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DESIGN OF MACHINE ELEMENTS:

Dr. Dinesh Kumar
Associate Professor
Dept. of Mech. Eng., MNIT Jaipur

Few major references are included here. Other references may be found in individual chapters.
1. Norton Robert L., “Machine Design: An Integrated Approach”,
Fourth Edition, Pearson Education Inc., New Jersey, 2011.
2. Shigley J. E. and Mischke C. R., Budynas R. G. and Nisbett K. J.,
“Mechanical Engineering Design“ McGraw Hill, 8th Edition, USA,
2008.
*It is impossible to write better than in the above mentioned references and paraphrasing any statements may lead to loss of technical
meanings /contents of the statements, and hence, many statements are quoted directly from these works.

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BEARINGS

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Bearings
• General term: two parts have relative motion,
regardless of their shape or configuration.
• Lubrication is needed to reduce friction and remove
heat.
• Bearings may roll or slide or do both
simultaneously.
• Classified as: radial or thrust, sliding/plain or rolling
(ball and roller) contact.
• Plain bearings are typically custom designed for the
application, while rolling-element bearings are
typically selected from manufacturers’ catalogs to
suit the loads, speeds, and desired life of the
particular application.

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Lubricants and Lubrication

• Any interposed substance that reduces friction
and wear is a lubricant
• Used as a third body at the contact of the two
surfaces
• Usually, lubricants are liquid (mineral or
synthetic) or solid (graphite, Molybdenum
disulphides)
• Liquid lubricants are characterized by viscosity –
fluid’s resistance to shear

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EP (Extreme Pressure) Lubricants add fatty acids or

other compounds to the oil that attack the metal
chemically and form a contaminant layer that protects
and reduces friction even when the oil film is squeezed
out of the interface by high contact loads

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Lubrication
• Lubrication is commonly classified
according to the degree with which
the lubricant separates the sliding
surfaces
• Three general types of lubrication can
occur in bearings:
– Full-film/Thick film
– Mixed film
– Boundary

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Lubrication: Thick-film
• As in Fig. (a) the surfaces are
separated by thick film of lubricant
and there will not be any metal-to-
metal contact.
• The film thickness is anywhere from 8
to 20 μm.
• Typical values of coefficient of friction
are 0.002 to 0.010.
• Hydrodynamic lubrication is coming
under this category.
• Wear is the minimum in this case.
• It is most desirable type of lubrication

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Lubrication: Thin-film
• Here even though the surfaces
are separated by thin film of
lubricant, at some high spots
Metal-to-metal contact does
exist , Fig. (b).
• Because of this intermittent
contacts, it also known as
mixed film lubrication.
• Surface wear is mild.
• The coefficient of friction
commonly ranges from 0.004
to 0.10.

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Lubrication: Boundary
• Here the surface contact is
continuous and extensive as
Shown in Fig.(c).
• The lubricant is continuously
smeared over the surfaces
and provides a continuously
renewed adsorbed surface
film which reduces the
friction and wear.
• The typical coefficient of
friction is 0.05 to 0.20.

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Concepts of Hydrodynamic Lubrication

• Fig. shows a loaded

journal bearing at rest
• At rest, the bearing
clearance space is filled
with oil, but the load F
has squeezed out the oil
film at the bottom.
• Metal-to-contact exists.
The axes of bearing and
journal are co-axial.

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Concepts of Hydrodynamic Lubrication

• When the journal starts

rotating slowly in clockwise
direction, because of
friction, the journal starts
to climb the wall of the
bearing surface.
• Boundary lubrication exists
now. However, the journal
rotation draws the oil
between the surfaces and
they separate.

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Concepts of Hydrodynamic Lubrication

• As the speed increases, more oil is drawn in
and enough pressure is built up in the
contact zone to float” the journal.
• Further increase in speed, additional
pressure of the converging oil flow to the
right of the minimum film thickness
position (ho) moves the shaft slightly to the
left of center. As a result full separation of
journal and bearing surfaces occurs.
• This is known as – Hydrodynamic
lubrication or full film/thick film lubrication.
• At this equilibrium condition, the pressure
force on journal balances the external load
F.

