Research Article The Influence of Amplitude- and Frequency ...downloads.hindawi.com/journals/sv/2016/7674124.pdfdistribution of vibrations created by subway in the fre-quency range

Research ArticleThe Influence of Amplitude- andFrequency-Dependent Stiffness of Rail Pads on the RandomVibration of a Vehicle-Track Coupled System

Kai Wei12 Pan Zhang12 Ping Wang12 Junhua Xiao3 and Zhe Luo4

1MOE Key Laboratory of High-Speed Railway Engineering Chengdu 610031 China2School of Civil Engineering Southwest Jiaotong University Chengdu 610031 China3MOE Key Laboratory of Road and Traffic Engineering Tongji University Shanghai 201804 China4Department of Civil Engineering University of Akron Akron OH 44325 USA

Correspondence should be addressed to Ping Wang 392173023qqcom

Received 4 February 2016 Revised 18 April 2016 Accepted 17 May 2016

Academic Editor Ivo Calio

Copyright copy 2016 Kai Wei et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The nonlinear curves between the external static loads of Thermoplastic Polyurethane Elastomer (TPE) rail pads and theircompressive deformations were measured A finite element model (FEM) for a rail-fastener system was produced to determinethe nonlinear compressive deformations of TPE rail pads and their nonlinear static stiffness under the static vehicle weight and thepreload of rail fastener Next the vertical vehicle-track coupled model was employed to investigate the influence of the amplitude-and frequency-dependent stiffness of TPE rail pads on the vehicle-track random vibration It is found that the static stiffness ofTPE rail pads ranges from 191 to 379 kNmm apparently different from the classical secant stiffness of 267 kNmm Additionallycomparedwith the nonlinear amplitude- and frequency-dependent stiffness of rail pads the classical secant stiffness would not onlyseverely underestimate the random vibration acceleration levels of wheel-track coupled system at frequencies of 65ndash150Hz but alsoalter the dominant frequency-distribution of vehicle wheel and steel rail Considering that these frequencies of 65ndash150Hz are thedominant frequencies of ground vibration accelerations caused by low-speed railway the nonlinear amplitude- and frequency-dependent stiffness of rail pads should be taken into account in prediction of environment vibrations due to low-speed railway

1 Introduction

Urban railway traffic influences the environment by emis-sions of ground-borne vibration (1ndash80Hz) and structure-borne noise (16ndash250Hz) Vibrations and noises can some-times reach such a high level that can hardly be tolerated byneighboring residents especially in heavily populated urbanenvironments Consequently the issue of train-inducedvibration has received increasing attention particularly aspeople become more aware of environmental issues More-over as new lines are proposed noise and vibration areimportant aspects that require careful consideration in theplanning stage and often form the basis of objections to newrail development

Numerous efforts have been made to accurately pre-dict environmental vibration and noise generated by urban

railway Nielsen et al presented a hybrid model for the pre-diction of ground-borne vibration due to discrete wheel andrail irregularities [1] Triepaischajonsak and Thompson alsointroduced a hybridmodelling approach to study the sleeper-passing effect [2] It is found that the sleeper-passing effectis less significant than excitation due to track unevennessKouroussis et al used a numerical prediction approach toinvestigate the main parameters affecting ground vibrationduring the passing of trams in Brussels and found thatthe calculated high ground vibrations stem from singularrail surface defects [3] Discontinuous irregularities havebeen shown to cause a significant increase in vibration incomparison to a smooth track [4] Hung et al establisheda 25D finiteinfinite element model to simulate the soilvibrations caused by subway trains [5] and they concludedthat velocity and acceleration responses of the soil are largely

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 7674124 10 pageshttpdxdoiorg10115520167674124

2 Shock and Vibration

amplified due to the presence of rail irregularity Liu and Zhaiinvestigated vertical dynamic wheelrail interaction resultingfrom a polygonal wheel and they found that influence of out-of-roundwheel ismainly related to thewheelset vibration [6]According to the previous experiences the wheel vibrationlevels have close relation with environment vibration andnoise Li et al presented a numerical procedure for thesimulation of the concrete bridge-borne low-frequency noisecaused by the coupled vibration of a train-track-bridgesystem One of the conclusions is that the dominant fre-quency range for trains traveling between 50 and 80 kmh isprimarily attributable to random rail roughness and dynamiccharacteristics of both the bridge and track structure [7]

As stated previously the dynamic component of thevertical wheel-rail contact force due to out-of-round wheelsand rail irregularities is an important source of groundvibration and structure-borne noise This is especially truewhen the design speed of railway is below the wave velocitiesin the soil such as a subway or tram where the free-fieldresponse can be dominated by the dynamic loads [8 9]Thus the precision of the forecasted ground vibration andstructure-borne noise induced by low-speed urban railwayprimarily depends on the accurate prediction of randomdynamic loads of vehicle-track coupled system

In recent decades a large proportion of models andalgorithms have been introduced and developed to calculatethe dynamic random loads of a vehicle-track coupled systemIn general a vehicle-track coupled model is composed of avehicle model a trackmodel and a wheel-track coupled rela-tion The vehicle model has evolved from a multibody model[10 11] of amass-spring system to a solid finite elementmodel(FEM) [12 13] The multibody model is easy to understandand its accuracy can meet the engineering requirements Thesolid finite element model can be used to show the vibrationcharacteristics of the detail structures but its computationefficiency is low The track models have been developedfrom the classical finitely long models to the infinitely longmodels such as Symplectic Method [14ndash16] An FEM ofa nonballasted track [17 18] and a discrete element model(DEM) of a ballasted track have also been proposed [1920] In the wheelrail coupled relations the nonlinear Hertzcontact model is universally used to calculate the wheelrailnormal force and the classical linear Kalker contact modelor the nonlinear ShenndashHedrickndashElkins contact model is alsooften employed to compute the wheelrail creep force [21 22]In addition some simple linear wheelrail coupled relationsare also proposed such as a 2D linear Hertz spring model[23] and a 3D linear wheelrail interaction model [24] Afterestablishment of the abovementioned models a series ofthe numerical integration algorithms in time-space domain[10 16] or the theoretical analyticalmethods in the frequency-wavenumber domain [15 25] can be adopted for calculationof the random vibration responses of a vehicle-track coupledsystem

Although there have been a large number of vehicle-track coupled models and the corresponding algorithmsthere remains a discrepancy between the predicted and mea-sured vibration responses for a vehicle-track coupled systemespecially in the frequency domain The problem probably is

related to the calculation parameters used in the vehicle-trackcoupled models In a vehicle-track coupled system there areinevitably polymer materials for vibration attenuation suchas rail pads under sleeper pads [26 27] and bed pads [2829] The parameters of these polymer materials have a closerelation with the environmental temperature the frequencyof external loads and the amplitude of external loads [30ndash32] A great quantity of experimental results demonstrate thatthe dynamic stiffness of polymer materials enhances with thedecrease of temperature or with the increase of frequencyHowever the variation of dynamic stiffness of polymermaterials with load amounts still remains under debate Forexample one study [31] found that the stiffness of rail padsdecreases as the load amount increases while another study[33] found that the stiffness of rail pads increases as the loadamount increases this discrepancy undoubtedly results fromthe chemical compositions of the polymer rail pads used inthe two studies

