AMM.110-116.186.pdf

7/21/2019 AMM.110-116.186.pdf

http://slidepdf.com/reader/full/amm110-116186pdf 1/11

Derailment Analysis of High-Speed Railway Vehicle Bogies

Yung-Chang Cheng1,Chern-Hwa Chen

2 ,Che-Jung Yang

3

1,3Department of Mechanical and Automation Engineering

National Kaohsiung First University of Science and Technology Kaohsiung, Taiwan2Department of Civil and Environmental Engineering

National University of Kaohsiung

Kaohsiung, Taiwan

[email protected], [email protected]

Keywords:component; derailment quotient; nonlinear creep model; twelve degrees of freedomsystem

Abstract:Based on the heuristic nonlinear creep model, the nonlinear coupled differential equationsof the motion of a 12 degree-of-freedom (12-DOF) bogie system which takes account of the lateral

displacement, vertical displacement, the roll angle and the yaw angle of the each wheelset and the

bogie frame, moving on curved tracks are derived. The nonlinear creep forces and moments are

constructed via the saturation constant of the nonlinear creep model in completeness. The effect of

the suspension parameters of a bogie system on the derailment quotient is investigated. Results

obtained in this study show that the derailment quotient of a bogie system increases as the vehicle

speed increases. In addition, the derailment quotient of a bogie system is generally decreased with

the increasing values of suspension parameters.

Introduction

Running safety, especially derailment behavior, of a high speed railway vehicle is a very important

topic. The problem of achieving high-speed operation without the derailment has always been of

interest to vehicle designers.

The studies on the dynamic stability of a bogie running on a curved track considering the linear

and nonlinear creep forces can been found in a number of literatures [1-4]. Based on the linear creep

model without considering the creep moments between wheels and rails, the curving performance

of the unsymmetric bogie was presented by Wickens [1].The dynamic stability of a bogie with

variable yaw constraint suspension was studied by Scheffel et al . [2]. Utilizing the nonlinear creep

model, Lóránt and Stépán [3] studied the relations between the nonlinear creep forces and the flange

clearance. Based on the heuristic nonlinear creep model, Dukkipati et al . [4] illustrated the steady

state curving behavior of a conventional bogie and an unconventional bogie. The comparative studyon the steady state curving performance and the dynamic stability of some unconventional bogies

designs is investigated. In the previous studies, the inertia forces of the bogie frame and the

nonlinear creep moments were not considered.

Running safety is an important subject in the dynamic behavior of railway vehicle. Mechanism

of derailment for railway vehicle has been investigated for many years [5-7]. Wu and Zeng [8] studied

the influences of the flange contact angle, friction coefficient and primary suspension forces on the derailment safety.

For the vehicle-bridge system, the theory of energy random analysis for train derailment on bridge is investigated by

Xiang and Zeng [9]. They examined the criteria of energy increment for judging train derailment and the whole

process of train derailment on bridge. A new criterion for prediction of wheelset is presented by Durali

and Jalili [10] studied and compared the capability for prediction of wheelset derailment by a new

derailment criterion and the derailment coefficient. Finally, Francesco et al . [11] presented andcompared the numerical analysis of mathematical models and the experimental results on

derailment of a full-scale vehicle model on roll rig.

Applied Mechanics and Materials Vols. 110-116 (2012) pp 186-195Online available since 2011/Oct/24 at www.scientific.net © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.110-116.186

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 130.130.37.84, University of Wollongong, Wollongong, Australia-18/06/13,14:38:08)

http://www.scientific.net/

http://www.ttp.net/

http://www.ttp.net/

http://www.scientific.net/

7/21/2019 AMM.110-116.186.pdf


The derailment analysis of vehicle models [5−11] is achieved via the linear creep model. In

practice, the nonlinear creep forces and moments are highly sensitive to the hunting stability and

dynamic response of railway vehicles [12]. Even though the equations of the motion of a ten

degrees of freedom system can be found in Lee and Cheng [12]; nevertheless, the vertical

displacement and roll angle of a bogie frame are not considered. In addition, the suspension force in

the vertical direction and the suspension moment in the longitudinal direction acting on a bogieframe were not provided.

