Supply to the Bearings .................... 52 Instrumentation ............................. ..... 55 Oil Circulating System ........................ 57 Evaluation of Cooling Capacity ................60 Seals ......................................... 64 4. THERMAL ANALYSIS Thermal Analysis of the Spindle Housing .......68 Friction in Rolling Bearings ................. 68 Heat Generation ............................... 70 Heat Removal ............................ ...... 82 Steady State Temperature Fields ............... 86 Thermally Induced Loads ...................... 89 Computation of Thermal Loads .................. 91 iii 5. EXPERIMENTAL RESULTS AND DISCUSSION Test Procedure ................................ 97 Curve Fitting of Experimental Data ............98 Temperature ................................... 99 Steady State Temperatures Versus Spindle Speed ......................... 100 Steady State Temperatures Versus Oil Flow Rate ......................... 102 Overall Temperature Equation ............ 120 Steady State Temperatures: Comparison ................................ 121 Power Measurements ........................... 127 Motor Power Losses ...................... 127 Mechanical Power Losses ................. 127 Hydraulic Power Losses .................. 128 Configuration Power Losses ................... 129 Mechanical Power Losses ................. 129 Hydraulic Power Losses ..................129 Hydraulic Power Losses Versus Spindle Speed ......................... 130 Hydraulic Power Losses Versus Oil Flow Rate ......................... 131 Overall Hydraulic Power Losses Equation .............................. 132 Power Losses: Comparison ................ 133 Bearing Loads ................................ 148 Externally Applied Load ................. 148 Bearing Thermal Loads ................... 149 Performance of the Seals ..................... 150 Bearing Failures ............................ 151 Radiax Bearing Failure .................. 152 High Speed Bearing Failure .............. 154 6. BEARING LOADS Load Deflection Relationships ................ 157 Radial Loads ................................. 162 Axial Loads ................................. 165 Combined Loading ............................ 168 Bearing Life Calculation ..................... 172 Bearing Preload ............................. 173 Preloading Methods .......................... 174 Case 1: Variable Preload ............... 174 Case 2: Constant Preload ................ 180 High Speed Loads ............................ 183 Cylindrical Roller Bearings ................. 183 Tapered Roller Bearings ...................... 187 Centrifugal Forces ...................... 187 Gyroscopic Moment ....................... 189 Combined Loading ........................ 191 7. CONCLUSIONS Spindle Configurations ....................... 193 Cylindrical Roller Bearings ............. 193 Tapered Roller Bearings ................. 194 Experimental Conclusions .................... 195 Empirical Equations ..................... 195 Bearing Preload ......................... 198 Recommendations .............................. 199 Design Modifications for Configuration I ....................... 199 Design Modifications for Configuration II ...................... 200 Final Comment ................................ 201 APPENDIX RADIAL LOAD COMPUTATION PROGRAM .............. 204 COMBINED LOAD COMPUTATION PROGRAM ............ 205 LOAD DEFLECTION COMPUTATION PROGRAM .......... 206 HIGH SPEED CYLINDRICAL ROLLER BEARING PROGRAM .................................... 207 BIBLIOGRAPHY ............................................. 208 BIOGRAPHICAL SKETCH .....................................212 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DESIGN OF HIGH SPEED, HIGH POWER SPINDLES BASED ON ROLLER BEARINGS By Ismael A. HernandezRosario May 1989 Chairman: Dr. Jiri Tlusty Major Department: Mechanical Engineering Department An experimental investigation was performed on two spindle configurations based on roller bearings to determine their potential for High Speed, High Power Machining applications. The type of roller bearings considered were super precision tapered roller bearings and double row cylindrical roller bearings. The idleload performance of each spindle was evaluated in terms of maximum operating speed, operating temperatures, lubrication requirements and required power to operate the spindle. The tapered bearing spindle was provided with a constant preloading mechanism. Neither spindle was operated at the target 1.0 million DN (DN is the product of the bearing bore diameter in mm times the spindle speed in rpm), although both spindles exceeded the speed capabilities of current machine tools with similar bearing arrangements. The spindle based on tapered roller bearings is strongly recommended for High Speed, High Power applications for its low power losses and low operating temperatures at 9,000 rpm, a DN value of 900,000 (the maximum speed achieved) at an operating temperature of 77 degrees Centigrade and 15 kW of power losses. The configuration based on cylindrical roller bearings is not recommended to operate above the speed of 6,000 rpm, DN value of 600,000 after which the operating temperature and power losses are above the practical limit. vii CHAPTER I HIGH SPEED, HIGH POWER MILLING Introduction The development of advanced cutting tools has drastically reduced the time required to perform metal removal operations. These new tool materials are capable of operating at speeds up to an order of magnitude higher than previously existing tools [1]. Thus, the use of these tools to the maximum capabilities is called High Speed Machining (HSM). These tools can be used for HSM of steel at operating speeds of 200 m/min. using Coated Carbides, or HSM of cast iron at 1000 m/min. using Silicon Nitrides or HSM of aluminum at speeds between 10005000 m/min. using High Speed Steels or Solid Carbides [2]. The main advantage of HSM is the capability to remove metal faster. The increased metal removal rates (MRR) is extremely attractive for such industries where machining accounts for a considerable portion of the processing time or the manufacturing cost, such as the aerospace industry, e.g. aircraft frames and engines, or the manufacturing of automotive engine blocks. Considering that by 1986 over 115 billion dollars were spent on metal removal operations [3], any increase in productivity would have substantial economic effects. Development of High Speed Milling The first person to investigate high speed metal removal was Dr. Carl J. Salomon in Germany from 1924 to 1931 [4]. Dr. Salomon investigated the relationship between cutting speed and cutting temperature. As a result of his investigation, Dr. Salomon concluded that as cutting speed increased, so did the cutting temperature, until a critical maximum temperature was reached. Once at this critical temperature, any further increase in the cutting speed would produce a decrease in cutting temperature. As the cutting speed was increased even further, the cutting temperature would drop to usual operating levels. Thus, around the critical temperature there is a range of very high temperatures at which tools can not operate. Below this range, usual metal removing operations were performed. Once above this critical temperature range, increased metal removal rates could be obtained if the necessary cutting speeds were achieved. The benefit of the region above the critical temperature is that the cutting speed could be increased such that infinite metal removal rates were theoretically possible. Unfortunately, Dr. Salomon's work and experimental data were mostly destroyed during World War Two, and limited information of his work is currently available. In 1958, R. L. Vaughn, working for Lockheed, started a research program sponsored by the United States Air Force to investigate the response of some high strength materials to high cutting speeds (152,400 surface meters per minute, smm) [4]. Some of the conclusions presented by Vaughn [4] which are of particular interest to this dissertation are 1. High speed milling could be used for machining high strength materials. 2. Productivity will increase with the use of HSM. 3. Surface finish is improved with HSM. 4. The amount of wear per unit volume of material removed decreased with HSM. 5. An aluminum alloy, 7075T6, was machined at 36560 smm with no measurable tool wear. 6. The increase in cutting force was between 33 to 70%, over conventional machining forces. 7. At the time of investigation, the technology available could not make maximum use of HSM. During the 1960's and 1970's various companies, such as Vought and Lockheed, experimented with HSM. In each case, the investigators agreed on the potential increase in productivity that HSM may yield [4]. Raj Aggarwal summarizes the results from published data on HSM investigations [5]. These investigations have shown that an increase in cutting speed will produce a reduction in power consumption per unit volume of metal removed (unit horsepower). Although the effect of increased cutting speed would depend on the chip load used during the investigation, in general, lower unit horsepower was obtained for higher speeds. On the effect of cutting speed on cutting forces, results varied, while some researchers measured some decrease in the forces; others found little or no change. At this point, it is important to note that companies involved in HSM are reluctant to publish their complete results and test conditions, based on commercial competitivity [ 6 ], which makes the comparison of results quite difficult. An area of agreement is the application of HSM to the end milling of thin aluminum ribs, where improved surface finish was obtained [5]. High Speed, High Power Machining T. Raj Aggarwal concludes that high speed capabilities alone will not produce a relevant increase in productivity [5]. To obtain significant improvements in productivity, high speed milling must be coupled to high power machining. High power machining refers to those machining operations where the power requirements are above the capacity of common machine tools (1020 kW). The combination of HSM and high power milling is called High Speed, High Power (HSHP) milling. F.J. McGee [7] directed a HSM program for the Vought Corporation. As part of the research program, he identified the ideal HSHP machine tool for their investigation as having a spindle rated at 20,000 rpm speed and 75 kW power; unfortunately such a machine tool was not available. The closest available spindle was a 20,000 rpm, 22 kW spindle by Bryant. McGee [7] stresses the fact that the spindles 5 currently available in the market do not have the power required to make optimum use of HSM. In an effort to correct this lack of HSHP milling machines the trend has been to retrofit existing machines with high speed spindles with improved power capabilities [8]. Although this procedure will