In recent years the influence of the frequency- andamplitude-dependent dynamic parameters of polymer mate-rials in vehicle-track coupled system has been investigatedHowever there are still some unsolved issuesWei et al used afrequency-domain algorithmof vehicle-track coupled systemand the existing experimental results of frequency-dependentstiffness of rail pads to investigate the influence of frequency-dependent stiffness of rail pads on the frequency-domaindistribution of vibrations created by subway in the fre-quency range of 0sim200Hz [34] In this study the amplitude-dependent characteristics of rail pads were neglected Zhuet al implemented a nonlinear and fractional derivativeviscoelastic (FDV) model into the time-domain dynamicanalysis of coupled vehicle-slab track (CVST) systems [35]Apart from the low calculation efficiency of the time-domainmodel considering the proposed model was verified onlyby the experiments with low-frequency loads (less than10Hz) the difference between the proposed model andthe ordinary model in high-frequency domain (more than10Hz) is worthy of further research Thus it is necessaryto comprehensively and efficiently consider the frequency-and amplitude-dependent dynamic parameters of polymermaterials in a vehicle-track coupled system

For the purpose of investigation into the effect of thenonlinear amplitude- and frequency-dependent dynamicstiffness of rail pads on the frequency-domain random vibra-tion responses of vehicle-track coupled system the rail padsof Thermoplastic Polyurethane Elastomer (TPE) typicallyused in Chinese subway fasteners were chosen as the focusof this study Firstly the nonlinear curves between the staticloads of TPE rail pads and their corresponding compres-sive deformations were measured with the universal testingmachine (Section 2) Secondly a finite element model (FEM)applied for nonlinear static analysis of a rail-fastener systemwas established to calculate the compressive deformations ofTPE rail pads and their corresponding static stiffness underthe static vehicle weight and the preload of rail fastener(Section 3) Finally based on the nonlinear static results of therail-fastener system the vertical vehicle-track coupled modelwas employed to investigate the influence of the nonlinearamplitude- and frequency-dependent stiffness of TPE rail

Shock and Vibration 3

(a) (b)

Figure 1 Load testing of TPE rail pads (a) universal testingmachineand (b) test setup

pads on the random vibration in a vehicle-track coupledsystem (Section 4)

2 Amplitude-Dependent Stiffness of Rail Pads

The nonlinear curves between the static loads of TPE railpads and their corresponding compressive deformationsweremeasured with a universal testing machine (Figure 1(a)) at anambient temperature of 25∘C

21 Experimental Procedure First a piece of the prototypeTPE rail pad was installed between the loading plate ofuniversal testing machine and the bearing plate (Figure 1(b))In order to ensure the uniform loading on the surface ofrail pad a section of 60 kgm steel rail longer than the testpad was placed on the surface of rail pad Next the test padwas preloaded twice or more times prior to the start of thetest so as to eliminate experimental error According to thestipulations in Chinese standard [36] the preload should bemore than the maximum static load on a piece of rail pad inservice Considering the sharing support by the neighboringfasteners the maximum static load on a piece of rail padis generally less than 18 of the vehicle weight In view ofChinese ldquoTypeArdquo subway vehicleweight of 640 kN (where 18of the vehicle weight is equivalent to 80 kN) the preload wasset to aminimumof 100 kN in this test Following completionof the preloading a piece of TPE rail pad was loaded from0 kN to 90 kN with the loading rate of 3 kNs [36] and theload-deformation curve of the test pad was recorded

The procedure outlined above represents the test proce-dure for a single piece of TPE rail pad In this research a totalof three pieces of TPE rail pads were measured to minimizethe experimental error from a single piece of rail pad

22 Experimental Results A load-deformation curve wasobtained from each of the three TPE specimens In orderto eliminate experimental error the three load-deformation

05 10 15 20 25 30 35 4000Compressed deformations of rail pads (mm)

0

10

20

30

40

50

60

70

80

90

100

Exte

rnal

load

s (kN

)

Figure 2 The average load-deformation curve of three TPE railpads

curves were averaged the resulting load-deformation curveis shown in Figure 2 It is clear that the relationship betweenthe external loads and the corresponding deformations of thetest pad is nonlinear

221 The Linear Constant Static Stiffness of Rail Pads Inprevious research the static stiffness of rail pads was simplyregarded as the linear secant stiffness which can be calculatedwith

119870 =

119865

2minus 119865

1

119878

2minus 119878

1

(1)

where 1198651is the preload of two springs in a fastener system

(generally about 20 kN in Chinese subway fasteners) 1198781is

the compressive deformation of rail pads under 1198651 1198652is the

total loads involving the vehicle weight and the preload oftwo springs in a fastener system and 119878

2is the compressive

deformation of rail pads under1198652 However it is apparent that

the linear secant stiffness of rail pads will not be applicable tothe rail pads with the strong nonlinear stiffness

222 The Nonlinear Amplitude-Dependent Static Stiffness ofRail Pads In order to accurately obtain the variation ofthe static stiffness of the test pads with the external loadamounts the load-deformation curve in Figure 2 was fittedwith a quartic equation (see (2)) and then the first-orderderivative of the fitting equation (2) is computed (ie (3)) Asshown in (3) the relation between the static stiffness and thecompressive deformation of TPE rail pads is also nonlinear

119865 = 00401119878

4

+ 01228119878

3

+ 3368119878

2

+ 5021119878 minus 04518 (2)

119870 = 01604119878

3

+ 03684119878

2

+ 6736119878 + 5021 (3)

where 119865 is the external load 119870 is the nonlinear amplitude-dependent static stiffness of rail pads and 119878 is the compressivedeformation of rail pads According to (3) it is easy todraw the stiffness-deformation curve of TPE rail pads as


0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)

1 2 3 40Compressed deformations of rail pads (mm)

Figure 3 The stiffness-deformation curve of TPE rail pads

shown in Figure 3 It can be observed from Figure 3 that thestatic stiffness of rail pads increases with the compressivedeformation and the rate of increase also gradually increases

3 Nonlinear Static Analysis ofRail-Fastener System

According to the test results presented in Section 2 thestiffness of a TPE rail pad depends on its compressivedeformation under the vehicle weight and the preload ofrail fastener Therefore the nonlinear static analysis of rail-fastener system should be firstly conducted to determine thecompressive deformation of rail pad and its correspondingstatic stiffness before investigating the random vibration ofthe vehicle-track coupled system

