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Hydrodynamic Lubrication in Journal Bearing 8

2

Hydrodynamic Lubrication in JournalBearing

This chapter provides background to hydrodynamic lubrication in journal bearings.

Starting with a brief introduction to lubrication in tribology, discussions then move

on to different lubrication regimes, fluid viscosity and how it relates to other

properties, journal bearing basic design, and hydrodynamic mechanism. A section is

dedicated to discuss different measurement techniques available for bearing oil films.

This is then followed by a section on theoretical solutions in hydrodynamic

lubrication. Three aspects were discussed: (1) historical advancement, (2) standardReynolds’ equation, and (3) cavitation boundary conditions.

2.1 Introduction: Lubrication in Tribology

Lubrication issues and concerns in science and engineering are addressed in a

specific school of knowledge known as Tribology (Szeri, 1998). Though the word

tribology was only introduced about 40 years ago (Jost, 2006), the work on machine

lubrication can be traced back to the civilization of Mesopotamia and Egypt

(Williams, 1994). In the early years, simple machines such as wheels were invented

to carry loads and lubrication was added to make the system more efficient

(Lansdown, 1996).

Machine requirements are continuously challenged and increasing demands are

placed on machine lubrication. Specific knowledge is required to implement best

practices in lubrication application, conditioning, and monitoring. Consequently,

better techniques are constantly investigated to gain new knowledge and improve the

present understanding.

2.2

Machine Lubrication and Lubrication Regime

Rubbing or sliding of machine parts can create friction and heat. If not treated,

machine parts will get worn out over time sooner than initially designed for.Lubrication is introduced to separate machine parts and hence reduce friction and

wear rate of the machine components (Szeri, 1998). Often, friction is an undesirable

process that could cause unnecessary power lost through heat produced during the

process.

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In practice, engineers encounter machine failures and poor performance that could be

traced back to improper lubrication practices. Before an attempt is made to solve any

lubrication problems, it is crucial to first identify the lubrication regime so that the

right models can be proposed in the solutions (Cameron, 1966; Williams, 1994).

Lubrication regimes may be defined based on the coefficient of friction, µ . In thecase of a journal bearing, the plot of coefficient of friction µ against η /P willgenerally produce a similar trend (often call Stribeck curve) in Figure 2-1. The three

terms in the plot are the absolute viscosity (η), the rotational speed in revolutions persecond ( ), and the projected load ( P ).

Figure 2-1: Stribeck curve showing coefficient of friction plotted against the

product of the absolute viscosity (η) and the rotational speed in

revolutions per second ( ) divided by the projected load ( P ).

The interest of the present study is in hydrodynamic lubrication regime (i.e. a full

film condition) where µ has an increasing relation to the η /P values. In this region,for a lightly loaded journal bearing, Petroff’s law can be used to model the relation

where a uniform shear stress is assumed acting on the journal. The flow in

hydrodynamic lubrication regime is partly driven by the virtue of the viscous drag

force (Couette flow) and the pressure difference due to the fluid wedge being

established (Pouseville flow). Therefore, the coefficient of friction is a measure of

the viscous drag force. For the limit of Petroff’s law application, a moderate

eccentricity ratio ε is recommended in the range up to 0.5 to 0.7. In Table 2-1, the

typical values of film thickness, h are shown along with the friction coefficient range

for three main lubrication regimes.

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Lubrication Regimes µ h (µm)

Boundary Lubrication 0.050 – 0.200 0.01 – 0.10

Mixed lubrication 0.001 – 0.010 0.10 – 1.00

Hydrodynamic lubrication 0.001 – 0.010 1.00 – 100

Table 2-1: Typical µ and h values for various lubrication regimes (Szeri,1998)

2.3 Fluid Viscosity

Fluid viscosity is a measure of fluid internal resistance to motion (Massey, 2005;

White, 2003). Scientifically, fluid viscosity is described in terms of shear stress and

shear strain at a given temperature. When shear stress is applied, fluid layers will

slide relative to the other at a rate determined largely by its viscosity. The flow will

subside only when the shear stress is removed or balanced. This resistance to flow or

simply viscosity is quantifiable by a resistance law in equation (2.1), which was first

postulated in 1687 by Isaac Newton as reported by White (2003).

dydu

τ η = (2.1)

where η is the dynamic viscosity, τ is the shear stress (i.e shearing force over

parallel area), and dydu / is the velocity gradient or shear rate. Dynamic viscosity is

often measured in Pascal.second (Pa.s) or centipoise (cP = mPa.s). A device used to

measure fluid viscosity is called a viscometer. Table 2-2 shows common types ofviscometer, which are differentiated by their measurement principle.

