ride comfort of high-speed trains travelling over railway bridges

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Ride comfort of high-speed trainstravelling over railway bridgesMH Kargarnovin a , D Younesian a , D Thompson b & C Jones ba Mechanical Engineering Department , Sharif University ofTechnology , Tehran, Iranb Institute of Sound and Vibration Research (ISVR), University ofSouthampton , Southampton, UKPublished online: 06 Aug 2006.

To cite this article: MH Kargarnovin , D Younesian , D Thompson & C Jones (2005) Ride comfort ofhigh-speed trains travelling over railway bridges, Vehicle System Dynamics: International Journal ofVehicle Mechanics and Mobility, 43:3, 173-197, DOI: 10.1080/00423110512331335111

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Ride comfort of high-speed trains travelling over railway bridges

M. H. KARGARNOVIN*{, D. YOUNESIAN{, D. THOMPSON{ andC. JONES{

{Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran{Institute of Sound and Vibration Research (ISVR), University of Southampton, Southampton,

UK

The ride comfort of high-speed trains passing over railway bridges is studied in this paper. Aparametric study is carried out using a time domainmodel. The effects of some design parametersare investigated such as damping and stiffness of the suspension system and also ballast stiffness.The influence of the track irregularity and train speed on two comfort indicators, namelySperling’s comfort index and the maximum acceleration level are also studied. Two types ofrailway bridges, a simple girder and an elastically supported bridge are considered.Timoshenko beam theory is used for modelling the rail and bridge and two layers of parallel

damped springs in conjunction with a layer of mass are used to model the rail-pads, sleepers andballast.A randomly irregular vertical trackprofile ismodelled, characterized by its power spectraldensity (PSD). The ‘roughness’ is generated for three classes of tracks. Nonlinear Hertz theory isused for modelling the wheel-rail contact. The influences of some nonlinear parameters in acarriage-track-bridge system, such as the load-stiffening characteristics of the rail-pad and theballast and that of rubber elements in the primary and secondary suspension systems, on thecomfort indicators are also studied. Based on Galerkin’s method of solution, a new analyticalapproach is developed for the combination between the rigid and flexural mode shapes, whichcould be used not only for elastically supported bridges but also other beam-type structures.

1. Introduction

The dynamic interaction between the bridge and the train travelling over it is animportant subject that must be considered not only in the design of railway bridges butalso in the design of suspension systems of trains. These types of studies are carried outmostly for one of the following design considerations:

1. Dynamic and impact influences on the bridge structures.2. Design of the bridge structures against fatigue.3. The noise generated from trains.

*Corresponding author. Email: [email protected]

Vehicle System DynamicsVol. 43, No. 3, March 2005, 173 – 199

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online ª 2005 Taylor & Francis Group Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/00423110512331335111

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4. Stability and running safety of trains.5. Ride comfort of trains.

Particularly, in the case of high-speed lines, the dynamic behaviour of the bridgestructure must be taken into account in its design. It is important for the design of thetrain suspension and also the bridge structure. A brief review of research on vehicle –bridge interactions has been compiled by Yang and Yau (1996) with emphasis onapplications to high-speed railway bridges. Also, Fryba (1996, 1999) has pointed outmore than 200 references in this area. However, few studies have been carried out whichexamine the ride comfort of trains passing over railway bridges. Yang et al. (1999)studied the ride comfort inside a railway vehicle represented as sequence of movingloads and also as two degree-of-freedom systems as it travelled over a simply supportedsingle and three-span bridge. They used maximum acceleration level as the comfortindicator. Using the finite element method, they showed that the vertical irregularity,ballast stiffness, suspension system stiffness and damping could each substantially affectthe ride comfort. They also concluded that the design of a high-speed railway bridgeshould be governed by criteria for passenger comfort rather than for strength alone. Thedynamic response of a high-speed train excited by the irregularity of short and longwavelengths has been analysed with a coupled vehicle – bridge dynamics theory in Lin etal. (2001). In that paper, the safety criterion for the irregularity has been set based onmanagement standards of train safety and passenger comfort. The steady-state responseand ride comfort of trains moving over a series of simply supported bridges has beenstudied by Wu and Yang (2003). Sperling’s comfort index and maximum accelerationlevel were used for ride comfort analysis. By using a variety of irregularity spectra tomodel the rail roughness, it was shown that the level of comfort remains nearlyindependent of the train speed in the moderate to high-speed range (150 – 400 km/hr).In some countries, bridges are designed with soft bearings at the supports in order to

