DSpace at VNU: High-speed trains subject to abrupt braking

High-speed trains subject to abrupt brakingMinh Thi Trana, Kok Keng Anga, Van Hai Luongband Jian Daia a Department of Civil and Environmental Engineering, National University of Singapor

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International Journal of Vehicle Mechanics and Mobility

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High-speed trains subject to abrupt braking

Minh Thi Tran, Kok Keng Ang, Van Hai Luong & Jian Dai

To cite this article: Minh Thi Tran, Kok Keng Ang, Van Hai Luong & Jian Dai (2016):

High-speed trains subject to abrupt braking, Vehicle System Dynamics, DOI:

10.1080/00423114.2016.1232837

To link to this article: http://dx.doi.org/10.1080/00423114.2016.1232837

Published online: 18 Sep 2016.

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High-speed trains subject to abrupt braking

Minh Thi Trana, Kok Keng Anga, Van Hai Luongband Jian Daia

a Department of Civil and Environmental Engineering, National University of Singapore, Singapore,

Singapore;bDepartment of Civil Engineering, Ho Chi Minh City University of Technology, Vietnam National University Ho Chi Minh City (VNU-HCM), Ho Chi Minh City, Vietnam

ABSTRACT

The dynamic response of high-speed train subject to braking is

investigated using the moving element method Possible sliding

of wheels over the rails is accounted for The train is modelled as

a 15-DOF system comprising of a car body, two bogies and four

wheels interconnected by spring-damping units The rail is

mod-elled as a Euler–Bernoulli beam resting on a two-parameter elastic

damped foundation The interaction between the moving train and

track-foundation is accounted for through the normal and tangential

wheel–rail contact forces The effects of braking torque, wheel–rail

contact condition, initial train speed and severity of railhead

rough-ness on the dynamic response of the high-speed train are

inves-tigated For a given initial train speed and track irregularity, the

study revealed that there is an optimal braking torque that would

result in the smallest braking distance with no occurrence of wheel

sliding, representing a good compromise between train instability

and safety.

ARTICLE HISTORY

Received 11 February 2016 Revised 24 July 2016 Accepted 20 August 2016

KEYWORDS

High-speed trains; braking torque; moving element method; wheel sliding

1 Introduction

The rapid increase in the use of high-speed trains for travels all over the world and theunfortunate occurrences of many catastrophic accidents involving high-speed trains arereasons research on railway dynamics is becoming more and more important The study

of the response of high-speed trains subject to braking is particularly critical, as it tributes directly to ensuring better operational safety and superior train design Heavybraking severely affects the dynamics of the train, which could result in safety and traininstability concerns The former relates to safe braking distance and the latter pertains tothe perilous occurrence of wheel sliding and potentially disastrous derailment

con-Analytical studies are limited to simplified cases where the train-track-foundation tem is typically modelled as a railway beam resting on a Winkler elastic foundation andthe moving train idealised as a single or a sequence of moving loads Most of these works

sys-in the literature are concerned with the uniform motion of trasys-in [1 3] and there were only

a few studies carried out on trains travelling at non-uniform speed Suzuki [4] derived thegoverning equation of a finite beam subject to travelling loads involving acceleration by

CONTACT Kok Keng Ang ceeangkk@nus.edu.sg Department of Civil and Environmental Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore117576, Singapore

using the energy method Yadav [5] investigated the analytical vibration response of a track-foundation system resulting from a vehicle modelled as a sprung-mass travelling atvariable velocities over a finite track.

train-The well-known finite element method (FEM) has been employed to solve various ing load problems However, the method suffers from difficulties when the moving loadapproaches the boundaries of the domain and eventually disappears when the load crossesthese boundaries To overcome these complications in the FEM, Krenk et al [6] proposed

mov-an FEM solution using convected coordinates, similar to the moving coordinate systemproposed by Timoshenko [1], to determine the response of an elastic half-space subjected

to a moving load By adopting moving coordinates, the load is always kept stationary tive to the boundaries of the mesh and thus overcomes the aforementioned problem faced

rela-by the FEM Koh et al [7] adopted the same idea to investigate train-track problems andnamed the numerical algorithm the moving element method (MEM) Subsequently, themethod was extended by Ang and Dai [8] to investigate the ‘jumping wheel’ phenomenon

in high-speed train motion at constant velocity over a transition region where there is asudden change of foundation stiffness A computational study using the MEM was car-ried out by Ang et al [9] to investigate the dynamic response of a high-speed train tracksystem Results obtained using the Hertz contact model and the linearised Hertz contactmodel were compared and discussed More recently, Tran et al [10] applied the MEM toinvestigate the non-uniform motion of a high-speed train, modelled as a 3-DOF system,travelling over a viscoelastic foundation The magnitude of deceleration considered was,however, not high and the sliding of the train wheels was thus not considered

