A new model in rail–vehicles dynamics considering nonlinear suspension component behavior

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International Journal of Mechanical Sciences 51 (2009) 222–232

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

A new model in rail–vehicles dynamics considering nonlinear suspensioncomponents behavior

H. Sayyaadi �, N. Shokouhi

Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9567, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 11 June 2008

Received in revised form

11 January 2009

Accepted 14 January 2009Available online 21 January 2009

Keywords:

Rail vehicle dynamics

Air spring model

Track geometrical irregularities

03/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ijmecsci.2009.01.003

esponding author. Tel.: +98 2166165682; fax

ail address: [email protected] (H. Sayyaadi

a b s t r a c t

In this paper, a complete four axle rail vehicle model is addressed with 70 degrees of freedom (DOFs)

including a carbody, two bogies, and four axels. In order to include the effects of the track irregularities

in vehicle dynamics behavior, a simplified track model is proposed and it is validated by some

experimental data and test results. As the performance of the suspension components, especially for air

springs, have significant effects on rail–vehicle dynamics and ride comfort of passengers, a complete

nonlinear thermo-dynamical air spring model, which is a combination of two different models, is

introduced and implemented in the complete rail–vehicle dynamics. By implementing Presthus

formulation [Derivation of air spring model parameters for train simulation. Master dissertation,

Department of Applied Physics and Mechanical Engineering, Division of Fluid Mechanics, LULEA

University, 2002], the thermo-dynamical parameters of air spring are estimated and then they are tuned

based on the experimental data. The results of the complete rail–vehicle field tests, show remarkable

agreement between proposed model and test data.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, speed-up in technology and its new features bringhigher speed with reliable safety and better ride comfort in railtransportation industries. Traffic jam in capital and big cities allaround the world, wasting passengers’ time at the air terminals,huge mass transportation, and so on brings a good opportunity forrail industries to attract more and more passengers and cargosto their services. In addition to safety, the other important factorfor the passengers to decide about their transportation type isride comfort. And that is why accessing better ride comfort forpassengers during their trip is essential.

To serve better ride comfort to the passengers, the secondarysuspension of most new EMU and DMU rail–vehicles is equippedwith air springs. Air springs are very important isolation com-ponent, which guarantee good ride comfort for the passengersduring their trip. In the most published rail–vehicle models,developed by researches [2–12], the thermo-dynamical effects ofair springs in the rail–vehicle dynamics are ignored and thesecondary suspension of vehicle is modeled by some simplesprings and dampers models.

In this paper, complete dynamics of one IRICo DMU trailer carwith nonlinear components behavior is addressed. The dynamicsbehaviors of all components are validated by some experimental

ll rights reserved.

: +98 2166000021.

).

results. In the proposed model, track rails assumed to be rigidwith viscoelastic bed in vertical and lateral directions [13,14]. Inorder to consider the effects of the track irregularities on thevehicle behavior, track data measured by EM120 machine areused. In the model of rail vehicle which moves along the straightline, effects of ballast and sleepers masses on the vehicledynamics are ignored. Four contact parameters introduced byShabana and Zaazaa [15,16] are used to define the contact pointbetween rail and wheel. In order to improve the simulationperformance, a feed-forward neural network (FFNN) is trainedand then it is used to compute the contact point parameters.Contact forces are calculated based on the Polach theory [17].Numerical and experimental results are summarized and com-pared at the end of this article to verify the proposed model andtechnique.

2. Vehicle description

IRICo DMU train has four cars; two motor cars at both ends andtwo trailer cars in the middle. The schematic diagram for onecomplete train is depicted in Fig. 1.

Each car is suspended by two bogies. The side view of twoaxle IRICo DMU trailer bogie is shown in Fig. 2. To attain properstability and good ride comfort for the passengers, bogies areequipped with primary and secondary suspension systems. Thesecondary suspension has two air springs to suspend the vehiclebody, four vertical and lateral dampers, and two connection links

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Fig. 1. Composition of IRICo DMU.

Fig. 2. Side view of IRICo DMU trailer bogie.