In a cup type viscometer, viscosity is measured by the time it takes for the fluid to

empty the container in seconds. This is known as the kinematic viscosity υ with SI

unit in meter squared per second (m2/s) or centistokes (cSt). Kinematic viscosity of

any fluid is related to dynamic viscosity by its density. Mathematically,

ρ η υ = (2.2)

In a non-convetional approach, viscosity is measured using acoustic waves. Figure

2-2 shows an acoustic viscometer called ViSmart(TM) (manufactured by BiODE and

released in November, 2005) about a size of a US quarter (2.3 mm in diameter).

ViSmart allows instantaneous measurement of viscosity of fluids from 10-10,000 cP.

It only requires about 120 microlitres of sample and can stand a temperature up to

135 ºC. It is based on the use of a shear wave whereby viscosity is measured fromthe difference in shear wave power before and after contact with the fluid.

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Types of Viscometer Measurement Principle

Cup-type viscometer Viscosity is measured by the time it takes a sample to flow out

of the orifice of the sample container

Capillary viscometer Viscosity is measured by letting a sample flow inside the

capillary and measuring the difference in pressures between

both ends of the capillary.

Falling-ball viscometer Viscosity is measured by measuring the time it takes for a

cylindrical or spherical object to fall through a sample over a

specific distance

Rotational viscometer Viscosity is measured by measuring the running torque of the

cylindrical rotors immersed in a sample.

Vibro viscometer Viscosity is measured by controlling the amplitude of the

transducer immersed in a sample and measuring the electric

current that drives the transducer.

Table 2-2: Viscosity measurement devices

Figure 2-2: An acoustic viscometer called ViSmart(TM) by BiODE

(www.biode.com)

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Shah and Balasubramaniam (2000) and Greenwood and Bamberger (2002) also used

shear waves to measure fluid viscosity. However, the viscosity measurement was

based on the amplitude reflection coefficient instead of power. In their work, a shear

transducer was used to transmit a shear pulse at normal incident to an interface and

receive the reflected pulse back. The amplitude reflection coefficient can then be

determined and correlated to the fluid viscosity using a specific model. Though bothearly studies involved viscosity measurement in bulk oil, a similar technique can be

adapted to the case of a thin oil film as shown later in Chapter 8 and 9.

Fluid viscosity is affected by other fluid properties such as temperature, pressure, and

the phase condition of the fluid flow. This has been studied in the past and

established mathematical models are available. More of these aspects are presented

next.

2.3.1

Viscosity and Temperature Relation in Hydrodynamic Lubrication

The viscosity of liquids decreases as the temperature increases. Conversely, theviscosity of gases increases as the temperature increases. The change in viscosity due

to a temperature variation for different fluids can be obtained experimentally. For

industrial fluids like oil lubricants, data on viscosity-temperature relations are often

available for users.

For higher pressure conditions (such as in journal bearing lubrication), the

relationship can be quantified by the Walther equation (Khonsari and Booser, 2001),

T B A log)7.0log(log −=+ν (2.3)

The constants A and B can be determined given the values of kinematic viscosity at

two temperatures (typically 40 and 100 °C). Another model is Vogel equation(Khonsari and Booser, 2001) which offers a better approximation of viscosity overan extended temperature range.

)( ref T T

ref e

−−= β η η (2.4)

Similarly, the constant β is approximated from the viscosities at two different

temperatures by,

12

12 )ln(

T T −=

η η β (2.5)

2.3.2 The Effect of Air Bubbles on Fluid Viscosity and the Speed of Sound

The presence of air bubbles in a fluid flow will change fluid properties, especially its

viscosity. Bubbly oil in a journal bearing is believed to have increased load carrying

capacity by enhancing the oil viscosity (Tonder, 1975; Nikolajsen, 1999). The size of

the bubbles and the air volume fraction are believed to affect oil viscosity and hence

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the bearing load carrying capacity (Choi and Kim, 2002). Based on the results of a

numerical analysis, Choi and Kim (2002) concluded that the load carrying capacity

of a journal bearing increases as the bubbles in the supplied lubricant become smaller

(i.e. the surface tension is getting larger). The load carrying capacity also increases

with the air volume fraction, but this effect is said to continue up to a critical volume.