31 Nonlinear Static FEM of Rail-Fastener System The non-linear static FEM of the rail-fastener system was establishedby using commercial software (ANSYS) In this FEMmodel asteel rail of 60 kgm with fasteners installed at the interval of06m was simulated using Beam4 and Combin39 elementsin ANSYS respectively Combin39 elements are capable ofsimulating the nonlinear relation between the loads and thedeformationsThe vertical stiffness of a rail-fastener system iscomposed of the spring stiffness and the pad stiffness Sincethe spring stiffness is only 05sim12 kNmm the stiffness ofthe rail pad can be approximately regarded as the stiffness ofthe entire fastener system Thus according to the nonlinearrelationship between the external loads and the compressivedeformations of rail pads Combin39 elements were usedto simulate the nonlinear mechanical behavior of the entirefastener system

In the nonlinear static analysis the loads imposed on arail involved the rail weight (60 kgm) the preload on eachfastener (20 kN) and the half of vehicle weight (320 kN)Thedistance between the four wheels is listed in Table 1

32 Nonlinear Static Results of Rail-Fastener System Based onthe nonlinear static results of the FEM for the rail-fastener

Table 1 The parameters of Chinese ldquoType Ardquo subway vehicle

Parameters MagnitudeMass of vehicle body119872

119888

(kg) 38500Mass of vehicle bogie119872

119905

(kg) 2980Mass of vehicle wheelset119872

119908

(kg) 1350The moment of inertia of vehicle body 119869

119888

(kgsdotm2) 25 times 106

Themoment of inertia of vehicle bogie 119869119905


The stiffness and of the primary suspension119870pz(Nsdotm

minus1) 21 times 106

The damping of the primary suspension119862pz(NsdotSsdotm


The stiffness of the secondary suspension119870sz(Nsdotm


The damping of the secondary suspension119862sz(NsdotSsdotm


The length between two bogie centers in avehicle(m) 18

The vehicle wheelbase(m) 24

10 20 30 40 500Length of steel rail (m)

minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00D

efor

mat

ions

of r

ail p

ads (

mm

)

Figure 4 The compressive deformations of TPE rail pads at thedifferent positions of a rail under a rail-fastener preload of 20 kNand the half of vehicle weight (320 kN)

system the compressive deformations of TPE rail pads at thevarious positions of a rail under the preload of the fastenersand the half of vehicle weight are shown in Figure 4

It can be observed from Figure 4 that a load equivalentto 18 of the vehicle weight only influences the compressivedeformations of three groups of the neighboring TPE railpads before and after loading by a single vehicle wheel andthe mutual interference of the two bogies can be disregardedIn addition it can be found from the calculated resultsthat the compressive deformations of TPE rail pads under afastener preload of 20 kN are 18mm which corresponds toa static stiffness of 191 kNmm After one half of the vehicleweight is imposed on a rail themaximum compressive defor-mation of TPE rail pads becomes 34mm this deformationcorresponds to a static stiffness of 379 kNmm which reflectsan increase by about 98 as compared to the static stiffnessof TPE rail pads under fastener preload However the linear


Table 2 The parameters of the monolithic nonballasted track

Components of track Parameters Magnitude

Steel railYoungrsquos modulus 119864

119903

(Nsdotmminus2) 206 times 1011

Area moment of inertia 119868119903

(m4) 322 times 10minus5

Themass in one meter119898119903

(kgsdotmminus1) 6064

Rail pad The stiffness 119870119901

(Nsdotmminus1) Variation with amounts and the frequencies of loadsThe damping 119862

119901

(NsdotSsdotmminus1) 75 times 104

12 14 16 18 20 2210Length of steel rail (m)

0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)

Figure 5 The static stiffness of TPE rail pads at different positionsunder a bogie

secant stiffness of TPE rail pads computed with (1) is only267 kNmm which is apparently not equal to the actual non-linear stiffness Therefore the linear secant stiffness of TPErail pads cannot accurately reflect the nonlinear variation ofstatic stiffness with the external load amounts

In light of the fact that the compressive deformationsof TPE rail pads under any one bogie will essentially havethe same distribution (see Figure 4) it is sufficient to onlydemonstrate the corresponding stiffness of TPE rail padsat different positions under a bogie as shown in Figure 5According to the calculated results in Figure 5 it is possibleto ascertain the static stiffness of rail pads in a vehicle-trackcoupled model

4 Influence of the NonlinearAmplitude- and Frequency-DependentStiffness on the Random Vibration ofa Vehicle-Track Coupled System

41 The Vertical Vehicle-Track Coupled Model and theKey Parameters The vertical vehicle-track coupled modelapplied for calculation of the random vibration of the vehicle-track system due to track irregularity has been reported byWei et al [34] In this model the whole vehicle can bemodelled as the one with two suspensions involving theup-and-down and nodding movement of vehicle body andbogies and the vertical movement of four wheels for a totalof 10 freedomdegrees (Figure 6)Thedistance among the four

Mc

Jc

Ksz

Kpz

Cp

Csz

MtJt

Cpz

Mw

Kp

ErIr mr

Figure 6 Vehicle-track coupled dynamic model

wheels and the key parameters of vehicle are listed in Table 1The rail can be regarded as an Euler beam supported by thediscrete fasteners with the interval spacing of 06m Due tothe rigid connection between the sleeper and the bed and thebig mass of the monolithic concrete bed the bottom of railfastener can be approximately simulated as a fixed constraint(Figure 6) The key parameters of track are in Table 2

Additionally in the vertical vehicle-track coupled modelthe linear wheelrail contact stiffness 119870

ℎis used to calculate

the vertical wheelrail contact force The linear wheelrailcontact stiffness 119870

ℎcan be derived by (4)sim(8)

119901 (119905) = [

1

119866

Δ119885 (119905)]

32

(4)

where 119901(119905) is wheelrail contact force (N) 119866 is a wheelrailcontact constant (mN23) and Δ119885(119905) is elastic compressionbetween wheel and rail (m) This compression value iscomposed of the static wheelrail compression Δ119885

1198950(119905) and

the wheelrail relative deformation Δ119885119895119908119903(119905) as shown in

Δ119885 (119905) = Δ119885

1198950(119905) + Δ119885

119895119908119903(119905) (5)

Δ119885

1198950(119905) = 119866119875

23

0

Δ119885

119895119908119903(119905) = 119885

119908119895(119905) minus 119885

119903(119909

119866119895 119905) minus 119885

1198950(119905)

(6)


In (6) 1198750is static wheel load (N) which is half of static

axle loadThen the dynamic wheelrail force can be obtainedwith application of

119875

119895119908119903(119905) = 119875

119895(119905) minus 119875

0

= [

1

119866

(Δ119885

1198950+ Δ119885

119895119908119903(119905))]