Fig. 3. Track model; parameters are based on the work of Jin and Wen [13]. Rail

disturbances: dleft,horizontal, dleft,vertical, dright,horizontal, and dright,vertical.

g(s1 )

f(s2)Yrp

Yw

S1Zrp

Q Zw

Xw

S1

S2

Qw

w

r

r r

S2w

Fig. 4. Four contact parameters introduced by Shabana and Zaazaa [15].

Fig. 5. Effective parameters dL, dV, cA, and yA for contact point extraction.

Fig. 6. Feed-forward neural network.

Fig. 7. Contact point position in left rail.

H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232 223

to connect the bogie frame to the carbody. The primarysuspension is made of two coil springs, two leaf springs and onevertical damper at each side of wheel set.

3. Dynamics model

3.1. Track model

In order to include effects due to the track geometricalirregularities in the rail–vehicle dynamics, simplified version ofJin and Wen [13,14] track model is in use here. In the proposedmodel, four track irregularities are introduced as bed distur-bances. As the vehicle is modeled while passing through straightline, the effects of ballast and sleepers masses in the vehicledynamics and coupling effects between left and right ballastmasses are ignored. Track stiffness and damping rates areaccording to the work of Jin and Wen [13]. Schematic diagramof the track model, with four track irregularities, is shown in Fig. 3.

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H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232224

3.2. Rail–wheel contact point and contact forces model

By using four geometrical contact parameters Sr1, Sr

2, Sw1 ,

and Sw2 , shown in Fig. 4, and the method proposed by Shabana

and Zaazaa [15], contact point between rail and wheel is deter-mined. According to Shabana, Sr

1 is found out by integrating speedcomponent of point Q along the rail, shown in Fig. 4, and Sw

2 isthe angle between the Z-axis of coordinate system fixed tothe axle and vertical line. The remaining two contact parametersare calculated by a searching algorithm which guarantees theperpendicularity of rail reaction force and tangent plane of thewheel at the contact point. To do this, four geometricalparameters dL, dR, yA, and cA, shown in Fig. 5, are investigated.By these assumptions and using four contact parameters andnormal reaction force of the rails, creep forces are calculatedbased on the Polach theory [17].

For improving the simulation capabilities and also decreasingthe simulation time, a feed-forward neural network is suggestedto compute these two contact parameters as shown in Fig. 6. Thisneural network has four inputs dL, dR, yA, and cA, two outputs Sr

2 &

Fig. 8. Contact point position in left wheel.

4 Leaf Springs4 Coil Springs

2 Vert. Damper

CGAxle

2 Lat Dam

2 Air Sprin

Traction Rods

Lat Dampers Lat Buffers,

Z-Link

CGcar

CGframe

CGAxle

Air SpringVert. Damper

Air SpringVert. Damper

Leaf Springs Coil Springs

Vert. Damper

Leaf Springs Coil Springs

Vert. Damper

Fig. 9. Vehicle mode

Sw1 and 14 neurons in two hidden layers. The activation function

for hidden layers and output layer is tangent sigmoid and linear,respectively. The network is trained based on the Levenberg–Marquardt back-propagation method.

Proposed network is trained according to the searchingalgorithm results, generated during sinusoidal variations of axlestates. The lateral and vertical axle states changed with 0.01 and0.003 mm amplitude and 0.5 and 0.25 Hz frequencies, respec-tively. At the same time, yaw and roll angle of axle varied with 0.41and 0.21, amplitude and 0.334 and 0.1667 Hz frequencies. Thesimulation time is selected so that each input signal has at leasttwo periods in the simulation time. Results are shown in Figs. 7and 8. Good agreement between network output and searchingalgorithm results guarantees the performance of trained network.Synaptic weights of the trained network are presented inAppendix 1.

4 Leaf Springs 4 Coil Springs

2 Vert. Dampers

CGcar

CGframe

CGAxle

s

4 Leaf Springs 4 Coil Springs

2 Vert. Dampers

2 Lat Dampers+2 Lat Buffers Z-Link

2 Air Springs+2 Vert. Dampers

CGframe

CGAxle

4 Leaf Springs 4 Coil Springs

2 Vert. Dampers

CGAxle

pers+2 Lat Buffers Z-Link

gs+2 Vert. Dampers

l with 42 DOFs.

Table 1Masses and inertia properties of the rail–vehicle.

