As earlier, angle OCA =

Let CA be produced to touch the bearing surface at B

Let the angle OBC be

Therefore

AB is the oil film thickness h to be found

OB is the radius of the bearing R1 and

CB = CA + AB = R2 + h = ecos+ R1cos

From the sine rule of triangles,

Therefore

1R

sin

e

sin

2

21

2

sinR

e1cos

e

R 1 h

R 2D

O

C

A

B

G

)23...(sinR

e1RcoseRh 2

21

2

12

Film thickness- accurate expression

Expanding the square root using binomial theorm and neglecting higher order terms we get

Now R1 - R2 = c and eccentricity ratio = e/c, then

c/R1 is usually of the order 10-3 and therefore h can be approximated as c(1+cos). The maximum error occurs at = 90o, when the simple relation gives h = c, while the more accurate one gives

The film thickness is exact at =0 and = and at these angles it equals c(1+e) and c(1-e) respectively

2

21

2

12 sinR2

e1RcoseRh

)24...(sinR

c.

2cos1ch 2

1

2

1

2

R

c.

21ch

Substitution into Reynold’s equation

• Reynold’s equation in one dimension is

• Where• U = surface speed• h = viscosity• h = film thickness• ho = film thickness when dp/dx = 0

We will replace the linear distance x by R2, the distance moved by the shaft periphery. For convenience we will also drop the subscript of radius

Therefore x = RThe oil film thickness was derived as h = c(1 + cos)

3o

h

hhU6

dx

dp

Unwrapping the oil film- circular to linear

e

R 1 h

R 2D

O

C

A

G

Suppose we unwrap the oil film splitting it at point G, we get a profile as below

Rotation in clockwise direction

0 /2 3/2 2

U

h

Substituting the values for x and h in Reynold’s equation we get

)cos1(c

)cos1(c)cos1(cU6

Rd

dp3

o

Where o is the position where dp/d = 0, so

c(1 + coso) = ho

= 0

=

Where (c2/6UhR)p is written as p*, the non-dimensional pressure.

The integrals of and are to be determined

To solve these intergrals, Sommerfeld used the following substitution. He defined a substitution angle such that

This has the property that at = 0, and 2, also is 0, , and 2.

Sommerfeld substitution

3o

2*

2

)cos1(

d)cos1(

)cos1(

ddp

RU6

dpc

2)cos1(

d

3)cos1(

d

The above equation can be written as

cos1

coscos

Equations based on Sommerfeld substitution

2/322 )1(

sin

)cos1(

d

2/52

22

3 )1(

2sin4/2/sin2

)cos1(

d

cos1

coscos

On solving we get

And

Now

Therefore

and

cos1

1cos1

2

cos1

1cos1

2

Non-dimensional pressure

It is now possible to write the equation for p* as

Where C is the constant of integration Therefore

We need 2 boundary conditions to evaluate o and C

C)cos1(

d)cos1(

)cos1(

dp

3o2*

C)1(

4/)2sin(2/sin2

cos1

1

)1(

sinp

2/52

22

o

2

2/32*

Boundary conditions for C and o

We put p = 0 when = 0, = 0. Therefore we get C = 0The pressure equation now reads

In order to evaluate o, 3 pressure conditions have been defined, 2 by Sommerfeld and 1 by Reynolds

p = 0At = 2 (Sommerfeld)At >= (Half Sommerfeld)p =0 and dp/d = 0 at a particular value of > (Reynolds)

2/52

22

o

2

2/32*

)1(

4/)2sin(2/sin2

cos1

1

)1(

sinp

0 /2 3/2 2

U

h

Applying Sommerfelds first condition

Sommerfelds 1st. Condition: p = 0 at = 2, = 2, sin2 and sin 4 are 0Therefore

Which gives and

If cos and sin are replaced by the corresponding relations in then it is found that and

2

22

)cos1(

120p

2

o

*

o

2

cos12

1

2

cos o

22*

)cos1)(2(

)cos2(sinp

2o 2

3cos

Pressure and force developed

Rd

Pressure curve

•Consider a small element of shaft of surface length Rd where R is the radius of the shaft and is the angle traced by the shaft while rotating.

•The pressure within this element is p

•The resultant force per unit axial length is pRd and will have a component along the line of centers equal to pRdcos and at right angles of magnitude pRdsin.