In this study, the nonlinear differential equations of motion of the bogie modeled by the 12-DOF

system are derived using the heuristic nonlinear creep model. The dynamic response of the bogie

with earthquake forces is evaluated by the fourth-order Runge-Kutta integration scheme. The

derailment quotients of the left wheel of front wheelset are calculated with various speeds of bogie

and suspension parameters. Finally, the running safety index of a bogie, the derailment quotient, is

presented and compared using the nonlinear creep model for various vehicle speeds and suspension

parameters.

Differential Equations of Motion

Equations of Motion of Bogie Frame.Consider a bogie, as shown in Figures 1 and 2, moving on

curved tracks. The governing equations of motion for lateral displacement y , vertical displacement

z , roll angle φ , and yaw angle ψ of the bogie frame are 2

t syt se t

Vm y F ( )m g

gR φ = + − , (1)

2

t szt se t

Vm z F 1 m g

gR φ

= − +

, (2)

x t sxtI Mφ = , (3)

z t sztI Mψ = , (4)

where V is the forward speed of the bogie and R is the radius of the curved track. The physical

quantities sytF , sztF , xI , zI , sxtM , sztM and m are defined in the Lee and Cheng [12] and Appendix.

Equations of Motion of Wheelsets.Following the notations used by Lee and Cheng [12], when the

inertia forces and the heuristic nonlinear creep forces and moments are considered, the governing

coupled differential equations of motion for the lateral displacementiy , vertical displacement

iz ,

roll angle iφ , and the yaw angle iψ , of the wheelsets are coupled differential equations

( )

( )

( )

2n

w i Lyi i i i i

n

Ryi i i i i Lyi Ryi

2

syi ext ext w se

Vm y F y , y , ,

R

F y , y , , N N

V F W W m ggR

ψ ψ

ψ ψ

φ

− =

+ + +

+ + − +

, (5)

( )

( )

( )

2n

i se Lzi i i i i

n

Rzi i i i i Rzi Lzi

2

szi ext se ext w

Vm z F y , y , ,

R

F y , y , , N N

V F W W m g

gR

φ ψ ψ

ψ ψ

φ

+ =

+ + +

+ − − +

, (6)

Applied Mechanics and Materials Vols. 110-116 187

7/21/2019 AMM.110-116.186.pdf


( )

( )

( )

( ) ( )

( )

wx i wy i

0

n

Ryi Rzi i i i i

n

Rzi Ryi i i i i

nLyi Lzi i i i i

n

Lzij Lyij i i i i Lyi Lzi Ryi Rzi

Rzi Ryi Lzi Lyi Lxi Rxi sx

V VI I

r R

R F y , y , ,

R F y , y , ,

R F y , y , ,

R F y , y , , R N R N

R N R N M M M

φ ψ

ψ ψ

ψ ψ

ψ ψ

ψ ψ

= −

+

−

+

− + +

− + + + +

i exiM+

, (7)

( )

( )

( )

( )

( ) ( )

n

wz i wy i Rxi Ryi i i i i

0

n

Ryi Rxi i i i i

n

Lxi Lyi i i i i

n

Lyi Lxi i i i i

nRxi Ryi Lxi Lyi Lzi i i i i

n

Rzi i

VI I R F y , y , ,

r

R F y , y , ,

R F y , y , ,

R F y , y , ,

R N R N M y , y , ,

M y ,

ψ φ ψ ψ

ψ ψ

ψ ψ

ψ ψ

ψ ψ

= − +

−

+

−

+ + +

+

( )i i i sziy , , Mψ ψ +

, (8)

where the subscript i, i 1, 2= , in the physical quantities in this paper represent the corresponding

physical quantities of the front and the rear wheelset, respectively. V is the forward speed of the

bogie. n

Rxi i i i iF (y , y , , )ψ ψ , n

Ryi i i i iF (y , y , , )ψ ψ , n

Rzi i i i iF (y , y , , )ψ ψ , n

Lxi i i i iF (y , y , , )ψ ψ , n

Lyi i i i iF (y , y , , )ψ ψ

and n

Lzi i i i iF (y , y , , )ψ ψ are the x, y and z components of the creep forces of the right wheel and the

left wheel, respectively, n

Rxi i i i iM (y , y , , )ψ ψ , n

Rzi i i i iM (y , y , , )ψ ψ , n

Lxi i i i iM (y , y , , )ψ ψ andn

Lzi i i i iM (y , y , , )ψ ψ are the creep moments in the z direction with respect to the right wheel and the

left wheel, respectively. The other physical parameters are all defined in the Lee and Cheng [12]

and Appendix.A heuristic nonlinear creep model, which combines the Kalker’s linear creep theory with a creep

force saturation representation, is used in the analysis. The nonlinear creep forces and the nonlinear

creep moments are given as (Lee and Cheng [12])*n *

jxi i i i i i jxiF (y , y , , ) Fψ ψ α = , (9a)