Parameters Values Parameters Values Parameters Values

maxle 1747 kg mframe 2841.3 kg mcarbody 33,142 kg

Ixx,axle 1098 kg m2 Ixx,frame 1030 kg m2 Ixx,carbody 30,000 kg m2

Iyy,axle 191 kg m2 Iyy,frame 1054 kg m2 Iyy,carbody 687,231 kg m2

Izz,axle 1098 kg m2 Izz,frame 2003 kg m2 Izz,carbody 687,231 kg m2

Kex

Kez

Czβ

β

z2

2M

Kvx Cx

x2

Kexχ

Ffx,maxFfz,max Kvz

z

1 1

x

Fig. 10. Berg air spring model [27].

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F

X1 X1

X2Felastic

Fviscoelastic C = c (|x1|+�)

X2

k

F

·

Fig. 11. One-dimensional Haupt and Sedlan air spring model.

Fig. 12. Proposed modified model for air spring.

Table 2Air spring parameters value.

Parameters Description Values

ls Connecting pipe length 3.2 m

As Connecting pipe cross section 0.001359 m2

Ae Effective area of air spring 0.291 m2

P0 Initial absolute air spring pressure 3.806 bar

Pg Gauge pressure –

r Density of air at P0 pressure 4.523 kg/m3

Vr0 Reservoir volume 0.04 m2

Vb0 Air spring volume 0.064 m2

kt Total lost coefficient of connection pipes 3.4727

Kauxiliary Auxiliary spring stiffness in air spring 8234 kN/m

M Air mass in the pipes, air bag and air reservoir 198.385 kg

Fig. 13. Test rig of air spring at ContiTech Company, Germany.

Fig. 14. Loss angle of air spring M ¼ 198.385 kg, Kez ¼ 461.629 N/mm,

Kvz ¼ 351.185 N/mm, Cz,1.8 ¼ 11.508 kN s/m1.8, khaupt ¼ 96 kN/m, and Chaupt ¼ 1.52

kN s/m, z ¼ 0.00063.

Fig. 15. Dynamic vertical stiffness of the air spring M ¼ 198.385 kg, Kez ¼ 461.629

N/mm, Kvz ¼ 351.185 N/mm, Cz,1.8 ¼ 11.508 kN s/m1.8, khaupt ¼ 96 kN/m, Chaupt ¼

1.52 kN s/m, z ¼ 0.00063.

H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232 225

3.3. Vehicle model

This work proposes a new method for studying suspensioncomponents effects on the vehicle dynamics performances.

Accordingly, real behaviors of all components are described inthis section. For this reason, and due to the complexity ofcomponents behaviors, especially for air springs, it is almostimpossible to describe complete vehicle dynamics with a singleset of equations. So, the vehicle is modeled as modular type andinternal forces of each component are calculated, using nonlineardescription functions and system states. Newtonian approach isimplemented for dynamics modeling of different parts and theinteracting forces and moments between them are investigated.The vehicle model with 7 lumped masses and 42 degrees offreedom (DOFs) is shown in Fig. 9 and some of its importantspecifications are tabulated in Table 1.

3.3.1. Air spring model

Air springs, which are made of carbon black filled naturalrubber (CBFNR), have long lifetime and can isolate the vehiclebody from the unpredictable noise, vibration, and disturbances.

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Fig. 16. Lateral hysteresis loop of the air spring Key ¼ 154 kN/m, Kvy ¼ 82.26 kN/m,

Cy ¼ 1.109 kN s/m, khaupt ¼ 96 kN/m, Chaupt ¼ 1.52 kN s/m, z ¼ 0.00063.

Fig. 17. Test results of IRICo DMU Dampers, done by SACHS Co., Germany.

Table 3Dampers parameters.

Type of damper Condition a b

Primary vertical dampers 0pj _Ddamper jp0:1 40800 0

0:1o _Ddamper19163 2163.7

_Ddampero� 0:1 19025 �2177.5

Secondary vertical/lateral dampers 0pj _Ddamper jp0:05 61375 0

0:05o _Ddamper18221 2157.7

_Ddampero� 0:05 14143 �2361.6

Stopper

Lateral(4.5) 17±1

40±1(3.5)

Fig. 18. Lateral buffer position in bogie.

