JOURNAL BEARINGSAND LUBRICATION SCOPE • Need • Types of lubrication • Viscosity • Petroff’s equation • Stable lubrication • Thick-film lubrication • Hydrodynamic theory • Design considerations • The relations of the variables • Steady-state conditions in self-contained bearings • Clearance • Pressure fed bearings • Load and materials • Bearing types • Boundary lubricated bearings • The most important are revolute pair and sliding pair • Objective of lubrication is to reduce friction, wear and heating of machine parts which move relative to each other. • Lubricant is exactly that substance which does the above when inserted between moving surfaces • Lubrication is needed everywhere, for example, sleeve bearings, antifriction bearing, cam and follower, gear teeth, piston in cylinder, crank shaft and connecting rod bearings • In applications requiring low load bearing capacity, nylon bearings requiring no lubrication, a powder metallurgy bearing with lubricant built-in, a bronze bearing with ring oiling, wick feeding, or solid lubricant film or grease lubrication may be satisfactory • Journal=shaft and sleeve=bushing Need drop in velocity. speed is necessary • Hydrostatic – Lubricant is introduced at high pressure. speed could be small • Elasto-hydrodynamic – Occurs in rolling contact. or increase in lubricant temperature lead to no thick film.Types of lubrication • Hydrodynamic – Thick-film. Hertzian contact theory and fluid mechanics are required • Boundary – Insufficient surface area. like in gears and rolling element bearings. fluid mechanics and viscosity are less important and chemical composition is more important • Solid-film – Necessary when operation is to be at extremely high temperatures because ordinary minerals oils degrade. composite bearing materials are being researched because liquid lubricants also proved to be environmentally non-sustainable . increase in bearing load. fluid mechanics. lessening of lubricant quantity. partial metal to metal contact occurs which is mixed hydrodynamicboundary lubrication before gross metal-to-metal contact occurs. graphite and molybdenum disulphide are often used. F du τ = =µ A dy Newton’s law of viscosity: VISCOSITY .F U τ = =µ A h Conversion: 1 reyn = 6. IPS system: lbf-sec/in2 or psi-sec = 1 reyn SI System: Pa-s or N-s/m2. du / dy = U / h µ is the absolute viscosity or dynamic viscosity.89 * kPa-s = 6890 Pa-s. If shear rate is constant. The units of absolute viscosity Then. kinematic viscosity – stoke (St) • Poise = dyn-sec/cm2 = 10-5*104 N-sec/m2 = 0.001 Pa-s ⇒ µ (Pa-s) = 103 Z (cP) • µ (reyn) = 68900 Poise =6.• CGS units have been banished but are still in widespread use • Force .89*106 Z (cP) The adorable outcasts! . dynamic viscosity .1 Pa-s = (1/68900) reyn • Centi-poise (cP) = Z= 10-2*(10-5*104 N-sec/m2) = 0.Dyne (dyn).poise (P). the kinematic viscosity of the lubricant in centiStokes (Stoke is cm2/sec) can be obtained by the following equation: Saybolt universal viscosimeter of ASTM . • This measured time is terms as Saybolt seconds.25 mm long.6 mm in a diameter and 12.22t − t • Kinematic viscosity is measured • Measured is the time in seconds required for 60 ml of lubricant at a specified temperature to run through a tube 17. • Using the Saybolt seconds.180 Z k = 0. Viscosity from Saybolt viscosimeter .22t − 10 ( ) t Since. Density in kg per m3.22t − (10 ) t Pa-s.−6 m2/sec. 2 υ (m / sec ) = 10 Z k (cSt ) 180 −6 υ = 0. The dynamic or absolute viscosity is density times the kinematic viscosity. 180 −6 µ = ρ 0. . Ts =T1=for both bearings. external sump type and selfcontained type T∞=Ambient temperature Tf = Tav= Film temperature which is also the average temperature. Also the temperature at which to T2= Outlet temperature T1= Inlet temperature Different temperatures of relevance . ∆T = − T T f i 2 •Petroff’s theory is developed for lightly loaded bearings •Nevertheless. petroff’s bearing theory explains the useful (i) dimensionless groups (ii) gives an equation for coefficient of friction that is good even for eccentric bearings •He assumed that the shaft is concentric hence his theory is not good for actual bearing which is eccentric when it is running •Petroff