上海交通大学：《Design and Manufacturing》课程教学资源（参考文献）Shigley's Mechanical Engineering Design ch12 journal bearing

Budynas-Nisbett Shigley's I Design of Mechanical 12 Lubrication and Joural T©The McGraw-Hil Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 12 Lubrication and Journal earings Chapter Outline 12-1 Types of Lubrication 598 12-2 Viscosity 599 12-3 Petroff's Equation 601 12-4 Stable Lubrication 603 12-5 Thick-Film Lubrication 604 12-6 Hydrodynamic Theory 605 12-7 Design Considerations 609 12-8 The Relations of the Variables 611 12-9 Steady-State Conditions in Self-Contained Bearings 625 12-10 Clearance 628 12-11 Pressure-Fed Bearings 630 12-12 Loads and Materials 636 12-13 Bearing Types 638 12-14 Thrust Bearings 639 12-15 Boundary-Lubricated Bearings 640 597

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 597 Companies, 2008 Lubrication and Journal Bearings Chapter Outline 12–1 Types of Lubrication 598 12–2 Viscosity 599 12–3 Petroff’s Equation 601 12–4 Stable Lubrication 603 12–5 Thick-Film Lubrication 604 12–6 Hydrodynamic Theory 605 12–7 Design Considerations 609 12–8 The Relations of the Variables 611 12–9 Steady-State Conditions in Self-Contained Bearings 625 12–10 Clearance 628 12–11 Pressure-Fed Bearings 630 12–12 Loads and Materials 636 12–13 Bearing Types 638 12–14 Thrust Bearings 639 12–15 Boundary-Lubricated Bearings 640 12 597

598 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 598 Mechanical Engineering Design The object of lubrication is to reduce friction,wear,and heating of machine parts that move relative to each other.A lubricant is any substance that,when inserted between the moving surfaces,accomplishes these purposes.In a sleeve bearing,a shaft,or jour- nal,rotates or oscillates within a sleeve,or bushing,and the relative motion is sliding. In an antifriction bearing,the main relative motion is rolling.A follower may either roll or slide on the cam.Gear teeth mate with each other by a combination of rolling and sliding.Pistons slide within their cylinders.All these applications require lubrication to reduce friction,wear,and heating The field of application for journal bearings is immense.The crankshaft and connecting-rod bearings of an automotive engine must operate for thousands of miles at high temperatures and under varying load conditions.The journal bearings used in the steam turbines of power-generating stations are said to have reliabilities approaching 100 percent.At the other extreme there are thousands of applications in which the loads are light and the service relatively unimportant;a simple,easily installed bearing is required,using little or no lubrication.In such cases an antifriction bearing might be a poor answer because of the cost,the elaborate enclosures,the close tolerances,the radial space required,the high speeds,or the increased inertial effects.Instead,a nylon bearing requiring no lubrication,a powder-metallurgy bearing with the lubrication"built in,"or a bronze bearing with ring oiling,wick feeding,or solid-lubricant film or grease lubri- cation might be a very satisfactory solution.Recent metallurgy developments in bearing materials,combined with increased knowledge of the lubrication process,now make it possible to design journal bearings with satisfactory lives and very good reliabilities. Much of the material we have studied thus far in this book has been based on fun- damental engineering studies,such as statics,dynamics,the mechanics of solids,metal processing,mathematics,and metallurgy.In the study of lubrication and journal bear- ings,additional fundamental studies,such as chemistry.fluid mechanics,thermody- namics,and heat transfer,must be utilized in developing the material.While we shall not utilize all of them in the material to be included here,you can now begin to appre- ciate better how the study of mechanical engineering design is really an integration of most of your previous studies and a directing of this total background toward the resolution of a single objective. 12-1 Types of Lubrication Five distinct forms of lubrication may be identified: 1 Hydrodynamic Hydrostatic Elastohydrodynamic Boundary 5 Solid film Hydrodynamic lubrication means that the load-carrying surfaces of the bearing are separated by a relatively thick film of lubricant,so as to prevent metal-to-metal contact, and that the stability thus obtained can be explained by the laws of fluid mechanics. Hydrodynamic lubrication does not depend upon the introduction of the lubricant under pressure,though that may occur;but it does require the existence of an adequate sup- ply at all times.The film pressure is created by the moving surface itself pulling the lubricant into a wedge-shaped zone at a velocity sufficiently high to create the pressure necessary to separate the surfaces against the load on the bearing.Hydrodynamic lubri- cation is also called full-film,or fluid.lubrication