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Journal position in hydrodynamic lubrication

In stable operating condition, the pressure

distribution on the journal is shown in Fig..

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Effect of three basic

parameters :
Viscosity (),
Rotating Speed in rps (n),
Bearing unit load (P),
on type of lubrication

• The operating regimes of different lubrication mechanisms

are depicted by Stribeck in Fig. by plotting coefficient of
friction verses the non-dimensional factor known as
bearing modulus.

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• The higher the viscosity, the

lower the rotating speed needed
to “float” the journal at a given
load.
• Any further increase in viscosity
produces more bearing friction
thereby increasing the forces
needed to shear the oil film.

• The higher the rotating speed, the lower

the viscosity needed to “float” the journal
at a given load. ( μn / P )
• The smaller the bearing unit load, the
lower the rotating speed and the viscosity
needed to “float” the journal.

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Aquaplane
•Aquaplaning or hydroplaning occurs
when a layer of water builds
between the wheels of the vehicle
and the road surface, leading to a
loss of traction that prevents the
vehicle from responding to control
inputs.
• If it occurs to all wheels
simultaneously, the vehicle becomes,
in effect, an uncontrolled sled.
• Aquaplaning is a different
phenomenon from when water on
the roadway merely acting as
a lubricant.
Source: https://en.wikipedia.org/wiki/Aquaplaning

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• The basic requirements for achieving

Hydrodynamic lubrication are :
1. Surfaces which are in relative motion to be
separated.
2. “Wedging,” as provided by the shaft
eccentricity.
3. The presence of a suitable fluid.

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Viscosity of a lubricant
• It is the internal friction that resists
the motion in fluids.
• The unit of viscosity in SI units is
Ns/m2 or Pa.s. Since this is a large
unit, it is normally expressed as
millipascal second mPa.s or centipoise
cp.
• 1 cP (centipoise) is 1 mPa-s.
• Kinematic viscosity (ν) – measured
with a viscometer
• Absolute viscosity ( - <Pa-s>) –
calculated: Force required to move the plate is
– =; Where  is the density of the given by 𝑈
𝐹=𝜇 𝐴
fluid at a test temperature. ℎ
• Typical absolute viscosities at 20º C
are 0.0179 cP for air, 1.0 cP for water
and 393 cP for SAE 30 engine oil.

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Viscosity and Temperature

• Viscosity varies inversely
proportional to
temperature
• The major concern when
operating at high
temperatures is the loss of
adhesion of the oil to the
metallic surface
• Certain oils (synthetic
mainly) maintain a uniform
viscosity over a large range Viscosity - temperature curves
of temperatures

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Material Combinations in Sliding Bearings

• The properties sought in a bearing material are
– relative softness (to absorb foreign particles),
– reasonable strength,
– machinability (to maintain tolerances),
– lubricity,
– temperature and corrosion resistance, and,
– in some cases, porosity (to absorb lubricant).

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Petroff’s Equation
• In 1883, Petroff published his work on
bearing friction based on simplified
assumptions.
– No eccentricity between bearings and
journal and hence there is no “Wedging
action”.
– Oil film is unable to support load.
– No lubricant flow in the axial direction.
• It defines groups of dimensionless
parameters and also the coefficient of
friction predicted by this law turns out Unloaded Journal
bearing used for
to be quite good even when the shaft Petroff’s analysis
is not concentric.

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Flow Between Parallel surfaces

Direction of motion
Velocity of top plate = u of top plate
Shear force F

Top layer of fluid moves with

y
Velocity profile same velocity as the plate
(same throughout)

Lubricant Velocity of bottom plate = 0

A is area of the plate
•There is no pressure buildup in the fluid due to relative motion
•It remains constant throughout influenced only by the load
•As load increases the surfaces are pushed towards each other until they
are likely to touch

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Hydrodynamic lubrication

Lift force
Top surface Force normal to surface

Drag force

Oil wedge
Direction of movement
of oil wedge

Bottom surface
•Surfaces are inclined to each other thereby compressing the fluid as it flows.
•This leads to a pressure buildup that tends to force the surfaces apart
•Larger loads can be carried

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Petroff’s Equation…
• Consider a journal and bearing
similar to sleeve bearing, but
concentric and with the axis
vertical.
• We can model this as two flat
plates, as shown below, because
the gap h is so small compared
to radius of curvature.