was the first person to explain the phenomenon of bearing friction Tf ∆T = + T T f i 2 PETROFF’S EQUATION: . c=radial clearance between the bearing and journal. r/c=radial clearance ratio. U=relative velocity of bearing and journal. l=length of bearing. r=radius of the journal.U 2π r µ N τ =µ = h c 2 3 2 r 2 µN r 2 = f = 2π 2 π S c P c P c r µN S = c P 2 τ=shear stress. P=pressure in the bearing. µ=coefficient of dynamic viscosity Petroff’s equation (1883) f = 2π 2 µN r 2 T = fWr = ( f )(2rlP)(r ) = 2r flP 4π r l µ N 2π r µ N T = (τ A)(r ) = (2π rl )(r ) = c c Contd. T=torque. f=coefficient of friction. W=load on the bearing. N=speed in rps. S= Sommerfeld number. . h=film thickness. .density Q& gen & p ∆T = lc ρπ rN j C p ∆T Q& gen = mC Because the bushing temperature is half-way between the film temperature and ambient temperature 2 T f = T∞ + 16π U 0 A0 c 2 3 µ N j lr 3 Q& gen U 0 A0 (T f − T∞ ) 2 ∆T = = & p mC lc ρπ rN j C p 2 µ N j ( 2rl ) c fr c r 3 rWN j = 4π rWN j = 2π TN j = 2π fWrN j = 2π c r W r c Heat generated due to work done on the lubricant by the journal Combined radiation & convention universal heat transfer coefficient Q& loss = U 0 A0 (Tb − T∞ ) = U 0 A0 (T f − T∞ ) Average velocity m& = lc ρ U 2 = lc ρπ rN j Bearing surface area area Contd. Stable Lubrication P µN j ≥ 0.362 (10 −6 ) . e ε= c Thick Film Lubrication . Hydrodynamic Theory . u = −U ⇒ ∂u 1 dp = y + C1 ∂y µ dx ⇒ dp ∂τ = dx ∂y . ∂u τ =µ ∂y dp ∂τ = + + − + τ τ F p dx dydz dxdz dy dxdz − pdydz = 0 ∑ dx ∂y Hydrodynamic lubrication equations: y = 0. y = h u = 0.dp ∂ 2u =µ 2 dx ∂y integrating 1 dp 2 u= y + C1 y + C2 2 µ dx Substituting. Integrating once more. 0 Q = ∫ udy h Uh h3 dp Q=− − 2 12µ dx When the pressure is maximum. dp/dx=0. hence U u=− y h dQ =0 dx If the lubricant is incompressible. C2 = 0 C1 = − − h 2 µ dx 1 dp 2 U ⇒u= y − hy ) − y ( 2 µ dx h Sommerfeld’s conclusion . then the volume flow rate is same for all crosssections. Hence Volume flow rate: U h dp . .3 d h3 dp dh = −6U dx µ dx dx 2 r µ N r f = φ c c P Normally difficult to solve this partial differential equation analytically. Sommerfeld gave a solution in 1904. which can be expressed as 3 ∂ h ∂p ∂ h ∂p ∂h − = −6U ∂x µ ∂x ∂z µ ∂z ∂x dQ U dh d h 3 dp = 0 ⇒ = − dx 2 dx dx 12 µ dx Contd. N = N j + Nb − 2 NW NW = N f 2 Nb NW r µ ( N j + Nb − 2 NW ) = S 1 + −2 S= P Nj c Nj 2 r µNj S = c P Effective journal speed . β There are two groups of variables. These are: 1) The viscosity. The first group of variables are independent variables and either given or under the control of the designer. The designer can not control these except indirectly by changing one or more of the first group. c. Design Considerations . P 3) The speed N 4) The bearing dimensions r. µ 2) The load per unit of projected bearing area.• • • 1) 2) 3) 4) The coefficient of friction f The temperature rise ∆T The volume flow rate of oil Q The minimum film thickness ho In the second group are the dependent variables. P C p ∆T = Temperature rise dimensionless variable θ P = Position of maximum film pressure Q = Flow variable rcNl Qs = Flow ratio Q P = Maximum film pressure ratio Pmax r f = coefficient of friction variable c h0 = Minimum film thickness variable c φ = Position of minimum film thickness. degrees The dimensionless variables and charts of Raymondi & Boyd . First: . .Contd. 8T +127 You can use the table 12-1 in conjunction with the following equation as an alternative to this chart.60 grades of lubricants. Absolute viscosity: .table 12 . T in oC µ = µ0e b & µ0 may be obtained from b 1.1 for SAE . Viscosity with respect to average temperature for various SAE grades of lubricants from 10 to 70.10 to SAE . . . . . . . . . . . 120∆Tc rf c = P 1 Qs Q rcN j l ) ( 1 − 2 Q The value of “P” must be substituted in MPa .0. . 12-11 of Shigley (6ed) Another example: Iterative Determination of Average Temperature of the Film . Contd. 6ed) . 12-11 of Shigley (6ed) 12-21 (Shigley. 12-21 (Shigley. 