Rail Vehicle Dynamics

1

Outline of talk

• Introduction.

• Objectives.

• Rail vehicle and track details.

• Mathematical models for rail vehicle

• Eigenvalue analysis and stability analysis.

• Wheel/rail irregularities.

• Dynamic response analysis.

- Frequency domain approach

- Time domain approach

• Experimental studies (on going)

2

Mathematical model

Rail vehicle dynamics

Vertical dynamics(Ride comfort and dynamic response analysis)

Lateral dynamics(Stability and dynamic response analysis)

Combined vertical and lateral dynamics(Stability and ride comfort analysis)

• 10 dof (degrees of freedom) vehicle model.

• 17 dof vehicle model.

• Vehicle/Track coupled model (FE)

• 31 dof full vehicle model(Ongoing)

• 2 dof wheelset model.

• 7 dof bogie/ truck model.

• 17 dof vehicle model.

Figure 1.1 Overview of work

3

Introduction to Railway VehicleDynamics

• Means of mass transportation in almost all countries.

• Development in respect of safety and speed is required.

• Modern railroad vehicles are required to carry more load per

axle

4

Railway vehicle system

Figure 1.2 General layout of railway vehicle system

5

Major problems associated withrailway dynamics system

• Lateral stability

• Ride comfort

• Derailment

• Curve negotiation

6

Lateral stability

• Lateral stability associated with the term ‘Hunting’.

• Conical wheel profile leads to creep forces from track.

• The friction or creep forces between rail and wheels provide

effective damping.

• The speed at which this effective damping between wheel and

rail becomes zero is called ‘critical speed’.

• At this speed, a sustained periodic oscillation or hunting occurs.

7

Ride comfort

• Ride quality : capability of the railroad vehicle suspension to

maintain the motion within the range of human comfort and or

within the range necessary to ensure that there is no damage to

the cargo it carries.

• The ride quality of a vehicle depends on displacement,

acceleration, rate of change of acceleration and other factors

like noise, dust, humidity and temperature.

8

Contd.

• The Sperling’s ride index (Wz) : used by Indian Railways. The

Sperling’s ride index is used to evaluate ride quality and ride

comfort.

• Ride comfort implies that the vehicle is being assessed

according to the effect of the mechanical vibrations on the

human body whereas in ride quality the vehicle itself is judged.

9

Objectives

1. To develop vertical dynamic model (rigid body) of the vehicle

focusing on ride dynamics.

2. To develop a rail vehicle model (rigid body) focusing on the

lateral dynamics. The development includes lateral stability

models of a single wheelset and of a truck.

3. To develop a coupled vehicle/track dynamic model, where

track is modelled using finite element method and vehicle as

rigid body to study the influence of track on ride behaviour.

10

Contd.

5. To develop combined vertical and lateral dynamic model (rigid

body) to predict both stability and Sperling ride index.

6. To experimentally validate the vertical and lateral dynamic

response predicted using mathematical modelling.

Track/wheel inputs considered

Track geometrical irregularities

Wheel flat

Weld defects

11

Track Inputs for Dynamic Studies1 Track geometrical irregularities acting as

random excitation

Figure 1.3 Different irregularities of track

12

Average vertical profile

0 5 10 15 20 25

-0.015

-0.010

-0.005

0.000

0.005

0.010

Vert

ical

pro

file

(m)

Time (s)0 5 10 15 20

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

Vert

ical

pro

file

(m2 /H

z)

Time (s)

Figure 1.4 Track vertical profile irregularities in time domain and its PSD

• Average vertical profile , 2l r

vZ ZZ +

=

For the present study these data on Indian rail tracks were

obtained from Iyengar and Jaiswal (1995).

13

Track alignment irregularities

0 5 10 15 20 25-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

Alig

nmen

t irr

egul

ariti

es (m

)

Time (s)0 5 10 15 20

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

Alig

nmen

t irr

egul

ariti

es (m

2 /Hz)

Frequency (Hz)

Figure 1.5 Track alignment irregularities in time domain and its PSD

• Average lateral profile or alignment, 2

l ra

Y YY +=

14

Cross level irregularities

0 5 10 15 20 25-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

Cro

ss le

vel (

m)

Time (s)

0 5 10 15 201E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

Cro

ss le

vel (

m2 /H

z)

Frequency (Hz)

Figure 1.6 Track cross level irregularities in time domain and its PSD

• Cross level, c l rZ Z Z= −

15

Gauge irregularities

0 5 10 15 20

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

Gau

ge (m

2 /Hz)

Frequency (Hz)

Figure 1.7 Track gauge irregularities in time domain and its PSD

• Gauge, g l rY Y Y= −

16

Track Inputs for Dynamic Studies2 Wheel irregularities

Some of the irregularities considered only in some cases are

2 Wheel flat irregularities act as periodic excitation

The imperfection considered in this study is the wheel flatness,

which is flat zone on the wheel tread caused by unintentional

sliding of the wheel on the rail when the brake locks.