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Petroff’s Equation…
• If the journal and bearing are concentric
modeled as two parallel plates, the oil film will
not support a transverse load.
• The shear stress τx acting on a differential
element of fluid in the gap is proportional to
the shear rate:

the constant of proportionality is the viscosity η.

• In a film of constant thickness h, the velocity
gradient du / dy = U / h and is constant.
• The force to shear the entire film is

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Petroff’s Equation…
• For the concentric journal and bearing, let the gap h = cd / 2
where cd is the diametral clearance,
• the velocity is U = πdn’ where n’ is revolutions per second, and
the shear area is A = πdl.
• The torque T0 required to shear the film is then

4 2 r 3l n'
Or , T0 =
h
This is Petroff ’s equation for the no-load torque in a fluid film.
• Now, If a small radial load W is applied to the shaft, the
frictional drag force can be considered equal to the product fW,
with friction torque expressed as
Or , T0 = fWr = f (2rlP )r
where P is the radial load per unit of projected bearing area.

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Petroff’s Equation…
• Then, neglecting the effect of little
eccentricity produced because of W, to
obtain
4 2 r 3l n'  n'  r 
f (2rlP )r =  f = 2 2   
h  P  h 
This is another form of the Petroff equation for a very small load.
• It provides a quick and simple means of obtaining
reasonable estimates of coefficients of friction of lightly
loaded bearings.
• The first quantity in the bracket stands for bearing
modulus and second one stands for clearance ratio.
Both are dimensionless parameters of the bearing.

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Stable and Unstable Lubrication

Unstable
Lubrication

Stable
Lubrication

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Example:

A 100-mm-diameter shaft is supported by a bearing of

80-mm length with a diametric clearance of 0.1 mm. It
is lubricated by oil having a viscosity ( at operating
temperature) of 50 mPa.s. The shaft rotates at 600
rpm and carries a radial load of 5000 N. Estimate the
bearing coefficient of friction and power loss using the
Petroff approach

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Example…
• Known: A shaft with known diameter, rotational speed, and
radial load is supported by an oil-lubricated bearing of
specified length and diametric clearance
• Find: Determine the bearing coefficient of friction and power
loss
• Assumption:
– No eccentricity between the bearing and journal, and no
lubricant flow in the axial direction, and the frictional drag
force is equal to the product of coefficient of friction times
the radial shaft load

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Solution
1 Coefficient Of friction:
600
𝜇𝑛 𝑅 50 × 10−3 𝑃𝑎. 𝑠 × ( 𝑟𝑝𝑠) 50𝑚𝑚
𝑓 = 2𝜋 2
= 2𝜋 2 60 = 0.0158
𝑃 𝑐 5000𝑁 0.10𝑚𝑚
0.08𝑚 × 0.1𝑚

2 Friction Torque: 0.1𝑚

𝑇𝑓 = 𝑓𝑊𝑅 = 0.0158 5000𝑁 = 3.95 𝑁. 𝑚
2

3 Power: 𝑃𝑜𝑤𝑒𝑟 = 2𝜋𝑇𝑓 𝑛 = 2𝜋 3.95𝑁. 𝑚 10𝑟𝑝𝑠 = 248 𝑊

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Hydrodynamic Lubrication Theory

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Terms used in Hydrodynamic Journal

Bearing
• Diametral clearance (cd)
• Radial clearance (cr)
• Diametral clearance ratio (cd/d)
• Eccentricity (e)
• Minimum oil film thickness
(hmin)
• Eccentricity ratio (e/cr)
• Short and long bearings (based
on l/d ratio)

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Reynolds’s Equation
• The present mathematical theory of lubrication is
based upon Reynolds’ theoretical work following the
experiments by Tower conducted on railroad bearings
in England during the early 1880s.
• This has provided a strong foundation and basis for
the design of hydro-dynamic lubricated bearings.
• Reynolds pictured the lubricant as adhering to both
surfaces and being pulled by the moving surface into a
narrowing, wedge-shaped space so as to create a fluid
or film pressure of sufficient intensity to support the
bearing load.

