The Mitchell tilting pad apparatus is a hydrodynamic measuring instrum ent [602199]

1
CHAPTER ONE
INTRODUCTION

The Mitchell tilting pad apparatus is a hydrodynamic measuring instrum ent
developed in the early 1880’ s in the laboratory of Beauchamp Tower in
England. Tower was employed to study the friction in railway journal bearings
and come up with the best method of lubricating them. The Mitchell tilting pad
apparatus is used broadly in two different experiments namely ;
1) Determination of the load carrying capacity of the slider bearing.
2) Confirming the theory of the hydrodynamic lu brication.

Tilting pad journal bearings are a source of both static support and dynamic
stiffness and damping. Tilting pad journal bearings have a number of pads,
typically four or five. Each pad in the bearing is free to rotate about a pivot and
cannot s upport a moment. As a result, the destabilizing forces are greatly
reduced or eliminated, and the bearings are no longer a potential source of
rotordynamic instability . This feature has made tilting pad journal bearings the
standard fluid -film bearing for most high -speed applications. High -speed
rotordynamic applications often have rotors that pass through one or two
bending critical speeds as the machines are accelerated to the operating speed.
The damping from the fluid film bearings is required to safely pass through
these bending critical speeds as the rotating element is accelerated. The
damping also helps suppress potentially destabilizing forces from sources such
as radial seals, balance pistons, impeller eye seals, internal friction fits, and
unbalan ced electromagnetic forces .

2

1.1 AIMS AND OBJECTIVES OF THE PROJECT

The project designing, constructing and testing on the Mitchell tilting pad
apparatus is aimed at achieving the following objectives:
1) To verify the hydrodynamic theory of lubrication as it was propounded by
Beauchamp Tower in 1880 AD.
2) To determine the load carrying capacity of the tilting pad slider bearing
3) To provide the fluid mechanics laboratory of the Mechanical Engineering
Department with a hydrodynamic fluid analyzing appa ratus.
4) To activate and motivate the students potentials into practically solving
problems facing mankind.
5) To run tests with the apparatus and compare the results obtained with the
established or ideal standards.

1.2 PROJECT JUSTIFICATION
Engineering is known to be to be practice -oriented discipline. Inotherwords, no
useful Engineering endeavor can exist in theory only, it must be applied to
touch and transform life through meaningful practice. Therefore, this project
task given to us is to ensure the knowledge we students gained throughout our
five year degree programme and channeled towards the construction of
Mechanical Engineering equipments .

1.3 APPLICATIONS AND USES
The Mitchell tilting pad apparatus is applicable in experiments which does not
require;

3 1) Hydrostatic lubrication
2) Boundary lubrication
3) Solid lubrication
4) Elastohydrodynamic form of lubrication.

It only finds its application useful in full film or fluid lubrications in motion, in
which there is a situation that the load carry ing surfaces of the bearing are
separated by an adequate supply at all times of a relatively thick film of
lubricant, so as to prevent metal to metal contact and that the stability thus
obtained can be explained by the laws of fluid mechanics. The Mitchell tilting
pad is a Mechanical Engineering apparatus with the above aims and objectives.

1.4 LIMITATIONS
The Mitchell tilting pad apparatus is limited to use with the specified oil
viscosity ( SAE20W/50) or Whitz oil which is typical automobile engine oil.
The Mitchell tilting pad apparatus is also limited to operation when there is
power failure or insufficient supply of power that drives the electric motor,
other limitations are stipulated below;
1) The use of lubricants that do not obey Newton’s law of vis cous flow.
2) The use of compressible lubricants
3) Fluid pressure vary ing in the axial direction.
From the limitations stated above, it is necessary to identify the Newton’s
viscous effect, which states that the shear stress in the fluid is proportional t o
the rate of change of velocit y with respect to ‘y’.

4 CHAPTER TWO
LITERATURE REVIEW
ORIGIN AND DEVE LOPMENT OF MITCHELL TILTING PAD
APPARAT US AND HYDRODYNAMIC LUBRICATION

As has been stated in chapter one, Mitchell tilting pad apparatus took its origin
from Beauchamp Tower’s research in 1880’s , when he was trying to confirm
the hydrodynamic theory of lubrication as was propounded by Osborne
Reynolds in the 1880s
Consequently to this, the apparatus has pass ed through series of modifications
to suit various tests being conducted .

2.1 Tilting Pad Bearing Modeling Development

The fundamental lubrication equation was originally formulated by Reynolds in
1886 . He developed the theory that explained the experime ntal results of Tower
and Petroff . Reynolds assumed that the flow could be treated as viscous and
laminar. These assumptions were well justified due to typical bearing operating
conditions in the late 1800s. He also assumed that since the lubricating film was
thin in the radial direction compared to the circumferential and axial directions
that there was no pressure gradient across the film radially.
The first attempts to quantify the dynamic response of the lubricating film itself
were made by Stodola and by Stodola’s student , Hummel . Stodola and Hummel
Were able to obtain a solution for the oil film stiffness and correctly obtained
analytical li nearized direct and crosscoupled stiffness terms based on the
Sommerfeld closed form solution . However, they did not recognize the
damping effect of the oil film, and their predictions indicated that an unstable

5 rotor would have vibration levels that increased without bound. This is a
limitation of linear analysis that does not consider the nonlinear behavior of the
oil film under large excursions or practical considerations such as contact
between the rotor and the stator. Hummel acknowledged in his thesis that the
rotor vibration remained finite but did not provide a specific mechanism .
Another early analysis that r ecognized the effect of bearing flexibility on
critical speeds was reported by Linn and Prohl . While not referring to oil film
flexibility explicitly, a general bearing flexibility was assumed and the resulting
lowering of the critical speeds compared to a rigid support assumption was
demonstrated.

