The document discusses lubrication theory and bearing design using dimensionless parameters. It covers Petroff's equation, which defines parameters governing friction properties in bearings. Reynolds' equation is derived, which governs fluid film pressure in bearings. The Raimondi and Boyd method solves Reynolds' equation numerically to analyze bearings with finite length using dimensionless parameters like Sommerfeld number and minimum film thickness variable. Performance is evaluated through coefficients like friction, flow, and maximum pressure.

1. Shroff S R Rotary Institut
e of Chemical Technology,
Vataria
Title Of Topic: Lubrication Theory
Design of Bearing with Raimondi And Boyd
Equation

2. Agenda Style
01
02
03
04
Lubrication Theory
Petroff’s Equation
Reynold’s Equation
Design Of Bearing With Raimondi And Boyd
Equation

3. Meet Our Team
Subject Teacher:
Mr. Satish Verma
Dhyey Shukla
170990119016
Vinay Patel
170990119014
Dhananjay Patel
170990119015

4. Petroff’s Equation
Petroff Equation is
used to determine the
coefficient of friction in
journal bearings.
The shaft is
concentric with
bearing.
The bearing is
subjected to light load.
• In Practice such conditions do not arise.
• Petroff’s Equation is important because it
defines the group of dimensionless parameters
that govern the friction properties of bearing.
Contents A
Assumptions - 1
Assumption - 2
Contents B

5. A vertical shaft
rotating in bearing
is shown in figure
r = radius of the journal (mm)
l = Length of the journal (mm)
c = Radial Clerance (mm)
𝑛𝑠 = Journal Speed (rev/sec)
The velocity at the surface of journal is given by
𝑈 = 2𝜋𝑟 . 𝑛𝑠
From Newtons law of viscosity
𝑃 = 𝜇𝐴(
𝑈
ℎ
)
• We will apply the above equation for viscous flow
through the annular portion between the journal and the
bearing in the circumferential direction.
A
B

6. P = tangential frictional force
A = area of journal surface =
(2𝜋𝑟)l
U = surface velocity = (2𝜋𝑟)𝑛𝑠
h = distance between journal
and bearing surfaces = c
Substituting above value in
Equation
𝑃 = 𝜇 2𝜋𝑟𝑙 2𝜋𝑟𝑛𝑠
1
𝑐
=
4𝜋2
𝑟2
𝑙𝜇𝑛𝑠
𝑐
The frictional torque is given by,
𝑀𝑡 𝑓 = Pr =
4𝜇2
𝑟3
𝑙𝜇𝑛𝑠
𝑐
Let us consider a radial force (W),
acting on the bearing as shown
in Figure
𝑃 =
𝑊
𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑒𝑎𝑟𝑖𝑛𝑔
𝑊 = 2𝑝𝑟𝑙
The frictional force will be (f W) and
frictional torque will be (f 𝑊
𝑟).
Therefore,
(𝑀𝑡)𝑓 = 𝑓𝑊
𝑟 = 𝑓 2𝑝𝑟𝑙 𝑟 = 𝑓(2𝑝𝑟2
𝑙)
Where f is the coefficient of friction,
From (D) and (F),
4𝜋2
𝑟3
𝑙𝜇𝑛𝑠
𝑐
= 𝑓(2𝑝𝑟2
𝑙)
𝑓 = 2𝜋2
𝑟
𝑐
𝜇𝑛𝑠
𝑝
C
D
E
F
Petroff’s Equation

7. Important
Petroff’s equation indicates that there are two
important dimensionless parameters, namely,
𝑟
𝑐
and
𝜇𝑛𝑠
𝑝
,
that govern the coefficient of friction
and other frictional properties like frictional torque,
frictional power loss and temperature rise in the
bearing.

8. Reynold’s Equation
6
The shaft and the bearing are rigid.
3
The inertia force in the oil film are
negligible
5
The effect of curvature of the film with
respect to film thickness is neglected. It
is assumed that the film is so thin that
pressure is constant across the film
thickness.
1
The lubrication obeys Newton’s law of
viscosity
4
The viscosity of lubricant is constant
7
There is a continuous supply are
lubricant.
2
The Lubricant is incompressible
Text Here

9. • An element having dimensions dx, dy and
dz is considered in this analysis,
• X is the axis in the direction of motion,
• Y is the axis in the radial plane and
• Z is the axis parallel to the axis of the
journal.
• u, v and w are velocities in X, Y, Z directions
respectively.
• 𝜏𝑋 and 𝜏𝑍 are shear stresses along X and Z
directions.
• p is the fluid film pressure.