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Reynolds’s Equation …
• One of the important simplifying assumptions
resulted from Reynold’s realization that the fluid films
were so thin in comparison with the bearing radius
that the curvature could be neglected.
• This enabled him to replace the curved partial
bearing with a flat bearing, called a plane slider
bearing.
• Other assumptions made were:
– The lubricant obeys Newton’s viscous effect.
– The forces due to the inertia of the lubricant are neglected.
– The lubricant is assumed to be incompressible.
– The viscosity is assumed to be constant throughout the film.
– The pressure does not vary in the axial direction.

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Reynolds’s Equation …
Additional Assumptions
• The bushing and journal extend infinitely in the z direction; this
means there can be no lubricant flow in the z direction.
• The film pressure is constant in the y direction. Thus the
pressure depends only on the coordinate x.
• The velocity of any particle of lubricant in the film depends
only on the coordinates x and y.

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Reynolds’s Equation …

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Reynolds’s Equation …
• Summing the forces in the x direction gives

But, we have

where the partial derivative is used because the velocity u depends upon both x and y.

Thus, we obtain

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Reynolds’s Equation …

Apply B.Cs To get:

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Reynolds’s Equation …
• When the pressure is maximum, dp/dx = 0
and the velocity is

• Define Q as the volume of lubricant flowing in the x

direction per unit time. By using a width of unity in the
z direction, the volume may be obtained by the
expression

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Reynolds’s Equation …
• The assumption of an incompressible
lubricant states that the flow is the same for
any cross section

Which is the classical Reynolds equation for one-dimensional

flow. It neglects side leakage, that is, flow in the z direction.

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Reynolds’s Equation …
A similar development is used when side leakage is not
neglected. The resulting equation is

Where: h – local oil film thickness, μ – dynamic viscosity of oil, p

– local oil film pressure, U – linear velocity of journal, x -
circumferential direction. z - longitudinal direction.

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Expression for the film thickness

• An approximate expression for the film
thickness h as a function of  is

Where the  is the eccentricity

ratio, given by:

• The film thickness h is maximum at θ = 0 and

minimum at θ = π, found from

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Solution of Reynolds’s Equation

• Close form (a general analytical) solution of
Reynolds’s equation can not be obtained;
therefore, finite elements method is used to
solve it.
• Analytical solutions of Reynolds’s equation exist
only for certain assumptions:
– Sommerfeld Solution under infinitely long bearing
assumption.
– Ocvirk Solution under infinitely short bearing
assumption

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Sommerfeld (Long Bearing) Solution

• Reynolds solved a simplified version in series
form (in 1886) by assuming that the bearing is
infinitely long in the z direction, which makes
the flow zero and the pressure distribution
over that direction constant, and thus makes
the term ∂p / ∂z = 0.
• With this simplification, the Reynolds’
equation becomes

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Sommerfeld (Long Bearing) Solution…

• In 1904, A. Sommerfeld found a closed-form
solution for the infinitely long bearing as

which gives the pressure p in the lubricant film as a function of

angular position  around the bearing for particular dimensions of
journal radius r, radial clearance cr, eccentricity ratio ε, surface
velocity U, and viscosity η.
• p0 accounts for any supply pressure at the otherwise
zero-pressure position at θ = 0.
• It is referred to as the half-Sommerfeld solution or the
long-bearing solution.

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Sommerfeld (Long Bearing) Solution…

• Sommerfeld also determined an equation for
the total load P on a long bearing as

• This equation can be rearranged in

nondimensional form to provide a
characteristic bearing number called the
Sommerfeld number S.
• First rearrange terms:

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Sommerfeld (Long Bearing) Solution…

• The average pressure pavg on the bearing is

• The velocity U = πdn’ where n’ is revolutions per

second, and cr = cd / 2.
• Substituting gives

• Note that S is a function only of the eccentricity

ratio ε but can also be expressed in terms of
geometry, pressure, velocity, and viscosity.

