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WORCESTER POLYTECHNIC INSTITUTEWORCESTER POLYTECHNIC INSTITUTEMECHANICAL ENGINEERING DEPARTMENT

DESIGN OF MACHINE ELEMENTSME-3320, C2005

Lecture 20-22-23

February 22, 2005

C. Furlong

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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

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Bearings and lubrication: frictionJournal bearings(short/long) -- sliding type

Friction

Relative velocity

Friction as a function of velocity

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Bearings and lubrication: materialsJournal bearings(short/long) -- sliding type

Note hardness difference between shaft and bearing material

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(Analogy)

Linear (tangential) velocity:

Film (lubricant) thickness

Bearings and lubrication: shear stresses in lubricantJournal 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:

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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 reyn 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

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Bearings and lubrication: shear stresses in lubricantReynolds equation for eccentric journal bearings

Definitions

(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

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Bearings and lubrication: shear stresses in lubricantReynolds equation for eccentric journal bearings

xhUz

phzxphx

=

+

33

61Reynolds equation:

Close-form solutions for Reynolds equation do not exist, however, it can be solved numerically (e.g., with finite element methods).

Particular close-form solutions for simplified Reynolds equation do exist

Involves solution of simplified Reynolds equation:

Neglecting pressure gradient in the axial direction (assumption): 0=

zp

xhUx

phx

=

63

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Bearings and lubrication: Reynolds equationLong bearing assumption: Sommerfeld solution

No-leakage of lubricant at both ends of journal

Pressure distribution, p, is: or

pCUrp +

+++

= 222 )cos1)(2()cos2)((sin6

Total load, P,is: 2/1222

)cos1)(2(12

++

=

rCUlrP

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)

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Bearings and lubrication: Reynolds equationShort bearing assumption: Ocvirk solution Incorporates leakage of lubricant at both ends of journal

Pressure distribution, p, is: 322

2 )cos1(sin3

4

+

= zlrC

Upr

Total load, P,is: 23

2

3 '4dr C

ldnKCUlKP ==

Coefficient, K (dimensionless), is:[ ]

22

2/1222

)1(416)1(

+=K

Important dimensionless quantity: Ocvrick number, ON

( ) [ ]22 2/122222 )1( 16)1('4 +=== dCldnpKO davgN

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Bearings and lubrication: Reynolds 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):

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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

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Bearings and lubricationPressure distribution in short journal bearings

%Pressure (Long-bearing solution)

(Short-bearing solution)

%P- distributions (approximations) in short and long bearings:

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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

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Examples

Review and Master: Example 10-1

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Reading

Homework assignment

Chapters 10 of textbook: Sections 10.0 to 10.13

Review notes and text: ES2501, ES2502, ES2503

Authors: 10-2

Solve: 10-9, 10-13, 10-19e