improve the HSM capabilities of the existing machine tools, there is still the need for a spindle capable of achieving spindle speeds of 10,000 rpm with power capabilities above 30 kW. Tlusty [2] defines the requirements for a HSHP spindle capable of high metal removal rates and without power limitations. For the face milling of cast iron and steel, Tlusty recommends [2] the use of a spindle based on 100 mm diameter roller bearings, tapered roller bearings (TRB) or double row cylindrical roller bearings (CRB), a 10,000 rpm and 115 kW rating. Tlusty shows that such a spindle could make optimum use of the new cutting tool materials. The high stiffness values which are inherent to roller bearings combined with the use of stability lobes would make the maximum use of the new cutting tools. A spindle based on 100 mm bore TRB or CRB which operates at 10,000 rpm, would operate well above the catalogue maximum for these types of bearings. Usually these bearings are operated below 4000 rpm [9,10]. In order to operate these spindles above such speeds, special lubrication and cooling systems should be provided. The consequences of thermal differential expansions must also be determined. If HSM is to be ever fully implemented, then HSHP spindles must be developed. For these spindles to be developed, the performance of large diameter CRB and TRB in machine tools operating at very high speeds must be researched and understood. Goals and Scope This dissertation is an experimental investigation on the HSHP performance of two types of large diameter (over 100 mm) bearings which are widely used in machine tools, double row cylindrical roller bearings and tapered roller bearings. The goal of this dissertation is to establish which of these bearings could best be used in HSHP spindles and what are their requirements for a successful spindle design. For each spindle configuration its lubrication and cooling requirements, its operating temperatures and its maximum operating speed must be determined. Parameters which characterize the performance of a spindle must also be defined or identified. The design and/or development of new bearing geometries is beyond the scope of this dissertation. The high cost of developing and producing a new, nonstandard bearing geometry is above the economic capabilities of the machine tool industry. However, redesigning of spindles is well within the economic bounds of the machine tool industry. The goal of this dissertation is to provide new and much needed knowledge on the HSHP performance of large diameter 7 bearings in machine tools and to identify those parameters which are essential for a successful HSHP spindle design based on roller bearings. CHAPTER II LITERATURE SEARCH Analytical Developments In 1963, Harris presented the first paper [11] in which an analytical method was used to predict the behavior of a bearing assembly. In this paper, Harris presents a method to estimate the operating temperature of rolling element bearings assuming steady state operation and using a finite difference scheme. The operating temperature at several different nodal points of a bearing assembly could be estimated since at each nodal point the net increase in energy is zero at steady state. By definition, at steady state, the amount of heat transferred into a nodal point equals the amount of heat transferred out of the nodal point. According to Harris [11], the heat generated in the bearings is due to a load torque which resist the rotation of the rolling elements plus a viscous torque induced by the lubricant surrounding the rolling elements. By comparing this generated heat to the heat dissipation capacity of the assembly, the operating temperature may be estimated. Since the generated heat is the result of power losses, it's computation is relevant to this dissertation. As presented by Harris, the heat generation depends on the type of bearing used (ball or roller bearing), the bearing geometry (contact angle), loading conditions (radial or thrust), bearing diameter and lubricant properties. The load torque can be estimated using equation (2.1) [11]. M, = 0.782 fx Pe d, (Nmm) (2.1) where Mr: Load torque (Nmm). f.: Load torque factor. P,: Equivalent applied load (N). d.: Bearing pitch diameter (mm). The load torque factor is a function of bearing design and the relative bearing load. Palmgren [11] experimentally determined relations for estimating f1 for most bearing types. For ball bearings the factor f. is given by 1 = z ( (2.2) where Po: Static load (N). Co: Static Load Rating (N). The coefficient z and the exponent y were determined experimentally and are given below in Table 2.1 from [11]. For roller bearings, the value of f. was also determined experimentally. The value of f, for several types of roller bearings is given in Table 2.2, 4lso from [11]. Table 2.1 Coefficient z and Exponent y for Ball Bearings Bearing Type Contact Angle (0) z y Deep Groove 0 0.0009 0.55 Angular Contact 30 0.001 0.33 Angular Contact 40 0.0013 0.33 Thrust 90 0.0012 0.33 Selfaligning 10 0.0003 0.40 Source: Harris, T.A., "How to Predict Temperature Increases in Rolling Bearings," Product Engineering, December 1963. Table 2.2 Load Torque Factor Values f, for Roller Bearings Bearing Type f. Cylindrical 0.00025 to 0.0003 Spherical 0.0004 to 0.0005 Tapered 0.0004 to 0.0005 Source: Harris, T.A., "How to Predict Temperature Increases in Rolling Bearings," Product Engineering, December 1963. The equivalent load P. is a function of the type of bearing, the geometry of the bearing, and the direction of the load [11]. For ball bearings, the equivalent load is given by either equation (2.3) or equation (2.4), whichever yields the larger value of P.. For radial roller bearings, P. is given by equation (2.5) or equation (2.6), whichever is larger. Equation (2.7) estimates the value of P8 for thrust bearings (ball or roller). P. = 0.9 F. cot(ac) 0.1 F,. (N) (2.3) P,= Fr (N) (2.4) P. = 0.8 F. cot(a) (N) (2.5) P, = Fr (N) (2.6) Pe = F. (N) (2.7) where F.: Axial load (N). a: Contact angle (o). F,: Radial load (N). The lubricant flowing inside the bearing cavity will induce drag forces on the rollers. These drag forces oppose the motion of the rollers, generating heat. The expressions presented in [11] to determine the viscous torque are given as equations (2.8) and (2.9). The viscous torque is a function of the bearing diameter, the kinematic viscosity of the lubricant, the lubrication method, and the rotational speed. M = 9.79x10fo(L*n)2f d.3 (Nmm) (2.8) when u*n > 2000 M, = 1.59 x 105fcod3 (Nmm) (2.9) when u*n 5 2000 where M,: Viscous torque (Nmm). fo: Viscous torque factor for circulating oil lubrication [11]: Angular Contact Ball Bearings (2 rows) fo=8.0 Tapered Roller Bearings fo= 8.0 Cylindrical Roller Bearings (1 row) fo = 6.0 4: Kinematic viscosity (cS). n: Rotational speed (rpm). The heat generation rate is then the sum of the two torques Mr and M, times the rotational speed in rpm, times a conversion factor. Thus, the heat generated at a given rotational speed under an opposing torque M is given by (2.10). Qc = 1.05 x 104 n M (W) (2.10) where Q.: Heat generation rate (W). M: Total opposing torque Mr + M,. (Nmm) The next step in the development of bearing analysis theory was to develop an understanding of the internal behavior of the bearings. To achieve this understanding, a large research effort was undertaken during the drive to develop more efficient and reliable aircraft engines. Faster and more powerful engines required the development of more reliable bearings which could operate at higher speeds for longer periods of time. To design bearings for these operating conditions, complicated bearing analysis methods and computer programs were developed. Some of the papers which developed the understanding and modeling of high speed bearings will now be discussed in chronological order. The next model was presented, also by Harris [12], in a paper which introduced a method to predict the occurrence of skidding in cylindrical roller bearings operating at high speed. Skidding occurs when a rolling element slides over the raceway surface instead of rolling over it. While in this condition, the cage speed is below the rotational speed of the bearing. Skidding is due to the fact that during high speed operation, centrifugal forces eliminate the normal load component acting between the rolling element and the raceway, causing the sliding of the rollers over the raceway surface. This deteriorates the roller and/or the raceway surfaces decreasing considerably the fatigue life of the bearing. To predict skidding, it is necessary to estimate bearing internal speeds and loads. The rotational speeds of the rolling elements, cage and rollers, must be known if the effects of centrifugal force on bearing behavior are to be approximated. Since the speed of the rolling elements is affected by the loading conditions we must solve simultaneously for the loading conditions and internal speeds. The equations needed to solve for bearing internal loads and for the cage speed must first be presented. The internal speeds of a roller are shown in Figure 2.1, from [12]. The rotational speed of the jI roller is given by w,j. The rotational speed of the cage is given as w,, while the speed of the inner ring is w. The model is presented by Harris for the case when the inner race is rotating while the outer race is static. This is just the case for spindle bearing systems investigated in this dissertation. The sliding velocities can be determined as Vj = 0.5(d. D.)(ww4) 0.5Dlww (2.11) Vj = 0.5(d,, + D,.,)w 0.5D.w, (2.12) where V.j: Sliding velocity at the inner contact of the j* rolling element (m/sec). Voj: Sliding velocity at the outer contact of the jt0% rolling element (m/sec). dn: Bearing pitch diameter (m). D,: Roller diameter (m). V/2( d4 Dv)Vc VcO UTERiNS L/2( d DV) V2 2dvVvj V/2( d,Dv( VVc) 2 ( dio Figure 2.1 Internal Bearing Speeds, from Harris, T.A., "An Analytical Method to Predict Skidding in High Speed Roller Bearings," ASLE Transactions, July 1966. In Figure 2.2, the loads acting on a roller are shown using the nomenclature used by Harris [12]. The i subscript refers to the inner race contact, the o subscript refers to the outer race contact, the j subscript refers to the j" roller, while y and z subscripts indicate horizontal and vertical components respectively. Thus, the load Qo, indicates a vertical load, acting on the outer race contact of the jl roller. The loads Q.o, and Q,., are the reactions to external applied loads acting on the j" roller. Load Fj is caused by the cage acting on the roller. Loads Qyoj and Qyj are loads caused by the fluid pressure acting on the rollers at each roller raceway contact, while the drag forces acting at each contact are given by Foj and Fj. The boldface version of the previous are the dimensionless forms of the corresponding loads. The effect of high speed operation on the roller, which induces a centrifugally oriented force, is F,. The elastohydrodynamic loads are introduced by Harris here. During steady state operation, the summation of the forces acting on each roller, in directions y and z, must equal zero. In dimensionless terms the force balance is given by [12] as 18 Ozoj L / ^"l 0zijj Figure 2.2 Loads Acting on a High Speed Roller Harris, T.A., "An Analytical Method to Predict Skidding in High Speed Roller Bearings," ASLE Transactions, July 1966. ___ (Q=ij + F.) Q.oj = 0 RL %tj + FVj  Ro _ (Qy.oj Fo.j Faj) = 0 RL (2.14) where Ro: Equivalent external radius of the cylinder (mm). R,: Equivalent internal radius of the cylinder (mm). Qyo.j = Fj = Q.IJ lw,, E'R Qzoj Qma,j lw E'Ro, Qyrj 1. E'R lw1 E'R. lw, E'Rc, F,, E' iw E'Ro (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) i: Roller length (mm). E 1 o2 (2.21) (2.13) QZ.o= E: Modulus of elasticity (N/m2). a: Poisson's Ratio. The lubricant induced loads Fjj,Foj, Qyoj and Qa.j are given next, in dimensionless form, as presented by Harris in [12]. Fi = 9.2G03 UjO7 + Foj = 9.2G03 Uoj0'7 + Vc~j Ic:j Q^ = 18.4 (1T)G3 U07 = 18.4 (1T)G3 U,10.7 (2.22) (2.23) (2.24) (2.25) where G = aE' a: Is the pressure coefficient of viscosity (mm2IN). Hj= 1.6 Hoj = 1.6 Go6 UijO7 0. 