Beyond the critical value, the load carrying capacity starts to decrease.

The speed of sound will change dramatically with the presence of air bubbles. This

is because air is a poor transmitter of sound. Urick (1947) correlated the speed of

sound c in a mixture to the mean density, ρ and mean compressibility, κ of the

mixture.

κρ

1=c (2.6)

For a two-phase flow where air bubbles are suspended in the oil, the compressibility

and density may be written as,

12 )1( κ φ φκ κ −+= (2.7)

12 )1( ρ φ φρ ρ −+= (2.8)

where 1 and 2 denote the continuous and dispersed phase respectively. The

volumetric content φ of air in the mixture can be rewritten in the quadratic form(Povey, 1997),

A

AC B B

2

42 −±−=φ (2.9)

where

−+

−=

1

22

2

2

12

1 11 ρ

ρ

ρ

ρ cc A (2.10)

−+

= 2

1

22

2

2

12

1 ρ

ρ

ρ

ρ cc B (2.11)

−=

2

2

12

2 1c

ccC (2.12)

2.3.3

Effective Viscosity

An average or effective viscosity is often used to predict bearing performance when

the temperature is relatively uniform throughout the film. This is adequate under

conditions of light load, low speed, and large bearing-thermal capacity (Szeri, 1998).

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The rationale is that the generated heat in the film due to viscous dissipation is

relatively small and will be lost to the surroundings by the lubricant as it exits the

bearing from the sides.

The effective viscosity of a lubricant can be approximated from the data sheet if the

average temperature of the lubricant leaking from the sides is known. The average

temperature, T s of the leaking lubricant from the sides can be quantified by

∆+=2

T T T i s (2.13)

where T i and ∆T are initial temperature and average temperature rise respectively.

In this work, the leaking temperature T s was approximated by measuring the outlet

oil temperature of the bearing because the temperature rise was not known. This

approximation is valid when the two values are closer such as when the journal

bearing speed is low and the temperature rise is relatively small (Khonsari and

Booser, 2001). In two cases investigated by Syverud (2001), a temperaturedifference of about 10˚ and 20˚C were reported for a bearing speed of 1500 and 2100

rpm respectively (Figure 9-1). In Frene et al. (1997), a temperature distribution for

eccentricity ratio of 0.8 at 2000 rpm obtained by numerical solution (Figure 9-2)

shows a temperature variation of 10˚C. In this work, a lower temperature variation is

to be expected since the bearing speed involved is relatively low (i.e. 300 rpm for

film thickness measurements around a journal bearing in Chapter 7 and 700 rpm for

viscosity measurements around a journal bearing in Chapter 9). The temperature

range involved in this work is between 25˚ to 50˚C, lower than a typical value (i.e.

71˚C) found in many bearing operations (Juvinall and Marshek, 2006). Therefore,

the approximation of the leaking temperature by the outlet temperature in this work

is believed to be reasonable.

2.4

Journal Bearing Basic Designs

The study of hydrodynamic lubrication in this thesis is related to its application in a

plain journal bearing. A schematic of the general features in a plain journal bearing is

shown in Figure 2-3. It consists of two main components. The shaft in the middle is

the journal and the part enveloping the shaft is the bearing, which is also known as

the bush. The housing or the sleeve is to support the journal bearing structure. The

journal diameter ( D), the bearing internal diameter (d ) and length ( L), and also radial

clearance (C ) are important criteria. The values and relative proportions of these

parameters play an important role in the capacity and performance of the journal

bearing in actual applications.

An important parameter in journal bearing operation is its eccentricity, e . It is the

distance between the centre of the journal (O j) and the centre of the bearing (O B) as

shown in Figure 2-4. The line passing through both O j and O B is called the centre

line. The positions of O j relative to O B for three initial stages in a journal bearing

operation are illustrated in Figure 2-4. An idle position is shown in Figure 2-4(a)

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where the journal is in contact with the bearing with the line of contact opposing to

the load and the distance between O j and O B is equal to the radial clearance, C . At

start up in Figure 2-4(b), mixed lubrication prevails and the journal is displaced and

slide in the interior of the bearing in the opposite rotational direction. As the fluid

wedge becomes established and lift speed is attained, the journal starts to assume the

position in Figure 2-4(c). The present study is concerned with this last condition.