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 598 © The McGraw−Hill Companies, 2008 598 Mechanical Engineering Design The object of lubrication is to reduce friction, wear, and heating of machine parts that move relative to each other. A lubricant is any substance that, when inserted between the moving surfaces, accomplishes these purposes. In a sleeve bearing, a shaft, or jour￾nal, rotates or oscillates within a sleeve, or bushing, and the relative motion is sliding. In an antifriction bearing, the main relative motion is rolling. A follower may either roll or slide on the cam. Gear teeth mate with each other by a combination of rolling and sliding. Pistons slide within their cylinders. All these applications require lubrication to reduce friction, wear, and heating. The field of application for journal bearings is immense. The crankshaft and connecting-rod bearings of an automotive engine must operate for thousands of miles at high temperatures and under varying load conditions. The journal bearings used in the steam turbines of power-generating stations are said to have reliabilities approaching 100 percent. At the other extreme there are thousands of applications in which the loads are light and the service relatively unimportant; a simple, easily installed bearing is required, using little or no lubrication. In such cases an antifriction bearing might be a poor answer because of the cost, the elaborate enclosures, the close tolerances, the radial space required, the high speeds, or the increased inertial effects. Instead, a nylon bearing requiring no lubrication, a powder-metallurgy bearing with the lubrication “built in,” or a bronze bearing with ring oiling, wick feeding, or solid-lubricant film or grease lubri￾cation might be a very satisfactory solution. Recent metallurgy developments in bearing materials, combined with increased knowledge of the lubrication process, now make it possible to design journal bearings with satisfactory lives and very good reliabilities. Much of the material we have studied thus far in this book has been based on fun￾damental engineering studies, such as statics, dynamics, the mechanics of solids, metal processing, mathematics, and metallurgy. In the study of lubrication and journal bear￾ings, additional fundamental studies, such as chemistry, fluid mechanics, thermody￾namics, and heat transfer, must be utilized in developing the material. While we shall not utilize all of them in the material to be included here, you can now begin to appre￾ciate better how the study of mechanical engineering design is really an integration of most of your previous studies and a directing of this total background toward the resolution of a single objective. 12–1 Types of Lubrication Five distinct forms of lubrication may be identified: 1 Hydrodynamic 2 Hydrostatic 3 Elastohydrodynamic 4 Boundary 5 Solid film Hydrodynamic lubrication means that the load-carrying surfaces of the bearing are separated by a relatively thick film of lubricant, so as to prevent metal-to-metal contact, and that the stability thus obtained can be explained by the laws of fluid mechanics. Hydrodynamic lubrication does not depend upon the introduction of the lubricant under pressure, though that may occur; but it does require the existence of an adequate sup￾ply at all times. The film pressure is created by the moving surface itself pulling the lubricant into a wedge-shaped zone at a velocity sufficiently high to create the pressure necessary to separate the surfaces against the load on the bearing. Hydrodynamic lubri￾cation is also called full-film, or fluid, lubrication

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill 59 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Journal Bearings 599 Hydrostatic lubrication is obtained by introducing the lubricant,which is some- times air or water,into the load-bearing area at a pressure high enough to separate the surfaces with a relatively thick film of lubricant.So,unlike hydrodynamic lubrication, this kind of lubrication does not require motion of one surface relative to another.We shall not deal with hydrostatic lubrication in this book,but the subject should be con- sidered in designing bearings where the velocities are small or zero and where the frictional resistance is to be an absolute minimum. Elastohydrodynamic lubrication is the phenomenon that occurs when a lubricant is introduced between surfaces that are in rolling contact,such as mating gears or rolling bearings.The mathematical explanation requires the Hertzian theory of contact stress and fluid mechanics. Insufficient surface area,a drop in the velocity of the moving surface,a lessening in the quantity of lubricant delivered to a bearing,an increase in the bearing load,or an increase in lubricant temperature resulting in a decrease in viscosity-any one of these-may prevent the buildup of a film thick enough for full-film lubrication.When this happens,the highest asperities may be separated by lubricant films only several molecular dimensions in thickness.This is called boundary lubrication.The change from hydrodynamic to boundary lubrication is not at all a sudden or abrupt one.It is probable that a mixed hydrodynamic-and boundary-type lubrication occurs first,and as the surfaces move closer together,the boundary-type lubrication becomes predominant. The viscosity of the lubricant is not of as much importance with boundary lubrication as is the chemical composition. When bearings must be operated at extreme temperatures,a solid-film lubricant such as graphite or molybdenum disulfide must be used because the ordinary mineral oils are not satisfactory.Much research is currently being carried out in an effort,too, to find composite bearing materials with low wear rates as well as small frictional coefficients. 12-2 Viscosity In Fig.12-1 let a plate A be moving with a velocity U on a film of lubricant of thickness h. We imagine the film as composed of a series of horizontal layers and the force F causing these layers to deform or slide on one another just like a deck of cards.The layers in con- tact with the moving plate are assumed to have a velocity U;those in contact with the stationary surface are assumed to have a zero velocity.Intermediate layers have velocities that depend upon their distances y from the stationary surface.Newton's viscous effect states that the shear stress in the fluid is proportional to the rate of change of velocity with respect to y.Thus F du T A=H (12-11 dy I Figure 12-1