Fixed geometry radial bearings were standard in the first half of the 20th
century, and tilting pad bearings only saw significant adoption begin during the
1960s. However, the tilting pad thrust bearing was inve nted independently by
Kingsbury and Mitchell . Mitchell also invented the tilting pad journal bearing
and installations of the tilting pad journal bea ring appear as early as 1916 . An
installation of a combined tilting pad journal and thrust bearing on the H .M.S.
Mackay of the British Royal Navy, which was launch ed in 1918 . Fixed bearings
remained the standard in the first half of the 20th century due to reduced cost
for fixed geometry installations, the higher parasitic losses associated with
tilting pad bea rings compared to fixed geometry bearings, lower load capacity
of tilting pad bearings , and lower operating speeds that could tolerate the
destabilizing effects of fixed geometry bearings. Boyd and Raimondi in
particular stated : “The plain journal bearing compares favorably with the
pivoted -pad bearing and by many criteria is somewhat superior to the latter.”
Because of these factors, the perceived drawbacks to tilting pad bearings
outweighed the benefits.

6 The advantages of tilting pad bearings in removing the bearings as a source of
self-excited vibrations was originally recognized by Hagg in 1946 . Hagg
presented experimental results for several tilting pad bearings, including 3 – pad,
4-pad, 5 -pad, and 6 -pad bearings. However, the fluids model that Hagg
employed to explain the stabilizing features of these bearings was
fundamentally incorrect. A linear flow profile was assumed that did not account
for the Reynolds equation.
Sternlicht presented a finite difference solution to the Reynolds equation based
on an isoviscous lubricant. The finite difference solution was used to calculate
the developed pressure field, which was then integrated to calculate forces. The
force solution was then perturbed to determine eight stiffness and damping
coefficients based on r otor motion at the journal. These eight coefficients are
widely accepted as a proper model for fixed geometry journal bearings where
temporal inertia is not important . Solutions to the perturbed Reynolds equation
also began to appear in textbooks, includin g the ones by Smith , Pinkus and
Sternlicht and Tondl . Smith’s treatment of the subject was brief, but did include
the eight stiffness and damping coefficients. Pinkus and Sternlicht investigated
the stability of rotors supported in plain journals, but the solutions were
performed in polar coordinates. Modern rotordynamic analyses are performed
in Cartesian coordinates for simplicity and for direct comparison to vibration
measurements. Tondl’s text was an investigation of various sources of
rotordynamic inst ability, of which fixed geometry bearings were a significant
contributor. Tondl’s treatment accounted for direct stiffness, direct damping,
and cross -coupled stiffness terms. His investigation included a treatment of the
perturbed Reynolds equation for bot h linear and nonlinear rotor vibrations.
Tondl also considered the benefits of noncircular fixed geometry bearing stator
profiles, including lobed bearings.

7 Even with the improvements to fixed geometry designs to enhance stability,
there is still a limit w here the destabilizing forces are high enough to overcome
the damping and drive the rotor unstable. Typically, the limit is reached when
the operating frequency is greater than twice the first bending natural
frequency . These limits began to be reached on a more consistent basis in the
1960s. The advantage of the stabilizing effect of the tilting pad bearing was
then seen to overcome the drawbacks of decreased load capacity and higher
parasitic losses.

2.2 FORMS OF HYDRODYNAMIC LUBRICATION AND ITS
APPLICA TION

Hydrodynamic lubrication is a form of lubrication in which the lubricant is in
constant motion to and fro the lubrication elements (components) ; the lubricants
being fluids. Some other forms of hydrodynamic lubrications include;
1Elasto hydrodynamic lubrication :
As pressure or load increases, viscosity of the oil also increases. As the
lubricant is carried into the convergent zone approaching the contact area, the
two surfaces deforms elastically due to the lubricant pressure developed in the
lubrican t causes a further increase in viscosity that is sufficient to separate the
surface at the leading edge of the contact area. Because of the high viscosity
and the short time required to carry the lubricant through the contact area, the
lubricant cannot esc ape and the surface will remain separated and this has little
effect on the film thickness because of the pressures involved, the oil film is
actually more rigid than the metal surfaces and increase the contact area, rather
than the metal surfaces and incr ease the contact area, rather than decrease the
film thickness.
2) Boundary lubrication occurs in insufficient surface area, a drop in the
velocity of the moving surface, a lessening in the quality of lubricant delivered

8 to a bearing, an increase in the be aring load, or an increase in lubricant
temperature resulting in a decrease in viscosity. When this happens, the highest
asperities may be separated by lubricant films with only several molecular
dimensions in thickness.
3) Solid-film lubricant occurs when bearings operate at extreme temperatures,
such as graphite or molybdenum disulfide must be used because the ordinary
mineral oils are not satisfactory
4) Hydrostatic lubrication is obtained by introducing the lubricant, which is
sometimes air or water, in to the load –bearing area at a pressure high enough
to separate the surfaces with a relatively thick film of lubricant. So, unlike
hydrodynamic lubrication, this kind of lubrication does not require motion of
one surface relative to another.