Fig. 19. Lateral buffer stiffness.

Table 4Haupt and Sedlan model parameters and stiffness coefficients of bushes.

Parameters Values Parameters Values

khaupt 9 kN/mm Kradial 2222.2 N/mm

Chaupt 3.8 kN s/m Ktorsionc 31.25 N m/1

z 0.00063 kcardanic 45.525 N m/1

H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232226

The air spring behaviors are so complicated that cannot bemodeled by simple equations. Air spring response is independentof the excitation frequency [18] and it behaves such as stressrelaxation function [19]. In addition, it has asymmetric hysteresisloop, which is independent of the excitation frequency [20]. Thesebehaviors bring difficulties to use frictional or columbic descrip-tion to approximate the air spring characteristics.

A lot of comprehensive researches have been done for real airsprings and CBFNR behaviors identification [21–26]. In the latest3D model, developed by Berg [27], the effects of elasticity, frictionand viscosity of air spring in vertical, lateral and longitudinaldirections are introduced. In this model, stress relaxation is notrepresented. The complete explanation about Berg model, shownin Fig. 10, is presented in [27]. Friction force in the Berg modelwhich is zero at turning points is

Ffriction /ðx� x0Þ

bþ ðx� x0Þsignð_xÞ (1)

In which, b is the constant, x the current displacement, and x0

the displacement at the previous turning point. As it is clear from

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H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232 227

Eq. (1), in this model, displacement at turning points should bedetected and assigned to the x0 variable, which is not a standardprocedure in the dynamic analysis and causes failure in solvingalgorithm.

For simulation of CBFNR behaviors, a model was developedby Haupt and Sedlan [19], which has elastic and viscoelasticelements. The viscoelastic element naturally produces asym-metric stress–strain response and is weakly time-dependent.These features are not presented in the other models and made itvery powerful to simulate the CBFNR behaviors. The simplifiedone-dimensional Haupt and Sedlan model with constant coeffi-cients is shown in Fig. 11.

The viscoelastic force in this simplified model is

Fvisco�elastic ¼c _x2

j_x1j þ z(2)

in which c and z are constant coefficients.In this research work, the Berg model which is validated by

some experimental data up to 16 Hz frequencies [27], is used tosimulate the air spring dynamics. However, because of difficulties

Bogie frame

Axle Box

AxleBox

Leaf Spring

Bushkz,leaf

kx,leaf

kx,bush

kz,bush

kty,bush

Fig. 20. Leaf spring schematic diagram in the x–z plane.

Table 5Primary suspension system, mechanical specifications.

Parameters Values (N/mm) Parameters Values

Kx,bush, Kz,bush 45000 Kty,bush 32 N m/1

Ky,bush 2118 Ktx,bush, Ktz,bush 1659 N m/1

Kx,leaf spring 44145 Kx,coil spring 365.69 N/mm

Ky,leaf spring 2118 Ky,coil spring 365.69 N/mm

Kz,leaf spring 71.9618 Kz,coil spring 682.77 N/mm

Table 6Suspension components of rail vehicle model (28 internal DOFs).

Components No. Identifier eq

Secondary Vert. damper 2 per bogie

Bi-linear funSecondary Lat. damper 2 per bogie

Primary Vert. damper 2 per bogie

Air spring 4 per car

Vertical

Lateral

Longitudina

Lateral buffer 4 per car Nonlinear-p

Bush–link 2 pairs per bogie Nonlinear, S

Coil spring 4 per bogie linear in sp

Leaf spring 4 per bogie linear in sp

comprises from assigning previous turning point displacementto the variable, the frictional part of this model is replaced bythe simplified viscoelastic model defined by Haupt and Sedlan.Schematic diagrams of the proposed modified models in verticaland lateral directions are shown in Fig. 12.