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Ocvirk (Short Bearing) Solution

• The long-bearing (Sommerfeld) solution
assumes no end leakage of oil from the
bearing, but at small l / d ratios (.25 to 2),
end leakage can be a significant factor.
• Ocvirk solution for infinitely short bearing
assumption neglects circumferential pressure
gradients (first term of Reynolds equation) to
get following differential equation:

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Ocvirk (Short Bearing) Solution

• The above equation can be integrated to give an
expression for pressure in the oil film as a function
of both θ and z:

This is known as the Ocvirk solution or the short-bearing

solution.

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Ocvirk (Short Bearing) Solution…

• The pressure distribution p with respect to z

is parabolic and peaks at the center of the
bearing length l and is zero at z =  l.
• Pressure p varies nonlinearly over  and
peaks in its second quadrant.

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Ocvirk (Short Bearing) Solution…

• The value of θmax at pmax can be found from:

• And the value of pmax can be found by

substituting z = 0 and  = max in equation:

to get

Ul 2 3 sin  max

pmax =
4rcr2 (1 +  cos  max ) 3

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Ocvirk (Short Bearing) Solution…

• Note the large error if the long-bearing solution were used for l/d < 1.
• At l / d = 1, the two solutions give similar results with the Ocvirk
solution predicting slightly higher pmax than the Sommerfeld solution.
• DuBois and Ocvirk found in tests that the short-bearing solution gave
results that closely matched experimental measurements for l / d ratios
from 1/4 to 1

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Ocvirk (Short Bearing) Solution…

• But what determines the angle of this
eccentricity line between the centers
Ob and Oj?
• This force P is shown vertical in the
figure and the angle between this force
and the θ = π axis is shown as φ.
•Angle φ can be found from

• The magnitude of the resultant

force P is related to the bearing
parameters as
Force Components in a Journal
Bearing

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Ocvirk (Short Bearing) Solution…

where Kε is a dimensionless parameter that is a function of the
eccentricity ratio ε:

Putting for U and cr to get

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Ocvirk (Short Bearing) Solution…

Torque and Power Losses in Journal Bearings
The shear force acting on each
member creates opposite-direction
torques, Tr on the rotating member
and Ts on the stationary member.

OR

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Ocvirk (Short Bearing) Solution…

Ratio of the stationary torque in an eccentric bearing
to the no-load torque for the concentric-journal:

The power Φ lost in the bearing:

Coefficient of Friction

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Meaning of Design of Hydrodynamic

Bearing
• Establish the operating and performance
parameters for the bearing while designing for
maximum load.
• Selecting suitable values for the following
parameters:
– length-to-diameter ratio;
– unit bearing pressure;
– start-up load;
– radial clearance;
– minimum oil film thickness; and
– maximum oil film temperature.

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Design of Hydrodynamic Bearings

• Usually the applied force P that the bearing is
expected to support and the speed of rotation n' are
known.
• The bearing diameter may or may not be known, but
often defined by stress, deflection, or other
considerations.
• The design of the bearing requires finding a suitable
combination of bearing diameter and/or length that
will operate with a suitable viscosity of fluid, have
reasonable and manufacturable clearance, and have
an eccentricity ratio that will not allow metal-to-
metal contact under load or any expected overload
conditions.

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Design of Hydrodynamic Bearings…

Design Load Factor—The Ocvirk Number
• A convenient way to approach this problem is to
define a dimensionless load factor, called The Ocvirk
Number, against which various bearing parameters
can be computed, plotted, and compared.
• Ocvirk Number:

because P = pavg × d × l

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Design of Hydrodynamic Bearings…

Design Load Factor—The Ocvirk Number…
• This number contains the parameters over which the
designer has control and shows that any combination
of those parameters that yields the same Ocvirk
number will have the same eccentricity ratio ε.
• The eccentricity ratio gives an indication of how close
to failure the oil film is, since hmin = cr(1 – ε).

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Design of Hydrodynamic Bearings…

It understates.

The calculation of load, torque, average and maximum pressures

in the oil film, the minimum film thickness and other bearing
parameters can be done using the empirical value of ε.