protect them from earthquakes. For this purpose, Yau et al. (2001) and Yang et al.(2004) have studied the impact response of elastically supported beams subjected to asequence of moving concentrated forces analytically. They calculated resonance andcancellation speeds in which moving forces amplify and reduce their own influences onthe beam response, respectively. In these two papers, the response of the elasticallysupported beam has been calculated using modal summation of a sine mode shape anda rigid body translation. These approximations, using sine mode shapes and ignoringthe rigid body rotation have been improved in the present paper.In the present paper, the ride comfort is studied using Sperling’s comfort index and

the maximum level of acceleration. The influence of some design parameters such asdamping and stiffness of the suspension system and also ballast stiffness on the comfortindicators is studied. For a conventional bridge, a comparison between the results of anonlinear, and an equivalent linear system is carried out. The nonlinearities consideredarise from two sources. On the vehicle, the rubber elements used in the primary andsecondary suspension systems, and on the track from the ballast and rail-pad load-stiffening characteristics. The effects of the rail ‘roughness’ and train speed on thecomfort indicators are also studied. The effect of support stiffness is also investigatedfor an elastically supported bridge.The track is modelled as a discretely supported Timoshenko beam on two

foundation layers representing the rail pad and ballast. The sleepers are representedas a continuous layer of mass. The bridge is also modelled as a Timoshenko beam and10 carriages, each modelled using 10 degrees of freedom, represent the train (figure 1).Galerkin’s time stepping method is used to solve the problem in the time domain after a

174 M.H. Kargarnovin et al.

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modal analysis. To represent the bridge structure, an exact method is developed for themode combination of elastically supported beams. This method, in which both therotational and translational rigid mode shapes are combined with exact flexural modeshapes of a free-free beam, is not only useful for dynamic analysis of elasticallysuspended bridges but also in the forced vibration analysis of similar beam-typestructures. The main purposes of this study can be stated as follows:

1. To examine the effects of nonlinearity in the track on the riding comfort and toexamine the error in an equivalent linear track model.

2. To study the effect of linearization of the behaviour of rubber elements in themodelling of the vehicle suspension systems of railway vehicles on the predictionof ride comfort.

3. To investigate how the rail roughness level influences the ride comfort.4. To study the influences of the damping and stiffness of the suspension system on

the ride comfort and looking for optimal values of these parameters in thepractical range.

5. To examine the effect of train speed on the ride comfort and also to look for asaturation phenomenon at high-speed range, in which comfort indicators remainunchanged with increasing speed.

6. To study the influences of the bridge support stiffness on ride comfort and howthe comfort indicators vary with it.

2. Mathematical modelling

In this section, the equations of motion of the vehicle, track and bridge are derived andpresented.

Figure 1. Vehicle, track and bridge models.

Ride comfort of high-speed trains 175

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2.1 Vehicle equations of motion

As shown in figure 1, the carriage body is connected to the two bogies via a secondarysuspension system. The coach body has vertical and pitch motions in the vertical plane,as do the bogies. The wheelsets are connected to bogies via the primary suspensionsystem. Only the vertical motion of the wheels is considered. According to the modelshown in figure 1, the equations of motion of the carriage components are derived asfollows.For the vertical and pitch motions of the coach body:

MBD €wBD þ FBDF þ FBDR þWBD ¼ 0 ð1Þ

JBD€jBD þ LcðFBDF � FBDRÞ ¼ 0 ð2Þ

For the vertical and pitch motions of the rear bogie:

MBG €wBGR � FBDR þ FBGR1 þ FBDR2 þWBG ¼ 0 ð3Þ

JBG€jBGR þ LtðFBGR2 � FBGR1Þ ¼ 0 ð4Þ

For the vertical and pitch motions of the front bogie:

MBG €wBGF � FBDF þ FBGF1 þ FBDF2 þWBG ¼ 0 ð5Þ

JBD€jBGF þ LtðFBGF2 � FBGF1Þ ¼ 0 ð6Þ

For the vertical motions of the first and second rear wheelsets:

Mw €wWR1 � FBGR1 þ FCR1 þWw ¼ 0 ð7Þ

Mw €wWR2 � FBGR2 þ FCR2 þWw ¼ 0 ð8Þ

For the vertical motions of the first and second front wheelsets:

Mw €wWF1 � FBGF1 þ FCF1 þWw ¼ 0 ð9Þ

Mw €wWF2 � FBGF2 þ FCF2 þWw ¼ 0 ð10Þ

In the equations (1) – (10) MBD, MBG and MW are the mass of coach body, bogie andwheelset and WBD, WBG and WW are their weights. JBD and JBG are the moments ofinertia of the body and bogie, respectively. wBD and jBG are the vertical and pitchdisplacements of the body, wBGR and fBGR are the vertical and pitch motions of therear bogie and wBGF and jBGF are the vertical and pitch motions of the front bogie.wWR1 and wWR2 are the vertical motions of the first and second rear wheelset and wWF1

and wWF2 are the vertical motions of the first and second front wheelset. FBDR andFBDF are the forces between the body and the rear and front bogies; in other wordsthese represent the secondary suspension system. Similarly, FBGR1, FBGR2 and FBGF1,FBGF2 are forces between bogies and wheelsets, representing the primary suspension.FCR1, FCR2, FCF1 and FCF2 are wheel-rail contact forces of the first and second rearwheelset and the first and second front wheelset. All of the mentioned forces will be


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fully defined in the next sections in the form of the linear and nonlinear force parts. Lc

is half distance between bogies and Lt is the semi-wheelbase.