6ed) . Contd. . must be a signed quantity Matlab code for the preceding example: . must be a signed quantity bdia=80.03 % unilateral tolerance (deviation) on bearing diameter. mm.clratio=r/c %clearance ratio r=jdia/2 c=(bmin-dmax)/2 %radial clearance %seeking the minimum sleeve diameter bmin=min(bdia. we look for maximum shaft (journal) and minimum hole (sleeve) % The radial clearance c is then equal to the cmin. for other grades the viscosity temperature relation must be appropriately substituted % The design is for minimum radial clearance scenario % this computer program is valid only for l/d=1. mm. the minimum clearance of the fit pause off %Iterative programme for the determination of the film temperature in journal bearning %from Raymondi and Boyd charts % This is for SAE30. jdia+jtol) % In minimum clearance design.01 % unilateral tolerance (deviation) on journal diameter. in mm btol=0. bdia+btol) %seeking maximum shaft dmax=max(jdia.08 % nominal diameter of the sleeve. in Newtons N=8 % journal speed in rps Ts=60 % sump temperature in degree centrigrade lbyd=1 % the l/d ratio jdia=80 % nominal diameter of the journal in mm jtol=-0. 1/2 or 1/4 W=3000 %load on the bearing. mu0=0. iteration=iteration+1 % From Table 12-1 for SAE30. a trial and error procedure is needed P=W/(jdia*l) % load per unit projected area of the journal.2) iteration=0 dis=Tav-Ts % or you may assign any value greater than 1 Tav=Ts+deltaT/2 % This is the trial # 1 for average temperature deltaT=42 % 2*(145 . Tav or Tf.8*Tav+127)) %absolute or dynamic viscosity in milli-Pacal-second.Ts) %iteration loop begins % maximum possible value in the Fig 12-11 is 145 degree centrigrade % For finding the average temperature of the film. .0971 % base viscosity. mPa-s bval=1360 % degree centrigrade mu=mu0*exp(bval/(1. MPa l=lbyd*jdia %length of the bearing Contd. mPa-s. for the current trial of Tav while (abs(dis)>0. the second estimate of temperature increase.009*S+0. we can consider the latest trail of Tav as the valid end pause deltaT=dT2 Tav=Tav-dis/2 % Now the next trial for Tav can be made as Tav-dis/2. dT2 is S=((clratio)^2)*(mu*0.001*N)/(P*10^6) %sommerfeld number deltaT=2*(Tav-Ts) %temperature rise in the bearing deltaT Contd.0475*(S^2)) %From Fig.12)*(0. sign willbe automatically taken care of dis=deltaT-dT2 %discrepancy in the estimation of temperature increase dT2=(P/0.T1=Tav-deltaT/2 %inlet temperature. 12-21. for l/d=1. . for SAE oils. same as sump temperature Ts T2=Tav+deltaT/2 % outlet temperature iteration %hence the T1 and T2 can be estimated as well % Now discrepancy being less than one.349+6. it is possible to use an equation to determine at the intermediate value of l/d: Problems with l/d ratio other than 1. ¼ or infinity .y= (l d ) 1 3 l l l l l 1 1 − 8 1 − d 1 − 2 d 1 − 4 d y∞ + 3 1 − 2 d 1 − 4 d y1 1 l l l l 1 − 1 − 1 − 4 y1 2 + 1 − 1 − 2 y1 4 d d d d 4 24 • With l/d ratio falling in between the marked values. ½. The value of alpha is usually taken as 1 otherwise. T f − Tb = α (Tb − T∞ ) & Qloss = UA (Tb − T∞ ) Steady-state conditions in self-contained bearings (also called as pillow block bearings and pedestal bearings) . Table 12-2 can be referred. for better accuracy. .Tb = Q& loss 1+ α U 0 A0 = T f − T∞ ) ( 1+ α T f + α T∞ Solving the preceding equation for Tb and substituting in the heat loss rate gives the following equation in proper variables Contd. 035 mm. Analyze the minimum clearance assembly and find the minimum film thickness. and the total oil flow if the average viscosity is 60 mPa-s. The busing bore is 50 mm in length.75 kN and the journal rotates at 1120 rpm. the coefficient of friction.• A full journal bearing has a shaft journal with a diameter of 30 mm and a unilateral tolerance of -0. The bearing load is 2. Another example: . Solution: . Contd. Contd. • A journal bearing ha a shaft diameter of 75 mm with a unilateral tolerance of -0.02 mm. The bushing bore has a diameter of 75.10 mm with a unilateral tolerance of 0.06 mm. The bushing is 36 mm and supports a load of 2 kN. The journal speed is 720 rpm. For the minimum clearance assembly find the minimum film thickness, the heat loss rate, and the maximum lubricant pressure for SAE20 and SAE40 lubricant operating at a an average film temperature of 60oC. example . Contd. . .Contd. . Fig.26 . 12. Fig. 12.27 . 12.32 .Fig. 12.33 .Fig. Fig.34 . 12. 12.Fig.35 . Fig.37 . 12. 38 .Fig. 12. 39 . 12.Fig.