1. 9/10/2018
Red Sea University Faculty of Engineering Department of Mechanical Engineering
JOURNAL BEARINGS
Moataz Abdelgadir Ali Abdelgadir

2. Journal bearings
They are very common engineering components and are used in almost all types of machinery. Combustion engines and turbines ( to obtain high efficiency and reliability).
It consists of two elements,
the journal (shaft), usually made of steel, and
the bearing (bush) made of white metal,

3. Journal bearings Analysis
There are two basic aspects of journal bearing analysis.
the basic analysis refers to load capacity, friction and lubricant flow rate as a function of load, speed and any other controlling parameters.
The second analysis relates to practical or operational problems, such as methods of lubricant supply, bearing designs to suppress vibration and cavitation or to allow for misalignment, and frictional heating of the lubricant.

4. Journal bearings Analysis
Evaluation of the Main Parameters (load capacity, friction and lubricant flow rate as a function of speed and any other controlling parameters)
3 approaches
The infinitely long bearing approximation is acceptable when L/B > 3
while the narrow bearing approximation can be used when L/B < 1/3.
For the intermediate ratios of 1/3 < L/B < 3, computed solutions for finite bearings are applied.

5. Journal bearings - Bearing Geometry
Geometry of the journal bearing;
R1 is the radius of the bush,
R2 is the radius of the shaft,
OB is the centre of the bush,
OS is the centre of the shaft.
Consider the triangle OSOBA
e is the eccentricity (i.e. distance between the axial centers of shaft and bush during the bearing's operation
h is the film thickness
shaft

6. Journal bearings - Bearing Geometry
shaft

7. The film thickness Consider the triangle OSOBA from last figure
It should be noted that the angle ‘α’ is very small. From inspection of the triangle, it can be written:
thus
(4)

8. The film thickness
By applying the sine rule and other geometrical relations, we obtain the film thickness equation
Where
ε is the eccentricity ratio = (e/c).
c is the clearance, the difference between the radii of bush and shaft (R1 - R2)
Equation (5) gives a description of the film shape in journal bearings to within 0.1% accuracy
(5)

9. Narrow Bearing Approximation
9/10/2018
Narrow Bearing Approximation
Also known as ‘Ocvirk's approximation’
It is assumed that the pressure gradient acting along the ‘x’ axis is very much smaller than along the ‘y’ axis,
i.e.: ∂p/∂x « ∂p/∂y
as shown in Figure
Pressure distribution in the narrow bearing approximation

10. Narrow Bearing Approximation
In this approximation since L « B and ∂p/∂x « ∂p/∂y, the first term of the Reynolds equation may be neglected and the equation becomes
Also, since h ≠ f(y) then (1) can be further simplified,
(1)
(2)

11. Narrow Bearing Approximation
By integrating equ. (2) two times and applying the following boundary conditions:
At the edges of the bearing
the pressure gradient is always zero along the central plane of the bearing
We obtain
(3)

12. Pressure Distribution
The narrow bearing appro-ximation can be used. it gives accurate results for L/D < 1/3. The one-dimensional Reynolds equation for the narrow bearing approximation is given by (3).
where ‘L’ is the length of the bearing along the ‘y’ axis. Substituting ‘x’ for angular displacement times radius gives:
x = Rθ with derivative dx = Rdθ

13. Pressure Distribution
Differentiating (3) and rearranging and then substituting for h in (3) gives the pressure distribution in a narrow journal bearing
It can be seen that this equation with the following boundary conditions gives:
p = 0 at θ = 0, π and 2π
They are called also “the Full-Sommerfeld and Half-Sommerfeld” conditions
(6)

14. Load Capacity
The total load is found by integrating the pressure around the bearing.
The Half-Sommerfeld condition was used for load calculations, i.e. the negative pressures in one half of the bearing were discounted.
Load is usually calculated from two components, one acting along the line of shaft and bush centers and a second component perpendicular to the first.
This method allows calculation of the angle between the line of centers and the load line.
The shaft does not deflect co-directionally with the load but instead always moves at an angle to the load-line.

15. Load Capacity
The angle is known as the ‘attitude angle’ and results in the position of minimum film thickness lying some distance from where the load-line intersects the shaft and bush.
To analyze and derive expressions for the load components ‘W1’ and ‘W2’ we consider a small element of area Rdθdy where the ‘y’ axis is normal to the plane

16. Load Capacity
The increment of force exerted by the hydrodynamic pressure on the element of area is resolved into two components:
· pRcosθdθdy acting along the line of shaft and bush centers
· pRsinθdθdy acting in the direction normal to the line of centers.

17. Load Capacity (7) (8) (9-a)
The load component acting along the line of centers is expressed by:
The component acting in the direction normal to the line of centers is
Substituting for ‘p’ from (6) and separating variables gives W1 component
(7)
(8)
(9-a)

18. Load Capacity
W2 component
(9-b)

19. Load Capacity
The individual integrals of (9-a & 9-b) can be evaluated separately from each other and they are
Substituting yields:
(10)
(11)

20. Load Capacity
Substituting for ‘W1’ and ‘W2’ gives the expression for the total load that the bearing will support
Where
- Δ is also known as the ‘Sommerfeld Number’ or ‘Duty Parameter’
- D = 2R is the shaft diameter
Equation (12) expresses the total load in terms of the geometrical and operating parameters of the bearing.
The Sommerfeld Number is a very important parameter in bearing design since it expresses the bearing load characteristic as a function of eccentricity ratio.
(12)

21. Computed values of Sommerfeld number ‘Δ’ versus eccentricity ratio ‘ε’
Load capacity
Computed values of Sommerfeld number ‘Δ’ versus eccentricity ratio ‘ε’

22. The modified Sommerfeld parameter
If the surface speed of the shaft is replaced by the angular velocity of the shaft then the left hand side of the graph shown in last figure can be used. When the shaft angular velocity is expressed in revolutions per second [rps] then the modified Sommerfeld parameter becomes
S = πΔ. Since:
U= 2πRN
(13)
where

23. The attitude angle
The attitude angle ‘β’ between the load line and the line of centres can be determined directly from the load components ‘W1’ and ‘W2’ from the following relation:
(14)

24. Friction Force
The friction force generated in the bearing due to the shearing of the lubricant is obtained by integrating the shear stress ‘’ over the bearing area ,
In journal bearings, the bottom surface is stationary whereas the top surface, the shaft, is moving, i.e. U1 = U and U2 = 0
And the friction force is equal to
(15)
(16)

25. Friction Force
In the narrow bearing approximation it is assumed that ∂p/∂x ≈ 0 since ∂p/∂x « ∂p/∂y and (16) becomes:
Substituting for ‘h’ and ‘dx = Rdθ’ gives
and integrating yields
(17)
(18)

26. Friction Force
Equation (1) is the friction in journal bearings at the surface of the shaft for the Half-Sommerfeld condition.
It can be seen from equation (18) that when:
· the shaft and bush are concentric then: e = 0 and ε = 0 and the value of the second term of equation becomes unity. The equation now reduces to the first term only. This is known as ‘Petroff friction’
· the shaft and bush are touching then: e = c and ε = 1 which causes infinite friction according to the model of hydrodynamic lubrication

27. Friction Force
Relationship between Petroff multiplier and eccentricity ratio for infinitely long
360° bearings

28. Assignment
Drive the Lubricant Flow Rate for this journal bearing