D

Qs

W

Ql

Ql L

Lubricant in,

Journal diameter

Leakage flow,

Leakage flow,

Bearing

Bearing Housing

Bearing

Housing

Clearance filledwith fluid

Shaft

D+2C

DC C

Figure 2-3: Journal bearing geometry and nomenclature (adapted from Khonsari

and Booser, 2001)

O

Oj

Oj Oj

ϕ W

e

ω

B

BOO B

(a)

(b) (c)

Sleeve (bushing) backing-metal

Bearing material (babbitt)

Journal at rest

Figure 2-4: Schematic diagram of bearing positions at (a) start-up, (b) no loadand (c) steady running conditions

The film thickness around the journal bearing is given by,

θ coseC h += (2.14)

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where C is the radial clearance, e is the eccentricty, θ is the angular position. The

eccentricity ratio, ε is often determined from the operating conditions of a journal bearing, represented in a dimensionless parameter called the Sommerfeld number.

More discussions on this relation follow in section 2.6.

2.5 Mechanism of Journal Bearing Lubrication

A solid-to-solid contact is eliminated completely when a full film bearing is

established. As the journal bearing continues to operate, the fluid wedge supports the

shaft and relocates it within the bearing clearance as in Figure 2-4(c). The fluid

wedge is achieved by the self-acting pumping action of a moving surface (Booser,

1995).

Typically, the minimum thickness of the oil film around the journal is very thin and

yet stiff. It was found in this study that a lubricant thickness of 10 µm (about onetenth of human hair) can carry 12 000 N load (the weight of 20 people). The thinner

the fluid wedge gets the higher the load-carrying capacity so long the application iskept within the operating limit.

The ability of a thin lubricant film to hold high loads is largely due to the pressure

build-up in the liquid. The pressure values associated with hydrodynamic lubrication

varies from case to case. The typical pressure range for different applications is given

in Table 2-3 (adapted from Khonsari and Booser, 2001). From the selection of the

journal geometry and the operating conditions, the desired pressure magnitude

relevant to the intended applications can be achieved. In this work, testing is intended

for pressure values in the range typical of automotive engine applications.

Applications Typical Pressure (MPa)

Steady Loading :

Water-lubricated bearings

Electric motors

Turbines and axial compressors

Railroad car axial

0.1-0.3

0.7-1.7

1.5-2.0

1.5-2.5

Dynamic Loading:

Automotive engine bearings

Automotive connecting rodSteel mill roll necks

15-25

25-2525-35

Table 2-3: Typical pressure values in various journal bearing applications

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2.6 Eccentricity Ratio and Sommerfeld umber

The eccentricity ratio represents the physical (geometry) conditions of a journal and

is important in predicting the bearing load-carrying capacity and the performance

parameters. The eccentricity ratio, ε is defined as the ratio of the eccentricity, e

over the radial clearance, C .

C

e=ε (2.15)

Using numerical solutions developed by Raimondi and Boyd (Raimondi and Boyd,

1958; Khonsari and Booser, 2001), the eccentricity ratio can be determined if the

Sommerfeld number is known. A Sommerfeld number can be computed using

equation (2.16) given the operating conditions of the journal bearing operation.

2

=C

R

W

LDS ηω (2.16)

where η is the oil absolute viscosity, ω is the rotational speed, L is the bearing

length, D and R are the bearing diameter and radius, C is the bearing radial clearance,

and W is the applied load.

The use of a Sommerfeld number to characterize the operating conditions of a

journal bearing with a given L/D ratio is valid within the assumptions that the

lubricant is incompressible and temperature variation is negligible. This is likely

when the journal bearing operates under conditions of light load, low speed, and

large bearing-thermal capacity (Szeri, 1998).

Other expressions may be used for different cases as listed in Table 2-4. These

approximate methods are used for analytical solutions (Frene et. al, 1997). There will

be more discussions given on these in section 2.8. The expression for cavitation

boundary conditions in Table 2-4 is more representative of the actual conditions

where observations of cavitated region have been reported with finger-like shapes

(Dowson and Taylor, 1979).