Mathematically,

Where λw is the wave length of the corrugation (60 to 90 mm)

and c0 is the amplitude (0.41 to 0.93 mm).

17

Track Inputs for Dynamic Studies3 Weld defects

• Irregularities acting as impulse excitation.

• Irregularities include the indentation on the railhead due to

the spalling or the raised joint and the dipped-joint.

H

• Raise on weld joint

00

2HV Vr

⎛ ⎞= ⎜ ⎟

⎝ ⎠

• Impulse velocity

(a)H = 0.002m

18

Track Inputs for Dynamic Studies

α2α1

• Dipped joint

( )0 1 2V V α α= +

Impulse velocity V0, α1 and α2 are dip angles

• Impulse velocity

(b)

Figure1.8 (a) and (b) Rail weld defects

α1 + α2 = 0.02 rad

19

Vehicle and track details

AC/EMU/T coach of Indian Railways is considered for modelling

Major components of the coach are

1. Carbody.

2. Bogie or truck assembly.

Bogie frame

Bogie bolster

Secondary suspension

Primary suspension

Axle box

Wheel and axle set

20

Carbody

20726

289614630

11734φ 952

3810

1197

36581676

3658

A

A

B

B

Section AA

Section BB

All dimensions in mm

Figure 1.9 View of full vehicle (Courtesy, ICF, Chennai)

21

Bogie or truck

Figure 1.10 View of bogie/truck (Courtesy, ICF, Chennai)

• The bogie consists of a 2 stage suspension and 2 pairs of wheels & axles

• There is primary suspension between the axles and the bogie constituted by 8 coil springs

• There is secondary suspension between the bogie and car body with 8 coaxial coil springs (one spring inside the other)

• The wheel and axle sets are mounted in spherical roller bearings on either end of the axle

22

Wheel and axle set

23

24

Vehicle parameters

mc Mass of car body (33700 kg)mb Mass of bogie (3150 kg)mw Mass of wheel set (1500 kg)Jcy Pitch moment of inertia of car body (7.67×105 kg.m2)Jby Pitch moment of inertia of bogie (2.92×103 kg.m2)Jcx Roll moment of inertia of car body (5.24×104 kg.m2)Jbx Roll moment of inertia of bogie (2.02×103 kg.m2)Jwx Roll moment of inertia of wheel set (7.13×102 kg.m2)Jcz Yaw moment of inertia of car body (7.36×105 kg.m2)Jbz Yaw moment of inertia of bogie (3.56×103 kg.m2)Jwz Yaw moment of inertia of wheel set (7.13×102 kg.m2)

25

Contd…

kpx Primary stiffness in longitudinal direction (58×106 N/m)kpy Primary stiffness in lateral direction (4.75×106 N/m)kpz Primary stiffness in vertical direction (0.7×106 N/m)ksy Secondary stiffness in lateral direction (351×103 N/m )ksz Secondary stiffness in vertical direction (334×103 N/m )

ksxSecondary stiffness in longitudinal direction (317×103

N/m )kh Hertzian stiffness of wheel and track (1.5×109 N/m)lc Half of bogie centre pin spacing (7.315 m)lb Semi wheel base of bogie (1.448 m)lp Half of primary spring spacing – lateral (1.127 m)

26

Contd…

ls Half of secondary spring spacing – lateral (0.794 m)lg Semi gauge length (0.864 m)hc Height of car cg from secondary spring (2.429m)hb Height of bogie cg from secondary spring (0.093 m)zc, zb1, zw1

Vertical displacement of car, bogie and wheel set

φc, φb1,φw1

Roll of car body, bogie and wheel set

ψc, ψb1,ψw1

Yaw of car body bogie and wheel set

V Linear velocity of wheellb Semi wheel base of bogie (1.448 m)

27

Contd…

ε Contact angle parameter (8)Roll coefficient (0.038)Conicity (0.05 rad)

Γλ

TRACK OR PERMANENT WAY• Track or permanent way is the railroad on which

the train runs.• Wheel load is directly transferred to the track.

Track consists of the following major components.

• Rails• Sleepers• Fittings and fastenings• Ballast• Formation• The rails standardised for Indian Railways

are 60 kg, 52 kg and 90R 28

Track

29

• The 52PSC and 60PSC track systems consist of 2 rails of an ‘I’ rail cross – section with masses of 52 and 60 kg respectively /m length of the rail.