5 33  Go06 UojO7 Q j0.2.13 4o VLj Vij = E' R (2.26) (2.27) (2.28) ILo Voj Vo = _____ (2.29) E' Ro )o Uij U 1j = (2.30) 2E' RL ILoC Ucj Uo= = (2.31) 2E' Ro f^3 Gqij[l(y/4qij)2]^2 I1L = 2 e dy (2.32) J0 f)= J Gqoj[l(y/4qoj)2] X/2 Ioj = 2 e" dy (2.33) J0 As it can be seen from above, the elastohydrodynamic loads are nonlinear functions of the roller speeds and lubricant properties. Note that the operating temperature is an input to the analysis and it is not corrected for each iteration. The model provides a method to solve for the cage speed we. If there is skidding, the cage speed will be below the expected value of w. = 1/2 w(l Dw/d.) (2.34) which is the cage speed during rolling motion. To determine the cage speed equations (2.13) and (2.14) are not enough. Torque balances must be performed at each bearing location and for the complete bearing. This would provide the necessary equations to solve for cage speed w. and roller rotational speed wj, cage load on the roller Fd, and outer race contact load Q,oj,. As it was noted before, the other loads are nonlinear functions of lubricant properties and roller speed. The inner race contact loads Qzj, are computed from static load analysis of the complete bearing. The solution method would require the computation of an initial cage speed from a known inner race speed. Using this cage speed, the conditions at each roller location are then computed. The loads are added up and they must balance the externally applied loads. Harris does not solve the model in this manner, due to the required computational tools which were not available to him at the time. Instead, he introduces some simplifying assumptions: 1. Since not all the rollers are loaded, Harris only considers the loaded contacts. 2. At steady state, the speed and load conditions at any loaded roller location is the same as in the most heavily loaded roller. The drag force acting on a loaded roller is determined by dividing the computed drag force by the number of loaded rollers. These simplifications drastically reduce the number of computations needed to solve for the roller speeds. Still, Harris's analysis yields a sufficiently close prediction of the occurrence of skidding. It shows that skidding does not exist in preloaded bearings. As soon as the centrifugal effects remove the preload, skidding starts. In those applications where out of roundness bearings are used, the geometry of the bearing improves skidding behavior. The simplifications do take a toll of the accuracy of the model. The simplified model can predict the occurrence of skidding but would not quantify it. The model is also limited for heavily loaded bearings. With the development of the digital computer, the complete analysis developed by Harris will later be used by other researchers in the development of more accurate models, as it will be shown below. Boness [13] provides some experimental data which corroborates the results obtained by Harris in his simplified model. At the same time, the experimental results aroused some doubt on the validity of Harris 's simplifying assumptions. The results presented in [13], show that for each roller, the rotational speed is different. The oil film thickness is also different at each roller location. This explains the limitations of Harris's model. Boness also found that by decreasing the amount of lubricant in the bearing cavity, the amount of skidding could be reduced by 75 percent. To obtain this amount of reduction in skidding, a very small amount of oil must be used; which is not always possible since at high speed applications oil provides the only reliable source of cooling. Poplawski [14] presents an analytical model which is based on the model developed by Harris [12] and the experimental results presented by Boness [13]. In his model, Poplawski considers the rotational speed of each roller as an independent variable which must be solved for in order to compute the operational conditions of the roller. Poplawski's model is quite similar to the complete analysis presented by Harris [12], but no simplifications are necessary thanks to the availability of powerful computers. It also includes the computation of the drag forces at each roller location. In Figure 2.3 the loads acting on a high speed roller, are shown according to Poplawski [14]. The similarity between this model [14] and the one presented by Harris [12] is obvious. In the model shown in Figure 2.3, there is an extra load acting on the roller, which is a drag force caused by the cage driving the roller and it is labeled fFFj. Therefore, rewriting equations (2.13) and (2.14) to include this term equations (2.35) and (2.36) are obtained. Ro __ (Q^j + F fvFm) Q.j = 0 (2.35) R. Ro QOj + F:j (Qwoj Fon Fj) = 0 (2.36) Ri Another difference between this model and the original is the computation of the deflection of the rollers. Harris [12] uses a load deflection behavior which ignores the lubricant film between the rollers and the raceways. Poplawski's model [14], does includes the deformation of the oil film between the rollers and the raceways. He modifies the deflection equation to 8j = 8,sinoj + 86ycos4j (Gx/2)+hj+hoj (2.37) 8j = 86j +68j (2.38) where 09 8hA 6j =) + P:j (2.39) =8 + Po ___ (2.40) (K'2 I6hEJ Pcij where h: Oil film thickness given by 8 C06(jO.u)0'7 E'0.3 R13 lW,0'3 h= _____________ (2.41) 3 po.3.3 6h 0.302 a'6(jiou)07 E'03 R013 1,.O.13 = (2.42) 6P p.3.3 The deflection behavior was used to determine the inner race contact loads as n F,, = EPij sin4> (2.43) j=l n Fy = EP1j cosoj (2.44) j=l which is the same method used by Harris. Figure 2.3 Loads acting on a High Speed Roller from Poplawski, J.V., "Slip and Cage Forces in a High Speed Roller Bearing," ASME Journal of Lubrication Technology, April 1972. One major development of Poplawski's model is the evaluation of the drag forces in more detail than in previous models. The equations used to determine the drag forces are as presented by Harris [12] but now evaluated at each roller location. The drag force acting on an unloaded roller with translator motion is given [14] as Fa^wm = Focj FJ (N) = 9.2(1+2t)G3Uou.'7 (2.45) There is also considerable friction between the cage and the guiding surfaces, either in the outer race or the inner race, depending on which side is used for cage riding. For inner race rotation and inner race guiding, Poplawski suggests that the force is given by Fp:xc = fs N (N) (2.46) where N is the normal force acting on the pilot. The last drag force component to be considered is due to the churning of the oil by the rollers and Poplawski introduces the following relationship Fo1h ,= 1/2 ?C.S.V.2 (2.47) where : Effective density of the mixture=%oil oil CD: Drag Coefficient S.: Effective Drag Area (mm2) V,: cage orbital velocity (m/sec.) As before, the force balance equations are not enough to solve for the unknown variables, namely F,, w., w,, and Poj. Torque balances are performed, based on an initial cage speed, for each loaded roller to estimate the rotational roller speed. Once the speed is computed for all rollers, the drag forces acting on the unloaded rollers and on the cage are estimated. Then a second torque balance is performed for the complete bearing assembly between the drag forces and the cage loads. If equilibrium does not exist, the cage speed is corrected and the roller conditions are computed once again. This procedure is iteratively repeated until equilibrium of the complete bearing is achieved. Poplawski's model has very good correlation with the experimental data presented by Boness [13]. It is a more complete model in the fact that it includes the speed of each roller as an independent variable. The incorporation of the individual roller drag forces makes of it a more realistic model. The work presented in [14] helped in the further development of the bearing analysis methods. In an effort to quantify the heat generation rates, Witte [15] derived some theoretical equations, which were later modified to accommodate experimental results for tapered roller bearings. A heat generation potential factor G was developed based on the geometry of a tapered roller bearing under pure thrust load. The author called this factor G; it is based strictly on the geometry of the bearing and it is a constant for a particular bearing series. The G factor can be obtained from a tapered bearing catalog or computed according to the equation given by Witte in [15] as D35 G = ___________________ (2.48) D0'7(nl)2/3 (sin a)X/3 The G factor is related to the resisting torque of the bearing. The lower the G factor is, the lower the heat generation for that bearing. The relationship between pure axial load, the G factor, and resisting torque is given by Witte [15] as M = l.lxlO"'G F. (Sut)05 (F.O)"3 (2.49) where M: Resisting torque (lb.in.). G: Bearing Geometry Factor. F.: Axial Load (lb.). S: Bearing Speed (rpm). 1: Lubricant viscosity at atmospheric pressure (cP). 