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Design of Hydrodynamic Bearings…

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Design of Hydrodynamic Bearings…

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Design Procedure
• Load and speed, and the shaft diameter (from
solid mechanics analysis) are typically known.
• A bearing length or l/d ratio should be chosen
based on packaging considerations.
• The clearance ratio is defined as cd / d. Clearance
ratios are typically kept in the range of 0.001 to
0.002.
• Larger clearance ratios will rapidly increase the
load number ON as cd / d is squared in equation
for Ocvirk number.
• Further, higher Ocvirk numbers give larger
eccentricity, pressure, and torque as can be seen
in Figures 10-10 and 10-11, but these factors
increase more slowly at higher ON.

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Design Procedure…
• An advantage of larger clearance ratios is
higher lubricant flow, which promotes cooler
running.
• Large l/d ratios may require greater clearance
ratios to accommodate shaft deflection.
• An Ocvirk number can be chosen and the
required viscosity of the lubricant found from
equations 10.7 to 10.11.
• Some iteration will usually be required to
obtain a balanced design.

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Design Procedure…
• If the dimensions of the shaft are not yet
determined, a diameter and length of bearing
can be found from iteration of the bearing
equations with an assumed Ocvirk number.
• A trial lubricant must be chosen and its
viscosity found for the assumed operating
temperatures from charts such as Figure 10-1.
• After the bearing is designed, a fluid flow and
heat transfer analysis can be done to
determine its required oil flow rates and
predicted operating temperatures.

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Design Procedure…
• G. B Dubois provided some guidance to chose
ON
– a load number of ON = 30 (ε = 0.82) be considered
an upper limit for “moderate” loading,
– ON = 60 (ε = 0.90) an upper limit for “heavy”
loading, and
– ON = 90 (ε = 0.93) a limit for “severe” loading.

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EXAMPLE 10-1

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Solution: Example 10-1

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Solution: Example 10-1…

1) Evaluate n’ and U:

2) Determine cd and cr:

3) Assume l/d ratio of 0.75 and calculate l:

4) Find the experimental eccentricity ratio from

equation 10.13b or from Figure 10-10 using the
suggested value of ON = 20.

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Solution: Example 10-1…

5) Find the dimensionless parameter Kε from equation
10.12c:

6) The viscosity η of lubricant required to support the

design load P can now be found by rearranging
equation 11.8b:

Enter Figure 10-1 to find that an oil of about ISO VG 100

will provide this value at 190°F. This oil is equivalent to an SAE
30W engine oil.

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Solution: Example 10-1…

7) Find the average pressure in the oil:

8) The angle θmax can be found from either equation

10.7c using the experimental value of ε = 0.747, or it
can be read from the experimental curve in Figure 10-
12:

9) Evaluate pmax from the equation (10.7b) or

experimental curve (Figure 10-11):

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Solution: Example 10-1…

10) Find the angle φ:

11) Evaluate stationary and rotating torques:

12) The power loss in the bearing is found from equation

10.10

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Solution: Example 10-1…

10) Determine the coefficient of friction in the bearing:

11) The minimum film thickness is found from equation

10.4b:

This is a reasonable value, since the composite Rrms (also Rq)

surface roughness needs to be no more than about a third to a
fourth of the minimum film thickness to avoid asperity contact
and a 30–40 μin Rq finish or better is easily obtainable by
precision milling, grinding, or honing.

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Solution: Example 10-1…

12) A safety factor against asperity contact can be
estimated by back-solving the model using a
minimum film thickness equal to the assumed
average surface finish of, say, 40 μin, and determining
what ON and load P would be required to reduce the
minimum oil-film thickness to that value.
This can easily be done in the model by switching hmin
and η to input status, P and ON to output status,
providing a guess value for ON, and iterating to a
solution. The result is:

which is an acceptable reserve for overloads.

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Solution: Example 10-1…

13) If this safety-factor calculation had indicated that a
small overload could put the bearing in trouble,
redesigning the bearing for a lower Ocvirk number
would give more margin against failure under
overloads.
Equation 10.12c, shown below, shows what could be
changed to reduce ON:

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