M A Sc_ thesis_ Uzzal

7/27/2019 M A Sc_ thesis_ Uzzal

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ABSTRACT

ROLE OF RAILWAY VEHICLE-TRACK SYSTEM AND DESIGN

PARAMETERS ON FLAT-INDUCED IMPACT LOAD

Md. Rajib Ul Alam

Wheel flats are known to cause high magnitude impact loads at the wheel-rail

interface, which can induce fatigue damage and failure of various vehicle and track

components. With demands for increased load and speed, the issue of wheel flats and a

strategy for in-time maintenance and replacement of defective wheels has become an

important concern for heavy haul operators. In this study, an analytical model of the

coupled vehicle-track system is developed by integrating a pitch plane model of the

vehicle with a two-dimensional model of the flexible track comprising 3-layers together

with a nonlinear rolling contact model. The track system model is periodically supported

by sleepers and ballasts characterized by their lumped parameters. The commonly used

Hertzian nonlinear contact model is utilized in analysis of the vertical vehicle-track

interactions. Generalized coordinate method is employed to solve for the coupled partial

differential equations of the track and ordinary differential equations of motion for the

lumped-parameter vehicle model. The validity of the coupled vehicle-track system is

demonstrated by comparing the simulation results with the reported measured data and

analytical solutions. The validated model is utilized to investigate the characteristics of

impact forces due to wheel flats and its effect on motions and forces transmitted to

vehicle and track components. The results are analyzed to examine the sequence of

events as the wheel flat enters the contact area. The magnitudes and predominant

frequencies of wheel-rail contact forces are examined in terms of system and operating


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parameters. A comprehensive parametric study is performed to study the effects of

selected vehicle-track design and operating parameters on the wheel-rail impact loads,

and forces transmitted to the bearing, railpad and ballast in the presence of single and

multiple wheel flats within either same or adjacent wheels. The study shows that the

magnitudes of cross wheel impact forces are larger when the phase angle between the two

flats is small. The impact force may be higher or lower than that caused by a single flat

depending on the relative positions of the flats. The study further revealed that factors

such as primary suspension stiffness, railpad and ballast stiffness, rail mass, sleeper

spacing, bending stiffness of rail and ballast mass have significant influence on the

impact load, whereas secondary suspension properties and ballast damping show

negligible effects.


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v

TABLE OF CONTENTS

LIST OF FIGURES ix

LIST OF TABLES xiv

NOMENCLATURE xv

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

1.1 INTRODUCTION 1

1.2 LITERATURE REVIEW 3

1.2.1 Wheel defects 5

1.2.2 Vehicle models 15

1.2.3 Modeling the track Components 23

1.2.4 Track system models 30

1.2.5 Wheel-rail contact models 35

1.2.6 Simulation methods 37

1.3 THESIS SCOPE AND OBJECTIVES 39

1.4 ORGANIZATION OF THE THESIS 40

CHAPTER 2

VEHICLE-TRACK SYSTEM MODEL AND METHOD OF ANALYSIS

2.1 INTRODUCTION 42

2.2 VEHICLE SYSTEM MODEL 43

2.2.1 Pitch-plane vehicle model 45

2.2.2 Equations of motion 46


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2.3 TRACK STRUCTURE 47

2.3.1 Track system model 49

2.3.2 Equation of motion 51

2.4 WHEEL-RAIL INTERFACE 53

2.5 METHOD OF ANALYSIS 58

2.6 SUMMARY 63

CHAPTER 3

MODEL VALIDATION AND VEHICLE-TRACK SYSTEM RESPONSE

3.1 INTRODUCTION 65

3.2 MODEL VALIDATION 66

3.3 RESPONSE ANALYSES OF THE VEHICLE-TRACK SYSTEMMODEL

72

3.3.1 Wheel-Rail Contact Force Response 74

3.3.2 Force Responses of the Vehicle and Track Components 79

3.3.3 Displacement Responses of the Vehicle-Track Components 84

3.4 SUMMARY 93

CHAPTER 4

PARAMETRIC STUDY

4.1 INTRODUCTION 95

4.2 SELECTION OF IMPORTANT MODEL PARAMETERS 96

4.3 INFLUENCE OF WHEEL FLAT 98

4.3.1 Effect of single wheel flat on wheel-rail impact force 98


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4.3.2 Effect of multiple wheel flats on W/R impact loads 103

4.3.3 Effect of flat length and depth 115

4.4 PARAMETRIC STUDY ON VEHICLE PARAMETERS 118

4.4.1 Effect of speed 118

4.4.2 Effect of wheel load 119

4.4.3 Effect of unsprung mass 120

4.4.4 Effect of suspension stiffness and damping on peak W/Rimpact load

121

4.4.5 Effect of suspension stiffness and damping on peak bearing

force

122

4.5 PARAMETRIC STUDY ON TRACK MODEL PARAMETERS 124

4.5.1 Effect of Rail Mass per unit Length 124

4.5.2 Effect of railpad stiffness and damping 125

4.5.3 Effect of ballast stiffness and damping 126

4.5.4 Effect of Sleeper mass 133

4.5.5 Effect of Ballast mass 133

4.5.6 Effect of sleeper spacing 135

4.5.7 Effect of bending stiffness of rail 136

4.6 SUMMARY 138

CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

5.1 GENERAL 140

5.2 HIGHLIGHTS OF THE PRESENT WORK 141

5.3 CONCLUSIONS 142


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5.4 RECOMMENDATIONS FOR FUTURE WORK 144

REFERENCES 146


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LIST OF FIGURES

Figure 1.1 The structure of the vehicle-track interaction models

Figure 1.2 An ideal chord type flat

Figure 1.3 A haversine type flat

Figure 1.4 A non-periodic OOR of wheel

Figure 1.5 Basic compositions of railway vehicle-track system

Figure 1.6 A single DOF one-dimensional model of the vehicle

Figure 1.7 A three-DOF one-dimensional vehicle model

Figure 1.8 A five-DOF pitch-plane vehicle model

Figure 1.9 A typical roll-plane vehicle model with several DOF

Figure 1.10 Two-dimensional 10-DOF pitch-plane vehicle model

Figure 1.11 A three-dimensional 10-DOF vehicle model

Figure 1.12 A double-beam rail model

Figure 1.13 Three-parameter pad models

Figure 1.14 Sleeper modeled as rigid mass resting on elastic ballast

Figure 1.15 A detailed model of ballast considering the stiffness and damping inshear

Figure 1.16 A lumped-parameter 3-layer track model

Figure 1.17 A single layer track model with continuous support

Figure 1.18 A double layer track model with continuous support

Figure 1.19 Rail beam on discrete supports (only sleeper mass is included)

Figure 1.20 A three-layer model of track system

Figure 2.1 A three-piece freight car truck


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Figure 2.2 A 5-DOF pitch-plane vehicle model

Figure 2.3 Various layers of the track structure

Figure 2.4 A three-layer railway track system model

Figure 2.5 Wheel/Rail contact model used in the present study

Figure 2.6 A wheel with an idealized flat

Figure 2.7 A wheel with haversine type flat

Figure 3.1 Comparison of wheel-rail impact force response of the present modelwith that reported by Zhai et al. [3]

Figure 3.2 Time-history of impact force response predicted by the current model: (a)

single impact; and (b) three-consecutive impacts (v = 27 km/h; fL = 52.8mm;

fD = 1 mm)

Figure 3.3 Variations in the measured dynamic contact force due to a wheel flat,reported by Zhai et al. [3]

Figure 3.4 Time history of rear wheel-rail impact force due to a flat on the rearwheel: (a) single cycle; and (b) three cycles.

Figure 3.5 Time history of the flat free front wheel-rail impact force in the presenceof a rearwheel flat at v = 70km/h: (a) single cycle; and (b) three cycles

Figure 3.6 Time histories of wheel-rail impact forces developed at front and rear

wheels due to a flat in the front wheel (v = 70 km/h; fL = 52 mm; fD =

0.4 mm): (a) front wheel-rail impact force; (b) rear wheel-rail impactforce.

Figure 3.7 Time history of front and rear wheel-rail contact force

Figure 3.8 Variations in the bearing force response due to a rear-wheel flat as afunction of static wheel load.