2.3 PRINCIPLE

The diagram below shows the principle with which hydrodynamic lubrication
works. To understand, consider a block sliding on a horizontal table (Figure 1a).
The force required to move it with constant velocity is equal to the frictional
force. That is; F = µW ; where µ is the coefficient of sliding friction and W is
the weight of the block.
If we now put a liquid film of thickness h between the block and the table
surface (Figure 1b) the force required to move the block with constant velocity
U is;
P = µUA
h

Where µ is the coefficient of viscosity of the liquid, h is the height of the film
and A is the bottom surface area of the block. Note that now the friction force
depends on the velocity of the block. But t his reduction in friction is useless as

9 the liquid film as shown in figure 1b cannot support the weight of the block.
The liquid will flow out of the sides under the weight.
What we need is not only that friction be reduced but that the weight is

support ed. Both are achieved if the block is tilted slightly forward as it moves
as shown in figure 1c. The block, as it moves forward, drags the liquid into the
narrow gap. As the liquid moves into the narrow gap, the pressure that builds up
supports the load.

10 2.4 HYDRODYNAMIC THEORY
Varying the leather belts in single, double or triple ply. Tower based his
investigations on a partial bearing, which has a bearing arc of 157 degrees. This
is shown diagrammatically as a bath type below;

Fig. 2.1 Schematic representation of partial bearing used by Tower
From the figure above, a schematic drawing of the journal bearing which Tower
analyzed having a diameter of 4in, a length of 6in and a bearing arc of 157
Lubricator Hole
N W
Partial Bronze
Bearing
Lubricant Level
journal

11 degrees, and having bath -type lubrication as shown. The coefficients of friction
obtained by Tower in his investigations on the bearing were quite low. After
testing this bearing, Tower drilled a 0.5 -in-diameter lubricator hole through the
top. In an effort to prevent this, a cork stopper was used, but this just popped
out. At this point, Tower undoubtedly realized that he was on the verge of
making a discovery. A pressure gauge connected to the hole indicated a
pressure of more than twice the unit -bearing lo ad. Finally, he investigated the
bearing f ilm pressures in detail throughout the bearing width and length and
reported a distribution similar to that of figure B below.

N
d=101.6mm P=0 P max
1=152.4mm

12 The results obtained by Tower had such regularity that Osborne Reynolds
concluded that there must be a definite equa tion relating the friction, the
pressure, and the velocity. The present mathematically theory of lubrication is
based upon Reynolds work following the experiment by Tower.
The original differential equation, developed by Reynolds, was used by him to
explai n Towers’ results. The solution is a challenging problem that has
interested many investigators ever since then, and it is still the starting point for
lubrication studies.
Reynolds pictured the lubricant as adhering to both surfaces and being pulled
by th e 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.
One of the important simplifying assumptions resulted from Reynolds’
realization that the fluid films were s o 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;
1) The lubricant obeys Newton’s visc ous effect.
2) The forces due to the inertia of the lubricant are neglected.
3) The lubricant is assumed to be incompressible.
4) The viscosity is assumed to be constant throughout the film.
5) The pressure does not vary in the axial direction.

2.5 HYDROD YNAMIC ACTION IN A JOURNAL BEARING
This occurs when a shaft is rotating in a journal bearing in the presence of a
lubricant, the phenomenon of hydroplaning is said to occur, which can actually
cause the shaft to lift off the surface of the bearing. The phe nomenon of
hydroplaning is represented in the figure below.

13 From the above figure, if the load (W ) is applied, the oil film is sustained due
to a pressure build -up in the oil film region. This oil pressure opposes the load
and is produced by the wedging a ction of the oil. As the incline load moves, the
oil will be forced to flow through a continually decreasing area. The greater the
plate velocity, the faster the pressure build -up and the load carrying capacity.
This phenomenon of hydroplaning is of seriou s concern.
Hydrodynamic lubrication finds a number of Engineering applications, the most
common being the journal bearing. When we slip on wearing rubber -soled
slippers on a wet bathroom floor, it is a manifestation of hydrodynamic
lubrication.