2.2 Track equations of motion

Timoshenko beam theory expresses the equations of motion for the vertical deflection(wR) and rotation (cR) of the rail at any point as follows:

rRAR@2wRðx; tÞ

@t2þ kRARGRð@cRðx; tÞ

@x� @2wRðx; tÞ

@x2Þ ¼ pðx; tÞ þ Fðx; tÞ ð11Þ

ERIR@2cRðx; tÞ

@x2� kRARGRðcRðx; tÞ �

@wRðx; tÞ@x

Þ ¼ rRIR@2cRðx; tÞ

@t2ð12Þ

ER , GR and rR are Young’s modulus, shear modulus and density of the rail and AR, IRand kR are the area, the second moment of area and the shear factor of the rail crosssection.

F(x,t) is the wheel-rail contact force per unit length of the rail and is defined as

Fðx; tÞ ¼Xi

FCiðtÞ � dðx� xWiðtÞÞ ð13Þ

in which FCi is the contact force between the rail and ith wheel and xWi is the position ofthe ith wheel. d is the Dirac delta function. P(x,t) is the foundation force per unit lengthof the rail and is defined as:

Pðx; tÞ ¼Xi

FPiðtÞ � dðx� xiðtÞÞ ð14Þ

in which FPi is the force between the rail and ith pad element and xi is the position of ith

pad element. For the motion of the ith sleeper element

MS €wSi þ FPi þ FBi ¼ 0 ð15Þ

in which MS is the sleeper mass. wSi and FBi are the ith sleeper displacement and theforce between the bridge and ith ballast element.

2.3 Bridge equations of motion

Using Timoshenko beam theory to model the bridge

rBRABR@2wBR

@t2þ kBRABRGBRð@cBRðx; tÞ

@x� @2wBR

@x2Þ ¼

Xi

FBiðtÞ � dðx� xiÞ

�D�KE�S ��½wBRð0; tÞ � dðxÞ þ wBRðLBR; tÞ � dðx� LBRÞ�ð16Þ

EBRIBR@2cBRðx; tÞ

@x2� kBRABRGBRðcBRðx; tÞ �

@wBRðx; tÞ@x

Þ ¼ rBRIBR@2cBRðx; tÞ

@t2ð17Þ


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in which

D ¼ 0 For simply supported bridge1 For elastically supported bridge

�

All parameters in equations (16) and (17) have been introduced already in the railequations except that here they assume the subscript BR to indicate that they are theparameters of the bridge beam and KE-S is stiffness of the elastic support.

3. Case study

In this section the physical parameters used for the vehicle, track and bridge model arepresented.

3.1 Bridge

The bridge, a real concrete high-speed rail bridge, typical of ones used in high-speedlines in Taiwan (Yau et al. 1999, 2001), is modelled as a Timoshenko beam.Geometrical and mechanical properties of the bridge are listed in table 1.

3.2 Track

Geometrical and mechanical properties of the UIC 60 rail section modelled areindicated in table 2.

Table 1. Properties of the bridge (Taiwan-HSR (Yau et al. 1999, 2001)).

Item Notation Value

Young’s modulus (concrete) EBR 28.2 GPaShear modulus (concrete) GBR 11.75 GPaMass density rBR 3954.7 kg/m3

Cross sectional area ABR 7.94 m2

Second moment of area IBR 8.72 m4

Shear coefficient kBR 0.41Length LBR 30 m

Table 2. Properties of the UIC60 rail (Esveld 1989).

Item Notation Value

Young’s modulus (steel) ER 210 GPaShear modulus (steel) GR 77 GPaMass density rR 7850 kg/m3

Cross sectional area AR 7.69 x 107 3 m2

Second moment of area IR 30.55 x 107 6 m4

Shear coefficient kR 0.40


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Using a full-scale measurement, Dahlberg (2002) showed that a nonlinear trackmodel can simulate the rail deflection fairly well whereas the linear model cannot forhigh-speed trains. His parameter values obtained from measured data are used for thelinear and nonlinear models of the track in this paper. The physical properties of thistype of track are given in table 3. In the linear case, the pad and ballast force can bedefined as