According to Berg and by taking into account of the Haupt andSedlan model, differential equations of air spring in vertical,lateral, and longitudinal directions are proposed as

Vertical direction:

M €ws ¼ Kvzðz�wsÞ � Czbj _wsjb signð _wsÞ; b ¼ 1:8

Fz ¼ ðp0 � paÞAe þ Kezzþ Kvzðz�wsÞ þc _x2

j_zj þ z(3)

Lateral and longitudinal directions:

Fw ¼ Kewwþ Kewwyþ Fvisco-elastic;w þ Kvwðw� uÞ

Cw _u ¼ Kvwðw� uÞ (4)

Fvisco-elastic;w ¼c_x2

j _w1j þ zw ¼ x; y (5)

The parameters in the above equations are estimated according tothe Presthus formulation as given in Eq. (6) [1]:

M ¼ lsAsrAe

As

Vr0

Vb0 þ Vr0

� �2

Kez ¼1

ðp0A2e nÞ=ðVb0 þ Vr0Þ þ pgðdAe=dzÞ

þ1

Kauxiliary

!�1

Kvz ¼1

ðp0A2e nÞ=ðVb0Þ þ pgðdAe=dzÞ

þ1

Kauxiliary

!�1

� Kez

Cz;b ¼1

2rktAs

Ae

As

Vr0

Vb0 þ Vr0

� �1þb

; b ¼ 2 (6)

uations Internal DOF

ction –

Nonlinear, Berg and Sedlan model x2,ws

Nonlinear, Berg and Sedlan model x2,u

l Nonlinear, Berg and Sedlan model x2,u

olynomial, order 4 –

edlan model x2

ace, based on experimental data –

ace, based on analytical calculations –

Table 7Accelerometers specifications.

KS77C.100 AS-2TG

Manufacturer Manfred Weber, Germany Kyowa, Japan

Type ICPs compatible Strain gauge

Range (g) 760 72

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KS77C.100KS77C.100

KS77C.100

AS-2TG

KS77C.100 AS-2TG AS-2TG

KS77C.100KS77C.100

KS77C.100

AS-2TG

Axle 2 Axle 4Axle 3Axle 1

Fig. 21. Accelerometers installation position: (a) front bogie, axle 2 and (b) rear bogie, axle 3.

Accelerometers on rear bogie

Accelerometerson front bogie

GPS Antenna installed over the roof

Fig. 22. Sensors installation position on the trailer car.

Fig. 23. Tehran-Ghazvin route data: (a) cant variation of Tehran-Ghazvin route and (b) speed and height profile, Tehran-Ghazvin route.

H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232228

The exact Cz,1.8 value is calculated based on the Presthus methodand the stiffness Kexw is approximated by Kexw ¼ 0.7(Kexh+load) [1].Numerical values of the above parameters are tabulated in Table 2.

To determine the exact values of the air spring parameters, oneIRICo DMU air spring was tested at ContiTech Company, Germany.Fig. 13 depicts the air spring test.

Based on the method introduced by Docquier et al. [30] andaccording to the test data and simulation results, the loss angleand vertical dynamics stiffness of air spring were investigatedwhich are shown in Figs. 14 and 15. As it can be concluded fromthese diagrams, good agreement between real test and simulationresults is achieved.

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Fig. 24. Axles 2 and 3 accelerations—validation of track model: (a) FFT of lateral acceleration—axle 2; (b) FFT of vertical acceleration—axle 2; (c) FFT of lateral

acceleration—axle 3; and (d) FFT of vertical acceleration—axle 3.

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For validation of air spring lateral behavior, hysteresis loopof air spring is investigated. Results are shown in Fig. 16. Goodagreement between test data and simulation results shows thatthe proposed equations can simulate the real behavior of airspring very well.

3.3.2. Dampers model

IRICo DMU primary and secondary suspension dampers weretested in Sachs Co., Germany. Results are shown in Fig. 17.According to the test results, damping rate of each damper isdescribed by bi-linear function:

F ¼ a _Ddamper þ b (7)

The coefficients in Eq. (7) are tabulated in Table 3 for differentconditions and installation locations of dampers.

3.3.3. Lateral buffer model

Lateral displacement of the carbody is restricted by four lateralbuffers, installed on the bogie frame as shown in Fig. 18. Eachlateral buffer has primary compression force equal to 100 N.Force–displacement diagram of the lateral buffer is shown inFig. 19. An air gap of 17 mm between carbody and lateral bufferlets the carbody moves freely in this range. Lateral displacementof carbody is restricted over 740 mm by two stoppers installed onthe bogie.