2.6 MITCHE LL TILTING PAD OPERATION
The Mitchell tilting pad has it’s mode or way that by adjustment and varying
few components like the speed regulator, adjustment of the oil film (adjusting
the slider) and also adjusting the micrometers to set the maximum height th e
slider can travel via the use of the rotating knobs. The zero setting of the
micrometer s indicates the altitude of the bearing pad from the belt. It is checked
by rotating the knob, which rotates the eccentric shafts to bring the pad into
uniform contact with the belt. As the power is turned on, the electric motor
drives the rear drum and this torque is transmitted to the front drum through the
flat leather belt that is a single ply and with a belt density of 970kg/m and an
allowable tensile stress of 1.7 mpa. The drums are serrated to minimize belt slip.
As the drum rotates, the belt carries oil from the bath (basin), which is the oil
reservoir through the slider bearing. The ‘oil film constant’ is determined by
adjusting the micrometer to get the ratio of the film thickness at the leading
edge divided by the film thickness of the trailing edge (k=h1/h2). The oil
carried by the belt rises through the manometers fixed both transversely and
longitudinally. These manometers are used to determine the pressure
distribution along the slider, and these pressure distributions are summed up to
give an equal and opposite reaction as regards to the gravitational effect on the
slider.

14

CHAPTER 3
APPARATUS DESCRIPTION, MODE OF OPERATION AND
SPECIFICATIONS
3.1 DESCRIPT ION OF APPARATUS
The Mitchell Tilting Pad Apparatus consist of a mild steel pad (or slider) of
which can be titled to the required angle and accurately positioned relative to a
mild steel casting which carries two steel roller drums and are in turn carryin g a
flat leather conveyor belt. One drum is driven by a variable speed A.C motor
via a V -belt made of rubber and the second drum is carried in a slotted
mounting for permissible adjustment of the belt tension. The surface of the
drum is serrated to minimi ze belt slip and also crowned to avoid derailing of
belt. The flat belt runs in a continuous loop beneath the pad carrying a thick oil
film over the pad surface from the oil reservoir. When pad is titled over the oil
pressure film, pressure is develope d. The oil distribution pressure is measured
over a multitude manometer installed on the pad/slider.
The apparatus is contained in a metal (mild steel ) tank filled with oil to such a
level that the lowest part of the belt is submerged. The belt travels along t he
machined top surface of the main casting, which is slotted to drain off the
excess oil. The tilting pad is supported on two eccentric shafts with hand wheels
in turn carried in side members well bolted to the frame of the apparatus.
The clearance betwe en the belt and the slider is measured by the two
micrometers located in -line respectively with the leading and trailing edges of

15 the slider. The clearance ranges from 0.5mm to 2.0mm and is several hundred
times as great as the clearance in real thrust bea ring and in consequence may be
easily measured with sufficient accuracy.
The oil pressure developed between the slider and the moving belt is indicated
by thirteen (15) graduated manometer tubes well secured to the slider. Seven (8)
of these tubes are equa lly spaced along the axis of the slider in the direction of
the motion while the remaining six (7) are located transversely in a plane
approximately coinciding with a point at which maximum pressure is
developed.
The o il used in this apparatus has a rating of SAE 20w/50 commonly known by
the roadside mechanics as Whiz oil and falls within a kinematic viscosity rating
of 100 -150 centistokes .

3.2 MODE OF OPERATION
As the power is turned on, the electric motor drives the rear drum. This
transmitted to the fro nt drum via the conveyor belt. These drums are serated to
minimize slip. As the drums rotate, the belt conveys oil from the reservoir
through the slider. The oil film constant is determined by adjusting the
micrometers to get the film thickness at the lead ing edge divided by the film
thickness at the trailing edge. The oil carried by the belt rises through the
manometer fixed transversely and longitudinally. These manometers are used to
determine the pressure distribution along the slider.

16 3.2 SPECIFICATI ONS
3.2.1 THE BELTS
For the CONVEYOR BELT
Fig 3:

Belt Material Leather
Number of Ply Single
Weight Density ( ρ) 1000kg/m3
Allowable Stress ( Sf) 2.0MPa
Width ( bf) 130mm
Thickness ( tf) 2mm
Length ( Lf) 1070mm
Coefficient of Frict ion (µf) 0.22
Quantity One (1)
TABLE 1
A leather belt was chosen because it has a high tensile strength and ability to
convey oil without slipping due to its rough surface.

17

For the V -BELT
Belt Material Rubber
Number of Ply Single
Weight Densi ty (ρ) 1100kg/m3
Allowable Stress (S) 2.0MPa
Width (b) 10mm
Thickness (t) 5mm
Calculated Length (L) 639mm
Coefficient of dynamic friction ( µ) 0.5
Quantity One (1)
TABLE 2

18
3.2.2 THE SLIDE R

Fig 4:

Material Cast alumin um with polished surface
Length 145m
Width 130mm
Thickness 16mm
Quantity One (1)
TABLE 3

19

3.2.3 THE MANOMETER TUBES
Fig 5:

Material Perspex
Outside Diameter 10mm
Inside Diameter 7mm
Length 500mm
Manometer Calibrat ion 0-400mm
Quantity Fifteen (15)
TABLE 4

20

3.2.4 THE OIL RESERVOIR
Fig 6:

Material Mild Steel
Length 750mm
Width 450mm
Height 300mm
Plate Thickness 2mm
Quantity One (1)
TABLE 5

21

3.2.5 THE ROLLER DRUMS
Fig 7:

The roller drums are serrated to prevent belt slip and one is attached
to a pulley system which is being driven by a rotor.
Fig 8:

Material Cast aluminum
Diameter 120mm
Length 162mm

22 Quantity Two (2)
TABLE 6
3.2.6 MICROME TER
Fig 9:

The micrometers are two (2) in number and has their anvil cut off in such a way
that only the spindle is used for getting the required height on the trailing edge
‘h0’ and the leading edge ‘ h1’. The micrometer ranges between 0 -10mm.