Figure 3.9 Variations in railpad force due to a rear wheel flat as a function of staticwheel load

Figure 3.10 Variations in ballast force due to a rear wheel flat as a function of staticwheel load


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Figure 3.11 Time histories of vertical displacements of the rear wheel and rail in thepresence of a flat

Figure 3.12 Time histories of vertical displacements of front wheel (no flat) and railat the wheel-rail contact point

Figure 3.13 Comparison of wheel and rail displacement responses of the presentmodel with those reported by Sun et al. [73]: (a) present model; and (b)

reported results (v = 70 km/h; fL = 40 mm; fD = 0.35 mm)

Figure 3.14 Displacement responses evaluated at (a) a point on the rail; (b) sleeper;and (c) ballast beneath the rail point in the presence of a rear wheel flat

Figure 3.15 Time history of car body vertical motion in the presence of a rear wheel

flat ( fL = 52 mm; fD = 0.4 mm)

Figure 3.16 Variation in the bounce and pitch responses of the bogie in the presenceof a rear wheel flat: (a) bounce motion; (b) pitch motion

Figure 4.1 Variations in radius of a wheel with single flat ( fL = 52 mm

andf

D = 0.4 mm) as a function of angular position of the contact

Figure 4.2 Influence of size of a front wheel flat on the impact force responses atthe wheel-rail interface: (a) front wheel; and (b) rear wheel

Figure 4.3 Effect of speed on peak front and rear wheel impact loads due to a single

flat on the front wheel ( fL = 52 mm and fD = 0.4 mm)

Figure 4.4 Effect of speed on the peak displacement of the rail at the front wheel-rail

contact point ( fL = 52 mm and fD = 0.4 mm)

Figure 4.5 Variations in radius of a wheel with two same size flats ( fL = 52 mm

and fD = 0.4 mm), which are 900 apart

Figure 4.6 Time response of rear wheel-rail impact force with two flats at 45 phaseangle

Figure 4.7 Time response of rear wheel-rail impact force with two flats at 90 phaseangle

Figure 4.8 Time response of rear wheel-rail impact force with two flats at 135phase angle


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Figure 4.9 Time response of rear wheel-rail impact force with two flats at 180phase angle

Figure 4.10 Time history of front and rear wheel impact force responsesDue to a single flat on both wheels in phase

Figure 4.11 Front and rear wheel impact force responses due to a single flat on bothwheels at 450 out-of-phase (rear wheel flat ahead 450)



Figure 4.14 Front and rear wheel impact force responses due to a single flat on both

wheels at 180

0

out-of-phase (rear wheel flat ahead 180

0

)

Figure 4.15 Effect of speed on wheel-rail impact load with single and multiple wheelsflats

Figure 4.16 Effect of flat length on W/R impact force for a constant flat depth of0.4mm

Figure 4.17 Effect of flat length on wheel-rail impact force

Figure 4.18 Effect of flat depth on W/R impact force with a constant flat length of 52mm

Figure 4.19 Effect of speed on wheel-rail impact force with three loads

Figure 4.20 Effect of static wheel load on peak wheel-rail impact force

Figure 4.21 Effect of unsprung mass on W/R impact load

Figure 4.22 Effect of primary suspension stiffness on peak wheel-rail impact force

Figure 4.23 Effect of primary suspension stiffness on peak bearing force

Figure 4.24 Effect of rail mass on wheel-rail impact force

Figure 4.25 Effect of railpad stiffness and damping on wheel-rail impact load

Figure 4.26 Effect of railpad stiffness and damping on peak pad force


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Figure 4.27 Effect of rail pad stiffness on peak rail displacement

Figure 4.28 Effect of ballast stiffness and damping on peak wheel-rail impact force

Figure 4.29 Effect of ballast stiffness on peak rail displacement

Figure 4.30 Effect of ballast stiffness on peak ballast displacement

Figure 4.31 Effect of ballast stiffness and damping on peak ballast force

Figure 4.32 Effect of sleeper mass on peak wheel-rail impact force

Figure 4.33 Effect of sleeper mass on peak pad force

Figure 4.34 Effect of ballast mass on peak wheel-rail impact force

Figure 4.35 Effect of sleeper spacing on peak wheel-rail impact force

Figure 4.36 Effect of sleeper spacing on peak rail displacement

Figure 4.37 Effect of bending stiffness on peak wheel-rail impact load

Figure 4.38 Effect of bending stiffness on peak rail displacement


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LIST OF TABLES

Table 3.1 Parameters used for examining validity of the model with single wheel flat

Table 3.2 Nominal simulation parameters

Table 4.1 Nominal simulation parameters used for parametric study


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NOMENCLATURE

SYMBOL DESCRIPTION

cM Car body mass (kg)

W Static wheel load (N)

tM Bogie mass (kg)

wM Wheel mass (kg)

tJ Bogie mass moment inertia (kg-m )

1sK Primary suspension stiffness (N/m)

1sC Primary suspension damping (N-s/m)

2sK Secondary suspension stiffness (N/m)

2sC

fl

rl

Secondary suspension damping (N-s/m)

Distance between the front wheel and mass center of bogie (m)

Distance between the rear wheel and mass center of bogie (m)

tl Wheelset distance (m)

R Wheel radius (m)

fL Flat length (mm)

fD Flat depth (mm)

HC Non-linear Hertzian spring constant (N/m )

rm Rail mass per unit length (kg/m)

E Elastic modulus of rail (N/ m )

I Rail second moment of area (m )

EI Rail bending stiffness (N-m )

sM Sleeper mass (kg)

bM Ballast mass (kg)

pK Railpad stiffness (N/m)

bK Ballast stiffness (N/m)


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wK Ballast shear stiffness (N/m)

fK Subgrade stiffness (N/m)

pC Railpad damping (N-s/m)

wC Ballast shear damping (N-s/m)

fC Subgrade damping (N-s/m)

l Length of the rail (m)

sl Sleeper distance (m)

( )cw t Car body displacement in vertical direction (m)

( )tw t Vehicle bogie displacement in vertical direction (m)

( )t t Pitch rotation of the car bogie (rad)

1( )ww t Front wheel vertical displacement (m)

2 ( )ww t Rear wheel vertical displacement (m)

( )cw t Car body velocity in vertical direction (m/s)

( )tw t Vehicle bogie velocity in vertical direction (m/s)

( )t t Pitch velocity of the car bogie (rad/s)

1

( )w

w t Velocity of front wheel in vertical direction (m/s)

2 ( )ww t Velocity of rear wheel in vertical direction (m/s)

( )cw t Car body acceleration in vertical direction (m/s )

( )tw t Vehicle bogie acceleration in vertical direction (m/s )

( )t t Pitch acceleration of the car bogie (rad/s )

1( )ww t Acceleration of front wheel in vertical direction (m/s )

2 ( )ww t Acceleration of rear wheel in vertical direction (m/s )

'jP t Wheel/rail contact force (N) (j =1-2)

1( )P t Front wheel-rail contact force (N)

2( )P t Rear wheel-rail contact force (N)

( )r t Wheel flat profile function


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( , )rw x t Vertical displacement of the rail (m)

( )siw t Vertical displacement of the sleeper (m)( i = 1, 2, 3,N )

( )biw t Vertical displacement of the ballast (m) ( i = 1, 2, 3,N )

( , )rw x t Vertical velocity of the rail (m/s)

( )siw t Vertical velocity of the sleeper (m/s) ( i = 1, 2, 3,N )

( )biw t Vertical velocity of the ballast (m/s) ( i = 1, 2, 3,N )

( , )rw x t Vertical acceleration of the rail (m/s )

( )siw t Vertical acceleration of the sleeper (m/s2) ( i = 1, 2, 3,N )

( )biw t Vertical acceleration of the ballast (m/s2) ( i = 1, 2, 3,N )

N Number of the sleepers/ballasts

( )rsiF t Rail/sleeper contact force ( i = 1, 2, 3,N )

( )sbiF t Sleeper/ballast contact force ( i = 1, 2, 3,N )

ix Position of the sleeper( i = 1, 2, 3,N )

j Number of wheels considered in the vehicle model (j=1-2)

Gjx Position of the wheel (j=1-2)

( )Y x Mode shape function

K Total number of modes of the rail

k Number of rail mode corresponding to the sleeper position ( k=1, K)

kY kth rail mode shape

z Wheel-rail overlap in vertical direction

( )kq t kth mode displacement of rail

( )kq t k th mode velocity of rail

( )kq t kth mode acceleration of rail

wrG Shear modulus of rail

wr Poissons ratio

w Wheel profile radius (m)


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tR Rail profile radius (m)

Natural frequency of the rail beam


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CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

1.1INTRODUCTIONRailway is the economical, environment friendly, safe and efficient transportation

mode, and it continues to play a very important role in the world commerce and

development. For about 200 years, railroad has been the most popular mode for

transportations of both the passengers and the freight. While much has been achieved, the

European Commission and American Association of Railway have expressed many

concerns that the rail sector has not shown the same progress towards a quieter operation

that the rival modes have achieved. Several research groups, and commissions have

contributed towards better solutions in order to improve safety and reliability, especially

in the freight sector, where such improvements have been least pronounced. A survey of

global market and investment trends has shown that annual investment in rail vehicles

has grown from about $US 18 billion in 1993 to $US 25 billion in 2000 and about $US

31 billion by 2004 [41]. North America accounts for about 20% of the world rail vehicle

market predicts strong growth as cities continue to expand their rail transit systems [41].