Train instability that may arise due to sudden braking is a serious problem A derailmentstudy [11] revealed that 30% of derailments in Russia occurred due to emergency brakingunder poor wheel–rail contact condition Lixin and Haitao [12] studied the 3-D dynamicresponse of heavy trains travelling at a low speed and subject to normal braking, in whichthe occurrence of wheel sliding was not investigated Handoko and Dhanasekar [13] pre-dicted the dynamics of simplified two-axle bogies of low-speed train both under constantspeed and under variable speed due to traction and braking Zhang and Dhanasekar [14]presented a low-speed train model under braking conditions in order to investigate carbody pitch, derailment and wheel-set skid The influence of wheel–rail contact conditionand track geometry defects on car body pitch was also discussed

Various multibody system dynamics (MBS) simulation models of a foundation system have been proposed in the literature Some models have been suc-cessfully incorporated into commercial software such as VAMPIRE, SIMPACK, GENSYS,ADAMS/Rail and NUCAR [15] However, these software tools cannot be employed toaccount for the pitching motion of the train wheel set and to account for possible wheelsliding Also, they have so far been limited in applications involving a train travelling

train-track-at a uniform velocity In dealing with railway vehicle subject to braking, these softwaretools require as input the speed profile of the train However, under realistic train brakingconditions, the speed–time history of the train is unknown in advance Thus, there are lim-itations in the use of commercial software for the study of railway trains subject to braking.The use of an inertial reference frame coordinate system proposed by Shabana et al.[16] can be employed to investigate the dynamics of trains subject to braking [17] Themethod relies on the use of an inertial reference frame coordinate system with origin fixed

in space and time This coordinate system is quite different from the convected coordinate

system employed in the MEM whose origin is attached to the centre of mass of the movingtrain load.

In the aforementioned research works relating to railway vehicle braking, researchersgenerally focused on a heavy train travelling at low speed under normal braking conditionswith the railhead assumed to be perfectly smooth However, real train-track systems arelikely to have various degrees of railhead roughness High-speed trains may undergo highdeceleration under emergency situations that require the train to come to a halt quickly

to avoid other possible catastrophes Such trains would then be subject to so-called mal braking Unlike normal braking, when a train decelerates under moderate-to-heavybraking conditions, instability due to train wheels sliding over the rails could occur Due

abnor-to track irregularity and the high speed of the train, it cannot be assumed that the wheel isalways in contact with the railway track The jumping wheel phenomenon, which describesthe condition in which there is a momentary loss of contact between wheel and rail, needs

to be accounted for Simple train models such as a moving load or moving sprung-masscannot be employed for the study of railway vehicle braking A more realistic train modelthat accounts for the effect of pitching moment arising from the longitudinal inertia effectsand wheel adhesion forces is necessary

This paper presents the results of a computational study to investigate the dynamicresponse of high-speed train due to braking using the MEM The effects of various factorssuch as braking torque, initial train speed, degree of track irregularity and wheel–rail con-tact condition on the dynamic response of the high-speed train were examined, includingthe occurrence of wheel sliding and the jumping wheel phenomenon

2 Formulation and methodology

In this study, the train is assumed to comprise of a car body, two bogies and four wheel setstravelling over the track-foundation subject to braking and propulsion resistance The trainand track-foundation are coupled through the interaction and friction between the wheelsand rail The railhead is assumed to have some imperfections termed as ‘track irregularity’

In view of the fact that the track gauge is large enough and that the track is straight, it isreasonable to assume that there is little interaction between the pair of wheels of each wheelset and there is negligible rolling displacement of the centroid of the wheel set Thus, a 2-Dmodel of the train-track system comprising of one rail and half the train is considered

tia about the pitch of each bogie are mband Jb, respectively The secondary suspension

consists of two spring-damping units, each modelled by a spring ksand dashpot cs The

bogies are supported through primary suspensions to the four wheels, each of mass mw

and moment of inertia about the pitch Jw The primary suspension system consists of four

Figure 1.Train model.

spring-damping units, each comprising a spring kp and dashpot cp The nonlinear Hertz

contact force and Polach adhesion force between the ith wheel and rail beam are F ci and

f i, respectively The positions of the secondary and primary suspension spring-dampingunits measured with respect to the centre of mass of car body and the bogies are specified

by l1and l2, respectively, as shown in Figure1

The governing equations of the car body, bogies and wheels may be derived fromNewton’s second law of motion

mc¨uc+ cs(2˙uc− ˙ubr− ˙ubf) + ks(2uc− ubr− ubf) = −mcg, (1)

mw¨uw2+ cp(˙uw2− l2˙θbf− ˙ubf) + kp(uw2− l2θbf − ubf) = Fc2− mwg, (8)

mw¨uw3+ cp(˙uw3+ l2˙θbr− ˙ubr) + kp(uw3+ l2θbr− ubr) = Fc3− mwg, (9)

mw¨uw4+ cp(˙uw4− l2˙θbr− ˙ubr) + kp(uw4− l2θbr− ubr) = Fc4− mwg, (10)

where uc andθc are the vertical and pitch displacements of the car body, respectively;