0: Lubricant pressureviscosity index (in2/ilb). and it is limited for (SI) values larger than 3000 and for axial loads which are less than twice the axial load rating of the bearing. For the case when radial loads are applied instead of an axial load, equation (2.49) should be modified to compensate for the different orientation of the load. Equation (2.50) gives the relation between M, G, and radial loads. M = l.lxlO4'G (Sut)0 (fTF,/K)X/3 (2.50) where fT: Equivalent thrust load factor K: Ratio of basic dynamic radial load rating to basic dynamic thrust load rating. and it is limited for (S.L) values larger than 3000 and for radial loads which are less than twice the radial load rating of the bearing. The fT and K factors can be obtained either from [15] or from the manufacturer of the tapered bearing. Witte obtained good correlation between his equations and experimental data. One shortcoming of his experiments was that he used less than 1.9 liters per minute of lubricating oil. This is quite low compared with what is commonly used in high speed bearing applications. Astridge and Smith [16] performed an experimental investigation in an attempt to quantify the power losses, and heat generation in high speed cylindrical roller bearings. They used bearings with bore diameters of 300 and 311 mm, operating them at 1.1 million DN. The bearings were operated with diametral clearance, simulating operating conditions in aircraft engine bearing applications. From their experimental results and other published data, Astridge and Smith [16] suggested 10 sources of heat generation: 1) Viscous dissipation between rollers and races. 2) Viscous dissipation between rollers ends and guide lips. 3) Elastic hysteresis in rollers and races. 4) Dissipation in films separating cylindrical end faces of rollers and cage. 5) Dissipation in films separating cage and traces. 6) Dissipation in films separating cage side faces and chambers wall. 7) Displacement of oil by rollers. 8) Flinging of oil from rotating surfaces. 9) Oil feed jet kinetic energy loss. 10) Abrasive wear and asperity removal. As it can be seen from the list, most of the sources are due to the drag forces acting between the rollers and the lubricant. The lubricant is displaced by the rollers as they move within the bearing cavity. According to Astridge and Smith [16] the single most important source of heat generation is due to the churning of the oil between the rollers and the raceways. In the case considered in [16] not all the rollers were loaded. Performing a parametric study, Astridge and Smith identified which of the parameters related to bearing operation have a larger influence on heat generation. The ones with a stronger effect on heat generation were found to be speed, oil flow rate, oil viscosity and pitch diameter. In [17] Rumbarger et al., presented a sophisticated computer analysis for single row high speed cylindrical roller bearings. The authors incorporate into a single model the loaddeflection behavior, the kinematic and the EHD behavior and the thermal behavior. Previous models did not consider the effects on bearing behavior of the interaction between these components of bearing performance. In the model presented by Harris in [12] a single overall bearing temperature is considered, while in [17] the temperature at each contact is computed based on the kinematics, the EHD conditions and the loads present at that contact. In contrast with Poplawski's model, in the model presented in [17] the drag forces are computed for each roller element using the estimated EHD conditions for the speed and temperature estimated for each contact. These loads are then compared with previous speed and temperature iterations, which if different are corrected. If the kinematic conditions in a roller location change, the overall load distribution may be affected, causing a change in the elastohydrodynamic conditions and in thermal performance. Due to the iterative solution method used, the computation needs are enormous. It is then necessary to limit the analysis to steady state operation, otherwise the required computation capabilities would make the codes too complex and expensive to use. In this model, the elastic, kinematic, and thermal analysis are similar to the ones used in the models presented by Harris and Poplawski which were discussed previously. The model presented by Rumbarger et al. in [17] is relevant since it introduces the use of a complete fluid analysis to evaluate the viscous effects of the lubricant on the rolling elements. Therefore, the discussion of the model would be concentrated into this new development presented in [17]. The authors in [17] identified two main viscous drag torque sources. The first source is the viscous drag caused by the rolling elements moving through the lubricant. As the rollers rotate within the bearing cavity, the lubricant flows around them and between the rollers and the guiding surfaces. The second source of viscous drags according to Rumbarger et al. is caused by the motion of the cage within the bearing. As the cage rotates, it is in contact with the lubricant at the inner and outer surfaces, at the lands and at the side surface. The total drag torque is the sum of the drag torques acting on each rolling element plus the drag torque acting on the cage. The total drag torque acting on a roller is the sum of the drag torque acting on the roller surface, plus the drag torque acting on the roller end, plus the retarding torque caused by the contact between the roller and the cage. The torque acting on the roller surface is computed by the authors of [17] as T = Tw A r (Nm) (2.51) where T: Drag torque acting over the element surface (Nm). T,: Wall Shear Stress (N/m2). A: Surface area of the roller (m2). r: Reference radius from the center of rotation (m). The authors [17] recommend for the computation of the shear stress acting on the rollers equation (2.52). T = f(1/2 ?U2) (N/m2) (2.52) where f: Friction factor computed from the Reynolds number assuming turbulent flow [17]. : Fluid mass density (oil and air mixture) (Kg/m3) U: Mass average velocity of the fluid (m/sec) To compute the drag torque acting on the roller ends the authors recommend equation (2.53). T..u = 0.5 w' r' C, (Nm) (2.53) where T,,.: Drag torque acting on the end of the roller (Nm) w: Rotational speed of the roller (rad/sec.) Cn: Correlation factor: 3.87/(NR.)5 for laminar flow Nn.<300,000 O.15/(NR.)0'7 for turbulent flow NR.>300,000 The last torque component acting on the rollers is due to the contact between the rollers and the cage. To estimate this torque the authors recommend the following equation Fj3 N Vt Vutil T.Afta E ( E _________A__..r A0 N=1 VR V.:j S/2 2* E Avkrvk] (2.54) k=l where Fj.: Contact force between the roller and the guiding surface (N). N: Number of horizontal lamina. Vn1: Velocity of the race at the ith horizontal lamina (m/sec.). V.j.: Velocity of the roller at the ilh horizontal lamina (m/sec.). S: Number of vertical lamina. Tma: Torque produced at the rollercage contact (Nm). c:: Friction coefficient between the roller and the guiding shoulder. AH.,: Area of the ith horizontal lamina (m2). rj: Distance from the i"" horizontal lamina to the center of the roller (m). Avjk: Area of the kh vertical lamina (m2). r.vj: Distance from the kt"h vertical lamina to the center of the roller (m). AQ: Total contact area between the rollers and the guiding surfaces (m2). which is obtained by dividing the contact area into various vertical and horizontal lamina. To compute the torque induced by the cage moving through the lubricant, equation (2.51) is used for the inner and outer surfaces of the cage and for the lands. For the sides, equation (2.53) is used. The main problem of the fluid model is it sensitivity to the amount of oil inside the bearing cavity. The authors used a volume percent of 15 to 20%; the percent of the total bearing cavity volume which the oil occupies. These values of volume percent yielded good correlation between the experimental results and the model computations. The accuracy of this procedure is questionable, since there is no reliable way to measure the amount of oil inside the bearing cavity. The density used for the mixture is computed based on an amount of oil present in the cavity, which is difficult to determined. The major contribution of the model presented in [17] is the use of an interdisciplinary approach to solve for the operating conditions of a high speed bearing [18]. Since the model in [17] was presented, several advanced computer codes have been developed for the analysis and design of high speed bearings. The driving force for the development of these codes have been the need for more reliable bearings for combat aircraft mainshaft bearings [19]. Two main types of bearing analysis codes have been developed, for quasistatic or steady state analysis and for dynamic or transient analysis. The first is represented by programs such as SHABERTH for the analysis of shaftbearing systems, and CYBEAN, for the analysis of cylindrical roller bearings [20]. The dynamic analysis codes are represented by the program DREB, which is used to analyze the transient behavior of ball and roller bearings [20]. A major shortcoming of these computer codes is that their results are seldom compared to experimental results as pointed out by Parker in [19]. Another problem pointed out by Parker [19], is that even if comparison to experimental results is intended, there are some computations which cannot be compared since there is no experimental way to obtain experimental data to match the computations. For example, some programs include in their output roller skew angles and element temperatures which are yet to be measured experimentally. Another problem with the computer codes is the dependency on the volume percent of oil in the bearing cavity to estimate the thermal behavior of the assembly. Those researchers which have attempted a comparison between the computer results and experimental data are required to chose such a volume percent such that their computations approximate the experimental results [17, 19,21,22,23]. Although the computer programs are still to be improved, they have facilitated the development of advanced bearing designs. The use of an interdisciplinary approach to the analysis of the behavior of high speed bearing can only be done using the computer. The problem is too complex to be solved by a single individual without the assistance of a high speed computer. The codes currently can only be used in high speed supercomputers, which means there are not available to most engineers involved in designs with bearing applications. High Speed Bearings: Experimental Results In 1974, Signer et al. [24] presented experimental data on high speed angular contact ball bearings. ACBB of 120 mm diameter, 20 and 24 contact angles were tested to 3 million DN. The test conditions were made to simulate the operating conditions in an aircraft turbine. It was found in this investigation [24] that power losses increased linearly with speed and with increased oil flow rate through the inner race. Inner race lubrication was more effective than other lubrication in reducing the operating temperature, for the same oil flow rate. It was interesting to find that when the oil flow rate was increased over 3.8x103 cubic meters per minute (1.0 gpm), the temperature increased, probably due to the increased quantity of lubricant within the bearing cavity and to the resultant churning. Parker and Signer [25] present the results of their investigation of high speed tapered bearings. The bearings used had 120.65 mm bore with capability to use either jet lubrication or conerib