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 pressure 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 dimensionless. 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 proportional 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 bearing 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 lubrication, is eccentrically located on the right or on the left side of the bearing.Visualize the journal 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 distance 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 concluded 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 pressuredistribution 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 simplifying 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