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 599 Companies, 2008 Lubrication and Journal Bearings 599 Figure 12–1 F u h y U A Hydrostatic lubrication is obtained by introducing the lubricant, which is some￾times air or water, into the load-bearing area at a pressure high enough to separate the surfaces with a relatively thick film of lubricant. So, unlike hydrodynamic lubrication, this kind of lubrication does not require motion of one surface relative to another. We shall not deal with hydrostatic lubrication in this book, but the subject should be con￾sidered in designing bearings where the velocities are small or zero and where the frictional resistance is to be an absolute minimum. Elastohydrodynamic lubrication is the phenomenon that occurs when a lubricant is introduced between surfaces that are in rolling contact, such as mating gears or rolling bearings. The mathematical explanation requires the Hertzian theory of contact stress and fluid mechanics. Insufficient surface area, a drop in the velocity of the moving surface, a lessening in the quantity of lubricant delivered to a bearing, an increase in the bearing load, or an increase in lubricant temperature resulting in a decrease in viscosity—any one of these—may prevent the buildup of a film thick enough for full-film lubrication. When this happens, the highest asperities may be separated by lubricant films only several molecular dimensions in thickness. This is called boundary lubrication. The change from hydrodynamic to boundary lubrication is not at all a sudden or abrupt one. It is probable that a mixed hydrodynamic- and boundary-type lubrication occurs first, and as the surfaces move closer together, the boundary-type lubrication becomes predominant. The viscosity of the lubricant is not of as much importance with boundary lubrication as is the chemical composition. When bearings must be operated at extreme temperatures, a solid-film lubricant such as graphite or molybdenum disulfide must be used because the ordinary mineral oils are not satisfactory. Much research is currently being carried out in an effort, too, to find composite bearing materials with low wear rates as well as small frictional coefficients. 12–2 Viscosity In Fig. 12–1 let a plate A be moving with a velocity U on a film of lubricant of thickness h. We imagine the film as composed of a series of horizontal layers and the force F causing these layers to deform or slide on one another just like a deck of cards. The layers in con￾tact with the moving plate are assumed to have a velocity U; those in contact with the stationary surface are assumed to have a zero velocity. Intermediate layers have velocities that depend upon their distances y from the stationary surface. Newton’s viscous effect states that the shear stress in the fluid is proportional to the rate of change of velocity with respect to y. Thus τ = F A = μ du dy (12–1)

00 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 600 Mechanical Engineering Design where u is the constant of proportionality and defines absolute viscosity,also called dynamic viscosity.The derivative du/dy is the rate of change of velocity with distance and may be called the rate of shear,or the velocity gradient.The viscosity u is thus a measure of the internal frictional resistance of the fluid.For most lubricating fluids,the rate of shear is constant,and du/dy =U/h.Thus,from Eq.(12-1), 【=A= (12-2 Fluids exhibiting this characteristic are said to be Newtonian fluids.The unit of vis- cosity in the ips system is seen to be the pound-force-second per square inch;this is the same as stress or pressure multiplied by time.The ips unit is called the reyn,in honor of Sir Osborne Reynolds. The absolute viscosity is measured by the pascal-second(Pa.s)in SI;this is the same as a Newton-second per square meter.The conversion from ips units to SI is the same as for stress.For example,multiply the absolute viscosity in reyns by 6890 to convert to units of Pa.s. The American Society of Mechanical Engineers (ASME)has published a list of cgs units that are not to be used in ASME documents.'This list results from a recom- mendation by the International Committee of Weights and Measures(CIPM)that the use of cgs units with special names be discouraged.Included in this list is a unit of force called the dyne (dyn),a unit of dynamic viscosity called the poise (P),and a unit of kinematic viscosity called the stoke (St).All of these units have been,and still are,used extensively in lubrication studies. The poise is the cgs unit of dynamic or absolute viscosity,and its unit is the dyne- second per square centimeter (dyn.s/cm-).It has been customary to use the centipoise (cP)in analysis,because its value is more convenient.When the viscosity is expressed in centipoises,it is designated by Z.The conversion from cgs units to SI and ips units is as follows: u(Pa·s)=(10)-3Z(cP) Z(cP) u(reyn)= 6.8910)6 u(mPa.s)=6.89 u'(ureyn) In using ips units,the microreyn (ureyn)is often more convenient.The symbol u'will be used to designate viscosity in ureyn such that u=u/(10). The ASTM standard method for determining viscosity uses an instrument called the Saybolt Universal Viscosimeter.The method consists of measuring the time in seconds for 60 mL of lubricant at a specified temperature to run through a tube 17.6 mm in diameter and 12.25 mm long.The result is called the kinematic viscosity,and in the past the unit of the square centimeter per second has been used.One square centimeter per sec- ond is defined as a stoke.By the use of the Hagen-Poiseuille law,the kinematic viscosity based upon seconds Saybolt,also called Saybolt Universal viscosity (SUV)in seconds,is Z=(0.22- 180 (12-3) where Zk is in centistokes(cSt)and t is the number of seconds Saybolt. ASME Orientation and Guide for Use of Metric Units,2nd ed.,American Society of Mechanical Engineers. 1972.p.13