lubrication. The use of conerib lubrication proved to be more efficient in limiting the operating temperature. It was also found that the use of conerib lubrication instead of jet lubrication reduced the power consumption. The experiment showed that the bearing temperatures and power losses increased with spindle speed. The effect of load on bearing temperature was insignificant. In [25], Parker and Signer presented results of their testing of TRB to DN values higher than one million. Since TRBs have a better loaddeflection characteristic than ACBBs or CRBs for the same envelope, they are preferred for some applications where weight or space are critical. It was also demonstrated that by providing the conerib/roller end contact with sufficient lubrication, TRB can be operated to very high speeds. The lubrication method recommended then was the use of holes drilled through the cone, through which oil was forced into the conerib area. Parker and Signer used specially designed TRB to investigate the high speed performance with conerib lubrication versus the performance with oil jet lubrication. The bearings used were of standard design but provided with conerib lubrication to improve their high speed operating performance. The bearing tested had a bore diameter of 120.6 mm, an outside diameter of 206.4 mm, a cup angle of 340, and it contained 25 rollers. The test speeds were 6,000, 10,000, 12,500 and 15,000 rpm. The oil flow rates used were 1.9 x 103 to 15.1 x 103 m3/min. The test results obtained in [25], showed that conerib lubrication plus jet oil lubrication was a better lubrication arrangement than oil jet lubrication alone. In fact, the higher speeds could not be achieved safely with oil jet lubrication alone. As for oil flow rate, by increasing the oil flow rate, temperatures decreased while power losses increased. It was also shown that for oil flow rates over 11.4 x 103 m3/min, a further increase in oil flow will not produce a significant temperature decrease. Observing the power losses induced by the increased oil flow rate, the use of oil flow rates larger than 11.4 x 103 m3/min do not seem justifiable. Spindle speed also produced considerable increase in temperature and power losses. The effects of load on bearing temperature were insignificant compared to the effects of the oil flow rate and spindle speeds tested. The authors of [25] used the equation derived in [15] to estimate the heat generation rates. The power losses estimated using the equation from [15], had good correlation with the experimental results. Parker et al. [26] presented results of computer optimized TRB bearings. These bearings were designed by first optimizing the standard TRB design, as the ones used in [25]. The optimized design was then presented to a leading TRB manufacturer who suggested changes which would allow the bearing to be economically manufactured. The bearings used in [26] used 23 rollers, it had a cup angle of 310, 120.65 mm bore diameter and outer diameter of 190.5 mm. The bearings were provided with conerib lubrication and instead of oil jet lubrication, the front of the bearing was lubricated through holes in the cone and through the spindle. Oil was forced centrifugally through these holes into the front of the bearing. Test speeds varied from 6,000 rpm to 20,000 rpm. Oil flow rates varied from 3.8 x 103 to 15.1 x 103 m3/min. The computer optimized bearing operated at lower temperatures, lower power losses and higher spindle speeds than the standard bearing. Effects of oil flow rate, spindle speed and load on bearing temperatures and losses were similar for the optimized bearing and the standard bearing. Currently, aircraft engines operate at a maximum DN value of 2.4 million [23,27]. The mean time between bearing removal is up to 3000 hours from 300 hours ten years ago [34]. Improvements in the lubrication methods have allowed researchers to operate ACBB and CRB to 3.0 million DN, while TRB have been operated to a 2.4 million DN [23,27]. The use of AISI M50, a vacuuminduction melted, vacuum arc melted alloy, has greatly improved the fatigue life of high speed bearings. CHAPTER III EXPERIMENTAL EQUIPMENT High Speed, High Power Milling Machine The Machine Tool Laboratory at the University of Florida is equipped with a HSHP milling machine, shown in Figure 3.1. The spindle is driven by a 115 kW, 3000 rpm, ASEA D.C. motor by means of a two stage flat belt transmission. The first stage is a belt from the motor to the intermediate shaft, located in the column of the milling machine. The second stage, is from the intermediate shaft to the the spindle. The speed ratio used for the high speed test between the motor and the spindle was 0.26. The spindle is mounted on the HSHP milling machine on the front, bolted to a mounting bracket. Lubrication connections and instrumentation are external to the HSHP machine, making the change of spindles a simple task. To change the spindle mounted on the machine, the current spindle is unbolted and removed using a hoist. The next spindle can then be mounted and bolted. The lubrication system can easily be modified to accommodate several spindle designs. AXIS SERVO Figure 3.1 HSHP Milling Machine This HSHP milling machine permits a complete investigation of the configurations under study. Each configuration is tested not only for idle operation performance, but also for cutting capabilities and chatter stability. Test Spindles The two spindle bearing configurations shown in Figure 3.2 and Figure 3.3 were tested for HSHP performance. Their operating temperatures, lubrication needs and power demands were investigated at several speeds, during idle, no load operation. Both spindles were equipped with circulating oil lubrication. The amount of oil circulated was varied from 1.5 liters to 3.8 liters per minute, per bearing. The spindles were tested for maximum operation speed. Configuration I is based on double row cylindrical roller bearings (CRB) NN 30K/SP manufactured by SKF. It has one NN3019K/SP on the drive side and a NN3022K/SP on the tool side. The CRBs support the radial loads while the thrust load is supported entirely by a Radiax, a 234420 BMI/SP series angular contact thrust ball bearing (ACTBB) by SKF, with a contact angle of 60. This configuration is sometimes referred to as Standard Configuration I by SKF researchers [28]. The preload in this configuration is provided, individually for each bearing. The radial bearings are preloaded radially by eliminating any clearance between the outer race and the rollers. As it can be seen from Figure 3.2, tightening the nut A pushes on the inner race of the lower CRB, moving the inner race and the rolling elements up the tapered. As the rolling elements are driven up the taper of the spindle, the diametral clearance between the elements and the outer race is reduced. Tightening the nut further, contact between all the rollers and the outer race is produced, completely eliminating any clearance. If nut A is tighten even more then interference is produced. The ACTBB is preloaded by tightening the nut B to press together the bearing assembly. As the nut B is tighten, any gap between the races and the bearing spacer C is eliminated. Once the nut B is completely tighten, the preload between raceways and balls is achieved. The preloading of this configuration is done during the assembly of the spindle and cannot be released, unless the spindle is completely disassembled. The maximum speed achieved by this configuration was 8,000 rpm. The operation temperatures were above the recommended for the type of oil used. The power losses were almost 14 kW, which means that for a 20 kW milling machine could only perform 5 kW of useful work at 8,000 rpm. When the spindle was driven over 8,000 rpm, the ACTBB failed within seconds of starting the test. This happened twice: at 9,000 and 10,000 rpm. The failure was too fast for the PROMESS sensor to detect any increase in the load of the bearings. After discussing the failure with SKF researchers, it was concluded that the cause of the failure was the loss of preload. The loss of preload induced skidding, which was the mode of failure of the bearing. To correct the problem, the mounting preload must be increased and a larger amount of lubricant must be provided to the upper raceway. To achieve this increase in preload, the spacer separating the two raceways, spacer C, must be ground, bringing the two raceways closer together. This increase in preload would also induce an increase in bearing temperature, which could not be permitted, since operating temperatures are already too high. Configuration II is based on TRB. This configuration operates under constant preload. A constant preload is maintained by the bearing in the drive side, the HYDRA RIB, by TIMKEN, Figure 3.4. The bearing is provided with a hydraulic chamber and piston mechanism which provide a load to the back of the rollers. As the chamber is pressurized, the piston displaces forward, pushing on the rollers. This forward displacement of the rollers produces the diametral interference or preload. The preload force is proportional to the hydraulic pressure in the chamber. If during the operation of the spindle the loads acting on the rollers increase, the piston would retract to a point where the load on the rollers equals the preset value. If on the other hand, the load on the bearing is reduced during the operation of the spindle, the piston would move forward until the preset load on the rollers is reestablished. Beaing NN3019 K Being 234420 ll Being M3022K Figure 3.2 Configuration I Test Spindle Hydra Rib Bearing >Oil Distribution Rings  High Speed Bearing Figure 3.3 Configuration II Test Spindle Oil Jet Hydraulic Oil Input \ Snap Ring , Rib Chamber Figure 3.4 HYDRARIBT Bearing Roller cone Piston Outer Race Inner Race Tapered Roller Cone Rib Lubrication Ring Cone Rib Lubrication Hole Figure 3.5 High Speed Bearing with ConeRib Lubrication The High Speed (HS) bearing, 100 mm diameter, in the tool side, Figure 3.5, is provided with conerib lubrication. The cone is lubricated through holes drilled from the back of the bearing to the conerib. At the back of the bearing, there is a ring which entraps the oil supplied by jets forcing it centrifugally into the holes. This configuration operated successfully up to 10,000 rpm. The only failure experienced with this configuration happened when lubrication to the conerib interface was interrupted. The operation temperature was at all times very acceptable with very low oil flow rates. The power losses were lower than those for Configuration I. Oil Supply to the Bearings As mentioned before, the configurations are equipped for circulating oil lubrication. Figures 3.2 and 3.3 show the oil inlet and outlet points for each configuration. Configuration I, is provided with two oil inlets per bearing, one at each side of the spindle housing. Once inside the housing the oil is forced around the