*n *

jyi i i i i i jyiF (y , y , , ) Fψ ψ α = , (9b)

*n *

zi i i i i i jziM (y , y , , ) Mψ ψ α = , (9c)

where the subscript j, j L, R = , in the physical quantities in this paper represent the corresponding

physical quantities of the right wheel and the left wheel, respectively. *

xiF , *

jyiF and *

ziM are the linear

creep forces and the linear creep moments evaluated from the Kalker’s linear theory

* 33 LLxi i

0

f r aF V(1+ ) a

V R r ψ

= − − −

, (10a)

* 11 12Lyi i L i i i L

0

f f V VF (y +r V ) ( )

V V R r φ ψ ψ δ = − − − − − , (10b)

* 12 22Lzi i i L i i L

0

f f V VM y V r

V V R r ψ φ ψ δ

= − + − − −

, (10c)

* 33 R Rxi i

0

f r aF V(1 ) a

V R r ψ

= − − − +

, (11a)

* 11 12Ryi i R i i i R

0

f f V VF (y +r V ) ( δ )

V V R r

φ ψ ψ = − − − − + , (11b)

* 12 12Rzi i i R i i R

0

f f V VM y V r

V V R r ψ φ ψ δ

= − + − − +

. (11c)

188 Mechanical and Aerospace Engineering

7/21/2019 AMM.110-116.186.pdf


By assuming that the roll angle and the yaw angle of each wheelset are small in this paper, the

nonlinear creep forces and moments with respect to the left wheel and the right wheel are given byn *n *n

Lxi Lxi Lyi iF F F ψ = − , *n *n

Lyi Lxi i LyiF F Fψ = + , ( )n *n

Lzi Lyi L iF F δ φ = +

(12a)

( )n *n

Lxi Lzi L i iM M δ φ ψ = + , *n

Lzi LziM M= (12b)

n *n *nRxi Rxi Ryi iF F F ψ = − , *n *n

Ryi Rxi i RyiF F Fψ = + , ( )n *nRzi Ryi R iF F δ φ = − − (13a)

( )n *n

Rxi Rzi R i iM M δ φ ψ = − − , *n

Rzi RziM M= (13b)

The saturation constant iα is (Lee and Cheng [12])

2 3

i i i i

i

i

i

i

1 1 1 for 3

3 27

1 for 3

β β β β β

α

β β

− + ≤

=

≥

(14)

where

Ri Li

i 2

β β

β

+=

,

( ) ( )2 2

* *

jxi jyi

ji

F F

N β µ

+

= (15)

From the static force equilibrium in the vertical direction, the normal forces of the left wheel and

the right wheel in the vertical direction,Lzi N and

Rzi N , normal forces of the left wheel and the right

wheel in the lateral direction, Lyi N and Ryi N , can be obtained and given in Lee and Cheng [12]. By

assuming that the lateral displacements of the contact points from their equilibrium position, the

position vectors of the contact points,LxiR ,

LyiR ,LziR ,

RxiR ,RyiR and

RziR , can be obtained. (Lee and

Cheng [12]) Therefore, summation moments in the longitudinal direction,Ryi Rzi Lyi LziR N R N+ and

Rzi Ryi Lzi LyiR N R N− − , summation moments in the vertical direction,Rxi Ryi Lxi LyiR N R N+ , can be obtained.