FP ¼ KpDp þ CpD�p ð18Þ

FB ¼ KBDB þ CBD�B ð19Þ

in which DP and DB are displacements of the pad and ballast elements and the dotabove the symbols represents time derivative and in the nonlinear case they can bedefined as (Dahlberg 2002, Wu and Thompson 2004):

FP ¼ KPLDp þKPND3P þ CpD

�p ð20Þ

FB ¼ KBLDB þKBND3B þ CBD

�B ð21Þ

All the above parameters have been defined in table 3. It should be noted the values ofthe track parameters are appropriate for the frequency range of 0 to 50 Hz (Dahlberg2002). In this study, the track irregularity is assumed to be a random functioncharacterized by the following power spectral density (PSD) function S(O) (Fryba1996, Yau et al. 2001):

SðOÞ ¼ AO22

ðO2 þ O21ÞðO2 þ O2

2Þð22Þ

in which O (=2p/l) denotes the spatial frequency (rad/m), l is the wavelength ofirregularity (m) and A(m), O1 (rad/m) and O2 (rad/m) are relevant parameters indicatedin table 4. This spectrum is based on measurements made on the railways of the USA(Garg and Dukkipati 1984). The rail roughness level is shown in figure 2 for three typesof tracks. ‘Class 6’ corresponds to the best quality. Using the spectrum defined in

Table 3. Properties of the rail foundation.

Item Notation Value

Equal Linear

Sleeper mass MS 125 kgPad stiffness KP 220 MN/mPad viscous damping CP 700 kNs/mBallast stiffness KB 90 MN/mBallast viscous damping CB 1125 kNs/m

Nonlinear

Pad stiffness (Linear part) KPL 52 MN/mPad stiffness (Nonlinear part) KPN 6.246108 MN/m3

Pad viscous damping CP 700 kNs/mBallast stiffness (Linear part) KBL 22.75 MN/mBallast stiffness (Nonlinear part) KBN 2.66108 MN/m3

Ballast viscous damping CB 1125 kNs/m


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equation (22) and the wavelength range from 0.03m to 30m, the method proposed byAu et al. 2002 has been used to generate an irregularity profile of the track r(x). Theresult is shown for three classes of track in figure 3.Nonlinear Hertzian theory is used to model the wheel-rail contact. According to

this theory the contact force between the rail and wheel can be defined as (Esveld1989):

FCW¼ �CHðwðxwðtÞ; tÞþrðxwðtÞÞ�wWðtÞÞ3=2 for wðxwðtÞ; tÞ þ rðxwðtÞÞ � wWðtÞ > 00 for wðxwðtÞ; tÞ þ rðxwðtÞÞ � wWðtÞ � 0

�;

ð23Þ

in which wW is the deflection at the wheel and w(xW(t),t) is the deflection of the rail atthe point beneath the wheel and r(xW(t)) is the rail roughness level at the position of thewheel. CH is the Hertzian contact spring constant, equal to 9.24 x 1010 Nm7 3/2

calculated for the SKS300 high-speed train wheelset and UIC60 rail.

Table 4. Track PSD model parameters (Yau et al. 2001).

Track Class 4 5 6

A (m) 2.396 107 5 9.356 107 6 1.56 107 6

O1 (rad/m) 2.066 107 2 2.066 107 2 2.066 107 2

O2 (rad/m) 0.825 0.825 0.825

Figure 2. Roughness spectrum level for three track-classes.


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3.3 Train

In this study, the series 300 mcarriage type of Japan’s Shinkansen (SKS) system ismodelled. The primary properties of the SKS300 are listed in tables 5 and 6.

Three types of suspension systems are considered in this paper. The first is a linearprimary and secondary suspension system. The mechanical properties of this kind ofsuspension system have been listed in table 6 from Wu and Yang (2003) and Yang andWu (2002). In this type of system, the primary and secondary suspensions are modelledby two sets of parallel linear springs and viscous dampers.

For the second system type, the secondary suspension is modelled by a nonlinearrubber element and the primary suspension system remains unchanged. For this type ofrubber spring, the relationship between the force and motion is based on asuperposition of three forces

FRub�Spr ¼ Fe þ Ff þ FM ð24Þ

Fe, Ff and FM are an elastic, friction and viscous force, respectively. This model isillustrated in figure 4 (Berg 1997).

In this type of rubber model, a linear spring, Coulomb friction and a Maxwellviscoelastic element are in parallel. It should be noted that a Maxwell element includesa linear spring and a dashpot in series. The data used for this type of rubber element arelisted in table 6.

The corresponding force-displacement loops are illustrated in figure 5 for threetypes of harmonic excitation. It can be seen that the equivalent linear systemincluding a spring and viscous damper in parallel has a hardening property versusfrequency this means that the slope of the major axis of closed loops increases with

Figure 3. Track irregularity profile generated for three track-classes.