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Approximate Method Mathematical Expressions

Infinitely Long Bearing

Approximation (ILA)

Full-Sommerfeld Condition:

ε π

ε ε 2

22/12

12

)2()1( +−=S

Half-Sommerfeld Condition:

( )( )( )

+−−+

=222

22

41

1

6

12

ε ε π πε

ε ε S

Cavitation Boundary Condition:

( )( )( ) ( )( )[ ] 2/12242

2

sincos14cos13

cos11

γ γ γ ε γ ε π

γ ε ε

−−+−

−−

=cavcavcav

cav

S

Infinite Short Bearing

Approximation (ISA)

Full-Sommerfeld Condition:

( ) 22322

1

2

1

−= L

DS

ε

ε

π

Half-Sommerfeld Condition:

( )( )[ ]

2

21222

22

161

1

+−

−=

L

DS

ε ε π πε

ε

Table 2-4: Sommerfeld number in different models

2.7 Measurement Techniques for Bearing Oil Film

There are different techniques developed for thickness measurement in bearings,

which have been reviewed by Sherrington and Smith (1986) and Spikes (1999). This

section describes some of the techniques as briefly introduced in Table 2-5 withreferences to specific papers.

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Technique/Paper Physical Principle Applications Minimum Thickness

(µµµµm)

Resistive

Chu and Cameron,

1967

Oil resistance due to applied

current or voltage is correlated

to film thickness.Bearing -

Capacitive

Luca and Wright, 1991

Charging and discharging

current is correlated to film

thickness.

Hydrodynamic contact 0.14

Inductive

Butcher, 1967-68

Change in oil impedance dueto interference of eddy current

field and the coil field is

correlated to film thickness

Bearings in a diesel

engine

2.5

Optical interferometry

Johnston et al. (1991)

A specific fringe produced by

interference of reflected

beams is correlated to film

thickness.

Rolling

contacts/bearings

0.002

Ultrasound

Dwyer-Joyce et al.

(2003)

Ultrasonic reflection from a

solid-fluid interface iscorrelated to film thickness.

Journal /thrust bearing 0.05

Table 2-5: A brief introduction to five techniques in bearing oil film measurements

2.7.1

Resistive, Capacitive, and Inductive Techniques

In early work, measurements were based on the electrical properties of the medium.

With all three cases, machine parts involved must be electrically isolated and thismay become a short fall in actual applications of journal bearings, which are often

embedded in the structure of a machine.

Resistive technique

The resistive technique, making use of the electrical property of lubricant, was

ventured in the 1950s. The electrical resistance of a lubricant film was found to

correlate to its thickness (Lane and Hughes, 1952; MacConochie and Cameron,

1960; Tallian et al., 1965; Dyson, 1966-7). The resistance measurement technique is

simple but there have been some variations. Chu and Cameron (1967) described two

main variations in the electrical systems: fixed-potential (Furey, 1961) and fixed

current electrical system (Siripongse et al., 1958). In the former version, a fixed potential is applied across the contact and the current produced is measured. Given

these parameters, the resistance of the oil film can be determined and correlated to its

thickness (Furey, 1961). In the later version, a fixed current is applied instead.

Along with the value of corresponding voltage, the oil resistance is calculated and

later correlated to its thickness.

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There are some weaknesses to this technique especially on the requirement of the

parts to be electrically isolated during the test. There is also a limitation in

quantitatively determining the thickness. In later studies, it has been combined with

other techniques. Zhang et al. (1991) reported the use of an R-C oscillation technique

to measure oil film thickness in EHL lubrication. In the technique, both resistor and

capacitor were used in one circuit, which produces sine signals. The signals changedas oil was introduced and the change was correlated to film thickness. It was

concluded that the technique is superior to that where resistance and capacitance are

used separately.

Capacitive Technique

Oil film thickness can be measured by a capacitive technique. Comparatively, the

outcome is better than that of the resistance technique. This technique came about in

1960s. The lubricant capacitance depends on the separation between the electrodes

on the bearing parts and the permittivity of the lubricant. If the permittivity of the

lubricant is known, the film thickness can be measured. Moore and Hamilton (1980)

found that the signals output from the capacitance bridge became larger as the filmthickness decreased. They also concluded that the results from the capacitive

transducer placed near the tip of the cam were not as good as the results from those

placed at the back and flank of the cam. The tip of the cam was associated with a

higher pressure and a smaller film thickness (Moore and Hamilton, 1980).