(ubr, θbr) and (ubf, θbf) are the vertical and pitch displacements of the rear and front

bogies, respectively;(u wi,θ wi ) are the vertical and pitch displacements of the ith wheel;

and s denotes the longitudinal displacement of the train at any instant of time Note that all train components are assumed to be rigidly connected in the longitudinal direction Fr

denotes the total running resistance force acting on car body, g the gravitational ation, Tbthe applied braking torque, h1the vertical distance between car body’s centre of

acceler-mass and longitudinal internal forces interlocking between car body and bogies, h2the tical distance between the longitudinal internal forces and the centre of mass of bogies, and

ver-h3the vertical distance between the centre of mass of bogies and the longitudinal internalforces connecting bogies to wheels

2.2 Running resistance

Running resistance generally includes both the aerodynamic drag and the rolling tance Based on an experimental study [18], the running resistance of a high-speed trainmay be written as

resis-R = c0+ cv˙s + ca˙s2, (16)

where the coefficients c0, cvand caare obtained from the wind tunnel test The third term

ca˙s2 denotes the aerodynamic drag and the first two terms are considered to be rolling

mechanical resistance The total running resistance force Fr, acting on the locomotive, is

obtained by multiplying the running resistance R with the total mass of the train (mc+

2mb+ 4mw).

2.3 Wheel–rail contact force

Based on the nonlinear Hertz contact model [19], the normal contact force F cibetween the

ith wheel and rail may be expressed as

where KHis the Hertzian spring constant given by

KH= 23



E2√

RwRr

in which Rwand Rrdenote the radii of the wheel and railhead, respectively,υ is the Poisson’s

ratio of the material, andy i the indentation at the contact surface at the ith wheel The

latter may be expressed as

y i = y ri + y ti − u wi, (19)

where y ri and y tidenote the vertical displacement of the rail and track irregularity at the

ith contact point, respectively, and u wi the vertical displacement of the ith wheel The track

irregularity describes the vertical unevenness in the railhead surface which arises due tovarious factors such as wear, tear and plastic deformation, and is widely assumed to takethe following sinusoidal form [20]

y ti = atsin2πx i

where atandλtdenote the amplitude and wavelength of the track irregularity, respectively

2.4 Wheel–rail frictional force

The longitudinal adhesion force f ibetween the wheel and the rail significantly affects theperformance of the drive dynamics A simplified wheel–rail contact model for the compu-tation of the adhesion force has been introduced by Polach [21] The model, which will beemployed in this study, is suitable for accounting the dynamics of wheel sliding resultingfrom heavy braking where there is an occurrence of large creep conditions

The simplified wheel–rail contact model for the adhesion force f iis given by

to the reduction factors in the adhesion and slip areas, respectively Based on Kalker’s lineartheory [22],ε imay be expressed as

ε i= Gπa i b i c11

in which a i and b i denote the semi-axes of the contact ellipse at the ith wheel, c11a

coeffi-cient from Kalker’s linear theory, G the shear modulus of rigidity, μ ithe friction coefficient

between the ith wheel and rail given by

μ i = μ0[(1 − A) e −B|˙s−Rw˙θ wi|+ A] (23)

and c li the longitudinal creep at the ith wheel

c li = |˙s − Rw˙θ wi|

In Equation (23), A denotes the ratio of limit friction coefficient μ∞at infinity slip velocity

to maximum friction coefficientμ0 at the zero slip velocity and B the coefficient of the

exponential friction decrease

2.5 Moving element method

The rail track is modelled as an infinite Euler–Bernoulli beam resting on a two-parameterelastic damped foundation The beam is subject to four moving loads arising from the

wheel contact forces F cias illustrated in Figure2 The track-foundation is discretised intofinite moving elements in which the formulation of the element equations is based on the

use of a convected coordinate r-axis with origin fixed at the centre of mass of the moving

car body as shown in Figure2 The differential governing equation of the motion of thetrack-foundation may be written as

rail beam and shear layer, respectively

The relationship of the fixed and moving coordinate systems is given by

Figure 2.Track-foundation model

In view of Equation (26), Equation (25) may be rewritten as

By adopting Galerkin’s approach, the element massMe, dampingCeand stiffnessKe

matrices for a typical moving element of length L may be written as

NT

,rN,rr dr − ˙sφEI

L0

NT

,rrN,rrr dr

− (m˙s2− ksm)

L0

NT

,rN,r dr − (m¨s + α˙s)

L0

NTN,r dr + k

L0

NTN dr, (30)

whereN is the element shape function based on Hermitian cubic polynomials; () ,rdenotes

partial derivative with respect to r.