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increasing frequency. It has been shown by Berg (1997) that equivalent linear stiffnessdecreases with decreasing amplitude of displacement. The equivalent linear damping,which can be calculated by equalization of the energy dissipated in one cycle, is alsofrequency dependent. It has a softening behaviour versus increasing displacementamplitude and a hardening behaviour versus increasing frequency (Berg 1997). Thisapproach to the modelling of rubber elements is valid in frequency range of 0 to 20Hz (Berg 1997). Using an averaging method in the intervals of 0 to 5 mm for theamplitude and 0 to 20 Hz for the frequency, the stiffness and damping have beencalculated for the equivalent linear system including a linear spring in parallel with alinear viscous damper. The equivalent calculated values are listed in table 6(Kargarvonin et al. 2004).

Figure 4. Rubber element model assumption.

Figure 5. Force versus displacement for harmonic excitation of rubber element with amplitude of 1 mmdifferent frequencies.


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In the third type of suspension system used in this study, nonlinear rubber springsmodel the primary suspension system and the secondary suspension remainsunchanged. For each case, the properties of the primary and secondary suspensionsystems are listed in table 6.

4. Numerical results

In this study, Galerkin’s method is used to solve the nonlinear time-dependentcoefficients of the coupled partial differential equations. In this method, the responsesof the rail and bridge assume to be in the form of

wRðx; tÞ ¼P

i jRiðxÞqRiðtÞCRðx; tÞ ¼

Pi CRiðxÞqRiðtÞ

�ð25Þ

for the rail response and

wBRðx; tÞ ¼P

i jBRiðxÞqBRiðtÞCBRðx; tÞ ¼

Pi CBRiðxÞqBRiðtÞ

�ð26Þ

for the bridge response. fR and cR are the mode shapes of the rail in vertical deflectionand rotation in the vertical plane and similarly fBR and cBR are the mode shapes ofbridge. Using the principle of orthogonality of mode shapes, the nonlinear equations of

Figure 6. The solution flow diagram.


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motion can be transformed into nonlinear sets of ordinary coupled differentialequations. The procedure of combining rigid and flexural modes is explained for theelastically supported bridge in the Appendix. A computer program has been developedin MATLAB software using Newmark’s method of integration. Using FFT and, aftercalculation of the response in the time domain, the response in the frequency domain iscalculated. Subsequently the response is weighted according to Sperling’s filter (Esveld1989).

HðfÞ ¼ 0:5881:911f2 þ ð0:25f2Þ2

ð1� ð0:277f2Þ2 þ ð1:563f� ð0:0368f3Þ2" #1

2

ð27Þ

with f representing the frequency in Hz. Finally the comfort index is calculated usingthe RMS value of the weighted acceleration as

Comfort Index ¼ 4:42 ðaWRMSÞ0:3 ð28Þ

in which aWRMS is the root-mean-square value of the weighted acceleration (m/s2)weighted with Sperling’s filter.The calculation procedure is shown in the flow diagram of figure 6. The maximum

acceleration and comfort index are calculated for the linear and nonlinear tracks andalso for three different types of suspension. Results are illustrated in figures 7 – 13versus train speed from 0 to 400 km/hr.For the maximum acceleration, two limiting values are considered; 0.05g imposed by

the SNCF (Grandil and Ramondence 1990), and a less strict limit of 1 m/s2 imposed by

Figure 7. Maximum acceleration of train under various train speeds for the first type of suspension system.


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Eurocode (European Committee for Standardization 1995). For the comfort index,each graph has been divided into regions for the descriptive labels given to variousranges of the index (see table 7). It can be seen from figures 7 and 8 that the ridecomfort can be significantly affected by the presence of the track irregularity. In orderto study the effects of two other sources of vibration – bridge-passing and sleeper-passing – another analysis has been carried out for two cases in three different speedsand the results have been illustrated in figure 9. In the first case, the bridge is carryingthe moving train without any track and in the second case the bridge assumes to berigid and in both cases no irregularity is assumed to be present.

A comparison between sleeper-passing and bridge-passing effects shows that the firstone is the dominant effect. Figure 9 also shows, in higher speeds these two effectsgetting closer.

It can also be seen from figures 7 and 8 that for the smooth tracks (no irregularprofile at all) both indicators –maximum acceleration and comfort index – remainindependent of train speed in the moderate to high-speed range (more than 150 km/hr).They are linearly increasing functions of the train speed in high-speed range. The slopesof the comfort indicators with speed are nearly independent of the class of track. This isan interesting result as it indicates that, in the high-speed range, these comfortindicators are predictable using a simple extrapolation or interpolation technique thatcould be based on empirical data.