There are many studies using the capacitive technique to measure oil thickness

(Butcher, 1967-68; New, 1974; Hopf et al., 1989; Luca and Wright, 1991; Glavatskih

et al., 2001; McCarthy et. al., 2004). In some of the studies (Butcher, 1967-68; New,

1974; Hopf et al., 1989; Glavatskih et al., 2001; McCarthy et. al., 2004), the

temperature effect was not an issue and thus not discussed. In the case of Luca and

Wright (1991), they examined experimentally the frictional heating and the lubricant

film thickness in the hydrodynamic contact of a shoe loaded against a rotating

lubricated disk. Temperature may affect the capacitance value measured and hence

the accuracy of thickness measurement if the temperature during measurements is

different from the calibration temperature of the capacitive sensor and the

temperature effect is not being compensated. Temperature effect was discussed in

Stiyer and Ghandhi (1997). Capacitance depends partly on the dielectric properties

of oil, which in turn may change due to temperature variation. Temperature related

errors in thickness measurement can also be due to expansion and contraction of

measurement fixtures.

McCarthy et. al. (2004) used capacitive sensors to measure oil thickness in PTFE and

Babbitt faced tilting-pad thrust bearings. They found that the oil film thickness

measured were influenced by the pad facing materials. With the use of PTFE coating

on the pad surface, a thinner film was established at the leading edge and thicker film

at the trailing edge. This technique has also been used to calibrate other techniques

such as the laser-induced fluorescence measurement (Stiyer and Ghandhi, 1997). In

the measurement of oil film thickness using this technique, Donahue et al. (1999)

found that shielding of the sensing electrode has resulted in the reduction of fringing

effects and stray capacitance and hence, better output signals.

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As in the resistive technique, the bearing parts also need to be electrically isolated in

the experiment. Another limitation is that it cannot be used in a mixed lubrication

condition when there is a metallic contact. However, this technique has been

improved over the years and combined with other techniques (Zhang et al.,1991).

Inductive Technique

The inductive technique involves the use of an eddy current sensor with a balanced

bridge circuit. The sensor coil produces an electromagnetic field when excited by a

crystal oscillator. As a result, an eddy current is induced on the surface of any

conducting targets. In turn, the induced eddy current will generate its own

electromagnetic field. The eddy current field and the coil field will interfere and

produce change in impedance, which then correlated to film thickness.

Just like the two cases aforementioned, the inductive technique also correlates the

variation of lubricant electrical property, namely inductance, to film thickness.

Likewise, electrical isolation of parts is necessary during the test and temperatureeffect is still a problem.

Glavatskih et al. (2001) used an eddy current sensors along with a temperature sensor

to simultaneously monitor both the temperature and the oil film thickness. The

temperature compensation was to promote a more accurate measurement of oil

thickness. It was suggested that this technique could be used to monitor the steady

and transient conditions of oil film thickness and temperature within bearings.

In a recent study, this technique was used with other technology. Yin et al. (2003)

have used the inductive technique coupled with a fibre optic transducer to design an

on-line condition monitoring of machine equipment. The inductive transducer was

designed specifically to detect wear particles and distinguish them into ferrous or

non-ferrous catergories.

2.7.2 Optical Technique

The optical technique provides an alternative that does not require electrical

isolation. Some of the early studies on hydrodynamic and elastohydrodymic

lubrication were carried out using optical means (Kirk, 1962; Gohar and Cameron,

1963; Cameron and Gohar, 1966; Higginson and Reed; 1967-68). Some of the

apparatus in the early study by Higginson and Reed was basic, involving a white

light source (monochromatic), lenses, prisms, beam splitter, thin film silvered

surfaces, iris, spectroscope slit, and spectroscope eyepiece. The arrangement of the

apparatus made it possible for the observation of fringes from the reflected light on

the surfaces. The fringes were due to some discrete wavelengths being cancelled out

upon reflection from the surfaces. The missing discrete wavelengths appeared as

black lines in the spectrum. From the nature of the fringes, the lubricant film

thickness was deduced. But there is a limitation in the number of fringes that can be

analysed. For the early work, the range of applicable thickness was 0.25-100 µm(Higginson and Reed, 1967-68). In later years, improvements have been made in

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other studies. Johnston et al. (1991), in the work on rolling contacts by low viscosity

lubricants, were able to accurately measure oil film thickness down to 15 nm. Further

improvements were made in this optical technique with the use of a spacer layer and

a spectrometer/image analyser. With the spacer layer, the optical technique can be

used with thickness less than half a wavelength of light. The used of a spectrometer

or an image analyser is to replace human eyes so that the interference colour can be precisely distinguished for better results. An improvement in the measurable film

thickness has been made up to about 2 nm with ±1 nm since.