In recent years, the operational safety has drawn considerable attention in order to

reduce operating cost as well as unscheduled service interruption that may arise from the

wheel and rail defects. These defects are mainly wheel flats, shelling, spalling,

corrugation of rails and wheels attributed to material fatigue, blocked brakes,

manufacturing flaws, etc. Such defects trigger severe repeated high frequency impact

forces in wheel-rail interface, which may cause failure of various components and lead to

derailments in extreme cases. Formation mechanism of these defects involves complex


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interactions between the wheel profile and the rail surface, thermo elastic instability and

development of unusual forces in relation to the wheel/rail adhesion [82,94]. A sudden

change in wheel-rail contact force due to these defects can lead to an accelerated

deterioration of the vehicle and track structure components, such as wheelsets, bearings,

rails, and sleepers. It is estimated that the railway industry in North America is currently

spending nearly $90 millions annually to replace 125,000 wheels due to wheel defects

[37]. Only for one type of wheel defect, namely spalling, North American railroads spend

$15 million annually to replace spalled wheels [64]. The associated unscheduled service

interruption costs are far more significant.

Among all the various wheel defects, a wheel flat is known to be quite common

[3, 38, 43]. This type of defect is generally caused by unintentional sliding of the wheel

on the rail, and occurs mainly when the braking force is too high in relation to the

available wheel/rail friction. The response of the railway vehicle system with a wheel flat

is strongly dependent on the vertical dynamics of wheel-rail interactions. The nonlinear

Hertzian point contact theory is most commonly used in the analysis of impact forces

attributed to the wheel-rail contact. The complexity of the analysis and severity of the

contact forces, however, increases manyfolds when wheel flats are present. A detailed

study on dynamic wheel-rail interactions is thus required in order to characterize the

contact impact forces between the wheel and the rail, which will permit timely detection

and removal of a defective wheel. The demands for high axle loads and operating speeds

in order to enhance the operation efficiency could further amplify the magnitudes of

wheel flat induced impact forces.


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In this study, an analytical model of the coupled vehicle-track system is

developed by integrating the pitch plane model of the vehicle-track system comprising a

two-dimensional model of the 3-layer flexible track system and a 5-DOF of the vehicle

coupled through a nonlinear rolling contact. Generalized coordinate method is employed

to analyze the vehicle-track interactions. The validity of the coupled vehicle/track system

is demonstrated by comparing the simulation results with the reported experimental data

and analytical solutions. The validated model is utilized to investigate the characteristics

of impact forces in the presence of single or multiple wheel flats. Finally, a detailed

parametric study on the effect of different vehicle-track design and operational

parameters on the wheel-rail impact loads due to wheel flats is presented.

1.2LITERATURE REVIEWA study of dynamic wheel-rail interactions in the presence of wheel defects

involves thorough understanding of various contributing factors. These include the

dynamic motion of the vehicle components, deflection of the multiple-layered continuous

track structure, nature of wheel defects, dynamic interactions of the moving wheels with

the flexible track, etc. Relevant reported studies are thus reviewed and briefly

summarized in this chapter in an attempt to build essential background and scope of this

dissertation research. The coupled vehicle-track system dynamics in the presence of

various types of wheel defects have been extensively investigated in the last few decades

to identify the sources of wheel-rail impact loads, and to define the threshold values of

impact loads for timely removal of defective wheels. These defects come into existence

either from the wheel or the rail, or from both. Owing to the complexities associated with

measurements of impact loads caused by a moving load, only a few studies have


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performed field measurements of impact loads caused by the wheel and rail defects under

limited operating conditions [5, 31, 35, 47, 90]. A vast number of studies have relied on

development of effective simulation models for predicting the dynamic impact loads [3,

17, 18, 32, 45, 53, 92, 95, 110]. An array of vehicle and track system models have been

developed in the past few decades for analyses of wheel-rail vertical interactions for

studies on the ride quality, safety, wear, curving, etc. Knothe and Grassie [42] have

presented a comprehensive review of advancements in the field of railway vehicle

dynamics including the development in dynamic models of study the vehicle-track

interactions due to wheel defects in high frequency range. A state-of-the art review of

rock and roll dynamics of railway vehicles has been reported by Sankar and Samaha [68].

Dahlberg [4, 55] and Taheri et al. [69] have reported review of studies on dynamic

interactions between the rail and track, and vehicle-guideway interactions, respectively.

The studies on dynamic interactions of vehicle and tracks generally involve a vehicle

model, wheel-rail contact model, track model, and irregularities of the wheel or the track.

These component models are integrated, as shown in Fig. 1.1, and are briefly discussed in

the following subsections.

Track Model

Vehicle Model

Wheel Defect Model W/R Contact Model

Wheel Displacement InputContact Force Output

Rail Displacement Input Contact Force Output

Output Results

Fig. 1.1: The structure of the vehicle-track interaction models


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In order to facilitate the analysis, it is necessary to study in details the modeling of

vehicle and track system combined with their individual components and wheel defects.

It is also required to survey the relevant reported studies to be acquainted with the nature

of effects of these factors on the wheel-rail impact load. The relevant literatures are thus

reviewed and discussed in the following sub-sections to build up the essential background

and to formulate the scope of this dissertation research. A detail review and discussion of

each of the components of the above Fig. 1.1 is also presented in the following sub

sections.

1.2.1 Wheel defects

The term wheel defects is used to describe various types of wheel imperfections

that may develop either during operation or in manufacturing, and wheel reprofiling

stages. There are different types of defects present in the railway wheels, which range

from short wavelength to large wavelength defects. These include flats, shelling, spalling,

corrugation, eccentricity, etc. Nielsen et al. [82] described the formation of these defects

and their effects on wheel-rail impact loads through a comprehensive review on

published studies. It was concluded that timely detection and replacement of a defective

wheel offers large economic returns by reducing the maintenance and repair cost. Barke

and Chiu [38] also presented a comprehensive review on published studies on the effects

of wheel defects and track design parameters on the dynamic forces transmitted to the

track and vehicle components. The reported studies have facilitated the developments in

wheel removal criteria and detection of defective wheels in order to prevent the failure or


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addition to safety and economic considerations, these defects reduce passenger comfort

and significantly increase the intensity of noise [43].

Various railroad organizations have set for the criteria for removal of wheels with

flats. The proposed threshold values of the wheel flats are primarily based on flat size and

the impact load produced by the flat. Different organizations, however, have defined

different threshold values for the wheel defects. The AAR [91] criteria for removal of

wheel from service imposed that a railway wheel with 50.8 mm long single flat or

38.1mm long two adjoining flats cannot be placed or continue to be in service. The AAR

also states the threshold value of wheel-rail impact loads due to a single flat. According

to AAR rule [101], a wheel should be replaced if the peak impact forces due to single flat

approaches in the 222.41 to 266.89 kN range.

According to Swedish Railway, the condemning limit for a wheel flat is based on

a flat length of 40 mm and flat depth of 0.35 mm [10]. Transport Canada safety

regulations [108] require that a railway company may not continue a car in service if

wheel has a slid flat spot that is more than 63.50 mm in length or two adjoining flat spots

each of which is more than 50.80 mm. According to UK Rail safety and standard board

[109], freight vehicle with axle load equal to or over 17.5 tones a wheel with flat length

exceeding 70 mm must be taken out of service. These removal criteria for defective

wheels are mostly based on the magnitude of the impact force induced by the flat, while

considerable discrepancies among the different criteria exist.

The assessment of potential damages caused by a wheel flats and development of

reliable criterion necessitate development of effective impact load prediction tools. A

wide range of mathematical descriptions have thus evolved to characterize the geometry


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of wheel flats in order to investigate the impact loads [3, 5, 7, 12, 14,30, 37]. On the

basis of the reported flat geometries, the wheel flats have been classified as chord type

flat, cosine type flat and combined flat. A chord type flat model considers the flat as a

newly formed fresh flat with relatively sharp edges, where the interacting force between

the wheel and the rail is estimated based on the assumption that one of two edges of the

flat is always in contact with the rail.

The chord flat models have been used widely in various studies on wheel-rail

impact load, rail acceleration, and noise [7, 14, 30]. A chord type wheel flat model is

shown in Fig. 1.2. The geometry of this type of flat is described by its length ( fL ) and

depth ( fD )

Mathematically, the wheel profile with chord type flat can be expressed as [14]

(1 )

0

R cosr

0

(1.1)

fD

o

fL

R

x

( )tr A

Fig. 1.2: An ideal chord type flat


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Where, is the angle subtended by the flat at the wheel centre, R is the radius of the

wheel, ris the variations in the wheel radius due to the flat, is the angle of an arbitrary

point within the flat zone or the cord, and expressed as:

1

1

sin ( / )

sin [( ) / ]f

x R

L x R

0 / 2

/ 2

f

f f

x L

L x L

(1.2)

Where, fL is the length of the flat, x is the longitudinal coordinate of an arbitrary point

within the flat curve.