By adopting Rayleigh damping [23], the damping matrixCemay be expressed as

Equation (29), the damping parameters are proposed as follows:

α = a0m + a1k; φ = a1; λ = a1ksm (32)

By assembling the element matrices, the equation of motion for the combined track-foundation (high-speed train) model can be written as

wherez denotes the global displacement vector; M, C and K are the global mass, damping

and stiffness matrices, respectively; andF is the global load vector The above dynamic

equation can be solved by any direct integration methods such as the Newmark-β

high-decelerating train of negligible mass is first considered The problem reduces to the case of

a track-foundation subject to four constant moving wheel loads In view of this simplifiedcase, there are no inertia effects and thus no consideration of sliding of wheels The foun-dation supporting the track is modelled as a two-parameter elastic damped foundation.Solutions obtained using the proposed MEM are verified through comparison with resultsobtained via the FEM

The train is assumed to be travelling initially at a constant speed˙s0= 70 m s−1 After

time t1, which is taken to be long enough for the vibration of the train-track-foundationsystem to attain steady state, the train is assumed to decelerate uniformly at¨s = −2 m s−2

and finally comes to a halt at time t2 This deceleration magnitude is typical during heavybraking For simplicity, results are obtained based on four moving wheel loads of 1 kNeach Values of parameters related to the properties of the track and foundation [7] aresummarised in Table1 Note that the value of shear modulus ksmis taken from [25]

In the FEM model, a sufficiently long segment of the railway track is discretised Thesegment may be divided into three sub-segments, a central portion and two end portions.The central portion, where the train travels during the period considered, is taken to be1278.5 m The central portion is padded by two end portions of sufficient lengths in order

to mitigate the erroneous boundary effects due to the moving train load approaching theboundaries of the FEM model Through a convergence study, the lengths of the end por-tions are taken equal to 25 m Due to the advantage enjoyed by the MEM in dealing withmoving load problems, a relatively shorter segment is required Also from a convergencestudy made, the length required for the truncated railway track in the MEM model is only68.5 m

Figure3shows the rail displacement profiles obtained by FEM and MEM in the vicinity

of the wheel–rail contact point after the train decelerates for 1 s As can be seen in Figure3,good agreement between both results is obtained In view of the fact that the FEM requires

a longer domain length as compared to the MEM, it is not surprising that the tional time required is substantially longer than that needed in the MEM This comparisonstudy clearly illustrated the fact that the MEM is accurate as well as computationallyefficient and more generally suited for the study of moving load problems as compared

computa-to the FEM

Next, a study is carried out to investigate the response of a more realistic 15-DOF trainmodel subject to braking using the proposed MEM The influence of various parameterssuch as the magnitude of the braking torque, initial train speed and wheel–rail contactcondition on the dynamic response is examined

Table 1.Parameters for track-foundation and train

Figure 3.Comparison of rail displacement profiles.

Table 2.Typical parameters for wheel–rail tact conditions

con-Wheel–rail contact condition

The MEM model adopted comprises a truncated railway track of 68.5 m length that

is discretised non-uniformly with elements ranging from a coarse 1 m to a more refined0.25 m size The properties of the track-foundation [7] and the train [26] are summarised inTable1 The train is assumed to be travelling at its operational speed when braking torque

is applied resulting in train deceleration The coefficients c0, cvand caused to computethe resistance force in Equation (16) are 1176× 10−5N kg−1, 77.616× 10−5N s m−1kg−1and 1.6× 10−5N s2m−2kg−1[18], respectively The nonlinear Hertz spring constant used

to model the contact between wheels and rail is computed from Equation (18) with the

radii of the wheel Rw, railhead Rr and Poisson’s ratio of the wheel/rail materialυ taken

to be 460, 300 and 0.3 mm, respectively Typical parameters for the wheel–rail contactcondition used to compute the adhesion force from Equation (21) are given in Table2

[21] The equations of motion are solved using Newmark’s method employing a time step

of 0.0002 s

3.2 Effect of braking on wheel sliding

The magnitude of the braking torque applied to decelerate a train will undoubtedly affectthe stability and safety of the train If the braking torque is high enough, some or all thewheels may slide, which is a cause for concern as possible rail derailment may occur Onthe other hand, if the braking torque is too low, all wheels tend to roll, but the braking