Wx

Wy

WLine of centers

Bearing

Shaft

Force developed

• If Wx is the total integrated force in the x-direction and Wy the total integrated force in the y-direction,

• Where L is the axial length considered• p does not vary with L

2

0x cospRdLW

2

0y sinpRdLW

It has been derived earlier thatp = 6U(R/c2)p*, it is possible to write

We can also define Wx* and Wy* such that

2

0

*2

2

x dcospLc

RU6W

2

0

**x dcospW

2

0

**y dsinpW

Total load and attitude angleThe resultant force on the bearing W which must be equilibrated by the applied load, is

Or

The angle between the line of centers and the resultant load line, which is called the attitude angle denoted by is given by

)WW(W 2y

2x

)WW(W2*

y

2*x

*

x

y

W

Wtan

Rd

Pressure curve

Wx

Wy

W-Wx

)25...(dsind

dpsinpW

2

0

*2

0

**x

)28...(dcosd

dpW

2

0

**

y

)27...(dsind

dpW

2

0

**

x

We have seen earlier that

and

These can be integrated by parts to give:

2

0

**x dcospW

2

0

**y dsinpW

Now, p = p* = 0 at = 0, therefore

and

)26...(dcosd

dpcospW

2

0

*2

0

**y

3o

2*

)cos1(

d)cos1(

)cos1(

ddp

3

o2

*

)cos1(

)cos1(

)cos1(

1

d

dp

2o 2

3cos

Earlier it was seen that

Therefore

Now (from Sommerfeld’s condition)

32

2

2

*

)cos1(

1

)2/1(

)1(

)cos1(

1

d

dpTherefore

h* = h/c = 1 + cosTherefore ho

* = 1 + coso = (1-2)/(1+2/2)

Substituting the value of in equations (27)

and (28) and using Sommerfeld’s substitution we get Wx

* = Wx = 0 and

ddp*

2

0

**

y dcosd

dpW

2

0

22/52

*0

2/32)

4

2sin

2()1)((sin

)1(

h

)1(

sin

• Finally we get

As , , therefore

)2/1()1(W

22/12

*y

0Wx tan

2/

This is valid only under Sommerfeld’s first condition i.e. p = 0 when = 0

Pressure curve

W=Wy

Bearing

Shaft

Rd

18

Lubricant properties

19

Lubricant properties

• The conditions and methods for testing and determining properties of lubricants are prescribed by the American Society for Testing Materials (ASTM)

• Lubricant property specifications are necessary in selection for a given requirement

• Cost effectiveness should also be considered

20

Specific and API gravitySpecific gravity =

Weight per unit volume of lubricant Weight per unit volume of water

(At a given temperature)

American Petroleum Institute (API) has instituted the term API gravity. The formula for API gravity is:

API gravity = 141.5Specific gravity

- 131.5 degrees

API gravity increases as specific gravity decreases

API gravity gives an indication of the type of crude

Barrels of crude oil per metric tonne = 1/ 141.5API gravity + 131.5

X 0.159

21

Flash point• It is the temperature at which an oil vaporizes sufficiently to sustain

momentary ignition when exposed to a flame under atmospheric conditions

• The lubricant is heated at a certain rate of temperature rise, until it is approximately a certain value below the expected flash point

• A flame is passed over the lubricant at small temperature rise intervals thereafter

• The heating rate is then reduced gradually until the flash point is reached

• The temperature at which a definite, self-extinguishing flash occurs on the surface of the oil is the flash point

• Flash point is found to increase with increase in viscosity

22

Fire point

• It is the temperature at which an oil will sustain ignition continually when exposed to a flame under atmospheric conditions

• The method for determining fire point is similar as for flash pt. the difference being that ignition is to be sustained for a minimum of 5s in the case of fire point

• The difference between flash and fire points for the same oil ranges from 10 to 100 oF and varies with viscosity

23

Pour point

• The lowest temperature at which an oil will flow under specified conditions

• The pour point is usually to be below the temperature of the operating environment so that it can flow and function properly

• Lab tests are done by raising the temperature of the lubricant above the pour point (so that it can flow), and the container is tilted to determine if it can flow

• The temperature is then decreased in stages until the lubricant stops flowing when the container is tilted

• This is the pour point temperature

24

Cloud point• The temperature at which wax precipitation starts

• The oil then takes on a cloudy appearance

• Is caused also due to the presence of moisture

• Refrigerants that are miscible in oil tend to lower the cloud point

• Operating temperature is to be kept above the cloud point

25

Floc point• It is the temperature at which flocculation begins to

occur (formation of flakes of solute)

• Usually occurs when the oil is chilled in the presence of a refrigerant

• A mixture of refrigerant R-12 and oil serves as the test sample

• Important where miscible refrigerants are used

26

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

27

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