The edges of the chord type flat tend to become more rounded as the wheel

continues to be service due to wear and/or deformation under repeated impact loads.

Subsequently, the chord type flat can be modeled as a cosine flat, which is also known as

a haversine flat or rounded flat. Haversine flat model is widely used for analysis of

dynamic behavior of rail vehicles and tracks together with the wheel-rail impact load due

to flat as used in [3, 5, 12, 37]. A haversine flat model with its mathematical expression is

shown in Fig. 1.3. A haversine flat is expressed as [37]:

1( ) [1 cos(2 / )]

2f fr t D x L (1.3)

Where fD is the flat depth that may be calculated as:

2 /(16 )f fD L R (1.4)


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Unlike the chord type flat, a haversine flat yield more uniform and continuous

contact between the wheel and the rail. The longitudinal position of wheel-rail contact

center is thus generally considered as the projection of the wheel center [7, 13, 37]. The

impact loads predicted by the haversine flat have generally shown reasonably good

agreements with the measured data [37].

It has been suggested that a wheel flat may not be truly represented by a chord or

the haversine function. A combined wheel flat model was introduced by Ishida and Ban

[30], where the characteristics of both chord type and cosine type flats are combined

together to analyze the dynamic behavior of the wheel and the rail. The model results,

however, did not show substantial advantage in enhancing the dynamic wheel-rail impact

load prediction ability when compared to the haversine flat. A comparative study on rail

acceleration response due to three types of flats was also presented in this study. From the

results, it was concluded that a chord flat model is more sensitive and cosine type flat

fD

o R

x

fL

( )tr

Fig. 1.3: A haversine type flat


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model is less sensitive to measured rail acceleration, while the combined flat model yield

better agreement with the measured data.

The operating speeds of railway vehicles have continued to increase to enhance

the operational efficiency. The potential effects of high-speed operation on the wheel-rail

interactions caused by the wheel flats have thus been emphasized in many studies [5, 14,

23, 35,37]. The majority of these studies have investigated the wheel-rail impact loads

due to single wheel flat using a non-linear Hertzian contact spring [3, 5, 23, 35]. Dong

[37] and Hou et al. [14] used finite element method to study the increase in impact force

due to a wheel flat with increasing speed. Thompson and Wu [8] also performed non-

linear analysis of wheel-rail impact loads under different operating speeds. These studies

have invariably concluded that the magnitudes of the contact force increase with

increasing speed. These studies have also identified the ranges of speed, where the

variations in wheel-rail impact load remain relatively insensitive to operating speed [3,

37]. Furthermore, the magnitudes of the wheelrail impact forces have a small peak in the

low speed range (3040 km/h), and after that speed the magnitudes increase with increase

in speed. A few studies have shown that the maximum contact force can decrease at

higher speeds, which are beyond the practical speed limit.

Sun et al. [10, 73] compared contact force due to wheel flats derived from the

model and measured data reported by Fermer and Nielsen [31]. The study concluded that

the magnitude and frequency of the impact forces derived from the model agree

reasonably with the measured data. The magnitude of wheel-rail impact forces derived

from both models and measured data were approximately 50 percent greater than the

static wheel load.


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Experimental and theoretical studies on impact loads due to wheel flats have

been described by Johansson and Nielsen [5, 31], and Newton and Clark [35]. These

studies have evolved into a number of significant findings, namely: (i) wheel defects

often lead to large impact loads but they are not always easily detected by visual

inspection of the wheel; (ii) for a constant velocity, the impact load increases with

increase in the flat length; and (iii) a nonlinear track model yields more accurate

prediction of the wheel-rail impact force due to wheel defects than the linear track model.

A freshly formed flat is known to grow as the wheel continues in service. Only minimal

efforts, however, have been made to study the rate of growth of a wheel flat and the

major contributing factors. Jergeus et al. [90] performed experiments to study the flat

growth and concluded that the rate of growth is very high at the beginning. It is thus

essential to take the wheelset out of service as quickly as possible when a wheel flat is

observed.

In general, wheels carrying any type of defects are referred to as out-of-round

(OOR) wheels. Nevertheless, many of the individual studies concerned with wheel-rail

impact load have considered a polygon shape of the wheel only as out-of-roundness, as

shown in Fig. 1.4 [5, 21, 89, 94]. Non-uniformity of the wheel profile is very common in

practice. When a perfectly round wheel is in use, the irregularities may develop in the

wheel trade to cause out-of-roundness (OOR) condition. The OOR may be characterized

by both periodic and non-periodic defect. The clamping of wheel during reprofiling may

cause periodic OOR, while a non-periodic OOR may be caused by unbalance in the

wheelset or by inhomogeneous material properties of the wheel [82]. Both of these types

OOR are usually found in disc-braked wheelsets. The vibration caused by OOR wheels


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are transmitted from the wheels via the bogies to the passenger compartments, which

often perceived as annoying coupled with humming noise. In addition to the deterioration

of comfort, it has been reported that maintenance costs associated with OOR wheels tend

to be considerable [21].

A comprehensive review of studies involving classifications of OOR, their

formations, wheel-rail impact loads attributed to OOR has been presented by Nielsen and

Johansson [82]. The study emphasized the need for development of improved wheel

removal criterion that will not only be based upon the geometry of the OOR but also on

the computed and/or measured maximum wheel/rail impact loads. The study further

concluded that a complete model of the vehicle-track system is required to study the

long-term wear behavior due to OOR, and more investigations on the formation and

control of OOR are needed.

Johansson and Nielsen [5], and Johansson and Andersson [94] have reported that

periodic wheel OOR leads to increase in ride vibration levels, especially at certain

speeds. The magnitude of vertical wheelrail contact force increases due to the wheel

Fig. 1.4: A non-periodic OOR of wheel [94]


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OOR, which contributes to reduce vehicle-track system fatigue life. Barke and Chiu [38]

have recently reported a review of studies on the effects of OOR on the wheel-rail impact

loads. The study concluded that a deeper understanding of the effect of impact loads on

fatigue lives of vehicle-track components is required using the comprehensive vehicle-

track models.

Apart from the wheel flat defects, a number of studies have investigated the

impact loads caused by various other types of surface defects and their propagation.

These include wheel spalling, wheel shelling, and wheel and rail corrugation that are

characterized by wavelength. Railway wheel spalling is assumed to occur as the result of

fine thermal cracks joining to produce the loss of a small piece of tread material. The

thermal cracks are developed owing to heating and rapid cooling of the wheel tread

during and after block braking. Stone [65] have presented an interpretive review on wheel

spalling and shelling. Some highlights of this review can be summarized as: (i) spalling

and shelling are dominated by rolling contact resistance; (ii) some factors like loss of

material strength, quasi-static thermal stresses must be considered in model to analyze

wheel spalling.

Shelling of wheel is another type of wheel defect that is assumed as the result of

rolling contact fatigue. It is manifested by loss of flakes of material from the wheel tread.

Eric et al. [96] have shown diagrammatically the relationship between shelling formation

and the related various operating and environmental parameters. A study on formation of

railway wheel shelling due to thermal effect is conducted by Moyar and Stone [66]. The

study concluded that periodic rail chill in the case of hot-braked treads has a strong effect


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on shelling and it can be minimized by uniformly distributed braking force, and

maintaining tread temperatures as low as possible.

Corrugation is characterized by an almost regular sequence of shiny peaks and

dark troughs generally spaced about 30-50 mm apart. A recent review on corrugation

characteristics treatments is presented by Sato et al. [67]. This study concluded that in

order to understand the formation of corrugation, a systematic study of the corrugation

phenomena including experimental as well as theoretical investigations is essential.

The effect of rail corrugation on wheel-rail dynamic impact load is investigated

by Jin et al. [17, 44, 85], Nielsen and Igeland [23], and Sun and Simson [74].

Experimental study of corrugation formation on the rail top surface is carried out by Suda

et al. [93] in order to investigate the entire process of corrugation and the phase relation

between corrugation profile, contact load, and slip. Several analytical models have been

developed to study the corrugation formation process as reported in [32, 45, 92, 95]. All

these studies have shown that once the corrugation has formed, it will lead to an

accelerated deterioration of the track structure and the vehicle and to the generation of

high frequency noise.

1.2.2 Vehicle models

The main components of a rail vehicle system are car body, bogie/side frame,

wheel, primary suspension, and secondary suspension etc. The car body rests on two

bogies each containing two wheelsets. The spring and damping elements connecting the

wheelset bearings and the bogie frame are referred to as the primary suspension. The

secondary suspension connects the bogie frame to the car body, as illustrated in Fig. 1.5.