• The rails in these tracks are supported by prestressed concrete sleepers with a constant sleeper spacing of 0.65 m.

• The track (rail and sleeper) super structure is covered with ballast, which in turn rests on the subgrade (soil).

• Many researchers have modeled the rail as a Beam (with mass and stiffness) On Elastic Foundation (BOEF), with the ballast being represented by a linear spring.

• Often the rail is described as an infinitely long beam discretely supported at rail/sleeper junctions by a series of springs, dampers and masses representing a Discretely Supported Model (DSM)

30

31

Rail parameters

A Cross sectional area of the rail (7.595×10–3 m2)E Modulus of elasticity of rail (2×1011 N/m2)I Rail second moment of inertia (7.595×10–3 m2)mrail Rail mass per unit length (60 kg/m)W Static load of the vehicle on track (202×103 N)υ Poisson’s ratio (0.3)r0 Nominal rolling radius of wheel(0.45 m)rrail Rail head radius (0.3 m)Kf Foundation stiffness (4×107 N/m)

32

Coordinate system

ψθ

ф

Z

Y

X • Translational degrees of freedom1. X – Longitudinal

2. Y- Lateral

3. Z –Bounce or vertical

• Rotational degrees of freedom1. θ – Pitch (about y-axis)

2. φ - Roll (about x-axis)

3. ψ - Yaw (about z-axis)

Figure 2.1Coordinate system

33

Lateral dynamic model of the wheelsetψψ

ψψ

ψ

a

y

ψCpy

Kpy CG

.Cpy

Kpy

Figure 2.4 Wheelset configuration

Two degrees of freedom

o lateral motion of the wheelset (y)

o yaw motion of the wheelset (Ψ)

34

Dynamic equation of motion

( ) 0w w sy s g y nw w wm y F M M F Fa aφ φΓ Γ⎛ ⎞ ⎛ ⎞+ + − + − − + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠&&

0z w s g zw w wI M M MΨ ΨΨ + + − =&&

• Lateral equation of motion is given by

• Yaw equation of motion is given by

35

Lateral dynamic truck/bogie model

b

b2

Kpy, Cpy

Kpx, Cpx

Cpx/2, Kpx/2

Ksy, Csy

Cpz, kpz b1

b3

Figure 2.5 Details of bogie vehicle / truck

• Bogies [1х3]

- Lateral displacement

- Yaw and Roll

• Wheel and axle [2х2]

- Lateral displacement

- Yaw

6 DOF MODEL

36

θc

θb1 θb2

zc

zb1 zb2

Car body

Bogie

Secondarysuspension

Primarysuspension

Wheels

mc Jcy

mb Jby

ks

kp

lc

lb

x

z

37

Vertical dynamic analysis -10 DOF MODEL

Zb

Track

Secondary suspension

Primarysuspension

Wheelset

θb

1

Bogie

Hertzian stiffness (Kh)

Zc

Zb

Zw

θc

θb

Z

Y

Ψ

X

Ф

θ

Figure 4.1 Model for analysis of vehicle system

38

Contd.

10 dof vertical dynamic model

Car body [1∗2]

-Vertical translation

-Pitch

Bogies [2∗2]

-Vertical translation

-Pitch

Wheel and axle [1∗4]

-Vertical translation

17 DOF MODEL

39

zc

zb1

zw1

θc

θb1

φw1

φb1

φc

40

Under frame

Rail

Wheel

AxlePrimary suspension

Bogie frame

X

Z

Y

Secondary suspension

Z

X

Z

Y

Y

X

Fig. 2. 1 Finite element – UF model

FE MODEL

41

Z

X

Z

Y

Y

X

42

Wheelset modeling

Msusp

MaxleY

Msuspen

Faxle

δrδl Mcreep

FNormal

Fcreep

Figure 2.2 Free body diagram of wheelset

43

17 dof lateral dynamic model

• Car body [1∗3]

- Lateral translation

- Roll and yaw

• Bogies [2∗3]

- Lateral translation

- Roll and yaw

• Wheel and axle [4∗2]

- Lateral translation

- Yaw

44

Equation of motion

Figure 3.1 Rigid body model of railroad vehicle

x

z

x

yz

φ

θ

ψ

θcφc

ψc

x

y

45

Contd.