From the Fig. 1, the suspension forces of wheelsets in the lateral direction,syiF , the suspension

forces in the vertical direction, sziF , the suspension moments in the longitudinal direction, sxiM , the

suspension moments of wheelsets in the vertical direction,sziM , acting on the wheelsets, the

external moment,exiM , due to the external forces acting on the each wheelset, can be derived as

i

syi py i py i py 1 t py T t

i

py 2 t py t py t py T t

F 2K y 2C y ( 1) 2K L 2K h

( 1) 2C L 2K y 2C y 2C h

ψ φ

ψ φ

= − − − − +

− − + + +

, (16a)

szi pz t pz t pz i pz iF 2K z 2C z 2K z 2C z= + − − , (16b)

2 2 2 2

sxi pz 1 t pz 1 t 1 pz i 1 pz iM 2K b 2C b 2b K 2b Cφ φ φ φ = + − − , (16c)

2 2 2 2

szi px 1 t px 1 t px 1 i px 1 iM 2K b 2C b 2K b 2C bψ ψ ψ ψ = + − − , (16d)

2

exi ext se i

V

M hW gR φ φ

= − − . (16e)

From Figure 1, the suspension forces of a bogie frame in the lateral direction, sytF , the

suspension forces of a bogie frame in the lateral direction, sztF , the suspension moments of the bogie

frame in longitudinal direction,sxtM , the suspension moments of the bogie frame in vertical

direction,sztM , and the suspension moments in the longitudinal direction,

sxtM , are given as

syt py i py i py i py i py T t

y sy t py sy t py T t

F 2K y 2C y 2K y 2C y 4K h

( 4K 2K )y ( 4C 2C )y 4C h

φ

φ

= + + + −

+ − − + − − −

, (17a)

( ) ( )szt sz pz t sz pz t

pz i pz i

F 2 K 2K z 2 C 2C z

2K z 2C z

= − + − +

+ +

, (17b)

2 2

sxt sz 2 t sz 3 t py T i

2

y T i py T i py T i pz 1 i

2 2

y T t py T t py T t pz 1 i

M 2K b 2C b 2K h y

2C h y 2K h y 2C h y 2K b

4K h y 4C h y 4K h 2C b

φ φ

φ

φ φ

= − − +

+ + + +

− − − +

, (17c)


7/21/2019 AMM.110-116.186.pdf


2 2 2

szt py 1 px 1 sx 2 t

2 2 2

py 2 px 1 sx 3 t

2 2

y 1 1 py 2 1 px 1 1 px 1 1

2 2

y 1 2 py 2 2 px 1 2 px 1 2

M ( 4K L 4K b 2K b )

( 4C L 4C b 2C b )

2K L y 2C L y 2K b 2C b

2K L y 2C L y 2K b 2C b

ψ

ψ

ψ ψ

ψ ψ

= − − −

+ − − −

+ + + +

− − + +

. (17d)

For simplicity, one assumes that the constraint functions is linear for a conical wheel on a knife-edged rail, so the assumptions with respect to the wheel and rail geometry are given in Lee and

Cheng [12]. After substituting the equations given above into equations (5)−(8) and neglecting the

high order terms and the influence of the vertical central displacement of the wheelset on the lateral

displacement, one obtains the following coupled nonlinear differential equations

( )

( )

( )

2

i 11w i i i

0 i 11i 12i i ext w se

2 2

ext extext w se i syi

2 f Vm y y V

R V

2r f 2 f V W m g

V R V

V W V W W m g F

gR gR

α ψ

α α ψ φ φ

φ φ

− = − −

− − − − +

− + + + +

, (18)

22

i 11 i 11i se i i i i szi

20 i 11i 12 i 12 i 12

i i i i i

0

2 f 2 f Vm z y y F

R V V

2r f 2 f 2 f 2 f

V V R r

λ α α φ φ φ

α α α λ α φψ φ φ φ

+ = − − +

− − + +

, (19)

( )

( )

2

i 12wx i wy i i

0 o

22ext

ext w se i

i 11 o i 12i 0 i

2

ext i 12ext w se i 0

2 o i 11i 12 i

2 f V VI I y

r R r

V W W m g y

gR

2 f (a r ) 2 f y a r V V

V W 2 f W m g a a r

gR R

2r f 2 f a

V

λ α φ ψ

φ λ

α λ α λ ψ

α φ λφ λ

α λ α φ λ

− − =

− + +

+

− − +

+ + + + +

+ − +

( )0 i exi

2

i 22i 11 o i sxi

o

r M

2 f 2 f (a r ) M

r

φ

α λ α λ ψ

+

+ + + +

, (20)

( )

( )

i 33 i 12wz i i i szi

o

o i 12wy i i 12 i

o

2

extext w se i

22i 33 i 22 i

i 33 22

2a f 2 f I y y M

r V

2r f V I 2 f

r V

V W W m g a

gR

2a f 2 f 2 a f f

V V R

λα α ψ

α φ α ψ

φ λψ

α α α ψ

= − + +

− − −

+ + +

− + + +

. (21)

As a result, equations (1)−(4) and (18)−(21) form the twelve governing differential equations ofmotion of the system.