Using FFT, a frequency analysis has been carried out in the case of peak speed. Fora peak speed of 54 km/hr, dominant frequency of 0.863 Hz is computed which is nearto the first natural frequency of the vertical vibration of the train (i.e. 0.847 Hz).Considering the frequency of the passing of the bogies (i.e. v/2Lc=0.856 Hz) one canobtain that this peak is related to the passing of bogies.

Figure 8. Comfort index of train under various train speeds for the first type of suspension system.


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For the smooth tracks, a comparison between results of the linear and nonlineartrack models reveals a big difference, however in the irregular track models, there isonly a small difference between these results observable only for very high train speeds.

4.1 Effect of the vehicle suspension

Results related to the effect of the suspension system are presented in figures 10 and11 in which the model of rubber is used as secondary suspension system and infigures 12 and 13 in which it is used as the primary suspension system. It can beseen that the conclusions about the variation of the comfort indicators versus trainspeed in the high-speed range still hold. Also a good agreement between nonlinearand linear models is observable in the high-speed range. These results also showthat using the equivalent linear system in design procedures is a safe approach forthe moderate speed range only when rubber elements are in the secondarysuspension system and for low speeds when they are used in primary suspensionsystem.The influences of the damping and stiffness of the secondary suspension system on

the comfort indicators are shown in figures 14 – 17.In these figures a damping and stiffness ratio are defined as:

bD ¼ Secondary suspension damping

Actual secondary suspension damping

bS ¼ Secondary suspension stiffness

Actual secondary suspension stiffness

ð29Þ

Figure 9. Bridge-passing and sleeper-passing effects on the comfort index.


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Figure 10. Maximum acceleration of train under various train speeds for the second type of suspensionsystem.

Figure 11. Comfort index of train under various train speeds for the second type of suspension system.


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Figure 12. Maximum acceleration of train under various train speeds for the third type of suspension system.

Figure 13. Comfort index of train under various train speeds for the third type of suspension system.


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Figure 14. Effect of secondary suspension damping on the maximum acceleration.

Figure 15. Effect of secondary suspension damping on the comfort index.


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Figure 16. Effect of secondary suspension stiffness on the maximum acceleration.

Figure 17. Effect of secondary suspension stiffness on the comfort index.


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in which, actual values of damping and stiffness are related to the first type ofsuspension system and their values can be found in table 6. These two parameters havebeen defined in order to search for optimal values of suspension damping and stiffnessover a practical range.

It can be seen from figures 14 and 15 that there is an optimum value for dampingthat lies close to the designated damping value of the SKS 300 (bD=1). The optimumdamping value increases with increasing train speed and using values higher than theoptimum damping value can drastically magnify the response of the carriage. Figures16 and 17 show that comfort indicators are monotonically increasing functions of thesecondary suspension stiffness and for high-stiffened suspensions the increasing slope isindependent of track class.

4.2 Effect of the ballast

The effects of the ballast stiffness on the maximum acceleration and also ride index areshown in figures 18 and 19. It can be seen that there is no optimum value of ballaststiffness that results in minimum value of comfort index. The saturation phenomenontakes place with increase of ballast stiffness. Softer ballast increases the values ofcomfort indicators significantly in the case of smooth rails. The linear model for thetrack has been used in this case and

xB ¼ Ballast Stiffness

Actual ballast stiffness ðEquivalent linearÞ ð30Þ

Figure 18. Effect of ballast stiffness on the maximum acceleration.


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in these two figures. This new parameter has also been defined to investigatewhether an optimal value of ballast damping exists in the practical range of ballaststiffness.

4.3 Effect of the bridge support stiffness

The analytical approach already described is used for the exact solution of the problemin the case of an elastically supported bridge. For a practical range of support stiffness[7,8], results are plotted in figures 20 and 21. In these figures the support stiffness ratiois defined by

gSup ¼ Reference Support Stiffness

Support Stiffnessð31Þ

in which

Reference Support Stiffness ¼ EBRIBRp3

L3BR

¼ 294:71MN

mð32Þ

gSup represents the ratio of support stiffness to a measure of bridge stiffness and a zerovalue of gSup indicates the special case of a simply supported bridge.These results indicate that the comfort indicators increase linearly with support

flexibility and this variation is independent of the track roughness level.

Figure 19. Effect of ballast stiffness on the comfort index.


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Figure 21. Effect of support stiffness on the comfort index.

Figure 20. Effect of support stiffness on the maximum acceleration.