32

minus 119875

0

= [119875

23

0

+

1

119866

Δ119885

119895119908119903(119905)]

32

minus 119875

0

(7)

Thus the linear wheelrail contact stiffness119870ℎis as shown

in

119870

ℎ=

120597119875

119895119908119903

120597Δ119885

119895119908119903

=

3

2

[119875

23

+

1

119866

Δ119885

119895119908119903]

12

[

1

119866

]

1003816

1003816

1003816

1003816

1003816

1003816

1003816Δ119885119895119908119903=0

=

3

2

1

119866

119875

13

(8)

In the dynamic analysis there is no relative movementbetween vehicle and track only irregularity movement ofthe track The positions between the vehicle wheels and therail fasteners in the vertical vehicle-track coupled model arethe same as those in the nonlinear static analysis of therail-fastener FEM (Section 3) The simulated train speed is80 kmh and the vertical track irregularity consists of the6th grade of the classical vertical track irregularity spectraof American Federal Railroad Administration (FRA) [37](in which the simulated wave lengths are 1sim100m) and thevertical short-wave measured track irregularity spectra ofChinese railway between Shijiazhuang Station and TaiyuanStation that is shown in (10) below (in which the simulatedshort wavelengths are 01sim1m) In (9)119882(119909) is power spectraldensity (PSD) of the vertical short-wave track irregularity(unit mm2mminus1) and 119909 is the spatial frequency of the trackirregularity (unit 1m)

119882(119909) = 0036119909

minus315

(9)

42 The Relation between the Stiffness of Rail Pads theAmounts and the Frequencies of External Loads Based onthe experimental results obtained in other studies [30 38 39]it has been found that the dynamic stiffness of rail pads isclosely linear with the frequencies of external loads under thelogarithmic coordinate [34] as shown in

log11987010

= 119896 times (log11989110

minus log119891010

) + log119870010

(10)

In (10) 119870 is the dynamic stiffness of rail pads at afrequency of 119891 119870

0is the initial dynamic stiffness of rail

pads at the lower frequency of 1198910 and 119896 is the slope of the

linear relation between the dynamic stiffness of rail padsand the frequencies of external loads under the logarithmiccoordinateThe index of 119896 ranges from 005 to 03 [30 38 39]and represents the extent of variation in the dynamic stiffnessof the rail pads with the frequencies of external loads Theinitial dynamic stiffness of 119870

0at the lower frequency of

Table 3 The calculation cases

Case 119870

0

119870

0

(119878) 119896

1 14 times 267 kN = 374 kN 02 14 times 267 kN = 374 kN 0153 14 times (191sim379 kN) = 267sim531 kN 015

119891

0depends on the compressive deformations of rail pads

induced by external loadsThus (10) should be modified into

log11987010

= 119896 times (log11989110

minus log119891010

) + log1198700(119878)10

(11)

where1198700(119878) is the initial dynamic stiffness of rail pads related

to their compressive deformations at the lower frequency of119891

0 In general the dynamic stiffness of Chinese subway rail

pads is less than 14 times their static stiffness at frequenciesof 3sim5Hz Therefore combined with the static stiffness ofTPE rail pads in Figure 5 the initial dynamic stiffness of TPErail pads can be approximately estimated with a ratio of 14 1between dynamic stiffness and static stiffness at 4Hz

In order to compare the influence of the linear secantstiffness the nonlinear frequency-dependent stiffness andthe nonlinear amplitude- and frequency-dependent stiffnessof rail pads on random vibration of a vehicle-track coupledsystem three calculation cases were designed on basis of thenonlinear static results of the rail-fastener system as shownin Table 3

In Case 1 the dynamic stiffness of all rail pads in thevehicle-track coupled model is considered to be 14 timesthe linear secant static stiffness calculated with (1) regardlessof the nonlinear amplitude- and frequency-dependent char-acteristics of rail pads In Case 2 the frequency-dependentdynamic stiffness of all rail pads in the vehicle-track coupledmodel is considered in accordance with (5) in which 14times the linear secant static stiffness of rail pads is takenas the approximate initial dynamic stiffness of rail pads ata low frequency of 4Hz without regard of the amplitude-dependent characteristics of rail pads In Case 3 (6) isused for comprehensive consideration of the amplitude- andfrequency-dependent dynamic stiffness of rail pads and theinitial dynamic stiffness of rail pads at the low frequency of4Hz is 14 times the nonlinear static stiffness of rail pads (asshown in Figure 5) in the vehicle-track coupled model

Considering that the study was mainly focused on theinfluence of the amplitude-dependent stiffness of rail pads onthe random vibration of a vehicle-track coupled system 119896 isuniformly defined as 015 in Cases 2 and 3

43 Effect of the Stiffness of TPE Rail Pads on RandomVibration of a Vehicle-Track Coupled System According tothe designed cases listed in Table 3 the vertical randomvibration of a vehicle body bogie wheel and rail is calculatedThe resulting vibrations of each of these structures are shownin Figures 7ndash10

431 PSD of the Vertical Random Vibration Accelerationof a Vehicle Body It can be observed from Figure 7 thatthe calculated dominant frequency domain of vehicle body


Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014

Figure 7 Power spectral density (PSD) of the vertical randomvibration acceleration of a vehicle body due to the linear secantstiffness the nonlinear frequency-dependent stiffness and the non-linear amplitude- and frequency-dependent stiffness of rail pads

A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3

Figure 8 Power spectral density (PSD) of the vertical randomvibration acceleration of a vehicle bogie due to the linear secantstiffness the nonlinear frequency-dependent stiffness and the non-linear amplitude- and frequency-dependent stiffness of rail pads

in this paper accords with the summary about the actualvibration generated by railway in [40]

It is also found from Figure 7 that the vertical randomvibration responses of vehicle body are identical among thethree cases which demonstrates that the influence of thevariation of the stiffness of rail pads on the random vibrationof vehicle body is very small and thus can be ignored

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3

Figure 9 Power spectral density (PSD) of the vertical randomvibration acceleration of a vehicle wheel due to the linear secantstiffness the nonlinear frequency-dependent stiffness and the non-linear amplitude- and frequency-dependent stiffness of rail pads

Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

Figure 10 Power spectral density (PSD) of the vertical randomvibration acceleration of a steel rail due to the linear secant stiffnessthe nonlinear frequency-dependent stiffness and the nonlinearamplitude- and frequency-dependent stiffness of rail pads

432 PSD of the Vertical Random Vibration Acceleration of aVehicle Bogie Similarly the calculated dominant frequencydomain of vehicle bogie in this paper also accords with thesummary about the actual vibration generated by railway in[40] It can be also observed from Figure 8 that the differencebetween the vertical random vibration responses of a vehiclebogie for the three cases is fairly small (within 20Hz) andyet the discrepancy for the responses in the three cases atfrequencies of 20sim150Hz cannot be ignored