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The type of vehicle model employed in a dynamic study mainly depends on the

objective of the formulation. According to Knothe et al. [42], in the low frequency range

the major issues in the vehicle system concern the curving, stability, and ride quality

performance. In the lower frequency range, up to 50 Hz, the track essentially behaves as a

relatively stiff spring and its effect on the vehicles behavior is small, especially in the

vertical direction [37]. The vehicle can thus be modeled as a lumped-parameter system.

A wide range of linear and nonlinear lumped-parameter models have been developed for

studies on lateral stability, curving, and ride comfort [13, 21, 35, 62, 63]. These includes

pitch-plane, roll-plane and three-dimensional models of the vehicle with several layers of

track.

When the vertical dynamic forces due to wheel and rail irregularities such as

wheel flats and rail joints are of concern, the wheel-rail interactions may be investigated

Fig. 1.5: Basic compositions of railway vehicle-track system [37]


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using a simple model of an effective wheel mass with a constant force acting on the

wheel. Such a simplified model is illustrated in Fig. 1.6, and has been employed in

numerous studies, where the contribution due to vehicle dynamics are assumed negligible

[5, 7, 8, 15, 22, 23, 31, 35, 39]. This simplifying assumption is justified by the very high

frequency components of the wheel-rail interactions, where the vehicle wheels remain the

most active component. Such model, however, do not permit the analyses of effects of

one wheels flat on the forces imparted by an adjacent wheel. Furthermore, analysis of

dynamic forces imposed on the bearings would require appropriate considerations of

dynamic motions of the bogie (with or without car body) with proper suspension

parameters [2, 6, 17, 19, 23, 37]. If the interaction between the two wheels on different

wheelsets is of concern, a half car model with several degrees-of-freedom would be most

appropriate. Such a model has been employed by Zhai et al. [3, 53], Schwab et al. [72],

Sun et al. [73], and Cai et al. [86] for analyses of wheel-rail impact loads under wheel

flat. The dynamic behaviors of railway wheelsets in the medium frequency range in both

vertical and lateral planes have also been investigated in [10, 12, 13, 14, 74]. A

comprehensive three-dimensional model, however, would be essential for investigating

both pitch and roll effect of the bogie and car body.

The analyses of very high frequency components of deflections and forces up to

20 kHz, for noise emissions studies, elastic wheel, and/or wheelsets models need to be

considered. Grassie et al. [36, 57] have investigated the behavior of railway wheelset

under high frequency excitation in vertical plane. A number of simulation models for an

elastic wheel have been developed using different approaches are developed and applied

in studies on vehicle-track interactions. Szolc [18] developed an elastic wheelset model to


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identify the sources of polygonalization of wheels. Similar models have also been used

by Clause and Schiehlen [78] to study vibration behavior of railway bogie, and by

Schneider et al. [79] to study wheel noise generation. Finite element methods have been

widely used to study the noise generation by elastic wheelsets with OOR defects [75, 76,

77, 78]. The elastic wheelset models, however, are meaningful only for stress, fatigue,

and failure analysis, and may not be significant for analyses of wheel-rail interaction

forces.

Apart from the vehicle components modeling, vehicle system models can be

primarily categorized into three groups. These are one-dimensional, two-dimensional,

and three-dimensional models. One-dimensional model is the simplest model that

considered a single wheel with static force representing the static load due to the car and

bogie, as shown in Fig. 1.6. The contact between the wheel and rail is maintained by

either linear or non-linear spring. Such a simple model has been widely used in many

studies on wheel-rail interaction forces due to wheel and rail defects [1, 5, 15, 20, 21, 22,

28]. Such a model would also be adequate for high frequency vibration analysis when the

interaction between the wheel and rail with irregularities is of concern. However, this

type of model is insufficient for analyzes of effects of impact forces due to wheel and rail

defects on various vehicle components. Moreover, the contributions due to pitch and roll

motions of the vehicle, and the presence of multiple defects in different wheelsets cannot

be evaluated by this model.

One-dimensional vehicle model with two- or three- DOF involving motions of

either the bogie or car or both are also quite common in the analysis of vehicle-track

system [2, 17, 30, 35]. A three-DOF one-dimensional model, as shown in Fig. 1.7,


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generally considers car, bogie, wheel, and primary and secondary suspensions. Such

models can effectively predict the dynamic forces between the bogie and wheel, i.e.

bearing force. The effect of car and bogie pitch and roll dynamic responses, and that of

one wheel flat to other wheels, however, cannot be evaluated.

Alternatively, two-dimensional models with several degrees-of-freedom have also been

developed for analyses of vehicle-track interactions. A two-dimensional model can be

either a pitch-plane or a roll-plane model depending on the type of analysis. A simplest

pitch-plane vehicle model consists of two wheels coupled to the sideframe through the

primary suspension and a static force acting at the center of the side frame due to car load

[6, 10, 23, 37, 86]. Some models also consider the car body coupled to the side frame

through secondary suspension, as shown in Fig. 1.8 [25, 27]. These models are sufficient

to study the vertical and pitch dynamic responses of the vehicle and dynamic coupling

between the two adjacent wheels in the presence of wheel and rail defects. Furthermore,

inclusion of the secondary suspension into the model allows investigation of dynamic

response of the car body. However, contributions due to roll dynamic response of the

W

HC

Wheel

Rail

Vehicle Load

Hertzian contact spring

Fig. 1.6: A single DOF one-dimensional model of

the vehicle


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wheelsets are ignored and effect of one wheels irregularity to the other wheel within the

same wheelset could not be obtained.

Bogie

Car bod

Secondarysuspension

Primary

Suspension

Wheel

Fig. 1.7: A three-DOF one-dimensional vehicle model [2, 35]

Fig. 1.8: A five-DOF pitch-plane vehicle model [37]


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A two-dimensional vehicle model in the roll plane consists of a wheelset, side

frame and car body connected together through the primary and secondary suspension, as

shown in Fig. 1.9 [25, 26]. Such models are useful for analyses of dynamic vertical and

lateral response of the vehicle system. The effect of one-wheel defect to the adjacent

wheel within the same wheelset can be effectively investigated. However, contributions

due to pitch dynamics response of the car body and wheelsets and effects of one wheel

defect to the other wheels within the same bogie or the car could not be investigated.

A few studies have developed a pitch-plane vehicle model of half of the vehicle

including half the car body and two bogies and four wheelsets [3, 13, 17, 53]. This model

includes not only the pitch motion of the car body but also that of the bogie. Such models

exhibit 10 to 12 DOF, as shown in Fig. 1.10. Such a model can permit analyses of

dynamic interaction between the leading and trailing bogie and wheels. Furthermore,

Wheel

Bogie

Car body

Rail

Fig. 1.9: A typical roll-plane vehicle model with several DOF


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impact force due to a wheel defect present in the front bogie wheels could also be

determined at the rear bogie wheels. Considering that the vehicle pitch mostly occurs at

low frequencies, a quarter-vehicle pitch-plane model would most likely be sufficient for

analyses of dynamic wheel-rail interactions in the presence of a wheel defects.

Comprehensive three dimensional vehicle models with relatively large number of

DOF have been increasingly used in recent years. The detailed representations of vehicle

components used in these models make them attractive and more realistic nature towards

the new researchers. A typical three-dimensional vehicle model incorporates half of the

car body, two bogie sideframes, and two wheelsets, while the total number of DOF may

range from 10 to 37. The primary and secondary suspensions are modeled by linear

spring-damper elements along the lateral and vertical directions. Such a model provides

all the advantages of roll, pitch plane models, and would be sufficient to investigate the

influences of coupled vertical, pitch, and lateral dynamics of the vehicle. Furthermore,

Fig. 1.10: Two-dimensional 10-DOF pitch-plane vehicle model [3]


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the cross effects of the four wheels of a bogie and the leading and trailing wheelsets can

be effectively investigated. Fig. 1.11 illustrates a three-dimensional model of the vehicle

that can be run over a flexible track.

1.2.3 Modeling the track Components

The railway track system comprises a periodically supported rail of infinite

length. The track system is a combination of multiple layers of the rail, sleepers, ballasts,

and sub-ballasts. A number of simulation models of the track of varying number of layers

and complexity have been reported in literature. The treatment of each layer is described

below together with the modeling considerations.