3.2.7THE OIL
Oil Type SAE 20w/50
Density 360kg/m3
Capacity 0.01m3
TABLE 7

23
3.2.8 BALL BEARINGS
This apparatus has about eight (8) ball bearings of which four (4) are fixed on
both roller drums, two (2) on each and the remaining four (4) are fixe d on the
cam like shafts which tilts the slider on rotation to achieve h0 and h1. They are
made up of high carbon steel.
Outside Diameter 45mm
Inside Diameter 20mm
Width 15mm
TABLE 8

24

CHAPTER 4

DESIGN ANALYSIS OF BELTS, DETERMINATION OF
REYNOLDS AND PRESSURE DISTRIBUTION EQUATION.
4.1 DESIGN ANALYSIS OF BELTS
The Mitchell tilting pad apparatus requires two (2) belts that are in constant
rotation to produce the required motion for effectiveness of the apparatus. The
belts are: a V -BELT whi ch connects the motor pulley and a pulley attached to
one of the roller drums and a FLAT BELT which connects the roller drums.
The design analysis below clearly shows how the specifications of the belts
were derived as well as other valuable parameters suc h as the tension on the
tight side, tension on the slack side, stress on the slack side, power transmitted
by the pulley governing the design, etc.

25

4.1.1 V -BELT DESIGN AND ANALYSIS
Fig 10: SCHEMATIC DIAGRAM OF A V -BELT
D1 = Diameter of the roller pulley
D2 = Diameter of motor pulley
ϴ = Groove angle
β1= Angle of wrap of the roller pulley
β2 = Angle of wrap of the motor pulley
X =Belt length of contact of the pulley on each side
Y = belt length in contact with the roller pulley
Z = Belt length in contact with the motor pulley
C = Cente r distance between the pulleys

26 A motor with an RPM of 1200 (N 2) and its pulley diameter, 50mm is required.
Also the required speed for the roller pulley is 600RPM (N 1).
RECALL:

–––- –––––––––––-
(1)

THUS:

––––––––––––––
(2)
Making D 1 the subject of the formula and imputing the values of each
parameter:

The required le ngth (L) of the belt that best fit the design of the pulleys, if we
recall, is given by the formula:

–––––––––––––-
(3)
We also know:

27

Designing with a center distanc e (C) of 200mm:

ALSO:

––––––––––-
(4)
RECALL THAT:

28

In RADIANS;

Substituting the value of “
” in radians into (4):

AND:

Substituting all va lues of “X”, “Y” and “Z” into (3):

(Approximately)

29 In determining the tension on the tight side (T 1), the tension on the slack side
(T2) and the stress on the slack side (S 2), we need to know the pulley that
governs the design by comparing their capacities using the expression:
–––––––––
(5)
Where: μ = coefficient of friction with a value of 0.5
β = angle of wrap

For the roller pulley (large pulley):

–––––––––––
(6)
Where: β 1 = angle of wrap for the large pulley and is given b y the formula

In RADIANS:

Substituting the value of “β 1” into (6):

30

For the motor pulley (smaller pulley):

––––––––––-
(7)
Where: β 2 = angle of wrap of small pulley and is given by the equation:

In RADIANS:

Substituting “β 2” into (7):

This shows that the motor pulley governs the design because its value is smaller
compared with that of the roller pulley.
“T1” and “T 2” has a relationship given by the fo rmula:

–––––––-
(8)

31 Since the smaller pulley governs the design:

It is also known that the tension on the tight side of a belt is a fu nction of the
allowable stress, also known as the stress on the tight side (S 1) and the cross
sectional area (A). It is given by the formula:

–––––––––––––––
(9)
Where:
S1 = 2.0MPa
A = belt width (b) × belt thickness (t)
= 0.01 × 0.005

Substituting values into (9):

m = mass of the belt = ρ × b × t
Where:
ρ = weight density having a value of 1100kg/m3
b = belt width with a value of 10mm = 0.01m
t = belt thickness with a value of 5mm = 0.005m

32

Since the smaller pulley controls the design, velocity (v) is given by:

Substituting al l values into (8):

The power (P) that should be transmitted by the pulley is:

––––––––––––
(10)

This means that the power rating recommended for the motor that best suits the
apparatus is 0.735kW, which is equivalent to 1hp.
The stress on the slack side (S 2) is determined by the formula:

–––––––––––––
(11)
Making S 2the subject if the formula and imputing their respective values:

33

4.1.2 CONVEYOR BELT DESIGN
Length of belt (L) is given by:

––––––––––––
(1)
For pulleys that are equal in size, th at is; D 1 = D 2 = D, each will have an angle
of contact of 180o also known as angle of wrap equal to each other. This simply
means that:

Hence, the groove angle ( ϴ) equals ZERO.
PROOF:
RECALL that;

––––––––––––
(2)
Substituting for “β 1” into (2):

Thus;

(Proven)

34 Since the groove angle ( ϴ) equals ZERO, it means that the belt length of
contac t of the pulley on each side (X) equals the center distance (C) between
the pulleys. That is:

––––––––––––––
(3)
And finally, since the belt length of co ntact of the pulleys (X) equals the center
distance between the pulleys (C), this simply tells us that the belt length in
contact with both pulleys are equal. Thus;

PROOF:
RECALL that;

Substituting ϴ = 0 and D 1 = D into the above equation:

ALSO;

Substituting also ϴ = 0 and D 2 = D into the above equation:

Comparing Y and Z, we will see that;

(Proven)

35 From (1):
Let;
––––––––––––-
(4)
Substituting for “Y” and “Z” into (4):

––––––––––––– –
(5)
Equation (1) can now be modified as:

––––––––––––-
(6)
Designing with a center distance (C) of 345mm, the belt length can be
determined thus:

Substituting values into (6):

“1070mm” is recomm ended due to adjustment of belt tension.

36 Determining “T 1” and “T 2” which is quite similar to the V -BELT is given by
the relation:

––––––––-
(7)
Solving for
:
β = 180o
In RADIANS:
β = 3.141rads
Coefficient of friction (μ) = 0.22 (greasy pulley)
Thus:

–––––––––––
(8)
We know that:

S1 = 2MPa = 2×10-6Pa
A = b × t
= 0.13 × 0.002
A = 2.6×10-4 m2
Substituting values into “T 1”:

37

VELOCITY (v) =

Substituting values into (9):

Solving for “S 1” which is the stress on the slack side from the relation:

–––––- –––––––-
(9)

38 4.1.3 INTUITIVE DESIGN FOR OTHER COMPONENTS
This approach is simply the faculty of knowing or sensing without the use of
any rational approach. In engineering design, not all components require
rational approach such as cal culation to get various specifications.
4.1.3.1 FOR THE CONVEYOR BELT
The length of the pulley is simply a determinant to the width of the belt which
is 130mm. And the thickness is simply as a result of its availability in the
market.
4.1.3.2 FOR THE SLIDER
The width of the slider is determined by the width of the conveyor belt.
Therefore 130mm is chosen as its width. Its length is gotten from mere
discretion, not too long and not too short. 145mm is chosen.
In other not to have a weighty slider, 10mm thickn ess is suitable. The standard
size for a manometer tube is 10mm. On that note, the fifteen (15) hole drilled on
the slider should be 10mm diameter each.
4.1.3.3 FOR THE FRAME
This is a component in the apparatus holding the roller drums in place. It is
designed using the center distance between the roller drums, 345mm.

39 4.2 DETERMINATION OF REYNOLDS AND
PRESSURE DISTRIBUTION EQUATION
Reynolds developed mathematically the pressure distribution (P) on
hydrodynamic effect on two surfaces. He considered the conditions necessary
for the viscous force and shear force as well as the indestructibility of matter
within the adjacent members of the slider bearing with converging fluid wedge.
Fig 12: DIAGRAM SHOWING A SLIDER BEARING
ASSUMPTIONS.
1. The fluid is incompressible.
2. The fluid is a Newtonian fluid.
3. Inertia and turbulence effect are negligible.
4. Fluid properties remain the same throughout the fluids.

40 5. Film of sufficient small thickness that the fluid pressure is constant
throughout the film.
6. Uniform pr essure in the Y -Direction.
7. Acceleration in the X -Direction.
8. Slider is at rest while the bearing plate is forced to move with constant
velocity (U) in the X -Direction.

Fig1 2: ISOLATED CONTROL ELEMENT
Isolating the control volume a nd considering the forces acting on the lubricant
within the fluid wedge (h) in both the X and Y direction as shown in the figure
above.
The forces due to pressure acts on the left and right sides of the control volume
and must balance the shear forces due to viscosity and velocity acting on the top
and the bottom of the control element. That is:
Sum of forward forces = sum of backward forces

–––––– (1)
Expanding and eliminating parameters i n (1) we have:

41
–––– ––––––––– (2)
From Newton’s law of viscosity (assumption 2):

Substituting into (2):

–––– ––––– (3)
Integrating (3) twice with respect to “y”, we obtain:

––––––– (4)
The plate is moving with a constant velocity (assumption 8). Therefore, we
apply boundary conditions to obtain values for the constants A and B.
AT: y = 0, v = U
AT: y = h, v = 0
Where: h = the film thickness at the wedge of the slider bearing:
and;

Substituting for “A” and “B” into (4),the velocity profile is given as:

42
––––––– (5)
As far as integration along Y is concerned,
is constant.
At maximum pressure point,
. Substituting into (5):

–––– –––––- (6)
Equation (6) depicts that at maximum pressure location in the bearing, the
velocity profile along the Y -axis is linear and the clearance at that location is
denoted by ‘h’.
The volume flow rate (Q) through the clearance is denoted by:

–––––––––- (7)
Substituting (4) into (7):

Integrating the above equation, it yields:

––––––––––– (8)
Making
the subject of the formula:

–––––––––––– (9)
“α” which is the angle of tilt of the pad is given by:

43

Where “x” is the point on the X -axis where the maximum pressure occurs.
Making “ h” the subject of the formula and substituting into (9), we obtain:

–––––––– (10)
Integrating (8) with respect to “x”:

––––––– (11)
“Q” and “C” are no w constants. Since the pressure must be the same at both
ends of the bearing, P 1 = P 2. Thus, applying the pressure boundary conditions
which are:
AT: x = 0, P =0
AT: x = L, P = 0, we obtain:

And

Subst ituting the above equations into (11):

–––––––––––- (12)
Where:
Q = Volume flow rate.
C = Clearance.
P = Pressure distribution.

44 From equation (8), we know that the flow rate a cross the sections of the pad is
the same. That is;
. Thus, differentiating (8) with respect to “x” and
equating it to zero:

–––––- (13)
OR:

–––––––––– (14)
Equation (14) is a 1 -dimensional REYNOLDS EQUATION. A 2 -dimensional
REYNOLDS EQUATION is shown below:

––––––– (15)
The solution of the Reynolds equation in a 2 -Dimensional pla ne for a Mitchell
tilting pad slider bearing is obtained through numerical analysis with the use of
a non -dimensional pressure “p” by an electronic computer as shown below:

–––––––––- ––– (16)
This gives a pressure at any surface of the pad which is in parabolic form.
BUT:
P = 9.81Hp
Where: H = manometer column height
Therefore, the pressure (P) given in pressure head term is:

45
––– –––––––– (17)
4.2.1 FOR THE LOAD CARRYING CAPACITY (F)
Equation (14) can be expressed in terms of a non -dimensional load carrying
capacity. Thus:

–––––––––– (18)
Since the t ransverse pressure distribution is parabolic, it implies that the mean
pressure on that transverse plane is two -thirds of the pressure on the centerline.
That is:

–––––––––– (19)
Where:
is the average height of the column of the oil on the longitudinal,
which may be calculated using SIMPSONS RULE. From the manometer
heights indicated by the manometer column at sections 1 -7:

–––––– (20)
Where: H 1, H 2, H 3, H 4, H 5, H6 and H 7 are the manometer heights of the sections.

46 CHAPTER 5
MATERIAL SELECTION AND COST ANALYSIS
5.1 MATERIAL SELECTION
The selection of the proper material for engineering purposes is one of the mos t
difficult problems of a designer. The best material is one which serves the
desired objective at minimum cost. There are three factors to be considered in
the selection of a material for a design;
1) Availability of the material
2) Suitability of the material
3) Cost of the material
Now the utility of the materials is determined by three major factors;
1) Physical properties .
These properties include: luster, color, shape and size, density, electric and
thermal conductivity, melting point, etc.
2) Chemical properties .
These include; resistance to corrosion and oxidation.
3) Mechanical properties.
These properties include; strength, stiffness, elasticity, ductility, plasticity,
brittleness, malleability, toughness, machinability, resilience, creep, fatigue,
hardness , etc.

47 Components such as the slider, the oil reservoir, the belt bed, the roller drum,
the frame and the handle are all made of low carbon mild steel containing o.5%
to 0.45% of carbon.
Steel as we know is an alloy of iron and carbon having a minimum content of
carbon up to 1.5%. The carbon occurs in the form of iron carbide because of its
ability to increase the hardness and strength of the steel. Most of the steel
produced now -a-days is carbon steel also know n as plain carbon steel. The
plain carbon is a steel w hich has it properties mainly due to its carbon contents
and does not contain more than o.5% of silicon and 1.55 of manganese. Though
other elements such as sulphur and phosphorus are also present to a greater or
lesser amount to impart certain desired pro perties.
For its physical properties; it has an appreciable melting point to withstand the
increase in temperature of the working fluid due to its good thermal
conductivity.
For it chemical properties; it has a very good resistance to corrosion and
oxidati on, two very important factors.
For its physical properties; it is machinable and has sufficient strength, rigidity
and stiffness.

5.2 COST ANALYSIS
Engineering projects cannot be restricted to the narrow confines of pure
technical works, because financia l estimation and expenditures are an integral
part of any useful and successful engineering Endeavour.