Upon further observation it is found that within thefrequency scope of 20sim50Hz the vertical random vibrationresponses of the vehicle bogie in Case 1 are highest followedby those in Case 2 with the responses in Case 3 being thelowest For example the PSD of the vertical random vibrationacceleration of the vehicle bogie is 0087m2s4Hz at 37Hz(point ldquoArdquo in Figure 8) for Case 1 while the PSDs of thevertical random vibration acceleration of vehicle bogie at thesame frequency in Cases 2 and 3 are 931 and 908 ofthe PSD in Case 1 respectively However in the frequencyrange of 50sim150Hz the results show the opposite trendwith the highest vertical random vibration responses of avehicle obtained in Case 3 and the lowest responses obtainedin Case 1 For instance the PSD of the vertical randomvibration acceleration of the vehicle bogie is 012m2s4Hz at737Hz (point ldquoBrdquo in Figure 8) for Case 3 while the PSDsof the vertical random vibration acceleration of a vehiclebogie at the same frequency in Cases 2 and 1 are 833and 370 of the PSD in Case 3 respectively Obviouslyin the case scenarios presented in this paper the nonlinearfrequency-dependent stiffness or the nonlinear amplitude-and frequency-dependent stiffness of rail pads principallyenhances the random vibration levels of the vehicle bogiein frequencies of 50sim150Hz slightly reduces these responsesat frequencies of 20sim50Hz and has no influence on theseresponses whatsoever at frequencies below 20Hz

Due to the little effect of the three types of rail pad stiffnesson the random vibrations at frequencies of 20sim50Hz therandom vibrations at frequencies higher than 50Hz are thesubsequent focus in this study

433 PSD of the Vertical Random Vibration Acceleration ofa Vehicle Wheel It is clear from Figure 9 that the verticaldominant random vibration accelerations of a vehicle wheelare mostly distributed in the frequency range of 0sim150Hz

It is found that the dominant frequency of verticalrandom vibration acceleration of vehicle wheel is 576Hzwhich has a corresponding maximal PSD of 42m2s4Hz infrequency domain in Case 1 In Case 2 the maximal PSDof vertical random vibration acceleration of vehicle wheelincreases to 70m2s4Hz which is 17 times the maximalPSD for Case 1 the corresponding dominant frequency alsorises to 667Hz which is 91 Hz higher than the dominantfrequency in Case 1 Compared with Case 2 the maximalPSD of vertical random vibration acceleration of the vehiclewheel further increases by about 10 to become 78m2s4HzinCase 3 and the corresponding dominant frequency furtherrises by 84Hz to become 751Hz It can be concluded thatthe nonlinear frequency-dependent stiffness or the nonlinearamplitude- and frequency-dependent stiffness of rail padsnot only increases the frequency-domain amplitudes ofrandom vibration responses of the vehicle wheel in a certainfrequency domain (50sim150Hz in the cases considered in thispaper) but also moves the frequency-distribution of randomvibration responses of the vehicle wheel to shift to the higherfrequencies

434 PSD of the Vertical Random Vibration Acceleration ofa Steel Rail It can be seen from Figure 10 that the vertical

dominant random vibration accelerations of a steel rail aremostly above 50Hz and the sensitive frequency domaininfluenced by three types of the stiffness of rail pads mainlyoccurs at frequencies in the range of 20sim200Hz particularlyin the frequencies between 65 and 185Hz

Similar to the variation of random vibration responses ina vehicle wheel with three types of the stiffness of rail padsthe influence of the nonlinear amplitude- and frequency-dependent stiffness of rail pads on the random vibrationresponses of steel rail is the most conspicuous within thefrequency scope of 65sim185Hz the influence of the nonlinearfrequency-dependent stiffness of rail pads is secondary andyet the influence of the linear secant stiffness of rail pads isrelatively minimal

In the three cases presented in this paper the maximalPSD of vertical random vibration acceleration of the steelrail is 41m2s4Hz with a dominant frequency of 761 Hzin Case 1 The maximal PSDs of vertical random vibrationacceleration of a steel rail and their corresponding dominantfrequencies increase to 62m2s4Hz at 847Hz in Case 2and to 69m2s4Hz at 850Hz in Case 3 respectively It isconcluded that contrasted with the linear secant stiffnessboth the nonlinear frequency-dependent stiffness and thenonlinear amplitude- and frequency-dependent stiffness ofrail pads significantly enlarge the random vibration of thevehicle wheel and steel rail in a certain frequency range(65sim150Hz in the calculation example presented in thispaper) Moreover the effect of the nonlinear amplitude- andfrequency-dependent stiffness of rail pads is obviously moresignificant than the influence of the nonlinear frequency-dependent stiffness of rail pads

5 Conclusions

Using rail pads of Thermoplastic Polyurethane Elastomer(TPE) usually used in Chinese subway fasteners as anexample the relationship between the static loads of TPErail pads and their corresponding compressive deformationswas measured with the universal testing machine The non-linear static analysis with application of a rail-fastener FEMwas performed to quantify the compressive deformationsof rail pads and their corresponding static stiffness underthe static vehicle weight and the preload of rail fastenersNext based on the nonlinear static results of the rail-fastener system the effect of the linear secant stiffness thenonlinear frequency-dependent stiffness and the nonlinearamplitude- and frequency-dependent stiffness of rail pads onthe frequency-domain random vibration responses of vehiclebody bogie wheel and steel rail was investigated Accordingto the results presented in this study some conclusions andsuggestions are summarized as follows

(1) The relationship between the static loads of TPE railpads and their corresponding compressive deforma-tions as measured with the universal testing machineis apparently nonlinear With application of the non-linear static FEM of a rail-fastener system it canbe found that the static stiffness of TPE rail pads is191 kNmm under a rail-fastener preload of 20 kN


and the maximum static stiffness of TPE rail pads is379 kNmm under both a vehicle weight of 640 kNand rail fasteners preload of 20 kN However inprevious research the linear secant static stiffness ofTPE rail pads was found to be 267 kNmm which isapparently different from the actual nonlinear staticstiffness Therefore since the rail pads have a strongnonlinear feature it is unreasonable to use the linearsecant stiffness without considering the variation ofrail pad stiffness with the external load amounts

(2) The random vibration responses in a vehicle-trackcoupled system demonstrate that the influence ofthe variation in stiffness of rail pads on the verticalrandom vibration of the vehicle body is negligibleCompared with the linear secant stiffness of rail padsfor the cases considered in this paper the nonlin-ear frequency-dependent stiffness or the nonlinearamplitude- and frequency-dependent stiffness of railpads has no influence whatsoever on the randomvibration levels of the vehicle bogie wheel and steelrail in the frequencies below 20Hz and it slightlyreduces these responses in the frequency rangebetween 20 and 50Hz But it drastically enhances therandom vibration levels at frequencies of 65sim150Hzwhich is none other than the dominant frequencydomain of the environment vibration accelerationcaused by low-speed urban railway