Rail

The flexible rail forms the most important components in studies involving

analysis of wheel-rail interactions. The simplest way to model a rail is to consider it as a

lumped mass, spring, and viscous damper [37]. The model parameters are generally

derived from theory of beams on elastic foundations. Li and Selig [110] developed a

Fig. 1.11: A three-dimensional 10-DOF vehicle model [14]


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lumped parameter rail model to study the wheel-rail impact load due to rail joints. The

study also presented a comparison between the lumped-parameter rail model and a finite

element model, and concluded that both models could provide consistent prediction of

peak magnitude of the impact force. The lumped-parameter rail model offers

considerable advantages, which include its simplicity for efficient computation and its

ability to deal with low frequency vehicle vibration. Furthermore, the non-linear

properties of the track components can be easily incorporated in the model. Such models,

however, cannot be considered sufficiently accurate for prediction of high frequency

wheel/ rail impacts, and contribution due to higher modes of the continuous elastic rail.

The rail is mostly modeled as a continuous beam, either as an Euler-Bernoulli

beam or a Timoshenko beam. The Euler beam rail model was used in earlier days for

static and stability analysis, which was considered to provide an accurate model for

representation of the rails response to vertical excitations at frequencies below 500 Hz

[42]. The Euler beam model of the rail has thus been used in many studies on dynamic

analysis of the vehicle-track system. Zhai et al. [3, 53], Jin et al. [13, 17], Morys [21], Cai

et al. [54] have employed Euler beam model to study the effects of rail dipped joints,

wheel flats, rail corrugation, and OOR defects under a moving force on the track. The

Euler beam model, however, assumes negligible shear deformation and rotational inertia

of the rail. The rail thus exhibits relatively higher stiffness, which may yields

overestimation of the dynamic force in the high frequency range. This model is thus not

suited for lateral dynamic analysis that involves the lateral flexibility of the rail web.

Alternatively, the rail has been widely modeled as a Timoshenko beam for predicting

dynamic loads due to the wheel and rail imperfections [5, 6, 7, 8, 9, 10, 12, 19, 22, 23, 28,


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31, 37, 39]. Timoshenko beam model of the rail has been used to study the effect of

nonlinearity and railpad stiffness on the wheel-rail impact and wheel-rail noise

generation. It has been reported that Timoshenko beam model is adequate for frequencies

up to 2.5 kHz for vertical responses [42]. However, for lateral and torsional modes,

railhead and foot need to be modeled as independent Timoshenko beams interconnected

by continuous springs.

Wu and Thompson [46] have developed a double Timoshenko beam rail model

suitable for very high frequency analysis (up to 5 kHz). In this model, the rail is divided

into two sections: the upper part represents the head and web, while the lower part

represents the foot, as shown in Fig. 1.12. These two parts are connected by springs, such

that the two beams vibrate together as a single Timoshenko beam at low frequencies. A

relative motion between the two beams, however, occurs at high frequencies, which

represents the cross-sectional deformation between the railhead and the foot. This model

has shown good agreement with results obtained from a FEM model and measured data

in terms of frequency-wavenumber relation. The model, however, poses considerable

difficulties in identifying cross-sectional and coupling parameters.

Fig. 1.12: A double-beam rail model [46]


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Railpad and fastener

Railpads, made of rubber, plastic or composite materials, are used to support the

rail, to protect sleepers from wear and damage, and provide electrical insulation of the

rails. The railpads and fasteners are usually modeled as a parallel combination of spring

and damping elements for analysis of vertical dynamics. A few studies have considered

structural damping with a constant loss factor[111]. The railpad is generally modeled as

an spring elements [9, 15, 16] or combined spring and damping elements to form either

continuous [2] or discrete [3,13, 17, 21] rail support. The discrete models consider the

visco-elastic pad model at a point on the rail foot at the center of sleeper support to

investigate the wheel-rail interactions [3,5, 6, 9, 19, 22, 23, 28, 31]. Such models have

also been used for analyses of noise emissions [7], and effect of rail corrugations [13, 17,

39, 74]. The deflection behavior of the rail pad at higher frequencies has been

experimentally measured by Thompson et al. [47, 50], and Fermer and Nielsen [31, 48].

The data was used to examine the validity of the analytical model developed by Fenander

[51]. Oscarsson [1] proposed a train-track interaction model to encompass non-linear

railpad stiffness to investigate its influence on rail corrugations.

Andersson and Oscarsson [49] have developed a state-dependent three-parameter

viscoelastic railpad model, as shown in Fig. 1.13, to study the dynamic behavior of pad

under high and low frequency excitations. The model provided higher pad stiffness at

higher frequencies. It was concluded that both low and high frequency behavior of

railpads could be accurately modeled with the inclusion of three parameters. The

influence of the state-dependent properties of the railpad on the wheel-rail contact force


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was, however, observed to be relatively small, while the model parameter identification

was considered to be tedious [49].

Sleeper

The sleepers support the rail and transmit vertical, lateral, and longitudinal forces

from the rail to the ballast. A sleeper can be considered either as a transverse beam with

either uniform or variable cross section or as discrete mass on an elastic foundation

representing the ballast, as shown in Fig. 1.14. This type of simple sleeper model has

been most widely used in analysis of vehicle-track interactions in the presence of wheel

and rail defects [1, 6, 8, 13, 17, 22, 23, 28, 31, 39]. These studies have invariably

concluded that sleeper modeled as a rigid mass is adequate for prediction of dynamic

vehicle-track interaction force, while the bending stiffness is neglected that may yield an

overestimation of impact force at higher speeds.

Sleeper

Ballast

Stiffness

Damping

(a) (b)

Fig. 1.13: Three-parameter pad models [49]

Fig. 1.14: Sleeper modeled as rigid mass resting on elastic ballast


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Alternatively, sleepers have been characterized by either an Euler-Bernoulli or

Rayleigh-Timoshenko beam model [56, 57]. These studies showed good agreements

between the model and measured responses of the rail and the sleeper at frequencies

below 700 Hz. Grassie and Cox [57, 58] have developed flexible sleepers as uniform

Timoshenko beam of prescribed flexural rigidity and mass per unit length, which

provided satisfactory correlation of computed sleeper strains and deflections with the

measured data acquired using a test train running over a sinusoidally corrugated rail. An

effective methodology for determining the sleeper parameters, however, has not yet been

formulated; the measured data have been used to identify a set of model coefficients [59].

A rigid mass representation of the sleeper has been considered adequate for

wheel-rail dynamic interactions. Knothe and Grassie [42] concluded that sleeper response

to contact forces at the railhead up to 1 kHz could be effectively predicted using the rigid

body sleeper model. Dahlberg [55] reported that a concrete sleeper could be adequately

modeled as a rigid mass under excitation below 100 Hz. the Euler-Bernoulli beam theory

would suffice under excitation up to 400 Hz, while Rayleigh-Timoshenko beam theory

should be used for accurate description of sleeper vibration at higher frequencies.

Ballast and subgrade

Ballasts are coarse stones placed under and around the sleepers to form a bed. It

limits sleeper movement by resisting vertical, transverse, and longitudinal forces

transmitted to the track. From a physical point of view, modeling of ballast materials and

its interactions with the sleepers is a very complicated task. The ballasts and subgrades,

however, do not greatly affect the wheel-rail contact, primarily due to their distant

placement from the rail. The specifications of various ballast materials have been


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reviewed by Peplow et al. [60], while Jacobsson [61] presented a review of ballast

materials with particular emphasis on constitutive and mathematical modeling. The

ballast and subgrade are generally grouped together as the foundation for the track, and

modeled as parallel spring-damper element [2, 5, 6, 7, 8, 9, 19, 21, 22, 23, 28, 29, 31, 35,

37, 39]. Zhai et al. [3, 53] and Jin et al. [13, 17] have considered the ballast as lumped

masses below ties, which are interconnected by the springs and dashpots in shear, as

shown in Fig. 1.15. This model permits for analysis of distributed ballast deflections

under excitations at the wheel-rail interface. Such models, however, require a large

number of system parameters and could lead to greater uncertainty of the simulation

results.

Although coupling the ballast and subgrade together as a single mass is very

common in practice, some researchers have modeled them as two individual masses

either connected in shear [10, 12], or through vertical springs and dampers without any

shear coupling [81] to investigate the influence of different parameters on dynamic wheel

loads. This type of model provided better correlation with the measured response, while

Sleeper

Ballast

Damper

Spring

Fig. 1.15: A detailed model of ballast consideringthe stiffness and damping in shear[3, 17]


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the difficulties associated with identification of adequate model parameter values have

been acknowledged.

1.2.4 Track system models

The reported track system models for analysis of dynamic train-track interactions

can be grouped into three categories, namely: (i) lumped parameter models; (ii) rail beam

on continuous supports; and (iii) rail beam on discrete supports. Dahlberg [55] presented

a detailed description of different types of track system models with their relative merits.