7/21/2019 AMM.110-116.186.pdf


Drailment Analysis

Running safety is one of important factors for dynamic behavior of railway vehicles. The

derailment quotient is adopted to investigate the running safety of railway vehicle. In this study,

because the lateral displacement, vertical displacement, roll angle and yaw angle of a wheelset are

considered, the derailment quotient can be derived completely. The derailment quotient of the left

wheel of the front wheelset of the front bogie, L

L

QP

, (Figure 3) is given as (Matsuo [6])

n

Ly1 Ly1L

nL Lz1 Lz1

F NQ

P F N

+=

+ (22)

where LQ is the contact force acting in lateral direction on the left wheel, LP is the contact force

acting in vertical direction on the left wheel. Additionally, Ly1F and Lz1F are creep forces in the lateral

direction and vertical direction, respectively. Ly1 N and Lz1 N are normal forces in the lateral direction

and vertical direction, respectively.

Numerical ResultBased on the system parameters in the Table I. [13, 14], this paper examines the derailment

quotients of a bogie system. Firstly, the Runge-Kutta fourth-order method is employed to

investigate the time responses for the wheelset. The time history of the nonlinear contact forces of

the left wheel of the front wheelset in the lateral and vertical direction evaluated by Equation (12)

can be obtained. Finally, the RMS (Root-Mean-Square) values of the derailment quotients evaluated

by Equation (28) are presented. For the present problem studied, the time steps of the dynamic

response of the bogie and the time history of earthquake forces are taken to be the same. As such,

the stable time step length of the Runge-Kutta fourth-order scheme is given by 0.005 s.

Figure 4 shows the effect of speeds on the derailment quotients without considering the

earthquake forces with the various primary suspensionxK and

yK . Generally, the derailment

quotients increase as the speeds increase. Moreover, the difference of the derailment quotients for

the variances of the stiffness xK and yK can be neglected as the speeds are lower than 400 km/h.

In Figure (4a), when the speeds exceed 400 km/h, the derailment quotients evaluated by the high

values of xK are greater than those obtained by the low values of xK . However, in Figure (4b), the

derailment quotients evaluated using the low values of yK are greater than those obtained by the

high values of yK as the speeds increase.

Figure 5 presents the effect of speeds on the derailment quotients without considering the

earthquake forces with the various secondary suspension sxC and syC . The derailment quotients are

increased when the vehicle speeds are increased. Furthermore, the longitudinal and lateral damping

of secondary suspension, sxC and syC , have only marginal influence on derailment quotients as thevehicle speed is small. In Figure (5a), the derailment quotients evaluated using the high values of

longitudinal damping sxC are greater than those obtained using the low values of sxC when the

vehicle speed exceeds a critical value of approximately 300 km/h. In addition, the derailment

quotients obtained from the low value of lateral damping syC are consistently higher than those

obtained using the high value of syC . From Figures 4 and 5, it is seen that the derailment quotients

can be reduced as the low values ofx

K and sxC , and the high values ofy

K and syC are applied in

the suspension system of a bogie.

ConclusionThis study has utilized a heuristic nonlinear creep model to analyze the derailment behavior of a

high-speed railway vehicle bogie system moving on curved tracks. The dynamics of the railway

vehicle bogie have been fully described utilizing a 12-DOF model comprising the lateral


7/21/2019 AMM.110-116.186.pdf


displacement, vertical displacement, roll angle and yaw angle of each wheelset and the bogie frame.

The effects of the suspension parameters on the derailment quotients are presented. Overall, the

results have shown that the derailment quotients can be reduced as the low values of xK and sxC ,

and the high values of yK and syC are used in the suspension system of the bogie.