(3) The nonlinear frequency-dependent stiffness or thenonlinear amplitude- and frequency-dependent stiff-ness of rail pads not only significantly enlarges thefrequency-domain amplitudes of the random vibra-tion responses of vehicle wheel and steel rail butalso shifts the dominant frequency-distribution of thevehicle wheel and steel rail to the higher frequenciesMoreover the effect of the nonlinear amplitude-and frequency-dependent stiffness of rail pads ismore significant than the influence of the nonlinearfrequency-dependent stiffness of rail pads

In summary it can be concluded that if there are polymermaterials with the strong nonlinear stiffness in a vehicleor track system the nonlinear amplitude- and frequency-dependent stiffness of thesematerialsmust be taken into con-sideration in order to precisely predict the random vibrationresponses of vehicle-track coupled system

Competing Interests

The authors declare that they have no competing interests

Funding

This research was supported by National Natural ScienceFoundation of China (Grant no 51578468) FundamentalResearch Funds for the Central Universities of China (Grantno 2682015CX087) National Outstanding Youth ScienceFoundation of China (Grant no 51425804) and Joint Fundsfrom both Chinese High-Speed Railway Company and the

National Natural Science Foundation of China (Grant nosU1234201 and U1434201)

Acknowledgments

The authors would like to thank Chang-sheng Zhou Ying-chun Liang and Zi-xuan Liu for their assistance in laboratoryexperiment and data processing

References

[1] J C O Nielsen G Lombaert and S Francois ldquoA hybridmodel for prediction of ground-borne vibration due to discretewheelrail irregularitiesrdquo Journal of Sound and Vibration vol345 pp 103ndash120 2015

[2] N Triepaischajonsak and D J Thompson ldquoA hybrid modellingapproach for predicting ground vibration from trainsrdquo Journalof Sound and Vibration vol 335 pp 147ndash173 2015

[3] G Kouroussis N Pauwels P Brux C Conti and O VerlindenldquoA numerical analysis of the influence of tram characteristicsand rail profile on railway traffic ground-borne noise and vibra-tion in the Brussels Regionrdquo Science of the Total Environmentvol 482-483 no 1 pp 452ndash460 2014

[4] G Kouroussis D P Connolly G Alexandrou and K VogiatzisldquoThe effect of railway local irregularities on ground vibrationrdquoTransportation Research Part D Transport and Environmentvol 39 pp 17ndash30 2015

[5] H H Hung G H Chen and Y B Yang ldquoEffect of railwayroughness on soil vibrations due to moving trains by 25Dfiniteinfinite element approachrdquo Engineering Structures vol 57pp 254ndash266 2013

[6] X Liu and W Zhai ldquoAnalysis of vertical dynamic wheelrailinteraction caused by polygonal wheels on high-speed trainsrdquoWear vol 314 no 1-2 pp 282ndash290 2014

[7] Q Li Y L Xu and D J Wu ldquoConcrete bridge-borne low-frequency noise simulation based on traintrackbridge dynamicinteractionrdquo Journal of Sound and Vibration vol 331 no 10 pp2457ndash2470 2012

[8] G Degrande M Schevenels P Chatterjee et al ldquoVibrationsdue to a test train at variable speeds in a deep bored tunnelembedded in London clayrdquo Journal of Sound and Vibration vol293 no 3ndash5 pp 626ndash644 2006

[9] D P Connolly P Alves Costa G Kouroussis P Galvin P KWoodward and O Laghrouche ldquoLarge scale international test-ing of railway ground vibrations across Europerdquo Soil Dynamicsand Earthquake Engineering vol 71 pp 1ndash12 2015

[10] W M Zhai K Y Wang and J H Lin ldquoModelling andexperiment of railway ballast vibrationsrdquo Journal of Sound andVibration vol 270 no 4-5 pp 673ndash683 2004

[11] S Falomi M Malvezzi and E Meli ldquoMultibody modelingof railway vehicles innovative algorithms for the detection ofwheel-rail contact pointsrdquo Wear vol 271 no 1-2 pp 453ndash4612011

[12] M Kassner ldquoFatigue strength analysis of a welded railwayvehicle structure by different methodsrdquo International Journal ofFatigue vol 34 no 1 pp 103ndash111 2012

[13] K D Vo H T Zhu A K Tieu and P B Kosasih ldquoFE methodto predict damage formation on curved track for various wornstatus of wheelrail profilesrdquoWear vol 322-323 pp 61ndash75 2015

[14] F Lu D Kennedy F W Williams and J H Lin ldquoSymplecticanalysis of vertical random vibration for coupled vehicle-track


systemsrdquo Journal of Sound and Vibration vol 317 no 1-2 pp236ndash249 2008

[15] Y-W Zhang Y Zhao J-H Lin W P Howson and F WWilliams ldquoA general symplectic method for the responseanalysis of infinitely periodic structures subjected to randomexcitationsrdquoLatinAmerican Journal of Solids and Structures vol9 no 5 pp 569ndash579 2012

[16] J Zhang Q Gao S J Tan andW X Zhong ldquoA precise integra-tion method for solving coupled vehicle-track dynamics withnonlinear wheel-rail contactrdquo Journal of Sound and Vibrationvol 331 no 21 pp 4763ndash4773 2012

[17] KD Vo A K TieuH T Zhu and P B Kosasih ldquoA 3Ddynamicmodel to investigate wheel-rail contact under high and lowadhesionrdquo International Journal of Mechanical Sciences vol 85pp 63ndash75 2014

[18] E Poveda R C Yu J C Lancha and G Ruiz ldquoA numericalstudy on the fatigue life design of concrete slabs for railwaytracksrdquo Engineering Structures vol 100 pp 455ndash467 2015

[19] G Saussine C Cholet P E Gautier et al ldquoModelling ballastbehavior under dynamic loading Part 1 a 2D polygonal discreteelement method approachrdquo Computer Methods in AppliedMechanics and Engineering vol 195 no 19ndash22 pp 2841ndash28592006

[20] H Huang and S Chrismer ldquoDiscrete element modeling ofballast settlement under trains moving at lsquocritical speedsrsquordquoConstruction andBuildingMaterials vol 38 pp 994ndash1000 2013

[21] E Meli S Magheri and M Malvezzi ldquoDevelopment andimplementation of a differential elasticwheel-rail contactmodelfor multibody applicationsrdquo Vehicle System Dynamics vol 49no 6 pp 969ndash1001 2011

[22] E Meli and A Ridolfi ldquoAn innovative wheelndashrail contact modelfor railway vehicles under degraded adhesion conditionsrdquoMultibody System Dynamics vol 33 no 3 pp 285ndash313 2013