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5. Conclusion

The ride comfort of trains passing over railway bridges has been studied. Two comfortindicators, maximum acceleration and Sperling’s comfort index, were calculated using atheoretical model for four different classes of tracks. The effects of train speed, railroughness and some nonlinear load-stiffening characteristics of the rail-pad, ballast andrubber elements in the primary and secondary suspension systems, on the ride comfortindicators were investigated. An exact analytical method of solution was developed forthe case of elastically supported bridges and the influences of the suspension stiffnessand damping, ballast stiffness and bridge support stiffness on the comfort indicatorshave been presented. The following conclusions are made from the numerical studies:

1. The influences of roughness level and also the train speed on the comfortindicators were studied. The ride comfort of the train can be significantlyaffected by the rail roughness level within the normal range of roughnessdisplayed by different classes of tracks in the wavelength range of 0.03m to30m. Between three sources of vibration (i.e. track irregularity, bridge-passingand sleeper-passing) the track irregularity is the dominant parameter on theride comfort.

2. For the smooth tracks, the differences between the results of linear andnonlinear track models are significant and the comfort indicators remainindependent of the train speed in the moderate to high-speed range.

3. There is a speed in which a peak value happens for maximum acceleration. Inthis speed, frequency of the passing of bogies is equal to the first naturalfrequency of the train.

4. Comfort indicators are approximately linearly increasing functions of the trainspeed in the high-speed range. The slopes of the comfort indicators with speedare nearly independent of the roughness level of track.

5. The effects of nonlinearity in the track and also in the primary and secondarysuspension on the riding comfort were examined. There is a good agreementbetween the results of a nonlinear track model and an equivalent linear trackmodel. Consequently the linear track model is adequate for comfort analysis oftrains on bridges.

6. In the case of rubber suspension systems, it is acceptable to use the linearsuspension model in which the equivalent stiffness and damping are calculatedusing an averaging method in design studies for the high-speed range and is arelatively safe approach in the moderate and low speed ranges for secondaryand primary rubber suspension systems, respectively.

7. The influences of the secondary suspension system parameters, the ballastproperties and the bridge support stiffness on the ride comfort were studied.There is an optimum value of secondary suspension damping and the designatedvalue of damping for the SKShigh-speed train is located in the optimum region.The value of optimum damping increases with increasing train speed.

8. Comfort indicators increase monotonically with increasing secondary suspen-sion stiffness and for very stiff suspensions the slope of the increase isindependent of track class.

9. A limit of comfort indicators is reached as the ballast stiffness increases.10. Comfort indicators are linear functions of support stiffness and magnified by

decreasing the value of support stiffness. The rate of this increase isindependent of the track roughness level.


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Acknowledgment

The first and second authors are grateful to the British Council in Tehran, Iran, forgranting the scholarship to the second author to conduct this research at the Instituteof Sound and Vibration Research (ISVR), and they extend their thanks to the caringstaff for their sincere help, especially Dr Fatemeh Ahmadi, the scholarships officer andDr Shahriar Shahidi, the education manager.

References

[1] Au, F.T.K., Wang, J.J. and Cheung, Y.K., 2002, Impact study of cabled- stayed railway bridge withrandom rail irregularities. Engineering Structures, 24, 529 – 541.

[2] Berg, M., 2001, A model for rubber springs in the dynamic analysis of rail vehicles. J. Rail Rapid Transit,112, 95 – 108.

[3] Dahlberg, T., 2002, Dynamic interaction between train and nonlinear railway model. In Proceedings of5th International Conference on Structural Dynamics, Munich, pp. 1155 – 1160.

[4] European Committee for Standardization. 1995, EUROCODE 1: basis of design and actions onStructures. Part 3: traffic loads on bridges, ENV 1991-3.

[5] Esveld, C., 1989, Modern Railway Track.[6] Fryba, L., 1996, Dynamics of Railway Bridges, (New York: Thomas Telford).[7] Fryba, L., 1999, Vibration of Solids and Structures under Moving Loads (Thomas Telford: New York).[8] Garg, V.K. and Dukkipati, RV., 1984 Dynamics of Railway Vehicle Systems (Canada: Academic Press).[9] Geist, B. and McLaughlin, J.R., 2001, Asymptotic formulas for the eigenvalues of the Timoshenko

beam. J. Math. Anal. Applns., 253, 341 – 380.[10] Grandil, J. and Ramondence, P., 1990, The Dynamic Behaviour of Railways on High-Speed Lines,

(SNCF: Paris).[11] Kargarnovin, M.H., Younesian, D., Thompson, D.J. and Jones, C.J.C., 2004, Nonlinear vibration and

comfort analysis of high-speed trains moving over railway bridges. In Proceedings of ESDA2004,Manchester.

[12] Lin, Y., Xin, L., Yang, C. and Zou, Z., 2001, Study on dynamic response of train excited by theirregularity of track on the high speed railway bridge. J. Vib. Shock, 3, 47 – 49.

[13] Yang, Y.B. and Yau, J.D., 1996, A review of researches on vehicle-bridge interaction with emphasis onhigh-speed rail bridges. In Proceedings of 20th National Conference on Theoretical and AppliedMathematics, Taipei.