Wheelset 1

1 1 1 1 1 11 1

w wy syw s w g w y ym y F M M F Qa aφ φ⎛ ⎞ ⎛ ⎞Γ Γ+ + − + − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

&&

1 1 1 1 1wz w s w g w zJ M M M Qψ ψ ψψ + + − =&&

Wheelset 2

2 2 2 2 2 21 1

w wy syw s w g w y ym y F M M F Qa aφ φ⎛ ⎞ ⎛ ⎞Γ Γ+ + − + − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

&&

2 2 2 2 2zw w s w g w zJ M M M Qψ ψ ψψ + + − =&&

46

Eigenvalue analysis

Predominant motion Frequency (Hz)

Carbody

Lateral 0.82

Roll 1.24

Yaw 1.11

Bogie 1

Lateral 6.95

Roll 1.63

Yaw 8.41

Bogie 2

Lateral 6.95

Roll 1.56

Yaw 8.41

Wheelset 1Lateral 44.64

Yaw 44.62

Wheelset 2Lateral 44.64

Yaw 44.62

Wheelset 3Lateral 44.64

Yaw 44.62

Wheelset 4Lateral 44.64

Yaw 44.62

Table 3.1 Natural frequenciesof vehicle (Hz)

47

Dynamic response analysis Frequency domain approach

Assumptions1. System with eight random disturbances due to track

irregularities at each of the eight rail wheel contact points.

2. The input from the left rail is considered to be completely

correlated with that from the right rail.

3. Input is space correlated between the successive wheels on

each rail.

4. p(t), q(t), r(t) and s(t) - random loads acting simultaneously on

the railroad vehicle at wheel rail contact points

5. αxp, αxq, αxr and αxs - corresponding receptances.

48

Contd.

b4 b6

b3

b1

b2

b5

p(t) q(t) r(t) s(t)

Car body

Bogie

Wheel

Figure 3.4 Points of application of random load

49

Contd.

The PSD Sx(f) of response x(t) is

{} )(cos2cos2cos2cos2

cos2cos2)(

6543

212222

fS

fS

pxsxrxsxqxrxqxpxs

xpxrxqxpxsxrxqxpx

φααφααφααφαα

φααφαααααα

+++

++++++=

Here Sp(f) = Sq(f) = Sr(f) = Ss(f) = PSD of p(t), q(t), r(t) and s(t)

Phase angles φ are

1 1 2 2

3 3 4 4

5 5 6 6

2 , 2 ,2 , 2 ,2 , 2

fb v fb vfb v fb vfb v fb v

φ π φ πφ π φ πφ π φ π

= == == =

• Time lags τ1, τ2 and τ3 corresponding to wheel bases

b1 (2.896 m), b2 (14.63 m) and b3 (17.526 m)

50

Vertical dynamic analysis

Assumptions

1. All displacements are considered to be small.

2. All characteristics like damping, stiffness etc. are linear.

3. The vehicle is moving at a constant speed on a straight track.

4. Since the vehicle is symmetrical about the centerline of the

track, only half of the coupled system is used for ease of

computation.

51

Contd.

10 dof vertical dynamic model

Car body [1∗2]

-Vertical translation

-Pitch

Bogies [2∗2]

-Vertical translation

-Pitch

Wheel and axle [1∗4]

-Vertical translation

52

Vertical dynamic analysis

Zb

Track

Secondary suspension

Primarysuspension

Wheelset

θb

1

Bogie

Hertzian stiffness (Kh)

Zc

Zb

Zw

θc

θb

Z

Y

Ψ

X

Ф

θ

Figure 4.1 Model for analysis of vehicle system

53

Sperling ride index

‘a’ is the amplitude of acceleration in cm/s2‘B’ is the acceleration weighting factor’f’ is the frequency in Hz.

For ride quality

For ride comfort

54

Contd.

Table 4.2 Ride indices for the different operating speeds (m/s)

AVP alone AVP +Wheel flat + Dipped jointSpeed (kmph)

Ride quality index

Ride comfort index

Ride quality index

Ride comfort index

45 2.2391 3.1782 3.4283 4.710450 2.1815 3.2251 3.5091 4.822955 2.1607 3.2958 3.5949 4.941060 2.2153 3.3946 3.6815 5.060665 2.3414 3.5250 3.7662 5.1770

55

Ride evaluation scales

Ride index Wz Ride quality

1 Very good

2 Good

3 Satisfactory

4 Acceptable for running

4.5 Not acceptable for running

5 Dangerous

Ride Index Wz Ride comfort

1 Just noticeable

2 Clearly noticeable

2.5 More pronounced but not unpleasant

3 Strong, irregular, but still tolerable

3.25 Very irregular

3.5 Extremely irregular, unpleasant, annoying; prolonged exposure intolerable

4 Extremely unpleasant ; prolonged exposure harmful

Table 4.3, Ride evaluation scales − ride quality and ride comfort