bearings through a groove in the outer surface of the outer race the bearing. The oil enters the bearing through three holes in the outer ring 120 degrees apart, provided for that purpose. Through these holes the oil is forced into the bearing cavity between the two rows of elements as shown in Figure 3.6. The oil is then forced out of the bearings, by the rolling element motion and centrifugal forces. The oil is then sucked out of the bearings through the exit ports. Configuration II is provided with three oil inlet points: two for the high speed bearing, and one for the HYDRARIBT. Once the oil enters the housing it is directed to the front of the bearings by the distribution ring. Both bearings are provided with rings at the front (small end of the rollers). The high speed bearing is provided with a second distribution ring which feeds three oil jets. These jets direct the flow to the back of the cone, which is provided with a special ring. This ring entraps the oil from the jets, which is then fed centrifugally into the conerib interface through holes drilled for that purpose in the cone. Configuration II was designed for horizontal use. When mounted in the vertical position, the upper bearing does not receive the required lubrication due to gravitational forces. Since the oil is sprayed up from the distribution ring, in vertical applications, it does not have the necessary pressure to force the oil through the bearing. To correct this problem, a screw type pump was provided above the HYDRARIBT. This pump supplied the necessary pressure drop to overcome gravity and provide an efficient flow of oil as long as a supply of 3.8 1pm is maintained to the top bearing. OIL\ Figure 3.6 Oil Supply to Double Row Bearings Instrumentation During the tests of configurations I and II, the temperatures were monitored using type K thermocouples placed at strategic positions in the test rig. The thermocouples were connected to a digital display thermometer. The thermocouples were located at the following positions: 1. In the oil supply line. 2. In the oil return line. 3. At the oil exit point of each bearing. 4. At the outer race of the bearings. 5. On the surface of the housing. The thermocouples at 1. and 2. measured the bulk oil temperatures before and after passing through the housing. The thermocouples used at 3. were in the suction line removing oil from each bearing. These thermocouples measured the exit temperature of the oil from each bearing, while 2. measures the temperature of the mixture of the oil from all bearings. Position 4. was measured for each bearing through a hole in the housing. Position 5. was measured at surface points above position 4. The thermocouples used in 1., 2., and 3. were in direct contact with the oil. The thermocouples used in 4. were encapsulated in a bayonet type assembly. The thermocouples used in 5. were in direct contact with the housing. Sensor Sensor Figure 3.7 PROMESS Sensor The load on the bearings was monitored using the PROMESS sensor. As shown in Figure 3.7, strain gages are located on the outer surface of the outer ring. As the loaded elements pass over the strain gages, these will provide an electrical signal proportional to the rolling element load. The PROMESS sensor is especially useful when monitoring the transient loads on the bearings. The spindle speed was measured by using a magnetic pickup and gear installed at the top of the spindle. The speed was displayed on a electronic counter at all times. This speed was compared against the speed measured using a handheld tachometer. The speed was monitored throughout the test. The input power to the motor was monitored using a set of current and voltage meters in the motor controller box. These meters measured the current and voltage supplied to the D.C. motor. The input power was computed from these measurements. Oil Circulating System There are several lubrication methods used in machine tools among them, grease lubrication, oil mist lubrication, airoil lubrication ("OL"), and circulating oil lubrication. Although the amount of oil required for lubrication is small, for high speed applications large amounts of oil must be used to provide the bearings with the necessary cooling. The oil circulating through the bearing cavity removes a large part of the heat generated. So far, circulating oil is the only lubrication method which provides the necessary cooling for high speed bearing applications. The oil used throughout the investigation was a SAE 10 equivalent oil, common in machine tools. A single type of oil was used. The use of a heavier oil will increase the hydraulic power losses and consequently, the operating temperature of the bearings. The power available for useful work (milling) will also be reduced due to an increase in hydraulic power losses. If on the other hand a lighter oil is used, the oil may exceed its operating range at high speeds and degrade. The friction between the rolling elements and the raceways would then increase, inducing an even larger operating temperature. Figure 3.8 shows the circulating loop for the cooling and lubrication of the housing. Since circulating oil lubrication is going to be used to cool the bearings, large quantities of oil are necessary. The oil must be kept at constant temperature, since the experimental investigation would be affected by a variable supply oil temperature. The oil is pumped from a 280 1 storage tank to the spindle by the supply pump. The supply pump is a variable vane pump with an operating range from 4 1pm to 53 1pm. Just before reaching the spindle, the oil flow is distributed into three streams. Each stream is controlled by a combination of needle valve and a flow meter. Here, the amount of oil going into each bearing is measured and controlled. If configuration I, is being tested, each of the three streams is then split in two, to supply the oil to the bearing from both sides of the housing. Once the oil has circulated through the bearings, removing heat from the bearing cavity, it is sucked out of the housing and returned into the storage tank by the suction pump. Due to the amount of churning within the bearings, the oil exiting the spindle is sucked out as foam. In the storage tank the oil is defoamed and cooled. To defoam the oil, it is passed through the screens, which removes the entrapped air. The oil is then pumped from the tank through the heat exchanger by the circulating pump. The cooling fluid in the heat exchanger is chilled water, from the laboratory's air conditioning system. After passing through the heat exchanger, the cold oil is returned back to the storage tank, near the warm oil return point, refer to Figure 3.8. It is a known fact that the larger the difference in temperature between the two fluids in the heat exchanger, the more efficient it works. The need to remove the foam from the oil before it passes through the heat exchanger limits the alternatives as where to locate the inlet to the heat exchanger. If the suction point of the cooling circuit is placed next to the warm oil return, all the foam coming into the tank will be pumped into the heat exchanger, reducing its efficiency. Therefore, the suction of the cooling circuit must be placed on the proper side of the screens, the closest possible to the warm oil return. Evaluation of Cooling Capacity In the initial stages of the investigation, it was observed that the temperature of the supply oil increased during the test. This increase in temperature significantly affected the investigation since the bearing temperature could not be related to a constant oil supply temperature. Therefore, an evaluation of the cooling system was performed. The question to be answered was if the circulating system was capable of providing the necessary cooling effect, removing from the warm oil all the heat it acquired from the bearings. The amount of heat removed by the oil, from the spindle is given by Qo = m c(To., T,) (kW) (3.1) where QoiB;: Heat removed by the oil from the bearings (kW). m: Oil flow rate (1pm). : Oil density (g/ml). c: Heat capacity of the oil (kJ/(kg C)). T.: Supply oil temperature (C). Tou.: Return oil temperature (C). The heat removed from the oil in the heat exchanger is given by QoLxmx = m c(TM. Toue) where The given by (kW) (3.2) QoCIHE: Heat removed from the oil (kW). m: Oil flow rate (1pm). z: Oil density (g/ml). c: Heat capacity of the oil (kJ/(kg C)). T,,: Oil temperature entering the heat exchanger ("C). Tout: Oil temperature exiting the heat exchanger (C). heat removed by the water in the heat exchanger is Qw.t.H = m c(Tou, T..) where (kW) (3.3) Qwf.m.3: Heat removed by the water from the oil (kW). m: Cooling water flow rate (1pm). n: Density of the water (g/ml). c: Heat capacity of the water (kJ/(kg C)). Tj,: Water temperature entering the heat exchanger (C). Tot: Water temperature exiting the heat exchanger (C). Experimental data was collected at steady state, it is listed in Table 3.1. With this data, the amount of heat removed from the oil in the heat exchanger, the amount of heat acquired by the oil from the bearings and the amount of heat acquired from the oil were computed. It was found that the heat exchanger did have the necessary capacity to cool the oil to the desired supply temperature. As it can be observed from Table 3.1, the temperature of the oil entering the heat exchanger is much lower than the temperature of the returning oil. Therefore, the problem was not that the heat exchanger could not supply the necessary cooling, but that the warm oil was not getting to the heat exchanger until it is too late. Upon inspection of the tank, it was found that the oil inlet to the heat exchanger was too far from the warm oil return point. This caused the warm oil to concentrate on one side of the tank, heating that side of the tank. This accumulation of warm oil increased until it reached the heat exchanger oil inlet. By that time, the amount of oil which needed to be cooled was above the cooling capacity of the heat exchanger, which in the mean time was circulating cool oil. To solve the problem the oil inlet point into the heat exchanger was moved closer to the oil return point. It could not be moved close enough since it must be placed after the screens, otherwise, the foam would make its way into the heat exchanger, reducing its cooling capacity. The final solution was to return the cold oil beside the warm oil return. This kept the return side cold and there was no chance for the warm oil to accumulate in that side. Table 3.1 Heat Exchanger Temperatures Water Oil Speed Oil Flow In Out In Out Ret.Oil 3000 1.9 ipm 100C 160C 310C 13C 440C 3000 3.0 1pm 110C 17C 33C 14C 410C 3000 3.8 1pm 10C 180C 33C 15C 38C 4000 0.8 1pm 8C 12C 140C 12C 660C 4000 1.9 1pm 110C 180C 36C 21C 530C 4000 3.0 1pm 11C 20C 370C 230C 490C 4000 3.8 1pm 110C 21C 370C 22C 490C 5000 1.9 Ipm 100C 220C 260C 220C 660C 5000 3.0 ipm 100C 25C 31C 250C 62C 5000 3.8 1pm 9C 27WC 330C 27C 