Tab1. Data of the System Parameters [13, 14]Parameters Value

Wheelset mass wm 1117.9= kg

Bogie frame mass tm 350.26= kg

Roll moment of the inertia of the wheelset wxI 608.1= kg-m2

Spin moment of the inertia of the wheelset wyI 72= kg-m2

Yaw moment of the inertia of the wheelset wzI 608.1= kg-m2

Roll moment of inertia of bogie frame txI 300= kg-m2

Yaw moment of the inertia of the bogie frame tzI 105.2= kg-m2

Wheel radius 0r 0.43= m

Half of the track gauge a 0.7175= m

Wheel conicity 0.05λ =

Half of the primary longitudinal spring arm 11.0= m

Half of the primary longitudinal damping arm 11.0= m

Half of the secondary longitudinal spring arm 21.18= m

Half of the secondary longitudinal dampingarm 3

1.4= m

Half of the primary lateral spring arm 1L 1.28= m

Half of the primary lateral damping arm 2L 1.5= m

Height of the external weight above the centerof gravity of the wheelset

h 1.4= m

Vertical distance from the wheelset center ofthe gravity to the secondary suspension Th 0.47= m

Longitudinal stiffness of the primarysuspension

5

pxK 9 10= × N/m

Lateral stiffness of the primary suspension5

pyK 3.9 10= × N/m

Vertical stiffness of the primary suspension5

pzK 6 10= × N/m

Vertical damping of the primary suspension4

pzC 4 10= × N-s/m

Longitudinal stiffness of the secondarysuspension

4

sxK 3.5 10= × N/m

Lateral stiffness of the secondary suspension4

syK 3.5 10= × N/m

Vertical stiffness of secondary suspension5

szK 3.5 10= × N/m

Longitudinal damping of the secondarysuspension

4

sxC 3.2 10= × N-s/m

Lateral damping of the secondary suspension 4

syC 1 10= × N-s/m

Vertical damping of secondary suspension4

szC 4 10= × N-s/m

Lateral creep force coefficient6

11f 2.212 10= × N

Lateral/spin creep force coefficient 12f 3120= N-m2

Spin creep force coefficient 22f 16= N

Longitudinal creep force coefficient6

33f 2.563 10= × N

Radius of curved tracks R 6250= m

Axle load 4W 5.6 10= × N

Coefficient of the friction 0.2µ =


7/21/2019 AMM.110-116.186.pdf


7/21/2019 AMM.110-116.186.pdf


Figure 1. Two-axle bogie model

Figure 2. The free body diagram of a single wheelset

Figure 3. Contact forces on left wheel


7/21/2019 AMM.110-116.186.pdf


100 200 300 400 500

Speed V (km/h

0

0.4

0.8

1.2

D e r a i l m e n t q u o t i e n t

(a)

K px = 900 kN/m

K px = 1800 kN/m

dangerous region

safe region

100 200 300 400 500

Speed V (km/h

0

0.4

0.8

1.2

1.6


(b)K

py = 390 kN/m

K py = 150 kN/m

dangerous region

safe region

Figure 4. The effect of vehicle speed on derailment quotient for the various (a) px

K (b) py

K of

primary suspension.

100 200 300 400 500

Speed V (km/h

0

0.4

0.8

1.2


(a)

Csx = 32 kN-s/m

Csx = 120 kN-s/m

dangerous region

safe region

100 200 300 400 500

Speed V (km/h

0

0.4

0.8

1.2


(b)

Csy = 10 kN-s/m

Csy = 50 kN-s/m

dangerous region

safe region

Figure 5. The effect of vehicle speed on derailment quotient for the various (a) sxC (b) sy

C ofsecondary suspension.


7/21/2019 AMM.110-116.186.pdf


Mechanical and Aerospace Engineering

10.4028/www.scientific.net/AMM.110-116

Derailment Analysis of High-Speed Railway Vehicle Bogies

10.4028/www.scientific.net/AMM.110-116.186

http://dx.doi.org/www.scientific.net/AMM.110-116

http://dx.doi.org/www.scientific.net/AMM.110-116.186

http://dx.doi.org/www.scientific.net/AMM.110-116.186

http://dx.doi.org/www.scientific.net/AMM.110-116

AMM.110-116.186.pdf

Documents

Transcript of AMM.110-116.186.pdf