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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