61C 6000 0.8 1pm 7C 110C 12C 10C 640C 6000 1.5 1pm 6C 12C 16C 12C 67C Seals As both configurations are lubricated using circulating oil and mounted in the vertical position, proper sealing is imperative. Any oil that leaks out of the housing, through the bottom, will fall on the workpiece. This oil may affect the life of the tool by exaggerating the thermal cycling of the tool, causing the failure of the tool. Also, it represents a hazard to the operator, since at high speeds, the oil is sprinkled onto the surroundings, making the area quite slippery. Due to the high rotational speeds, noncontact seals must be used. Noncontact seals have the extra advantage that they do not contribute to the friction torque, thus reducing the amount of heat generated. A similar arrangement of labyrinth seal was used for both configurations. A section view of the seal, for Configuration I and Configuration II, is shown in Figure 3.9 and in Figure 3.10, respectively. Both configurations were effectively sealed for most of our operating conditions. Spindle Suction Pump ...... Screens Variable Output Pump Figure 3.8 Circulating Oil System Spindle Housing Oil Suction Points Figure 3.9 Seal for Configuration I Oil Jet Housing CneRib Lubriction Inpuit il i /ints Oil Su.,,tion Poin H50 Tper Figure 3.10 Seal for Configuration II CHAPTER IV THERMAL ANALYSIS Thermal Analysis of the Spindle Housing To estimate the heat generation rates of bearings, researchers and bearing manufacturers have developed several empirical and theoretical equations. These equations relate heat generation to bearing geometry, operating conditions and lubrication parameters. In this chapter these relationships will be presented and compared among themselves and to experimental results. Also in this chapter, thermal profiles are presented, showing temperature distribution along the housing. The presence of thermal gradients between the bearings and the spindle housing may induce an increase of the original preload, which in some instances may cause bearing seizure. The thermal gradient is induced by the faster increase in rolling element temperature compared to the housing during the acceleration of the spindle to the operating speed. Friction in Rolling Bearings The heat generated in the bearings is the product of frictional power losses. The sources of these frictional losses as identified in [16,29,30] are: 1. Elastic hysteresis in rolling. As the bearing rolls there are deformations in the raceways and in the rolling elements. The energy consumed in producing this deformation is partly recovered when the element rolls to the next position. 2. Sliding in rollingelement/raceway contacts due to the geometry of the contacting surfaces. 3. Sliding due to deformation of contacting elements. 4. Sliding between the cage and the rolling elements, and between the cage and the guiding surfaces. 5. Sliding between roller ends and inner and/or outer ring flanges. 6. Viscous drag of the lubricant on the rolling elements and cage. The viscous friction is produced by the internal friction of the lubricant between the working surfaces. Also the churning of the oil between the cage and the rolling elements, between the raceways and the rolling elements and flanges. These losses increase with speed and amount of lubricant in the bearing cavity. In the experimental investigation, the effect of the above power losses were grouped into two measurable amounts, Mechanical Power losses and Hydraulic Power losses. The mechanical power losses are the consequence of mechanical friction in the bearing cavity, without oil being circulated through the bearing. The hydraulic power losses are the results of viscous friction between the oil in the bearing cavity and the rolling elements. These two main sources of heat are discussed in Chapter V. Heat Generation The increase in temperature during the operation of the bearings is the result of friction losses, which are manifested as heat. The sources of friction in a bearing, as mentioned above, include the friction at the contact between rolling element and each raceway, friction between the cage and the rolling elements and viscous drag between the circulating oil and the rolling elements. Several empirical relations have been developed to estimate the amount of heat generated in a bearing. The frictional power consumed by a bearing is given by [9,11] as Hf = 1.05x104 n M (W) (4.1) where Hf: heat generated (W) n: spindle speed (rpm) M: friction torque (Nmm) Also from [9,11], the bearing manufacturer estimates the friction torque as M = 0.5 Vj, F d (Nmm) (4.2) where Vif: friction coefficient for the bearing F: bearing load (N) d: bore diameter of the bearing (mm) The friction coefficient IL is given in [9,11] for several types of bearings for cylindrical roller bearings .f = 0.0011 for thrust ball bearings = 0.0013 for tapered roller bearings = 0.0018 These friction coefficients are for single row bearings operating at average speed and at a load for a life of 1000 million revolutions. The loads acting on the bearings are reactions to the belt tension. The magnitude of this tension is computed following the procedure suggested by the manufacturer in [31] for the type of belt used. For Configuration I, the tension load is 8600 N, while for Configuration II, the tension load is 3600 N. With the tension load known and using load equilibrium, the bearing reactions for configuration were determined. For Configuration I, the load on the lower bearing (NN3022 K) was estimated at 3000 N, for the top bearing (NN3019 K) it was 12000 N and the center bearing, the Radiax, 400 N which is the weight of the spindle. For Configuration II, the load acting on the lower bearing, the High Speed Bearing, was estimated as 1600 N, while at the HydraRibT the belt tension component was 5200 N and an axial component of 400 N due to the weight of the spindle. Using equation (4.2) to compute the friction torque for both configurations, using double the friction coefficient for the double row bearings, the following estimates were obtained: for Configuration I M30O22. = 360 Nmm Mmo K = 1232 Nmm M234420Moi = 46 Nmm for Configuration II Mm. 3xIB = 489 Nmm M.s = 146 Nmm The heat generated, computed using equation (4.1), at the different test speeds, for each configuration are listed next. Configuration I @ 3,000 rpm 516 Watts @ 5,000 rpm 859 Watts @ 7,000 rpm 1204 Watts @ 8,000 rpm 1376 Watts Configuration II @ 3,000 rpm 781 Watts @ 5,000 rpm 1312 Watts @ 7,000 rpm 1837 Watts @ 9,000 rpm 2362 Watts A more accurate way to compute the friction moment is by dividing it into two parts: an idling torque M, and a load torque M.. The sum of the two is the friction torque. The idling torque represents the friction torque during idle operation of the bearing and is given by [9,11] as Mo, = fxlO'8(vn)2'3d,3 vn>= 2000 (4.3) Mo, = foX16OxlOd,3 vn< 2000 (4.4) where fo,: factor depending on bearing design and lubrication method, for vertical spindles and oil jet lubrication: for double row ACBB............. 9 for CRB ...................... 46 for TRB ....................... 810 v: oil viscosity at working temperature (cS) d,: mean diameter of the bearing (mm) The friction torque due to the applied load can be computed using an equation recommended by Palmgren, [11]. Mi = fx Fed. (Nmm) (4.5) where M.: friction torque due to the load (Nmm) f,: factor dependent on the geometry of the bearing and relative load. Fe: equivalent force, as described below (N) d,: mean bearing diameter (mm) Recalling equation (2.2), for ball bearings, the factor f. is given by f = z( ) (4.6) for angular contact ball bearings, z=0.0001 and y=0.33 [11]. For roller bearings, f. will be for cylindrical roller bearings: f1= 0.00020.0003 for tapered roller bearings: f1=0.00030.0004 Fa for ball bearings is given by the following equations, also from [11]. Fe = 0.9F. ctna 0.1F, (4.7) or Fe = F, (4.8) whichever is larger, (4.7) or (4.8). For radial roller bearings, F, is given below as Fa = 0.8F. ctn a (4.9) or F. = F, (4.10) whichever is larger, (4.9) or (4.10). In Figures 4.1 and 4.2, the computed generated heat is plotted at different test speeds and oil flow rates for Configurations I and II, respectively. The generated heat was computed by adding the idle friction torque and the applied load friction torque and substituting the sum into equation (4.1). In Figures 4.3 and 4.4, the power losses determined experimentally for Configurations I and II, respectively, are plotted. The experimental power losses shown in the figure represent the sum of the Mechanical Power Losses and the Hydraulic Power Losses, which are defined in Chapter V. As it can be observed by comparing Figures 4.1 and 4.2 against Figures 4.3 and 4.4, equations (4.2) to (4.3) predicted a heat generation much lower than the measured during the test. The supply oil temperature and the return oil temperature were used to compute an average oil temperature for the computation of the viscosity of the oil inside the bearing cavity. For their tapered bearings, TIMKEN recommends in [10] the equations that follow to estimate the friction torque and the heat generation. M = kx G (SA)"5 (F.))"'3 (Nm) (4.11) where M: bearing operating torque (Nm) k.: conversion factor = 7.56x 106 (metric units) G: bearing geometry factor as given in the TIMKEN bearing catalog [32,33]. for HYDRARIB. = 152.7 for High Speed Bearing = 129.5 S: spindle speed (rpm) 4: oil viscosity (Centipoise) F.a: equivalent axial load (N) if the bearing is under combined loading, the equivalent load F.,q is determined as K if __ F., > 2.5 then F., = F,. (4.12) F, otherwise 1 F.a = f, F, (N) (4.13) K where F.: axial load (N) F,: radial load (N) K: bearing K factor, from the TIMKEN bearing catalog [32]: for HYDRARIBT = 1.63 for High Speed Bearing = 1.23 f,: axial load factor, function of (KF.)