2.7.3 Ultrasonic Technique

This technique is long established in other fields such as medicine, structures, and

material analysis as well as contacting surfaces. However, as far as lubricant film

measurement in bearings is concerned, the technique is relatively new, starting in the

late 1990s. Some early publications described the principle of reflection from

contacting surfaces (Tattersall, 1973; Arakawa, 1983; Krolikowski and Szczepek,

1991; Pialucha and Cawley, 1994; Drinkwater et al., 1994; Drinkwater et al., 1996;Hodgson et al, 2000). Earlier work by Wittig et al. (1994) shows the development of

an ultrasonic technique to measure oil film thickness inside an aero-engine bearing

chamber.

In the tribology group at the Univeristy of Sheffield, ultrasonic reflections have been

used to investigate different aspects of solid-solid interfaces for example in machine

elements (Drinkwater et. al., 1997; Robinson et al., 1999; Hodgson et al., 2000;

Quinn et al., 2001; Hankinson et al., 2001). Relatively recent studies based on the

reflection coefficient measurement investigated the applications of ultrasound to

determine film thickness in bearings (Harper et al., 2003; Dwyer-Joyce et al., 2003;

Dwyer-Joyce et. al., 2004). The advantages of this technique over electrical methods

are in its non-intrusive nature and no requirement for electrical isolations of the bearing parts involved. Because of the advantages, this technqiue has been extended

to investigate matters of concerns in the present study. Background to this method is

presented in Chapter 3 while the technical details in Chapter 4.

2.8 Related Theories in Hydrodynamic Lubrication

2.8.1

Historical Advancement

Advancement in the analysis of hydrodynamic lubrication specifically in fluid film

bearings has developed over many years by many great people. In this section, theadvancement is summarised in Figure 2-5 (Cameron, 1966; Frene et al., 1997; Szeri,

1998; Khonsari and Booser, 2001). The initial work on fluid film bearings could be

traced back as early as 1849. Figure 2-5 also shows the link of a particular approach

to other technology. For example, numerical solutions became an option only with

the advancement achieved in computing technology.

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Figure 2-5: Chronological progress in fluid film bearings

1954

Years People

Fluid Film Bearings

1883

1849

1883 Beauchamp Tower

Nikilay Petroff

G.A. Hirn

F.A. Von Pauli

Contributions

Observed pressure generation

Considered friction and power lost

Early experimental work

Early experimental work

1886 Osborne Reynolds’ Developed differential equation for pressure build-up

1904 ArnoldSommerfeld

Developed direct integration for

‘infinite length’ analyses

1949Alastair Cameron

W.L. WoodExtended Reynolds’ equation

solved on calculators

1958Oscar Pinkus

Albert Raimondi& John Boyd

Introduced numerical solutions of

Reynolds’ equation on computers

1931Herbert Swift

W. StieberConsidered cavitation conditions

1979T. Suganami

Andras Z. SzeriConsidered simultaneous solution

of energy equation

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As far as the present work is concerned, two most related theories are (1) Reynolds’

equation and (2) cavitation boundary conditions. In numerical solutions, both

theories are coupled so that the actual conditions with cavitation phenomena are

better represented. Details of these theories are given in Section 2.8.2 and 2.8.3

respectively.

2.8.2 Standard Reynolds’ Equation

Consider equation (2.17), which is obtained from both Navier-Stokes Equations

(conservation of momentum) and Continuity Equation (conservation of mass).

Details of derivation are referred to Khonsari and Booser (2001).