According to the experimental data, mechanical behavior ofthe lateral buffer is formulated as

F ¼ 0; 0oxo17

F ¼ 5� 10�5ðx� 17Þ4 � 0:0019ðx� 17Þ3

þ0:0144ðx� 17Þ2;þ1:1525ðx� 17Þ þ 0:100;17pxp40

8>><>>:

(8)

3.3.4. Connection link model

The carbody is linked to the bogie frame by two connectionlinks through center pivot device. Each link has a rubber bushat each end. Static stiffness of the rubber bush in radial, torsional,and cardanic movements are presented according to the testsdone by GMT Co., Germany. Because these rubber bushes aremade from CBFNR, for exact modeling of connection link behaviorin the vehicle dynamics, the simplified one-dimensional Hauptand Sedlan model, described in Fig. 11, is used for radialmovement of the bushes with the following equation:

Fvisco-elastic;radial ¼c _x2

j_r1j þ z(9)

The Haupt and Sedlan parameters, tabulated in Table 4, areaccording to the Allen results [28].

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Fig. 25. Validation of complete vehicle model: (a) FFT of carbody lateral acceleration; (b) FFT of carbody vertical acceleration; (c) FFT of front bogie frame lateral

acceleration; (d) FFT of front bogie frame vertical acceleration; (e) FFT of rear bogie frame lateral acceleration; and (f) FFT of rear bogie frame vertical acceleration.

H. Sayyaadi, N. Shokouhi / International Journal of Mechanical Sciences 51 (2009) 222–232230

3.3.5. Primary suspension, stiffness coefficients model

Four leaf springs in each axle, direct the wheel-set along thecurves. Each leaf spring has one rubber bush at each end.Schematic diagram of the leaf spring in the x–z plane is shownin Fig. 20. The same configuration is developed for the otherplanes.

Considering boundary condition of leaf spring at two ends, thestiffness of leaf spring in bending is determined as follows:

k�1leaf vertical ¼

1

EI

L4kty;bush

4ðLkty;bush þ EIÞ�

L3

3

" #(10)

Based on the manufacturer test results, coil spring stiffness isapproximated linearly. By using Eq. (10), stiffness coefficientsof the primary suspension are calculated. Numerical value ofprimary suspension parameters are tabulated in Table 5.

3.4. Complete vehicle model

All components model used in the rail–vehicle dynamics withrelated internal DOFs are listed in Table 6. It can be seen thatsuspension components add up 28 additional internal DOFs to the

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model. Whereas the vehicle masses have totally 42 DOFs, thecomplete rail–vehicle will be a model of 70 DOFs.

4. Model simulation and test results

Dynamics test of the IRICo DMU was performed in Tehran-Ghazvin route to validate the performance of the proposeddynamics model. VBOX III GPS, manufactured by Racelogic, UK,is used to measure the exact position and speed of the train.Measured speed is monitored and recorded by one of the 16channels of CRONOS PL data acquisition system manufactured byimc, Australia, which was connected to the portable computer.The accelerations of the trailer car masses are recorded by 15remaining channels by means of accelerometers with thespecifications presented in Table 7. These accelerometers areinstalled on two axles, two bogie frames and on the floor of thecarbody inside the car according to Figs. 21 and 22.

Cant variations of the track, recorded by EM120 machine,vehicle speed and height profile of the line are shown in Fig. 23.Referring to this figure, it is perceived that cant parameter in theinterval between 110 and 115 km does not have any significantvariation that means the track line is almost straight withoutsignificant curvature. Constant traveling speed and smooth heightprofile make this section of the test route suitable for validation ofthe proposed vehicle model.

As track data and geometrical irregularities recorded by EM120machine are not up-to-date and were recorded 25 months beforethe test, by using track data, poor agreement between test andsimulation results is achieved. For this reason, the vehicle modelis studied in two different phases. In the first phase, the modelwas executed by considering old track irregularities and axleaccelerations are investigated to validate track model. In thesecond phase, measured axle accelerations and their integrals intime domain are used as reference states to validate completevehicle model.

