Expression for load

• We have seen that under Sommerfeld’s condition W = Wy

• In non-dimensional terms, W* = Wy* =

• This is a function solely of the eccentricity ratio . Putting back the dimensional quantities we get

)2/1()1( 22/12

)2/1()1(c

LRU6

c

WLRU6W

22/122

2

2

*2

Sommerfeld number•Which gives

WhereW = total loadL = axial lengthU = surface speed = viscosityC = radial clearanceR = bearing radiusThe variable on the left hand side is known as the Sommerfeld number and is often designated by S or . It is more usual to work with the reciprocal

)29...()2/1()1(

6

R

c.

U

L/W22/122

2

• Sommerfeld’s number is used as abcissa for a number of design curves.

• The ordinate can be selected to allow the friction value, film thickness, oil leakage, temperature rise etc to be determined.

• Design curves have been produced of various variables against the Sommerfeld number using computer techniques by A.A Raimondi and J.Boyd of Westinghouse Research Labs(ASLE Transactions Vol 1 No 1 April 1958).

• These graphs include compensation for end leakage and eccentricity.

Ref: http://www.roymech.co.uk/Useful_Tables/Tribology/Liquid_Lubrication.htm

Sommerfeld number

Fric

tion

coeffi

cien

t

5

Reynold’s condition

• p = 0, dp/d = 0, at some value of > • The start of the curve is assumed at the point of

maximum oil thickness = 0• The pressure equation obtained earlier is

• If the pressure starts at =0 (= 0), then C = 0• If p*= 0 at any other value of , C will not be zero

C)1(

4/)2sin(2/sin2

cos1

1

)1(

sinp

2/52

22

o

2

2/32*

6

• We know that h = c(1 + cos) and therefore ho = c(1 + coso) where ho is the film thickness when dp/d = 0 at = o

• This equation is symmetrical about = 180o, hence o can have 2 values, one when pressure p is maximum and the other when p is minimum

• Therefore we can write

• On the scale we can write

• Where on the scale corresponds to on the scale

• p* = 0 at = + , sin(+) = -sin and cos( + ) = -cos

o

o

7

Reynold’s condition, p = 0, dp/d = 0, at some value of >

Pressure curve

W

Bearing

Shaft

= -

(max. pressure)

=+

(min. pressure = 0)

Rd

WyWx

=

Start of pressure curve

8

in terms of

• Substituting the above in the pressure equation we get p* = 0

• Expanding and multiplying out we find that

• Or

• This equation can be used to determine

cos1

4/)2sin(

2/)(sin2)(

sin)()1(

1 2

2

2/32

sin2)(cos)(2cossin0

)30...(

)(cossin

cos)(sin2c/e

9

Values of for different (obtained by Cameron and Wood)

(rad.)

0.8871 0.9 1.0 1.1 1.2 1.3 1.352

1.0 0.9727 0.7574 0.5383 0.3204 0.1073 0

= e/c = 1 when e = c (eccentricity = radial clearance)

= 0 when eccentricity = 0, i.e. the shaft and bearing are coaxial

The values of can be inserted into the pressure equation and integrated to give the loads Wx and Wy

10

Now

0

0

x

y

dcospRL

dsinpRL

W

W

cosW

sinWtan

0 00dcos

d

dpcospdsinp

The first term is zero as p = 0 at = 0 and (+).

Using Sommerfeld substitution the required integral in terms of comes out to be

0 0 2/32

d)1(

)(cos

cos1

cos11dcos

d

dp

dcosp

dsinp

0

0

11

)cos1()1(2

sincos)(2)(cossin2/32

2

)cos1()1(

sincos)(2/32

000

dcosd

dpsinpdcosp

P = 0 at = 0 and + , therefore the 1st. Term disappears. Using Sommerfeld’s substitution we get

020 )cos1(

)cos1(1

1

1dcosp

)cos1()1(

sincos)(sin*W

2/32

12

)cos1)(1(2

)cos1(2

2

2

2/12

)cos1(

cos)(sin)1(2tan

Therefore

Once has been found we can find W from Wsin or Wcos

)cos1)(1(2

)cos1(*Wcos*W

2

2

x

13

• As 1, The eccentricity radial clearance, therefore the infinite journal bearing approaches the value for 2 discs with internal contact.

• The expression is

• Where

• Now R1 - R2 = c, the radial clearance, so 1/Rred = c/R2, where R is taken as the radius of the shaft

ored h/UR447.2L

W

21

21

12red RR

RR

R

1

R

1

R

1

14

• The minimum film thickness ho = c(1-)

• Hence the expression for load carried can be written as

• Where is the Sommerfeld number

)1(R

c)1(

U

L/W

UR

h)L/W(447.2

2

2

red

o

15

Dielectric strength• A measure of the electrical insulating strength

• Measured as the maximum voltage it can withstand without conducting (expressed as volts/thickness)

• Less moisture- better insulators

• Dehydrating techniques are used to improve the dielectric strength

16

Carbon residue• Carbon residue is formed by evaporation and oxidation of

lubricant

• The test of the tendency of a lubricant to form carbon residue is called the “Conradson” test

• The test sample is heated until it is completely evaporated (cannot ignite)

• The residue is cooled and weighed

• Result interpreted as weight ratio of residue to oil sample

17

Lubricant additives

18

Lubricant additives or agents

• Added to preserve, improve and/or provide additional useful properties to a lubricant

• Protect the surface forming a film• Keep surfaces and lubricant passageways

clean• Inhibitors prevent the formation of harmful

products• Some are consumed (sacrificial), others are

not (non-sacrificial)

19

Lubricant additives- classification

• Oxidation inhibitors• Viscosity index improvers• Boundary and extreme pressure additives• Rust inhibitors• Detergents• Dispersants• Pour point depressants• Anti-foaming agents• Friction modifiers

20

Oxidation products

• Sludge: Black tar-like substance consisting of water, carbon, engine oil, organic residue and dirt

• Engine gum: Acts as a binder causing residue to stick to machinery components

• Varnish: Petroleum gum exposed to high temperature and ironed out on surfaces

• Laquer: Thin layer of reacted varnish• Carbon deposits: Combination of soot from fuel

burning and oxidation of lubricating oil

21

Oxidation prevention additives

• Preferential oxidation: Additive is more susceptable to oxidation than oil

• Oil particles are therefore prevented from oxidising

Oxygen

Additive

Oil

Oxygen reacts preferentially with additive molecules

Oil

22

Oxidation prevention additives

Metal deactivators-• Metal particles in the oil act as catalysts for the

oxidation reactions• The additives either react with the metal particles

or form a coating over them

Metal catalyst

Additives

Oil

Additive reacts with metal particles

Additive covers metal particles by forming a coating

Oil

23

Oxidation prevention additives- peroxide decomposers

Hydrocarbon + oxygen

Hydroxyperoxides

Hydroxyperoxides DecomposeMaterials susceptible to

oxidation by decomposed peroxides

Additives + Hydroxyperoxides

Non-oxidizing product

24

Rust Inhibitors- effect of water

• Below boiling point, water is present in a lubricating system.

• Water contaminant can lead to formation of rust

• Water enters by condensation and/or leakage from coolers or steam heating coils

• Some oils are hygroscopic and therefore physically absorb moisture

25

Rust prevention additives

• Rust inhibitors neutralize acids formed by oxidation• Polar additives form a protective layer on the metal surface

due to attraction by the surface• Chemically react with the metal surfaces to form a protective

film• E.g. metal sulphonates, fatty acids, phosphates

Rust prevention additive

Acids formed by oxidation

Harmless products

Polar additive layer or chemically reacted layer

Metal

Shield from air/water