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 600 © The McGraw−Hill Companies, 2008 600 Mechanical Engineering Design where μ is the constant of proportionality and defines absolute viscosity, also called dynamic viscosity. The derivative du/dy is the rate of change of velocity with distance and may be called the rate of shear, or the velocity gradient. The viscosity μ is thus a measure of the internal frictional resistance of the fluid. For most lubricating fluids, the rate of shear is constant, and du/dy = U/h. Thus, from Eq. (12–1), τ = F A = μ U h (12–2) Fluids exhibiting this characteristic are said to be Newtonian fluids. The unit of vis￾cosity in the ips system is seen to be the pound-force-second per square inch; this is the same as stress or pressure multiplied by time. The ips unit is called the reyn, in honor of Sir Osborne Reynolds. The absolute viscosity is measured by the pascal-second (Pa · s) in SI; this is the same as a Newton-second per square meter. The conversion from ips units to SI is the same as for stress. For example, multiply the absolute viscosity in reyns by 6890 to convert to units of Pa · s. The American Society of Mechanical Engineers (ASME) has published a list of cgs units that are not to be used in ASME documents.1 This list results from a recom￾mendation by the International Committee of Weights and Measures (CIPM) that the use of cgs units with special names be discouraged. Included in this list is a unit of force called the dyne (dyn), a unit of dynamic viscosity called the poise (P), and a unit of kinematic viscosity called the stoke (St). All of these units have been, and still are, used extensively in lubrication studies. The poise is the cgs unit of dynamic or absolute viscosity, and its unit is the dyne￾second per square centimeter (dyn · s/cm2). It has been customary to use the centipoise (cP) in analysis, because its value is more convenient. When the viscosity is expressed in centipoises, it is designated by Z. The conversion from cgs units to SI and ips units is as follows: μ(Pa · s) = (10) −3Z (cP) μ(reyn) = Z (cP) 6.89(10)6 μ(mPa · s) = 6.89 μ (μreyn) In using ips units, the microreyn (μreyn) is often more convenient. The symbol μ will be used to designate viscosity in μreyn such that μ = μ /(106). The ASTM standard method for determining viscosity uses an instrument called the Saybolt Universal Viscosimeter. The method consists of measuring the time in seconds for 60 mL of lubricant at a specified temperature to run through a tube 17.6 mm in diameter and 12.25 mm long. The result is called the kinematic viscosity, and in the past the unit of the square centimeter per second has been used. One square centimeter per sec￾ond is defined as a stoke. By the use of the Hagen-Poiseuille law, the kinematic viscosity based upon seconds Saybolt, also called Saybolt Universal viscosity (SUV) in seconds, is Zk = 0.22t − 180 t (12–3) where Zk is in centistokes (cSt) and t is the number of seconds Saybolt. 1 ASME Orientation and Guide for Use of Metric Units, 2nd ed., American Society of Mechanical Engineers, 1972, p. 13.

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hil 601 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Joumnal Bearings 601 Figure 12-2 10- A comparison of the viscosities of various fluids. 104 Castor oil 10s SAE30 oil 106 0 Water Gasoline 10- Air 50 100 150 200 Temperature,F In SI,the kinematic viscosity v has the unit of the square meter per second(m2/s). and the conversion is v(m2s)=10-6Zk(cS0) Thus,Eq.(12-3)becomes =(2-g9 (10-6) (12-4 To convert to dynamic viscosity,we multiply v by the density in SI units.Designating the density as p with the unit of the kilogram per cubic meter,we have u=p0.22t- 180 (10-6) (12-5) where u is in pascal-seconds. Figure 12-2 shows the absolute viscosity in the ips system of a number of fluids often used for lubrication purposes and their variation with temperature. 12-3 Petroff's Equation The phenomenon of bearing friction was first explained by Petroff on the assumption that the shaft is concentric.Though we shall seldom make use of Petroff's method of analysis in the material to follow,it is important because it defines groups of dimen- sionless parameters and because the coefficient of friction predicted by this law turns out to be quite good even when the shaft is not concentric. Let us now consider a vertical shaft rotating in a guide bearing.It is assumed that the bearing carries a very small load,that the clearance space is completely filled with oil,and that leakage is negligible (Fig.12-3).We denote the radius of the shaft by r

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 601 Companies, 2008 Lubrication and Journal Bearings 601 Figure 12–2 A comparison of the viscosities of various fluids. Air 0 50 100 150 200 10−9 10−8 10−7 10−6 10−5 10−4 10−3 Temperature, °F Absolute viscosity, reyn Castor oi SA l E 30 oil Water Gasoline In SI, the kinematic viscosity ν has the unit of the square meter per second (m2/s), and the conversion is ν(m2 /s) = 10−6Zk (cSt) Thus, Eq. (12–3) becomes ν = 0.22t − 180 t (10−6 ) (12–4) To convert to dynamic viscosity, we multiply ν by the density in SI units. Designating the density as ρ with the unit of the kilogram per cubic meter, we have μ = ρ 0.22t − 180 t (10−6 ) (12–5) where μ is in pascal-seconds. Figure 12–2 shows the absolute viscosity in the ips system of a number of fluids often used for lubrication purposes and their variation with temperature. 12–3 Petroff’s Equation The phenomenon of bearing friction was first explained by Petroff on the assumption that the shaft is concentric. Though we shall seldom make use of Petroff’s method of analysis in the material to follow, it is important because it defines groups of dimen￾sionless parameters and because the coefficient of friction predicted by this law turns out to be quite good even when the shaft is not concentric. Let us now consider a vertical shaft rotating in a guide bearing. It is assumed that the bearing carries a very small load, that the clearance space is completely filled with oil, and that leakage is negligible (Fig. 12–3). We denote the radius of the shaft by r,