( )434214 4 34 4 21

4 4 4 4 4 34 4 4 4 4 21squeezegeometric

shear toduerateflownet:termsCouette

gradients pressuretodue

rateflow:nettermsPoiseuille

33

2

1

1212 x

hU

x

hU U

x y

ph

y x

ph

xbba ∂∂

−∂∂

−∂∂

=

∂∂

∂∂

+

∂∂

∂∂

ρ ρ η

ρ

η

ρ

( ){

expansion localsqueezenormal

t

ρhww ab ∂∂

+−+43421

ρ (2.17)

Equation (2.17) has some limitations which are due to the assumptions made to

reduce complexity of analysis. This equation is valid in the following conditions

(Frene et. al, 1997; Khonsari and Booser, 2001);

• The medium is continuous.• The fluid is Newtonian.

•

Inertia and body forces are negligible compared to viscous forces.• Pressure variation across the bearing gap is negligible.• Flow is laminar.• Curvature effects are negligible.• No slip occurs between the fluid and the wall.

Terms in equation (2.17) are grouped into different categories according to common

factors. Each category is elaborated to get the physical meaning and hence add to

more understanding of the journal bearing mechanism.

The fluid velocity field in journal bearing is supported by both the pressure gradient

and the shearing force. The Poiseuille terms represent the effect of pressure gradient.The physical wedge formed in the journal bearing gap as it rotates will automatically

induce pressure difference, which then contributes to fluid motion.

The Couette terms considers the effect of the shear forces initiated by the journal

bearing rotation. So the shear effect is measured in terms of the velocity of the

moving parts. Khonsari and Booser (2001) have further discussed the possible

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components of Couette terms which include physical stretch, physical wedge, and its

density wedge.

The geometric squeeze and the normal squeeze account for squeeze effect due to

geometric configuration and the relative motion of the journal bearing surfaces. The

normal squeeze term is important when load-carrying capacity is considered. The lastterm, local expansion, measures the volume change of fluid mass due to bearing

heating.

In many applications, equation (2.17) can be simplified. A simplified form of

Reynolds’ equation is equation (2.18) obtained for impermeable walls and an

incompressible fluid. Equation (2.19) is a standard format applicable in most journal

bearing applications that satisfy these,

t

h

x

hU

y

ph

y x

ph

x ∂∂

+∂∂

=

∂∂

∂∂

+

∂∂

∂∂

ρ η

ρ

η

ρ

2

1

1212

33

(2.18)

However, equation (2.18) has no analytical solutions. Alternatively, twoapproximation methods were established to make analytical solutions possible: (1)

the infinitely long journal bearing approximation (ILA) and (2) the infinitely short

journal bearing approximation (ISA).

ILA is often used for large L/D values (> 2) and the ISA for small L/D values (

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study, observation of cavitation length is expected to be different from predicted

values as the oil is supplied at a different location.

With cavitation boundary conditions applied, the location of a cavitation initiation

(which was unknown before) can now be solved by the Sommerfeld substitution

(Pinkus and Sternlicht, 1961),

−

−= −

1cos

coscos

1

cav

cavcav

γ ε

γ ε θ (2.20)

where cavγ is the angle corresponds to cavθ . The dimensionless pressure, 0= P for

cavγ and the following relation is valid

( ) cavcavcavcavcavcav γ γ γ γ γ γ ε cos4sin4cossin22 −=− (2.21)

If the eccentricity ratio ε is known, then cavγ and cavθ can be determined by

equation (2.21) and equation (2.20) respectively. Solutions of Reynolds’ equation

with considerations of cavitation boundary conditions (also known as Reynods

boundary condition or Swift-Stieber boundary condition) by numerical methods have

been tabulated for different L/D ratios in Khonsari and Booser (2001). These values

are used to validate experimental results presented in later chapters.

2.9 Concluding Remarks

Lubrication is a subset of tribology, a study that considers two other important issues

in machine operation; friction and wear. Lubrication is defined by different regimes

that exist in real applications. Lubrication is characterised by its properties such as pressure, temperature, viscosity, phase condition, and thickness. This thesis focuses

on the hydrodynamic lubrication in a journal bearing. Three aspects related to journal

bearings were presented: (1) journal bearing basic design, (1) mechanism of journal

bearing, and (3) eccentricity ratio and Sommerfeld number. Methods in thin film

measurement were also discussed including resistive, capacitive, inductive, optical

and ultrasonic techniques. The ultrasound technique in particular has been extended

and used as the research method in this study. Related theories in hydrodynamic

lubrication were also given as they become the basis in analytical and numerical

solutions used for data comparison.

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