48 Adequate completion of an engineering job requires more than the system just
performing the intended function satisfactorily. It must accomplish this at a
minimum cost consistent with the degree of safety and aesthetic value. No
matter the kind of job manufacturing or repairs and reactivation cost is
involved, and since for every situation, alternative approaches are available, it
therefore behooves on the engineer to select the one that provides the least cost
maximum economy.
5.2.1 COSTING
Costing is simply the calculation of the financial involvement in any job or
project. It is required for the purpose of submitting a quotation or tender for the
job so as to ascertain the profitability, or otherwise of embarking on the job.
Cost in engineering is broadly grouped under:
1) DIRECT COST
These are costs of factors that can be directly attributed to the manufacturer of a
specific product. They include cost of ma terials and labor.
Material cost is the cost of material that goes into the finished product.
Labo r cost is the product of the number of pieces produced and the piecework
rate or the product of the time spent in manufacturing the product by the direct
shop floor works and the wage rate.

2) INDIRECT COST
These are cost of factors, which can only be indirectly attributed to the
manufacturer of a specific product. They are also called overhead.

49 Overheads are generally classified into works overhead, office ov erheads and
seals overhead.
For the purpose of estimati ng the financial involvement in this project, costing
is done component wise.

5.2.2 ANALYSIS OF COST

TOOLING COST
S/N DESCRIPTION QTY UNIT
COSTS TOTAL( N)
1 A set of spanners
(8”-19”) 1 650 500
2 Pliers 1 250 250
3 Screw driver 1 80 100
4 8” adjustable
spanner 1 800 800
5. Abrasive papers 4 400 1,600
3,400
TABLE 9
Total tooling cost = N3,400.00

50 MATERIAL COST
a) Painting
S/N DESCRIPTION QTY UNIT
COSTS TOTAL( N)
1 Medium tin of
oil pa int 1 350 350
2 Small tin of
aluminum paint 1 200 200
3 Thinner 2
bottles 200 400
4 Painting brush 2 100 200
1150
TABLE 10
Total painting cost = N 1150

51 b) Cost of Materials
S/N DESCRIPTION QTY UNIT COSTS TOTAL( N)
1 2mm sheet
(1 full sheet) 1 8000 8,000
2 Roller drum 2 3000 6,000
3 Leather belt 1 1500 1,500
4 Micro meter 2 750 1,500
5 25mm mild steel (170x100mm) 1 2000 2,000
6 Bolts, nuts and washers 16 30 480
7 Slider 1 2000 2000
21,480

Total material cost=N21 ,480
Labor cost = N30 ,000
Total material + labor cost = N51,480
Direct cost = Tooling Cost + Painting Cost + (Material Cost + Labor Cost )
=3,400 + 1,150 + 51,480
=N56 ,030
INDIRECT COSTS
i. Transportation =N6,000.00
ii. Miscellaneous Expe nses = N10,500.00
Total indirect cost = N 16,500.00
Total cost borne = Direct + Indirect Cost
= (N56,030 + N16,500 )
= N72,530.00

52 CHAPTER SIX
CONCLUSION, OBSERVATION AND RECOMMENDATION
6.1 CONCLU SION
In conclusion, we were pleased to see that the purpose of embarking on this
project is achieved. This purpose which is confirmatory to Tower ’s theory of
hydrodynamic effect of lubrication on a slider bearing. This project has proved
that the higher th e oil head on the manometer, the higher the load carried by the
slider. This project is not easy to embark on, considering the financial, labor
and time requirements involved in undertaking the various process gone
through.
6.2 OBSERVATION
1)In running th e apparatus, we observed that the maximum pressure Is
developed where the longitudinal ends.
2) We also observed that the oil level at the transverse section of the manometer
is parabolic.
3) Running the machine at high speed of 1430rpm results in a high v ibration,
thus the oil in the manometers fluctuates and getting the actual manometer
reading becomes difficult.

53 6.3 RECOMMENDATION
As I undergo the different stages of this project work, more skills, knowledge
and ideas were acquired on the process of des igning and construction.
The following recommendations should be made so that the efficiency of the
machine can be increased tremendously;
1)More of the apparatus should be provided so that the students in the
department can have enough of itfor their pra ctical experiments.
2)The department should not allow the project to be kept in the laboratory for
exhibition but should be put to use.
3)Due to the time limit, the department should allow for optimization of the
project by the successive grauduants .

54 REFERENCES
Bansal , R. K. (2006). A Textbook of Fluid Mechanics a nd Hydraulic Machines .
India: R. K. Gupte Publishers.
Chukwujekwu, S. (2013). Lecture Note on Mechanical Engineering Design .
Enugu: (Unpublished) ESUT .

Ezewe S. (2013). A lecture note on Mechanical Engineering Design . Enugu:
(Unpublished) Caritas University.

Khurmi, R. S., & Gupta, J. K. (2006). A textbook on Machine Design s. India: S.
Chand Publishers.
Mechanical Engineering Laboratory Practice Manual. (2012). Enugu:
(Unpublished) ESUT .

Odukwu, A. O. (2013). A lecture note on Advanced Fluid Mechanics. Enugu:
(Unpublished) ESUT.
Rajput, R. K. (2008) . A textbook of fluid mechanics and hydraulic machines .
India: S. Chand Publishers.

Richard, E. B. (2008) . Shigleys Mechanical Engineering Design , (8th Edition).
Singapore: McGraw Hill Pu blishers.
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