602 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hil Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 602 Mechanical Engineering Design Figure 12-3 “Keyway” sump Petroff's lightly loaded journal Oilfill bearing consisting of a shaft bole journal and a bushing with an axial-groove internal lubricant Bushing(bearing） reservoir.The linear velocity gradient is shown in the end Journal (shaft) view.The clearance c is several thousandths of an inch and is grossly exaggerated for presentation purposes. w Side leakage negligible A Section AA' the radial clearance by c,and the length of the bearing by l,all dimensions being in inches.If the shaft rotates at N rev/s,then its surface velocity is U =2xrN in/s.Since the shearing stress in the lubricant is equal to the velocity gradient times the viscosity, from Eq.(12-2)we have U2πruN (a) where the radial clearance c has been substituted for the distance h.The force required to shear the film is the stress times the area.The torque is the force times the lever arm r.Thus T=(A)(r)= 2πruN 4π2r3luN (2πr)(r)= 6 c If we now designate a small force on the bearing by W,in pounds-force,then the pres- sure P,in pounds-force per square inch of projected area,is P=W/2rl.The frictional force is fW,where fis the coefficient of friction,and so the frictional torque is T=fWr=(f)(2rlP)(r)=2r2fIP (c Substituting the value of the torque from Eq.(c)in Eq.(b)and solving for the coeffi- cient of friction,we find f=272UN r P c (12-61 Equation (12-6)is called Petroff's equation and was first published in 1883.The two quantities uN/P and r/c are very important parameters in lubrication.Substitution of the appropriate dimensions in each parameter will show that they are dimensionless. The bearing characteristic number;or the Sommerfeld number,is defined by the equation s=)'w c P (12-7刀 The Sommerfeld number is very important in lubrication analysis because it contains many of the parameters that are specified by the designer.Note that it is also dimen- sionless.The quantity r/c is called the radial clearance ratio.If we multiply both sides

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 602 © The McGraw−Hill Companies, 2008 602 Mechanical Engineering Design Figure 12–3 Petroff’s lightly loaded journal bearing consisting of a shaft journal and a bushing with an axial-groove internal lubricant reservoir. The linear velocity gradient is shown in the end view. The clearance c is several thousandths of an inch and is grossly exaggerated for presentation purposes. A N A' W W r c W l Section AA' W U “Keyway” sump Oilfill hole Bushing (bearing) Journal (shaft) Side leakage negligible the radial clearance by c, and the length of the bearing by l, all dimensions being in inches. If the shaft rotates at N rev/s, then its surface velocity is U = 2πr N in/s. Since the shearing stress in the lubricant is equal to the velocity gradient times the viscosity, from Eq. (12–2) we have τ = μ U h = 2πrμN c (a) where the radial clearance c has been substituted for the distance h. The force required to shear the film is the stress times the area. The torque is the force times the lever arm r. Thus T = (τ A)(r) = 2πrμN c (2πrl)(r) = 4π2r 3lμN c (b) If we now designate a small force on the bearing by W, in pounds-force, then the pres￾sure P, in pounds-force per square inch of projected area, is P = W/2rl. The frictional force is f W , where f is the coefficient of friction, and so the frictional torque is T = f Wr = ( f )(2rlP)(r) = 2r 2 flP (c) Substituting the value of the torque from Eq. (c) in Eq. (b) and solving for the coeffi- cient of friction, we find f = 2π2μN P r c (12–6) Equation (12–6) is called Petroff’s equation and was first published in 1883. The two quantities μN/P and r/c are very important parameters in lubrication. Substitution of the appropriate dimensions in each parameter will show that they are dimensionless. The bearing characteristic number, or the Sommerfeld number, is defined by the equation S = r c 2 μN P (12–7) The Sommerfeld number is very important in lubrication analysis because it contains many of the parameters that are specified by the designer. Note that it is also dimen￾sionless. The quantity r/c is called the radial clearance ratio. If we multiply both sides

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill 603 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Joumal Bearings 603 of Eq.(12-6)by this ratio,we obtain the interesting relation =2π2S (12-8) 12-4 Stable Lubrication The difference between boundary and hydrodynamic lubrication can be explained by reference to Fig.12-4.This plot of the change in the coefficient of friction versus the bearing characteristic uN/P was obtained by the McKee brothers in an actual test of friction.The plot is important because it defines stability of lubrication and helps us to understand hydrodynamic and boundary,or thin-film,lubrication. Recall Petroff's bearing model in the form of Eq.(12-6)predicts that f is pro- portional to uN/P,that is,a straight line from the origin in the first quadrant.On the coordinates of Fig.12-4 the locus to the right of point Cis an example.Petroff's model presumes thick-film lubrication,that is,no metal-to-metal contact,the surfaces being completely separated by a lubricant film. The McKee abscissa was ZN/P (centipoise x rev/min/psi)and the value of abscissa B in Fig.12-4 was 30.The corresponding uN/P(reyn x rev/s/psi)is 0.33(10-6).Designers keep uN/P 1.7(10-6),which corresponds to ZN/P 150. A design constraint to keep thick film lubrication is to be sure that uN ≥1.7(10-6) d Suppose we are operating to the right of line BA and something happens,say,an increase in lubricant temperature.This results in a lower viscosity and hence a smaller value of uN/P.The coefficient of friction decreases,not as much heat is generated in shearing the lubricant,and consequently the lubricant temperature drops.Thus the region to the right of line BA defines stable lubrication because variations are self-correcting. To the left of line BA,a decrease in viscosity would increase the friction.A temperature rise would ensue,and the viscosity would be reduced still more.The result would be compounded.Thus the region to the left of line BA represents unstable lubrication. It is also helpful to see that a small viscosity,and hence a small uN/P,means that the lubricant film is very thin and that there will be a greater possibility of some Figure 12-4 The variation of the coefficient of friction f with uN/P. Thin film Thick film (stable) ic B Bearing characteristic,uN/P 2S.A.McKee and T.R.McKee,"Journal Bearing Friction in the Region of Thin Film Lubrication." SAEJ,vol.31,1932.Pp.(T371-377