The simplest lumped parameter track model can be described by a single effective rigid

mass supported on track bed by linear spring and damping elements. The simple track

model can also be extended to include two or more layers models of the sleepers and

ballasts, as shown in Fig. 1.16. Such a model has been used by Ahlbeck and Daniels [32]

to study the formation of rail corrugations and wheel/rail impact forces due to wheel

OOR defects. Despite the computational simplicity, the lumped parameter models are

adequate for low frequency vibration analysis of the train-track system. The higher

Fig. 1.16: A lumped-parameter 3-layer track model [14]


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deflection modes and stresses in the track, however, can be evaluated using beam models

of track on continuous elastic support. A number of beam models based on either Euler

or Timoshenko beam theory have been developed. The most basic form of the beam

model consists of a rail beam seated on a bed of continuous layer of visco-elastic

elements or springs only [2, 16, 35]. The mass of the sleeper is often lumped with that of

the rail beam, assuming uniform distribution of sleepers over the track length. Newton

and Clark[35] developed Timoshenko beam model of the rail on continuous foundation

to investigate the wheel-rail impact load due to a wheel flat, as shown in Fig. 1.17.

Arnold and Joel [16] and Wen et al. [24] applied similar models to study the wheel-rail

impact load due to rail joints. Sing and Deepak [33] performed experiments to estimate

the track stiffness and damping properties.

Euler beam on elastic foundation with random stiffness and damping has been

recently used to investigate deformation under a moving load [34]. Such a model is

considered to represent the track system fairly well, and can provide a close form

analytical solution. Furthermore, there is a possibility to replace the subsoil foundation by

the frequency and wave number dependent stiffness. The model, however, presents

certain limitations, mainly: (i) the sleeper mass could not be adequately distributed over

Fig. 1.17: A single layer track model with continuous support [35]


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the rail; (ii) sleeper bending effect are neglected and dynamic behavior of sleeper can not

be investigated; (iii) the contribution of elastic rail pads are ignored; (iv) the contributions

due to discrete rail supports from individual sleeper are ignored; and (v) for too many

simplifications used in the model could overestimate the wheel-rail impact loads.

Grassie et al. [36] developed a two-layer track model, which consists of two

Timoshenko beams supported on continuous spring-damper elements. Similar models

have also been applied by Tassilly and Vincent [95] to study rail corrugations, Thompson

[75] to study wheel-rail noise generations, and Bitzenbauer and Dinkel [2] to investigate

the dynamic interactions problems between a moving vehicle and substructure. Wu and

Thompson [15] have also developed a similar model (Fig. 1.18) with inclusion of rigid

sleeper mass to study the wheel-rail impact loads due to rail joints. The model permits for

analysis of dynamic behavior of the sleeper considering both symmetric and asymmetric

bending modes, while only limited information could be derived for dynamic behavior of

the total track system.

The rail beam models on discrete supports have been most widely used for

analysis of impact forces caused by wheel flaws [5, 7,9,19, 21,29]. The supports are

usually characterized by either only sleeper masses or both sleeper and ballast masses

connected by spring-damping elements periodically. These models are also referred to as

two-layer, three-layer, and four-layer models depending on the number of layers

Fig. 1.18: A double layer track model with continuous support [2]


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considered. The sleepers and ballasts can be modeled as either beams or rigid masses. A

two-layer model generally employs the sleeper masses alone, while the ballast and

subgrades are modeled as spring-damping elements, as shown in Fig. 1.19 [6, 8, 22, 23,

28,31,39]. Similar track models have been employed to study impact load and noise

emissions caused by wheel and rail defects [5, 7, 8, 22, 23].

Similar models have also been used by Morys [21] to study the effect of OOR

wheel profiles. Lei [19], and Andersson and Abrahamsson [28] investigated the dynamic

response of vehicle and track under high-speed conditions. The primary advantage of

this type of track models lies in the fact that responses corresponding to three resonance

frequencies of the track structure, the rail and the sleeper can be adequately captured [55].

Furthermore, it includes the effect of discrete sleeper support into the impact analysis.

Euler-beam sleeper models have also been used to study the impact load due to

wheel flat and the track behavior due to corrugated rail [35, 56]. Both rail and sleepers

have been considered as Timoshenko beams by Nielsen and Igeland [23] to study the

impact load due wheel flat and rail corrugation, by Dong [37] to investigate the impact

load due to wheel flats and rail joints, and by Morys [21] to study the growth of out-of-

roundness of wheels.

Fig. 1.19: Rail beam on discrete supports (only sleeper mass is included) [55]


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Alternatively, three-layer track models have been developed where the ballast is

modeled as a mass in addition to the sleeper (Fig. 1.20). The figure shows additional

ballast masses interconnected by means of shear spring-damper for analysis of distributed

ballast deflections. Zhai and Cai [3, 53] developed a three-layer model to study the

wheel-rail impact loads due wheel flats and rail joints and the influence of the ballast

density on the wheel-rail contact force. Oscarsson [1] has used a similar model to

simulate the train-track interactions with stochastic track properties. Jin et al. [13, 17]

used the same model to study the effect of rail corrugations on vertical dynamics of the

rail and track. Sun et al. [10, 12] proposed a four-layer track model incorporating two

different masses for ballast and subgrade to investigate the wheel-rail impact load due to

flat. A five-layer model was also developed by Ishida and Ban [30] to analysis the effect

of different types of wheel flats in terms of impact loads and rail acceleration.

Newton and Clark [35] studied the effect of speed on wheel-rail impact force

considering three different track models, namely, Euler beam on elastic foundation

(EBOEF), Rayleigh-Timoshenko beam on elastic foundation (TBOEF) and Discrete

support model (DSM). The study concluded that at low speeds (up to 50 km/h), the

Fig. 1.20: A three-layer model of track system [53]


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EBOEF yields an underestimate of the impact force, while the TBOEF track model yields

similar to the experimental result. The EBOEF model, however, resulted in

overestimation of the impact force at higher speeds. The DSM model resulted in

considerably lower impact force than the EBOEF and TBOEF models in the entire speed

range.

1.2.5 Wheel-rail contact models

The accurate description of contact between the wheel and rail is a necessary

condition to obtain reliable prediction of not only the impact forces in the presence of

wheel and rail profile defects, but also for curving and lateral stability analysis. The

rolling contact problem in railway vehicles has been extensively investigated over past

several decades. These studies have evolved in numerous methods for analysis of the

vehicle-track contact problem. Elkins [100] presented a state-of-the-art review of various

wheel-rail dynamic contact models, which were divided in two categories: Hertzian and

non-Hertzian contact models. The relative merits and demerits of these two types of

contact models together with those of other contact models are also described in [100].

Hertzian contact model is perhaps the simplest and most widely used to

characterize the rolling contact in railway vehicle. The model uses a single point contact

that lies under the wheel geometric center, where the contact region is described by a

very small ellipse. A Hertzian contact spring is also widely used to represent the stiffness

between the wheel and rail at the point of contact. The contact force is described as a

nonlinear function of the wheel-rail overlap or relative deflection, such that:

HP C z (1.5)


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Where Pis the contact force, z is the wheel-rail overlap in the vertical

direction, exponent is a constant (=1.5) and HC is the Hertzian wheel/rail contact

coefficient that depends on wheel-rail geometry and materials.

Yan and Fischer[103] have analyzed applicability of the Hertz contact theory to

wheel-rail contact problems and concluded that the wheel-rail contact dynamics can be

described with reasonable accuracy for purely elastic contact when the surface curvature

of the rail within the contact area remains unchanged. A major disadvantage of non-linear

Hertzian contact model is that it underestimates the impact force at low speed and

overestimates the impact force at a higher speed [22]. It has been suggested that a

linearized contact spring could adequately represent the wheel-rail contact when

variations in the overlap are very small [70]. The linearized contact spring has been

widely used to study the rail corrugations, vibration due to high frequency irregularities

on wheel-rail tread, and noise generations [45, 70,75, 97]. The linearization, however,

yields overestimation of the contact stiffness in the vicinity of discontinuity and thereby

the impact loads [37].

Non-Hertzian contact models based on predicted contact area and shape have

been proposed by Kalker [99]. In the presence of a wheel flat, this approach requires

extensive computation as the contact area must be established as a function of wheel

angular position relative to the rail. A study of vehicle-track interaction by Baeza et al.

[88] has used the non-Hertzian contact model for a wheel with flat. The study stated that

it is not viable to solve this contact model simultaneously with the integration of

differential equation of motion. The study therefore, proposes pre-calculation of the

contact model for a set of relative wheel flat positions with respect to the track for


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different normal loads. Interpolation of the contact force model is then used for the

dynamic simulation. Pascal and Sauvage [102] have shown alternate methods to calculate

the wheel-rail contact forces using nonHertzian contact patch. The solutions of contact

problems for an elliptical contact zone are available from the works by Kalker[98] and

Shen et al. [107], and later mostly used by others for analysis of wheel-rail squealing

noise [80], corrugation studies [92, 95] etc.