[23] P J Remington ldquoWheelrail rolling noise I theoretical analy-sisrdquo Journal of the Acoustical Society of America vol 81 no 6pp 1805ndash1823 1987

[24] N Zhang H Xia W W Guo and G De Roeck ldquoA vehiclendashbridge linear interactionmodel and its validationrdquo InternationalJournal of Structural Stability and Dynamics vol 10 no 2 pp335ndash361 2010

[25] K Abe Y Chida P E Balde Quinay and K Koro ldquoDynamicinstability of a wheel moving on a discretely supported infiniterailrdquo Journal of Sound and Vibration no 15 pp 3413ndash3427 2014

[26] A Johansson J C O Nielsen R Bolmsvik A Karlstrom andR Lunden ldquoUnder sleeper padsmdashinfluence on dynamic train-track interactionrdquoWear vol 265 no 9-10 pp 1479ndash1487 2008

[27] M Sol-Sanchez F Moreno-Navarro and M C Rubio-GamezldquoViability of using end-of-life tire pads as under sleeper pads inrailwayrdquo Construction and Building Materials vol 64 pp 150ndash156 2014

[28] T Xin and L Gao ldquoReducing slab track vibration into bridgeusing elastic materials in high speed railwayrdquo Journal of Soundand Vibration vol 330 no 10 pp 2237ndash2248 2011

[29] H Saurenman and J Phillips ldquoIn-service tests of the effective-ness of vibration control measures on the BART rail transitsystemrdquo Journal of Sound and Vibration vol 293 no 3ndash5 pp888ndash900 2006

[30] J Maes H Sol and P Guillaume ldquoMeasurements of thedynamic railpad propertiesrdquo Journal of Sound and Vibrationvol 293 no 3ndash5 pp 557ndash565 2006

[31] I A Carrascal J A Casado J A Polanco and F Gutierrez-Solana ldquoDynamic behaviour of railway fastening setting padsrdquoEngineering Failure Analysis vol 14 no 2 pp 364ndash373 2007

[32] Y Luo Y Liu and H P Yin ldquoNumerical investigation ofnonlinear properties of a rubber absorber in rail fasteningsystemsrdquo International Journal of Mechanical Sciences vol 69pp 107ndash113 2013

[33] S Kaewunruen and A M Remennikov ldquoAn alternative rail padtester formeasuring dynamic properties of rail pads under largepreloadsrdquo Experimental Mechanics vol 48 no 1 pp 55ndash642008

[34] K Wei P Wang F Yang and J Xiao ldquoThe effect of thefrequency-dependent stiffness of rail pad on the environmentvibrations induced by subway train running in tunnelrdquo Proceed-ings of the Institution of Mechanical Engineers Part F Journal ofRail and Rapid Transit vol 230 no 3 pp 697ndash708 2014

[35] S Zhu C Cai and P D Spanos ldquoA nonlinear and fractionalderivative viscoelastic model for rail pads in the dynamicanalysis of coupled vehicle-slab track systemsrdquo Journal of Soundand Vibration vol 335 pp 304ndash320 2015

[36] GBT ldquoThe state administration of quality supervision inspec-tion and quarantine of Peoplersquos Republic of China nationalstandardization management committee of China Elastic platein fastener system of Rail transitrdquo Grant No GBT 21527-2008Standards Press of China Beijing China 2008 (Chinese)

[37] A Hamid and T L Yang ldquoAnalytical description of track-geometry vibrationsrdquo Transportation Research Record vol 838pp 19ndash26 1981

[38] D J Thompson and J W Verheij ldquoThe dynamic behaviour ofrail fasteners at high frequenciesrdquo Applied Acoustics vol 52 no1 pp 1ndash17 1997

[39] A Fenander ldquoFrequency dependent stiffness and damping ofrailpadsrdquo Proceedings of the Institution of Mechanical EngineersPart F Journal of Rail and Rapid Transit vol 211 no 1 pp 51ndash621997

[40] G Kouroussis D Connolly and O Verlinden ldquoRailway-induced ground vibrationsmdasha review of vehicle effectsrdquo Inter-national Journal of Rail Transportation vol 2 no 2 pp 69ndash1102014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



amplified due to the presence of rail irregularity Liu and Zhaiinvestigated vertical dynamic wheelrail interaction resultingfrom a polygonal wheel and they found that influence of out-of-roundwheel ismainly related to thewheelset vibration [6]According to the previous experiences the wheel vibrationlevels have close relation with environment vibration andnoise Li et al presented a numerical procedure for thesimulation of the concrete bridge-borne low-frequency noisecaused by the coupled vibration of a train-track-bridgesystem One of the conclusions is that the dominant fre-quency range for trains traveling between 50 and 80 kmh isprimarily attributable to random rail roughness and dynamiccharacteristics of both the bridge and track structure [7]

As stated previously the dynamic component of thevertical wheel-rail contact force due to out-of-round wheelsand rail irregularities is an important source of groundvibration and structure-borne noise This is especially truewhen the design speed of railway is below the wave velocitiesin the soil such as a subway or tram where the free-fieldresponse can be dominated by the dynamic loads [8 9]Thus the precision of the forecasted ground vibration andstructure-borne noise induced by low-speed urban railwayprimarily depends on the accurate prediction of randomdynamic loads of vehicle-track coupled system

In recent decades a large proportion of models andalgorithms have been introduced and developed to calculatethe dynamic random loads of a vehicle-track coupled systemIn general a vehicle-track coupled model is composed of avehicle model a trackmodel and a wheel-track coupled rela-tion The vehicle model has evolved from a multibody model[10 11] of amass-spring system to a solid finite elementmodel(FEM) [12 13] The multibody model is easy to understandand its accuracy can meet the engineering requirements Thesolid finite element model can be used to show the vibrationcharacteristics of the detail structures but its computationefficiency is low The track models have been developedfrom the classical finitely long models to the infinitely longmodels such as Symplectic Method [14ndash16] An FEM ofa nonballasted track [17 18] and a discrete element model(DEM) of a ballasted track have also been proposed [1920] In the wheelrail coupled relations the nonlinear Hertzcontact model is universally used to calculate the wheelrailnormal force and the classical linear Kalker contact modelor the nonlinear ShenndashHedrickndashElkins contact model is alsooften employed to compute the wheelrail creep force [21 22]In addition some simple linear wheelrail coupled relationsare also proposed such as a 2D linear Hertz spring model[23] and a 3D linear wheelrail interaction model [24] Afterestablishment of the abovementioned models a series ofthe numerical integration algorithms in time-space domain[10 16] or the theoretical analyticalmethods in the frequency-wavenumber domain [15 25] can be adopted for calculationof the random vibration responses of a vehicle-track coupledsystem