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 603 Companies, 2008 Lubrication and Journal Bearings 603 of Eq. (12–6) by this ratio, we obtain the interesting relation f r c = 2π2μN P r c 2 = 2π2 S (12–8) 12–4 Stable Lubrication The difference between boundary and hydrodynamic lubrication can be explained by reference to Fig. 12–4. This plot of the change in the coefficient of friction versus the bearing characteristic μN/P was obtained by the McKee brothers in an actual test of friction.2 The plot is important because it defines stability of lubrication and helps us to understand hydrodynamic and boundary, or thin-film, lubrication. Recall Petroff’s bearing model in the form of Eq. (12–6) predicts that f is pro￾portional to μN/P, that is, a straight line from the origin in the first quadrant. On the coordinates of Fig. 12–4 the locus to the right of point C is an example. Petroff’s model presumes thick-film lubrication, that is, no metal-to-metal contact, the surfaces being completely separated by a lubricant film. The McKee abscissa was Z N/P (centipoise × rev/min/psi) and the value of abscissa B in Fig. 12–4 was 30. The corresponding μN/P (reyn × rev/s/psi) is 0.33(10−6). Designers keep μN/P ≥ 1.7(10−6), which corresponds to Z N/P ≥ 150. A design constraint to keep thick film lubrication is to be sure that μN P ≥ 1.7(10−6 ) (a) Suppose we are operating to the right of line B A and something happens, say, an increase in lubricant temperature. This results in a lower viscosity and hence a smaller value of μN/P. The coefficient of friction decreases, not as much heat is generated in shearing the lubricant, and consequently the lubricant temperature drops. Thus the region to the right of line B A defines stable lubrication because variations are self-correcting. To the left of line B A, a decrease in viscosity would increase the friction. A temperature rise would ensue, and the viscosity would be reduced still more. The result would be compounded. Thus the region to the left of line B A represents unstable lubrication. It is also helpful to see that a small viscosity, and hence a small μN/P, means that the lubricant film is very thin and that there will be a greater possibility of some Figure 12–4 The variation of the coefficient of friction f with μN/P. B A C Thick film (stable) Thin film (unstable) Bearing characteristic, N⁄P Coefficient of friction f 2 S. A. McKee and T. R. McKee, “Journal Bearing Friction in the Region of Thin Film Lubrication,” SAE J., vol. 31, 1932, pp. (T)371–377.

604 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 604 Mechanical Engineering Design metal-to-metal contact,and hence of more friction.Thus,point C represents what is probably the beginning of metal-to-metal contact as uN/P becomes smaller. 12-5 Thick-Film Lubrication Let us now examine the formation of a lubricant film in a journal bearing.Figure 12-5a shows a journal that is just beginning to rotate in a clockwise direction.Under starting conditions,the bearing will be dry,or at least partly dry,and hence the journal will climb or roll up the right side of the bearing as shown in Fig.12-5a. Now suppose a lubricant is introduced into the top of the bearing as shown in Fig.12-5b.The action of the rotating journal is to pump the lubricant around the bear- ing in a clockwise direction.The lubricant is pumped into a wedge-shaped space and forces the journal over to the other side.A minimum film thickness ho occurs,not at the bottom of the journal,but displaced clockwise from the bottom as in Fig.12-5b.This is explained by the fact that a film pressure in the converging half of the film reaches a maximum somewhere to the left of the bearing center. Figure 12-5 shows how to decide whether the journal,under hydrodynamic lubrica- tion,is eccentrically located on the right or on the left side of the bearing.Visualize the jour- nal beginning to rotate.Find the side of the bearing upon which the journal tends to roll. Then,if the lubrication is hydrodynamic,mentally place the journal on the opposite side. The nomenclature of a journal bearing is shown in Fig.12-6.The dimension c is the radial clearance and is the difference in the radii of the bushing and journal.In Figure 12-5 Formation of a film. ↑w (a)Dry (b)Lubricated Figure 12-6 Line of centers Nomenclature of a partial journal bearing. Bushing Lho c=radial clearance