/F, as given in the bearing catalog [33]. To compute the heat generation rate for tapered roller bearings, equation (4.14), which is recommended by TIMKEN for their bearings was used. The computed generated heat is plotted in Figure 4.5, versus spindle speed at constant oil flow rate. Q = k2 G S"5 A5 FaQ ./3 (4.14) where Q: heat generation (W) k2: conversion factor (metric)= 7.9x107 As it can be observed by comparing Figure 4.5 against the experimental measurements in Figure 4.4, equation (4.14) predicted quite well the generated heat for Configuration II. o 1.5 LPM * 3.8 LPM A 2.3 LPM 1.2[ 3o.8 I, 4J L C9 M A I a 3.0 LPM 0 1000 2000 3000 4000 5000 8000 7000 8000 M000 10000 Spindle Speed RPM Figure 4.1 Computed Generated Heat vs. Spindle Speed Configuration I  I  o 0.8 LPM * 2.3 LPM S1.5 LPM S3.0 LPM 1.6 1.2 .8 0.0 0 Figure 4.2 Computed Generated Heat vs. Spindle Speed Configuration II I 000 2000 3000 4000 5000 8000 7000 8000 9000 10000 Spindle Speed RPM m .. .. ... if, S ... . m I o 1.5 LPM 3.8 LPM A 2.3 LPM a 3.0 LPM I 0 1000 2000 3000 4000 5000 6000 7000 8000 0W00 10000 Spindle Speed RPM Figure 4.3 Experimentally determined Heat Generation Configuration I 20 v a 15 01 I "3 0 L. 10 a, o 0.8 LPM * 3.0 LPM A 1.5 LPM o 2.3 LPi 1000 2000 3000 400) 5000 000 7000 8000 8000 10000 Spindle Speed RPM Figure 4.4 Experimentally Determined Heat Generation Configuration II 15 10 4j E 5 o 0.8 LPM o 2.3 LPM A 1.5 LPM * 3.0 LPM I000 2000 3000 4000 5000 8000 7000 8000 0000 10000 Spindle Speed RPM Figure 4.5 Computed Heat Generation Configuration II 12 I 8 03 I4 u ..... i t Heat Removal During high speed operation of rolling elements, the heat generated within the bearings is considerable, as it will be shown in Chapter V. This heat must be removed to avoid excessive thermal loads on the elements. If the temperature rises too much, the lubricant may exceed its operating range and the oil film between raceways and elements could be eliminated. Circulating oil lubrication has the largest heat removal capacity, due to the amount of oil which is forced through into the bearing cavity. The amount of heat removed by the oil can be computed by multiplying the mass flow of the circulating oil, by its heat capacity, times the change in temperature. Pol x= (m c)o1(TotT.jxy) (kW) (4.15) where Po1x: power removed by the oil (kW) m: oil flow rate (1pm) c: specific heat time the density of the oil, 1566 (KJ/(m3 C)) Tout,: oil temperature at the exit of the housing (C) T=uBP.y: oil temperature at the inlet of the housing (C) Table 4.1 Removed Heat/Generated Heat ~_______ Configuration I________ Speed Oil Flow Generated Removed Percentage RPM Rate (LPM) Heat (kW) Heat (kW) Removed 3000 1.5 3.6 2.7 76 3000 2.3 4.9 4.0 81 3000 3.0 5.5 4.5 82 3000 3.8 5.6 5.0 90 5000 1.5 6.6 4.6 70 5000 2.3 8.1 6.3 78 5000 3.0 9.0 7.0 78 5000 3.8 11.1 9.2 83 6000 1.5 9.0 5.4 61 6000 2.3 10.8 7.4 68 6000 3.0 11.8 9.2 77 6000 3.8 12.9 10.7 83 7000 1.5 12.5 5.26 42 7000 2.3 14.3 8.3 58 8000 1.5 14.4 5.8 40 8000 2.3 18.3 9.4 51 8000 3.8 20.7 17.3 83 Table 4.2 Removed Heat/Generated Heat ________Configuration II_____ Speed Oil Flow Generated Removed Percentage RPM Rate (LPM) Heat (kW) Heat (kW) Removed 3000 0.8 2.9 2.0 70 3000 1.5 2.9 2.4 82 3000 3.0 2.9 2.7 93 3000 3.8 2.9 2.6 89 5000 0.8 4.3 3.4 80 5000 2.3 4.6 3.6 79 5000 3.0 4.8 3.9 82 5000 3.8 4.8 4.0 82 7000 1.5 7.0 4.8 64 7000 2.3 7.0 4.9 70 7000 3.0 7.0 5.5 79 9000 0.4 10.6 5.5 52 9000 0.8 11.4 6.0 53 9000 1.5 12.4 6.3 54 9000 2.3 13.3 7.7 58 As the amount of oil increases, so does the cooling capacity, removing more heat from the bearings. The ratio of the heat removed to the heat generated increases with increased oil flow rate. In Tables 4.1 and 4.2, the percentages of generated heat removed by the oil are listed, for each configuration, at each speed and oil flow rate. As it can be seen from the table, as the oil flow rate increases, the percentage of the generated heat which is removed increases. Also from the table, as speed increases, for the same flow rate, the percentage of the generated heat removed by the oil decreases. This can be explained by the fact that as the speed increases, so does the temperature of the bearing, as it will be shown later. A higher bearing temperature will produce a larger heat conduction rate through the housing due to a larger temperature gradient between the housing and the environment. Thus, less heat is convected away by the oil. Also with an increase spindle speed, the oil in the bearing cavity traps a larger amount of air, changing itself into foam and hence reducing its convection capacity. An increase in oil flow rate will also produce an increase in power losses, as it will be shown in Chapter V. The increase in power losses is, in some cases, large enough to nullify the increased cooling capacity that a larger oil flow rate produces. Therefore, the net effect may be an insignificant decrease in temperature and a significant increase in power losses. From the experimental results, such as power losses, operating temperature and oil flow rate, design recommendations will be made for each configuration. Steady State Temperature Fields The steady state thermal fields were computed for both configurations. The analysis was performed using finite difference methods. The housing was divided into ring elements as shown in Figure 4.6 for Configuration II. The initial temperature was taken as room temperature except for those elements covered by the boundary conditions. The program was stopped when the surface temperature of the model approximated the experimentally measured surface temperature. Forced convection at the housing surfaces was assumed since the spindle rotation produces a considerable flow of air around the spindle. The equations used to estimate the amount of heat conducted radially from one element to the next are given by [34] as 2nkl Qa.L=_____ (TjT) (Watts) (4.16) Ro ln R1. where Qr.ei.x: Heat transmitted in the radial direction (Watts) k: thermal conductivity of the housing material (cast iron= 52 W/(m2C) [34] 1: axial length of the element (m) Tj: temperature of the j^ element (C) Tj.: temperature of the ijh element (C) Ro: outer radius of the housing (m) R.: inner radius of the housing (m) The equation used to compute the heat conducted from one element to the next in the axial direction is 2nkrdr Q^.__.= (TjT.) (Watts) (4.17) x where Q...L: Heat transmitted in the axial direction (Watts) r: radius of the i* element (m) dr: radial width of the element (m) x: axial distance between nodes (m) The equation used to estimate the heat conducted away by the air surrounding the spindle housing is given by [34] as Qoov o. = h A (T,, T.) (Watts) (4.18) where Qoov.oi.: Heat removed away by convection (Watts) h: convection coefficient = 9 W/m2 [34] A: heat transfer area (m2) T2: housing surface temperature (C) T.: temperature of the surroundings (C) Equation (4.19) was used to estimating the radiation heat transfer. Qrmimn = o 6 F A (Th4 T.4) (Watts) (4.19) where Qd.L.o: Heat removed away by radiation (Watts) a: StefanBoltsman constant = 5.66961x 10" (W/(m2K') [34] e: emissivity (.8) [34] F: shape factor = 1.0 [34] The boundary conditions used for the analysis of each housing were: 1. The bearings are represented as elements with constant temperature. The temperature assigned is the temperature of the bearing at steady state measured in the test. 2. The temperature at the inside surface of the housing is assumed to be equal to the average between the surface temperature of the center and the average bearing temperature, for the given speed and oil flow rate. 3. At the outer surface the housing loses heat to the environment through convection and radiation. 4. There is forced convection and radiation at the top surface. 5. The temperature of the environment was assumed constant at 220C. The computed thermal profiles for Configuration II at 5,000 rpm, 7,000 rpm and 9,000 rpm, and an oil flow rate of 2.3 1pm are shown in Figures 4.7, 4.8 and 4.9. The computed thermal profiles will be used to compute thermally induced loads on the rolling elements. Thermally Induced Loads As heat is generated in the bearings, a temperature gradient is developed between the bearings and the outer surface of the housing. Since the bearings and the housing are heating at different rates, their expansions occur at different rates. These differential expansions induce loads on the bearings. These loads will be proportional to the difference in thermal expansions between the bearings and the housings. Let's assume that the inner race, the rolling elements and the outer race are all at the same temperature. The thermal expansions of the inner ring, the outer race and the ith ring of the housing model are given respectively by 6. = F d n (T T=) (m) (4.20) 6io = F do Tx (To T.) (m) (4.21) 85k = F dh,, n (Thi T.) (m) (4.22) where 68.: thermal expansion of the inner ring (m) 6,T,: thermal expansion of the outer ring (m) 68,H: thermal expansion of the ilh housing element (m) r: thermal expansion coefficient 10.6xl06 C' [30] d: inner ring diameter (m) do,: outer ring diameter (m) dh.: diameter of the il housing element (m) T: temperature of the inner ring (C) To: temperature of the outer ring (C) Th,: temperature of the ill housing element (C) T.: starting temperature (C) The thermal expansion of the outer race is prevented by the much slower expansion of the housing. It is at this joint that the thermally induced interference happens, increasing the bearing preload. To determine the induced load, the expansion of the housing must first be computed. Using the thermal fields computed above, the expansion of each ring element in the housing can be computed. The expansion of the element in contact with the bearing can therefore be computed, and after computing the expansion of the outer ring of the bearing, the increase in interference could be determined. By using the loaddeflection relationships developed in Chapter VI, the thermally induced load could be computed. Computation of Thermal Loads Following the procedure described above for computing the thermal loads, a sample calculation will now be provided for the 7,000 rpm test of Configuration II. The thermal deflection at each concentric ring surrounding the lower bearing is first computed using the temperature distribution as shown in Figure 4.8. The outer diameter of the bearing element is 0.158 m. The next element is .012 m larger and the rest are divided using 0.026 m increments. Using equations (4.20) to (4.22), the thermal deflections are computed next using T. as 295K. For the outer race element, the thermally induced deflection is 6,o = 10.6 x 106*(0.158)*n*(340295) (m) 6T0 = 2.37x 104 (m) This 68, would be the deflection of the outer ring if it was not constrained by the other ring elements. To determine then the actual deflection, the deflections of all the rings must be computed. Once the thermal deflection of the outermost ring is estimated the deflection of the outer race of the bearing is determined. The minimum deflection computed for any of the rings surrounding the outer race was of 2.26 x 10' m. Thus, the maximum deflection of the outer race of the bearing is that of the ring which deflected the less or 2.26 x 104 m. To determine the increase in preload, the thermal deflection of a roller must first be computed. It is given by equation (4.22) using the diameter of the roller instead of the element diameter. 68, = 10.6 x 106*(0.013)*n*(340295) (m) 6,0 = 2.37x 10 (m) The increase in preload can now be estimated by subtracting the roller thermal deflection from the outer ring thermal deflection. This difference is multiplied by the stiffness of the bearing to obtain the increase in load. Thus, the difference in thermal deflections is 1.1 x 101 m. Using equation (6.12) and a roller stiffness value of 1.00 x 10' N/m [30] the load was computed as 3.1 x 102 N. This load is negligible for the type of bearing used. This coincides with the PROMESS sensor measurements. 93 Figure 4.6 Thermal Model for Configuration II 