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CHSLT - NEST WPI Mechanical Engineering Department
WORCESTER POLYTECHNIC INSTITUTEWORCESTER POLYTECHNIC INSTITUTEMECHANICAL ENGINEERING DEPARTMENT
DESIGN OF MACHINE ELEMENTSME-3320, C’2003
Lecture 20-22-23
February 17, 2003
C. Furlong
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubricationJournal bearings(short/long) -- sliding type
Short journal bearingsJournal bearings (short/long) and bushings
(Typically used in low-speed applications)
Thick-wall journal bearing
Hard material
Soft material
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: frictionJournal bearings(short/long) -- sliding type
Friction
Relative velocity
Friction as a function of velocity
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: materialsJournal bearings(short/long) -- sliding type
Note hardness difference between shaft and bearing material
CHSLT - NEST WPI Mechanical Engineering Department
(Analogy)
Linear (tangential) velocity:
Film (lubricant) thickness
Bearings and lubrication: shear stresses in lubricant
Journal bearings(short/long) -- sliding type
and for constant thickness, h,
hU
xy
where, - is absolute viscosity
- is linear (tangential) velocity
- is film (lubricant) thickness
Uh
dydu
xy Shear stresses:
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: viscosityAbsolute/Relative Absolute viscosity:
Units of absolute viscosity in English system: lb-sec/in2 (reyn) Units of absolute viscosity in SI system: Pa-sec Typical values are expressed in µreyn and mPa-sec (1 centipoise (cP) = 1mPa-sec) Kinematic viscosity ( ) and absolute viscosity are related as: Units of kinematic viscosity are: in2/sec (English) and cm2/sec (stoke - SI)
The term “viscosity”
without modifiers refers to absolute
viscosity
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: shear stresses in lubricant
Reynold’s equation for eccentric journal bearingsDefinitions
(Analogy: used to derivemathematical models)
2/dr CC Radial clearance:dCDiametral clearance:
Eccentricity: e (Maximum value of eccentricity is: )
rC
Eccentricity ratio:rC
e
)cos1( rChFilm thickness as a function of angular position ():
Minimum and maximum film thickness: )1( , )1( maxmin rr ChCh
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: shear stresses in lubricant
Reynold’s equation for eccentric journal bearings
xhU
zp
hzx
ph
x
33
61Reynold’s equation:
Close-form solutions for Reynold’s equation do not exist, however, it can be solved numerically (e.g., with finite element methods).
Particular close-form solutions for simplified Reynold’s equation do exist
Involves solution of simplified Reynold’s equation:
Neglecting pressure gradient in the axial direction (assumption):
0
zp
xhU
xp
hx
63
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: Reynold’s equationLong bearing assumption: Sommerfeld solution
No-leakage of lubricant at both ends of journal
Pressure distribution, p, is: or
pCUr
p
222 )cos1)(2(
)cos2)((sin6
Total load, P,is: 2/122
2
)cos1)(2(12
rC
UlrP
Average pressure, Pavg,is: ldPPavg
Important dimensionless quantity: Sommerfeld number, S
2'2/122
12)1)(2(
davg Cd
pnS
( n’ is rotational speed in revolutions per second)
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: Reynold’s equationShort bearing assumption: Ocvirk solution
Incorporates leakage of lubricant at both ends of journal
Pressure distribution, p, is: 322
2 )cos1(sin3
4
zl
rCU
pr
Total load, P,is: 2
3
2
3 '4
dr Cldn
KCUl
KP
Coefficient, K (dimensionless), is:
22
2/1222
)1(416)1(
K
Important dimensionless quantity: Ocvrick number, ON
22
2/122222
)1(16)1(
'4
dC
ld
np
KO davgN
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: Reynold’s equationShort bearing assumption: Ocvirk solution
Incorporates leakage of lubricant at both ends of journal
)60(0008.0)log(38517.021394.0)( NNcorrectedx OOCorrecting eccentricity ratio (correction based on experimental observations):
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubricationPressure distribution in journal bearings
Power loss: )(2 '1
'2 nnTr sinPeTT srwith
2/12
2'1
'2
3
)1()(
ds C
nnldT
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubricationPressure distribution in short journal bearings
%Pressure (Long-bearing solution)
(Short-bearing solution)
%P- distributions (approximations) in short and long bearings:
CHSLT - NEST WPI Mechanical Engineering Department
Bearings and lubrication: design considerationsJournal bearings(short/long) -- sliding type
If shaft has been designed (for stress and/or
deflection)Chose ratio l/d of bearing(based on packaging
considerations)
Larger l/d ratios give lower film pressures
Clearance ratio (Cd/d) are typically in the range of 0.001 to 0.002
Larger clearance ratios increase load number ON
Higher ON numbers give larger eccentricity, pressure, and torque
Larger clearance ratios used for better lubricant flow (increasing cooling conditions)
An Ocvirk number is chosen and lubricant is then selected
If shaft has not been designed
Diameter and length of bearing can be found from iteration of bearing equations with and assumed Ocvirk number
A trial lubricant must be chosen and its viscosity defined After bearing has been designed, heat and momentum (flow) analysis can be done… not in this course...
Note:ON 30 ( =0.82) moderate loading (recommended)
ON 60 ( =0.90) heavy loading
ON 90 ( =0.93) severe loading
CHSLT - NEST WPI Mechanical Engineering Department
Examples
Review and Master: Example 10-1
CHSLT - NEST WPI Mechanical Engineering Department
Reading
Homework assignment
Chapters 10 of textbook: Sections 10.0 to 10.13
Review notes and text: ES2501, ES2502, ES2503
Author’s: 10-2
Solve: 10-9, 10-13, 10-19