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 604 © The McGraw−Hill Companies, 2008 604 Mechanical Engineering Design Figure 12–5 Formation of a film. W W h0 (a) Dry W Q (flow) W (b) Lubricated metal-to-metal contact, and hence of more friction. Thus, point C represents what is probably the beginning of metal-to-metal contact as μN/P becomes smaller. 12–5 Thick-Film Lubrication Let us now examine the formation of a lubricant film in a journal bearing. Figure 12–5a shows a journal that is just beginning to rotate in a clockwise direction. Under starting conditions, the bearing will be dry, or at least partly dry, and hence the journal will climb or roll up the right side of the bearing as shown in Fig. 12–5a. Now suppose a lubricant is introduced into the top of the bearing as shown in Fig. 12–5b. The action of the rotating journal is to pump the lubricant around the bear￾ing in a clockwise direction. The lubricant is pumped into a wedge-shaped space and forces the journal over to the other side. A minimum film thickness h0 occurs, not at the bottom of the journal, but displaced clockwise from the bottom as in Fig. 12–5b. This is explained by the fact that a film pressure in the converging half of the film reaches a maximum somewhere to the left of the bearing center. Figure 12–5 shows how to decide whether the journal, under hydrodynamic lubrica￾tion, is eccentrically located on the right or on the left side of the bearing.Visualize the jour￾nal beginning to rotate. Find the side of the bearing upon which the journal tends to roll. Then, if the lubrication is hydrodynamic, mentally place the journal on the opposite side. The nomenclature of a journal bearing is shown in Fig. 12–6. The dimension c is the radial clearance and is the difference in the radii of the bushing and journal. In Figure 12–6 Nomenclature of a partial journal bearing. h0 O e O' N r Journal Line of centers Bushing c = radial clearance

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill 605 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Joumnal Bearings 605 Fig.12-6 the center of the journal is at O and the center of the bearing at O'.The dis- tance between these centers is the eccentricity and is denoted by e.The minimum film thickness is designated by ho,and it occurs at the line of centers.The film thickness at any other point is designated by h.We also define an eccentricity ratio e as e The bearing shown in the figure is known as a partial bearing.If the radius of the bushing is the same as the radius of the journal,it is known as a fitted bearing.If the bushing encloses the journal,as indicated by the dashed lines,it becomes a full bearing. The angle 8 describes the angular length of a partial bearing.For example,a 120 partial bearing has the angle B equal to 120. 12-6 Hydrodynamic Theory The present theory of hydrodynamic lubrication originated in the laboratory of Beauchamp Tower in the early 1880s in England.Tower had been employed to study the friction in railroad journal bearings and learn the best methods of lubricating them. It was an accident or error,during the course of this investigation,that prompted Tower to look at the problem in more detail and that resulted in a discovery that eventually led to the development of the theory. Figure 12-7 is a schematic drawing of the journal bearing that Tower investigated. It is a partial bearing,having a diameter of 4 in,a length of 6 in,and a bearing arc of 157,and having bath-type lubrication,as shown.The coefficients of friction obtained by Tower in his investigations on this bearing were quite low,which is now not surprising.After testing this bearing.Tower later drilled a -in-diameter lubricator hole through the top.But when the apparatus was set in motion,oil flowed out of this hole. In an effort to prevent this,a cork stopper was used,but this popped out,and so it was necessary to drive a wooden plug into the hole.When the wooden plug was pushed out too,Tower,at this point,undoubtedly realized that he was on the verge of discovery.A pressure gauge connected to the hole indicated a pressure of more than twice the unit bearing load.Finally,he investigated the bearing film pressures in detail throughout the bearing width and length and reported a distribution similar to that of Fig.12-8.3 The results obtained by Tower had such regularity that Osborne Reynolds con- cluded that there must be a definite equation relating the friction,the pressure,and the Figure 12-7 Lubricator hole- Partial bronze bearing Schematic representation of the partial bearing used by ↓↓↓ Tower. Lubricant level Beauchamp Tower,"First Report on Friction Experiments,"Proc.Inst.Mech.Eng.,November 1883. pp.632-666:"Second Report."ibid.,1885,pp.58-70:"Third Report,"ibid.,1888.pp.173-205; "Fourth Report,"ibid.,1891,pp.111-140

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 605 Companies, 2008 Lubrication and Journal Bearings 605 3 Beauchamp Tower, “First Report on Friction Experiments,” Proc. Inst. Mech. Eng., November 1883, pp. 632–666; “Second Report,” ibid., 1885, pp. 58–70; “Third Report,” ibid., 1888, pp. 173–205; “Fourth Report,” ibid., 1891, pp. 111–140. Fig. 12–6 the center of the journal is at O and the center of the bearing at O . The dis￾tance between these centers is the eccentricity and is denoted by e. The minimum film thickness is designated by h0, and it occurs at the line of centers. The film thickness at any other point is designated by h. We also define an eccentricity ratio  as  = e c The bearing shown in the figure is known as a partial bearing. If the radius of the bushing is the same as the radius of the journal, it is known as a fitted bearing. If the bushing encloses the journal, as indicated by the dashed lines, it becomes a full bearing. The angle β describes the angular length of a partial bearing. For example, a 120◦ partial bearing has the angle β equal to 120◦. 12–6 Hydrodynamic Theory The present theory of hydrodynamic lubrication originated in the laboratory of Beauchamp Tower in the early 1880s in England. Tower had been employed to study the friction in railroad journal bearings and learn the best methods of lubricating them. It was an accident or error, during the course of this investigation, that prompted Tower to look at the problem in more detail and that resulted in a discovery that eventually led to the development of the theory. Figure 12–7 is a schematic drawing of the journal bearing that Tower investigated. It is a partial bearing, having a diameter of 4 in, a length of 6 in, and a bearing arc of 157◦, and having bath-type lubrication, as shown. The coefficients of friction obtained by Tower in his investigations on this bearing were quite low, which is now not surprising. After testing this bearing, Tower later drilled a 1 2 -in-diameter lubricator hole through the top. But when the apparatus was set in motion, oil flowed out of this hole. In an effort to prevent this, a cork stopper was used, but this popped out, and so it was necessary to drive a wooden plug into the hole. When the wooden plug was pushed out too, Tower, at this point, undoubtedly realized that he was on the verge of discovery. A pressure gauge connected to the hole indicated a pressure of more than twice the unit bearing load. Finally, he investigated the bearing film pressures in detail throughout the bearing width and length and reported a distribution similar to that of Fig. 12–8.3 The results obtained by Tower had such regularity that Osborne Reynolds con￾cluded that there must be a definite equation relating the friction, the pressure, and the Figure 12–7 Schematic representation of the partial bearing used by Tower. N Journal Lubricant level Lubricator hole Partial bronze W bearing