For analysis of vehicle system dynamics, instead of using single-point contact,

multiple point wheel-rail contact models were used in [104, 105, 106]. Several multipoint

contact models based on elliptic and non-elliptic profile are cited in [106]. A multiple

point contact model based on Hertzian static contact theory has been developed by Dong

[37] to study the wheel-rail impact load due to wheel flat. This model has also been

employed later by Sun [10] for the same purpose. Both of these studies reported that

multiple contact model shows good correlation between the predicted and experimental

data. The study, nevertheless, assumes that contact region is symmetric about the vertical

axis, and the results obtained are very similar to those predicted by Hertzian point contact

model. A very recent study by Zhu [52] developed a multipoint adaptive contact model to

account for the asymmetric contact as the as the flat enters the rail. Further study is,

however, required to establish the spring stiffness for the model.

1.2.6 Simulation methods

There are two different techniques widely used in the analysis of wheel-rail

interaction forces. These are frequency domain technique and time domain technique. In

frequency domain technique, an imaginary strip containing the wheel/rail irregularities is

pulled at a steady speed between the vehicle and the track, while the vehicle model is


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held in a fixed position on the rail. Frequency domain technique has been used to study

the impact load due to wheel defects and the formation of corrugations and noise

generations [5, 71, 83, 92 ]. This technique takes less time to analyze and is effective for

the prediction of the frequency response related to excitation. However, this technique is

limited to investigation of the linear models only. When nonlinearity is present either in

vehicle-track system model or in contact model, it is necessary to adopt the solution

process in time domain. Two methods are generally employed in the dynamic analysis of

vehicle-track interaction in time domain, namely, modal analysis method and finite

element method.

In modal analysis method, a mode shape function is assumed depending on the

type of beam and type of support considered in the track model. All the partial

differential equations those depict the dynamic motion of continuous beam are converted

into ordinary differential equations. Rayleigh-Ritz approximation is employed in this

method to approximate the finite number of the lowest natural frequencies of a

continuous system. This method has been used to study the wheel-rail impact load due to

wheel flats and rail joints, and rail corrugations [3, 13, 17, 53, 73]. Sun et al. [10] carried

out an extensive study with three-dimensional coupled vehicle-track model represented

by 2077 equations and the simulation took 95 min on a Pentium 4 machine. Generalized

coordinate method has also been employed to study wheel-rail impact loads due to flat,

rail joints, rail corrugations, and noise generations [23, 35, 44, 80, 85, 86].

With the recent progress of computer capacity, use of finite element method for

solution of train-track interaction has become more attractive. Finite element method has

been applied to study the impact load due to wheel and rail defects and formation of OOR


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in wheel profiles [5, 7, 19, 22, 23, 28]. The extreme adaptability and flexibility of finite

element method have made it a powerful tool to solve PDE over complex domain.

However, the accuracy of the obtained solution is usually a function of the mesh

resolution and the solution often requires substantial amounts of computer and user time,

particularly when extensive parametric study is required.

1.3 THESIS SCOPE AND OBJECTIVES

From the review of relevant literature, it is evident that considerable research

efforts have been made to predict the impact loads caused by wheel flats. Although the

presence of multiple flats in one wheel or different wheels within the same bogie is quite

common in practice, vast majority of the efforts focus on the interactions due to a single

flat. The presence of multiple flats may create larger impact forces at the wheel-rail

interface depending on the size and position of the flats. Increase in demand for higher

speeds and haul loads may further increase magnitude of wheel-rail impact loads due to

multiple wheel flats. The development of an adequate model of coupled vehicle and track

system is thus desirable for simulation of impact loads caused by wide range of wheel-

rail irregularities and to predict the dynamic responses of individual components of both

the vehicle and the track systems. The track model must be in sufficient details as it plays

a strong role in development of wheel-rail interaction forces, especially at higher speeds.

The primary objective of this dissertation research is thus formulated to develop a

comprehensive general-purpose dynamic model of the railway vehicle/track system

capable of predicting dynamic responses of each individual components of the coupled

vehicle-track system in the presence of single as well as multiple flats. The model needs

to be validated based on available results, which can then be applied in the analysis of


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vertical dynamic forces due to wheel defects and parametric excitation of the track in the

high frequency range. The specific objectives of the present work are as follows:

Develop a two-dimensional pitch-plane model of the vehicle to study the

interactions of two wheels, taking into account the contribution of vehicle pitchmotion.

Develop a two-dimensional three-layer model of the track using Euler beamformulation of the rail, supported on discrete elastic supports, where the sleepersand ballasts are considered as rigid bodies, and rail pad and subgrade as vsico-elastic elements.

Formulate a haversine model for the single as well as multiple wheel flats to studythe wheel-rail impact forces.

Formulate a simulation methodology for simultaneous solution of partialdifferential equations representing the motion of the continuous rail and ordinarydifferential equations for motion of the vehicle.

Evaluate the impact forces arising from single as well as multiple wheel flats andinfluences of one wheel flat on the forces imparted at the interface of the adjacentwheel.

Investigate the influences of variations in various design and operating parameterson the magnitudes of the impact force, as well as forces and responses of vehicleand track system components.

1.4 ORGANIZATION OF THE THESIS

In chapter 2, a two-dimensional pitch-plane vehicle model is developed together

with a two-dimensional three-layer track model. The two models are coupled by the non-

linear Hertzian wheel-rail contact model. Rayleigh-Ritz method is used to analyze the

coupled continuous rail track and lumped-parameter vehicle system models.

In chapter 3, the vehicle-track system model is validated under wheel flat

conditions using both theoretical and experimental results obtained from the literature.

The responses of individual components of the vehicle and track system are also derived


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in terms of displacement together with the wheel-rail impact force as functions of

operating speed and flat geometry.

In chapter 4, the impact forces due to a single wheel flat as well as multiple flats

are investigated with both in phase and out of phase conditions. Effects of a flat in one

wheel on the forces developed at the interface of the other wheel within the same bogie

are also investigated. A comprehensive parametric study is conducted for better

understanding of the roles of various design and operational factors affecting the impact

loads induced by wheel flats.

In chapter 5, important conclusions drawn from this study and a list of

recommendation for further studies in this area are presented.


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CHAPTER 2

VEHICLE-TRACK SYSTEM MODEL AND METHOD OF ANALYSIS

2.1 INTRODUCTION

This research is primarily concerned with the vertical wheel-rail impact load

associated with vertical dynamics of the coupled railway vehicle and track structure

system. In general, a vehicle-track system model for simulation of vertical dynamic

interactions is composed of vehicle model, track model and the contact model with rail

and wheel irregularities. It has been suggested that the vertical dynamics of the vehicle

alone contributes only little to the wheel-rail impact force [37, 42]. The vehicle model

may thus be greatly simplified to ensure representative wheel loads and their variations.

In this study, the vehicle system is modeled as a 5-DOF lumped mass model comprising a

quarter of the car body and half of the bogie coupled to two wheels through the primary

suspension. The analysis of dynamic impact loads, however, necessitates a most

comprehensive modeling of the continuous track system [4, 37]. A number of track

models of varying complexities have been developed, as described in chapter 1 [10, 12,

13, 17, 53]. In this study, a multiple-layer track system model comprising the rail pads,

the ballasts, and the subgrade is considered to study the coupled rail-vehicle system

dynamics in the presence of wheel defects.

Analyses of impact loads caused by a defective wheel in the vicinity of the

primary contact under study can be effectively investigated by considering the roll plane

vehicle model [25, 26]. The impact load caused by a defective wheel within the adjacent

bogie, however, requires a pitch plane model of the vehicle moving along the track. The

modal coordinates method is used to analyze of vertical dynamic interactions between


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railway vehicle and the track system. The rail is modeled as an Euler Bernoulli beam that

considers transverse deflections only; this method is considered to be sufficiently

accurate for such analysis [3, 13, 17].

Considering that, the dynamic wheel-rail interactions concerned with the wheel

flaws are mostly dependent upon vertical dynamics of the vehicle-track system of

interest, the vehicle is assumed to be traveling on a straight track. This assumption is well

justified since the lateral and longitudinal relative motions between the wheel and rail are

small, and creep forces at the wheel-rail contact interface have little effect on the

dynamic vertical forces. The forward speed of the vehicle is also assumed to be constant,

while the contribution due to the track roughness is considered small in relation to forces

that may be caused by wheel defects.

In the present study, the dynamic wheel-rail impact load together with individual

system responses are investigated in the time domain. A pitch plane model of the vehicle-

track system is formulated to evaluate the effects of the wheel flats on the dynamic forces

developed at the wheel-rail interface. The generalized coordinates method is used to

convert the partial differential equations (PDE) describing the deflections of the

continuous track to ordinary differential equations (ODE). The deflection response of the

rail is determined from the theory of Euler simply supported beam.

2.2 VEHICLE SYSTEM MODEL

M A Sc_ thesis_ Uzzal

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