10. The forces acting on the element in X direction
are shown in Fig. Considering equilibrium
of forces,
The product (dx dy dz) indicates the
volume of
the element. Since the element has
positive volume
Therefore, Eq. (a) is written as,
According to Newton’s law of
viscosity
From (b) and (c),
Integrating twice,
The constants C1 and C2 of
integration are
evaluated from following two
boundary conditions
u = 0 when y = 0
u = U when y = h
Substituting these values in Eq.
(d),
The forces acting on the element in Z
direction
are shown in Fig. Considering equilibrium
of forces,
Since, (dx dy dz) ≠ 0
Equation (g) is written as,
A
B
C
D
E
F
G
H

11. According to Newton’s law of viscosity
From (j) and (h),
Integrating twice,
The constants C3 and C4 of integration are
evaluated from following two boundary conditions:
w = 0 when y = 0
W = 0 when y = h
Substituting these boundary conditions in Eq. (k),
Substituting the above values in Eq. (k),
The general continuity equation for incompressible
fl ow is given by,
Despite there is no fl ow in Y direction; the local
continuity equation in three directions must be
satisfied. Therefore
Integrating the above equation with respect to y,
within limits 0 to h,
The left hand side of the above equation is
expressed as,
Figure shows the fluid film in the X–Y
plane. When (y = 0), it indicates stationary bearing
surface and velocity in Y direction (v) is zero.
When
(y = h), it indicates journal surface and velocity in
Y
direction (v) is given by,
J
K
N
M
L
O

12. In the above expression, the curvature effect
is
neglected. Substituting the above values in
Eq. (o),
From Eqs (n) and (p)
We will apply Leibnitz’s theorem1 for
interchanging the signs of integration and
differentiation of the first term of the above
equation, because the upper limit h is a function
of
x. According to Leibnitz’s theorem
Substituting following values,
h1(x) = 0 h2(x) = h u(x, y) = u
u [h1 (x), x] = u at [h1 (x), x] = 0
u [h2 (x), x] = u at [h2 (x), x] = U
We get
Therefore, the fi rst term of Eq. (q) is given by,
In the second term of Eq. (q), the upper limit
h is constant with respect to y or z. Therefore,
the signs of integration and differentiation can be
interchanged. Or
Substituting Eqs (r) and (s) in Eq. (q),
Substituting the value of u from Eq. (f) in the
first expression of Eq. (t),
Substituting the value of w from Eq. (l) in the
second expression of Eq. (t),
P
R
S
P
Q
T
U

13. Substituting Eqs (u) and (v) in Eq. (t),
or,
 The above equation is known as Reynold’s
equation.
 There is no exact analytical solution
for this equation for bearings with finite length.
 Theoretically, exact solutions can be obtained
if the bearing is assumed to be either infinitely
 long or very short.
 These two solutions are called
Sommerfeld’s solutions.
 Approximate solutions
using numerical methods are available for bearings
with finite length.
V

14. Raimondi And Boyd Method
There is no exact solution to Reynold’s
equation for a journal bearing having a finite
length
01
AA Raimondi and John Boyd of Westinghouse
Research Laboratory solved this equation on
computer using the iteration technique
02
In the Raimondi and Boyd method, the
performance of the bearing is expressed in
terms of dimensionless parameters
03

15.  O and O’ are the axes of bearing and journal respectively.
 The distance OO’ is called
eccentricity and denoted by the letter e.
 The radial clearance c is given by,
c = R – r
 where,
c = radial clearance (mm)
R = radius of bearing (mm)
r = radius of journal (mm)
The eccentricity ratio (e) is defined as the ratio
of eccentricity to radial clearance.
 Therefore,
 where, e is the eccentricity ratio.
 Referring the figure
R = e + r + ℎ0
where,
ℎ0 = minimum fi lm thickness (mm)
Substituting Eq. (16.16) in expression (a),
16.16
A

16. The quantity
ℎ0
𝑐
is called the
minimum filmthickness variable.
The Sommerfeld number is given by
 where,
S = Sommerfeld number (dimensionless)
𝜇 = viscosity of the lubricant (N-s/mm2) or
(MPa-s)
𝑛𝑠 = journal speed (rev./s)
p = unit bearing pressure, i.e., load per unit of
the projected area (
𝑁
𝑚𝑚2)
 The Sommerfeld number contains all variables,
which are controlled by the designer.
 The angle ∅ shown in Fig. is called the
angle of eccentricity or attitude angle.
 It locates the position of minimum film thickness with respect to the
direction of load.
 The values of ∅ given in Table are in degrees.
 The coefficient of friction variable (CFV) is given by
16.17
16.18

17. Dimensionless
performance parameters
for full journal bearing with
side flow

18. where f is the coefficient of friction. The frictional
torque is given by,
Frictional power
Therefore,
The fl ow variable (FV) is given by
where,
 l = length of the bearing (mm)
 Q = flow of the lubricant (mm3/s)
 In this case, Q represents the total flow of the
lubricating oil, a part of which is circulated around
the periphery of the journal, while the remaining
oil flows out as side leakage.
 Qs represents the side leakage, which can be
calculated from the values of parameter
𝑄𝑠
𝑄
given in the table
 The maximum pressure (𝑃𝑚𝑎𝑥) developed in the
film is calculated from the ratio
𝑃
𝑃𝑚𝑎𝑥
given in
the last column of the table.
 This value is based on the assumption that the
oil is supplied at the atmospheric pressure.
 If the oil is supplied at a higher pressure, the
maximum pressure (𝑃𝑚𝑎𝑥) will also increase by
the corresponding value.

19. Thank you
Question And Suggestions Are Accepted