606 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hil Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 606 Mechanical Engineering Design Figure 12-8 Approximate pressure- distribution curves obtained by Tower. =6in =4n velocity.The present mathematical theory of lubrication is based upon Reynolds'work following the experiment by Tower.The original differential equation,developed by Reynolds,was used by him to explain Tower's results.The solution is a challenging problem that has interested many investigators ever since then,and it is still the starting point for lubrication studies Reynolds pictured the lubricant as adhering to both surfaces and being pulled by the moving surface into a narrowing,wedge-shaped space so as to create a fluid or film pressure of sufficient intensity to support the bearing load.One of the important sim- plifying assumptions resulted from Reynolds'realization that the fluid films were so thin in comparison with the bearing radius that the curvature could be neglected.This enabled him to replace the curved partial bearing with a flat bearing,called a plane slider bearing.Other assumptions made were: 1 The lubricant obeys Newton's viscous effect,Eg.(12-1). 2 The forces due to the inertia of the lubricant are neglected. 3 The lubricant is assumed to be incompressible. 4 The viscosity is assumed to be constant throughout the film. 5 The pressure does not vary in the axial direction. Figure 12-9a shows a journal rotating in the clockwise direction supported by a film of lubricant of variable thickness h on a partial bearing.which is fixed.We specify that the journal has a constant surface velocity U.Using Reynolds'assumption that curvature can be neglected,we fix a right-handed xyz reference system to the stationary bearing.We now make the following additional assumptions: 6 The bushing and journal extend infinitely in the zdirection:this means there can be no lubricant flow in the z direction. 7 The film pressure is constant in the y direction.Thus the pressure depends only on the coordinate x. 8 The velocity of any particle of lubricant in the film depends only on the coordinates x and y. We now select an element of lubricant in the film (Fig.12-9a)of dimensions dx. dy,and dz,and compute the forces that act on the sides of this element.As shown in Fig.12-9b,normal forces,due to the pressure,act upon the right and left sides of the Osborne Reynolds,"Theory of Lubrication,Part I,"Phil.Trans.Roy.Soc.London,1886

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 606 © The McGraw−Hill Companies, 2008 606 Mechanical Engineering Design Figure 12–8 Approximate pressure￾distribution curves obtained by Tower. pmax p = 0 N l = 6 in d = 4 in velocity. The present mathematical theory of lubrication is based upon Reynolds’ work following the experiment by Tower.4 The original differential equation, developed by Reynolds, was used by him to explain Tower’s results. The solution is a challenging problem that has interested many investigators ever since then, and it is still the starting point for lubrication studies. Reynolds pictured the lubricant as adhering to both surfaces and being pulled by the moving surface into a narrowing, wedge-shaped space so as to create a fluid or film pressure of sufficient intensity to support the bearing load. One of the important sim￾plifying assumptions resulted from Reynolds’ realization that the fluid films were so thin in comparison with the bearing radius that the curvature could be neglected. This enabled him to replace the curved partial bearing with a flat bearing, called a plane slider bearing. Other assumptions made were: 1 The lubricant obeys Newton’s viscous effect, Eq. (12–1). 2 The forces due to the inertia of the lubricant are neglected. 3 The lubricant is assumed to be incompressible. 4 The viscosity is assumed to be constant throughout the film. 5 The pressure does not vary in the axial direction. Figure 12–9a shows a journal rotating in the clockwise direction supported by a film of lubricant of variable thickness h on a partial bearing, which is fixed. We specify that the journal has a constant surface velocity U. Using Reynolds’ assumption that curvature can be neglected, we fix a right-handed xyz reference system to the stationary bearing. We now make the following additional assumptions: 6 The bushing and journal extend infinitely in the z direction; this means there can be no lubricant flow in the z direction. 7 The film pressure is constant in the y direction. Thus the pressure depends only on the coordinate x. 8 The velocity of any particle of lubricant in the film depends only on the coordinates x and y. We now select an element of lubricant in the film (Fig. 12–9a) of dimensions dx, dy, and dz, and compute the forces that act on the sides of this element. As shown in Fig. 12–9b, normal forces, due to the pressure, act upon the right and left sides of the 4 Osborne Reynolds, “Theory of Lubrication, Part I,” Phil. Trans. Roy. Soc. London, 1886