As in the vehicle–track system, track has the highest stiffness,the high-frequency responses of the vehicle are related to thetrack behavior. To validate dynamic model of track system,responses of the vehicle in frequency domain are investigated[29]. Fig. 24 shows the fast Fourier transform (FFT) of measuredsignals and simulation results for 100 m track length. It can beseen that in the frequency range between 20 and 60 Hz, goodagreement between test and simulation results is achieved.

For validation of the proposed vehicle model, the measuredvertical and lateral acceleration of axles within 200 m of the trackwith their integrals are used as reference inputs. Whereas theaccelerometers are not installed on the axle nos. 1 and 4, theaccelerations of axles 2 and 3 are used for axles 1 and 4 with aconstant lead/lag equal to axle base. FFT of test and simulation

results are shown in Fig. 25. Because each car is equipped with anauxiliary power unit (APU) which works with an internalcombustion engine, there is a peak response in carbody accelera-tions signals at 50 Hz frequency. To simulate the effect of APU onthe system, a sinusoidal force with relevant amplitude andfrequency is added to the vehicle model as a disturbance.

It can be seen that the vehicle test results and the proposedvehicle model have almost the same behavior. Little deviations inthe diagrams can be judged by the following items:

1.

The reference axle states only cover vertical and lateralmovement of the axles. During validation, other DOFs areconstrained constant.

2.

Only accelerations of axle nos. 2 and 3 are recorded. The othertwo axle states are generated based on the measuredaccelerations.

3.

Measured accelerations contain some sources of errors. Byintegrating accelerations, errors of speed and position signalshave been accumulated.

5. Conclusion

This research work proposes a new model for studdinginfluences of suspension system components behavior on railvehicle dynamics. For studding the behavior and performance ofsuspension components, complete track–vehicle model with 70DOFs is addressed as a modular type. In this new model, behaviorof each component is defined and validated using real test datafrom field experiments. Complete nonlinear air spring model, withtaking into consideration of thermo-dynamical effects, is devel-oped and model coefficients are tuned based on the real test data.This model can be easily used in dynamic modeling of air springs.For validation of the proposed track–vehicle model, dynamics testof the vehicle was carried out. Comparison of the results showgood agreement between proposed model and test results thatsays this new model can be used for simulation of the vehicleperformances very well and then it is a good model for furtherapplications such as improvement in ride quality and comfortindex, passive and/or active control and so on.

Acknowledgments

Authors acknowledge Sharif University of Technology, gratefulfor the excessive support from Irankhodro Rail TransportIndustries Co., and also like to express their sincere thanks toMr. M.S. Ghorashi for his help in managing field tests.

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Appendix 1. Neural network synaptic weights

y ¼ f ðW � xþ bÞ; f hidden layersðxÞ ¼2

1þ e�2x� 1; f output layerðxÞ ¼ x

W1 ¼

0:4879 �143:29 153:58 296:96

440:24 �1618:3 �922:66 �1334:3

31:414 61:038 58:806 286:63

�24:646 �3:7075 �68:557 168:59

�90:293 204:3 141:32 326:46

�25:573 �141:03 �86:023 �164:47

�74:14 5:2036 �304:34 �42:909

19:799 90:972 249:49 �199:13

�29:727 �80:828 �67:982 �241:81

266666666666666664

377777777777777775

; b1 ¼

�1:7844

�343:02

�21:85

18:173

71:443

16:336

58:029

�11:211

20:332

266666666666666664

377777777777777775

W2 ¼

3:8078 0:34866 �81:023 13:621 �0:1302 28:59 �1:6923 10:078 �107:8

�23:264 �6:9369 �340:7 �27:582 6:3036 210:98 10:203 �1:4993 �556:53

3:8083 0:34666 �80:939 13:612 �0:12951 28:561 �1:6881 10:077 �107:69

�1:464 �0:20806 �51:506 3:1561 �2:0031 23:906 �3:4124 3:1543 �76:946

�0:76366 �0:01884 �9:4441 �0:49465 �0:0555 6:2466 �0:25575 0:15104 �16:544

26666664

37777775; b2 ¼

�4:2292

�2:5291

�4:2332

�1:0352

0:5185

26666664

37777775

W3 ¼7:73 0:003244 �7:7259 0:022746 �0:00677

7:5261 0:003135 �7:5231 0:021734 0:020275

� �; b3 ¼

0:014172

�0:00269

� �

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