[14] Yang, Y.B. and Wu, Y.S., 2002, Behavior of moving trains over bridges shaken by earthquakes, InProceedings of 5th International Conference on Structural Dynamics, Munich, pp.509 – 514.

[15] Yang, Y.B., Lin, C.L., Yau, J.D. and Chang, D.W., 2004, Mechanism of resonance and cancellation fortrain-induces vibrations on bridges with elastic bearings. J. Sound Vib., 269, 345 – 360.

[16] Yau, J.D., Yang, Y.B. and Kuo, S.R., 1999, Impact response of high-speed rail bridges and ridingcomfort of rail cars. Engineering Structures, 25, 251 – 265.

[17] Yau, J.D., Wu, Y.B. and Yang, Y.B., 2001, Impact response of bridges with elastic bearings to movingloads. J. Sound Vib., 248, 9 – 30.

[18] Wu, Y.S. and Yang, Y.B., 2003, Steady-state response and riding comfort of trains moving over a seriesof simply supported bridges. Engineering Structures, 25, 251 – 265.

[19] Wu, T.X. and Thompson D.J., 2004, The effects of track non-linearity on wheel/rail impact. J. RailRapid Transit, 218, 1 – 12.

Appendix. Mode summation procedure for elastically supported bridge

Using equations of free motion of a Timoshenko beam

rBRABR@2wBR

@t2þ kBRABRGBRð@cBRðx; tÞ

@x� @2wBR

@x2Þ ¼ 0

EBRIBR@2cBR

@x2� kBRABRGBRðcBRðx; tÞ �

@wBR

@xÞ ¼ rBRIBR

@2cBR

@t2

and free-free boundary conditions of


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x ¼ 0:0;LBR )@wBR

@x � cBR ¼ 0:0dcBR

dx ¼ 0:0

(

one can get

DET

0 s2

a2�s20 s2

B2�s2

ba 0 bB 0s2SinðbaÞa2�s2

s2CosðbaÞa2�s2

s2SinðbBÞB2�s2

s2CosðbBÞB2�s2

baCosðbaÞ �baSinðbaÞ bBCosðbBÞ �bBSinðbaÞ

2664

3775 ¼ 0:0

For the frequency equation and

jBRðxÞ ¼ c1 Cosðba x

LBRÞ þ c2 Sinðba x

LBRÞ þ c3 CosðbB x

LBRÞ þ c4 Sinðba x

LBRÞ

cBRðxÞ ¼ d1 Cosðba x

LBRÞ þ d2 Sinðba x

LBRÞ þ d3 CosðbB x

LBRÞ þ d4 Sinðba x

LBRÞ

for the mode shape functions (Geist and McLaughlin 2001). In the above equations

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ s2

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr

2 � s2

2Þ2 þ 1

b2

rs

B ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ s2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr

2 � s2

2Þ2 þ 1

b2

rs

b2 ¼ rBRABRL4BRo

2

EBRIBR; r2 ¼ IBR

ABRL2BR

; s2 ¼ EBRIBR

kBRABRGBRL2BR

in which o is the natural frequency of a related free-free beam. Using orthogonalityprinciple of mode shapes

ZLBR

0

rBRABR fBRiðxÞfBRjðxÞ þ rBRIBR cBRiðxÞcBRjðxÞ ¼ 0 i 6¼ jMBRi i ¼ j

�

and equations (16), (17) and (26) leads to

For o1 ¼ 0:0 ) jBR1 ¼ x� L=2 ; cBR1 ¼ 1:0 Rigid RotationjBR1 ¼ 1:0 ; cBR1 ¼ 0:0 Rigid Displacement

�

€qBR1ðtÞ ¼~F

~A~F� ~B2

ZLBR

0

ðx� LBR

2Þpðx; tÞdx

q BR1 ðtÞ ¼~A

~A~F� ~B2

ZLBR

0

pðx; tÞdx

in which


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~A ¼ZLBR

0

jBR1ðxÞjBR1ðxÞdx ; ~B ¼ZLBR

0

jBR1ðxÞj BR1 ðxÞdx ;

~F ¼ZLBR

0

j BR1 ðxÞj BR1 ðxÞdx

and

qBRnðtÞ þ o2nqBRnðtÞ ¼

ZLBR

0

jBRnðxÞPðx; tÞdx n ¼ 2; 3; ::

In the above equation for an elastically supported bridge

Pðx; tÞ ¼XNi�1

FBiðtÞ � dðx� xiÞ �KE�S � ½wBRð0; tÞ dðxÞ þ wBRðLBR; tÞ dðx� LBRÞ�

In which

WBRð 0LBR

� �; tÞ ¼

Xj

jBRjð 0LBR

� �Þ qBRjðtÞ:


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ride comfort of high-speed trains travelling over railway bridges

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