Although there have been a large number of vehicle-track coupled models and the corresponding algorithmsthere remains a discrepancy between the predicted and mea-sured vibration responses for a vehicle-track coupled systemespecially in the frequency domain The problem probably is

related to the calculation parameters used in the vehicle-trackcoupled models In a vehicle-track coupled system there areinevitably polymer materials for vibration attenuation suchas rail pads under sleeper pads [26 27] and bed pads [2829] The parameters of these polymer materials have a closerelation with the environmental temperature the frequencyof external loads and the amplitude of external loads [30ndash32] A great quantity of experimental results demonstrate thatthe dynamic stiffness of polymer materials enhances with thedecrease of temperature or with the increase of frequencyHowever the variation of dynamic stiffness of polymermaterials with load amounts still remains under debate Forexample one study [31] found that the stiffness of rail padsdecreases as the load amount increases while another study[33] found that the stiffness of rail pads increases as the loadamount increases this discrepancy undoubtedly results fromthe chemical compositions of the polymer rail pads used inthe two studies

In recent years the influence of the frequency- andamplitude-dependent dynamic parameters of polymer mate-rials in vehicle-track coupled system has been investigatedHowever there are still some unsolved issuesWei et al used afrequency-domain algorithmof vehicle-track coupled systemand the existing experimental results of frequency-dependentstiffness of rail pads to investigate the influence of frequency-dependent stiffness of rail pads on the frequency-domaindistribution of vibrations created by subway in the fre-quency range of 0sim200Hz [34] In this study the amplitude-dependent characteristics of rail pads were neglected Zhuet al implemented a nonlinear and fractional derivativeviscoelastic (FDV) model into the time-domain dynamicanalysis of coupled vehicle-slab track (CVST) systems [35]Apart from the low calculation efficiency of the time-domainmodel considering the proposed model was verified onlyby the experiments with low-frequency loads (less than10Hz) the difference between the proposed model andthe ordinary model in high-frequency domain (more than10Hz) is worthy of further research Thus it is necessaryto comprehensively and efficiently consider the frequency-and amplitude-dependent dynamic parameters of polymermaterials in a vehicle-track coupled system

For the purpose of investigation into the effect of thenonlinear amplitude- and frequency-dependent dynamicstiffness of rail pads on the frequency-domain random vibra-tion responses of vehicle-track coupled system the rail padsof Thermoplastic Polyurethane Elastomer (TPE) typicallyused in Chinese subway fasteners were chosen as the focusof this study Firstly the nonlinear curves between the staticloads of TPE rail pads and their corresponding compres-sive deformations were measured with the universal testingmachine (Section 2) Secondly a finite element model (FEM)applied for nonlinear static analysis of a rail-fastener systemwas established to calculate the compressive deformations ofTPE rail pads and their corresponding static stiffness underthe static vehicle weight and the preload of rail fastener(Section 3) Finally based on the nonlinear static results of therail-fastener system the vertical vehicle-track coupled modelwas employed to investigate the influence of the nonlinearamplitude- and frequency-dependent stiffness of TPE rail


(a) (b)









0

10

20

30

40

50

60

70

80

90

100

Exte

rnal

load

s (kN

)




119870 =

119865

2minus 119865

1

119878

2minus 119878

1

(1)





2is the compressive




119865 = 00401119878

4

+ 01228119878

3

+ 3368119878

2

+ 5021119878 minus 04518 (2)

119870 = 01604119878

3

+ 03684119878

2

+ 6736119878 + 5021 (3)



0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)











119888


119905


119908


119888















minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00D

efor

mat

ions

of r

ail p

ads (

mm

)








119903








119901



0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)






Mc

Jc

Ksz

Kpz

Cp

Csz

MtJt

Cpz

Mw

Kp

ErIr mr







119901 (119905) = [

1

119866

Δ119885 (119905)]

32

(4)


1198950(119905) and


Δ119885 (119905) = Δ119885

1198950(119905) + Δ119885

119895119908119903(119905) (5)

Δ119885

1198950(119905) = 119866119875

23

0

Δ119885

119895119908119903(119905) = 119885

119908119895(119905) minus 119885

119903(119909

119866119895 119905) minus 119885

1198950(119905)

(6)




119875

119895119908119903(119905) = 119875

119895(119905) minus 119875

0

= [

1

119866

(Δ119885

1198950+ Δ119885

119895119908119903(119905))]

32

minus 119875

0

= [119875

23

0

+

1

119866

Δ119885

119895119908119903(119905)]

32

minus 119875

0

(7)


in

119870

ℎ=

120597119875

119895119908119903

120597Δ119885

119895119908119903

=

3

2

[119875

23

+

1

119866

Δ119885

119895119908119903]

12

[

1

119866

]

1003816

1003816

1003816

1003816

1003816

1003816

1003816Δ119885119895119908119903=0

=

3

2

1

119866

119875

13

(8)


119882(119909) = 0036119909

minus315

(9)


log11987010

= 119896 times (log11989110

minus log119891010

) + log119870010

(10)







Case 119870

0

119870

0

(119878) 119896


119891



log11987010

= 119896 times (log11989110

minus log119891010

) + log1198700(119878)10

(11)











Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014


A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3




PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3


Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8












5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)









0

10

20

30

40

50

60

70

80

90

100

Exte

rnal

load

s (kN

)




119870 =

119865

2minus 119865

1

119878

2minus 119878

1

(1)





2is the compressive




119865 = 00401119878

4

+ 01228119878

3

+ 3368119878

2

+ 5021119878 minus 04518 (2)

119870 = 01604119878

3

+ 03684119878

2

+ 6736119878 + 5021 (3)



0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)











119888


119905


119908


119888















minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00D

efor

mat

ions

of r

ail p

ads (

mm

)








119903








119901



0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)






Mc

Jc

Ksz

Kpz

Cp

Csz

MtJt

Cpz

Mw

Kp

ErIr mr







119901 (119905) = [

1

119866

Δ119885 (119905)]

32

(4)


1198950(119905) and


Δ119885 (119905) = Δ119885

1198950(119905) + Δ119885

119895119908119903(119905) (5)

Δ119885

1198950(119905) = 119866119875

23

0

Δ119885

119895119908119903(119905) = 119885

119908119895(119905) minus 119885

119903(119909

119866119895 119905) minus 119885

1198950(119905)

(6)




119875

119895119908119903(119905) = 119875

119895(119905) minus 119875

0

= [

1

119866

(Δ119885

1198950+ Δ119885

119895119908119903(119905))]

32

minus 119875

0

= [119875

23

0

+

1

119866

Δ119885

119895119908119903(119905)]

32

minus 119875

0

(7)


in

119870

ℎ=

120597119875

119895119908119903

120597Δ119885

119895119908119903

=

3

2

[119875

23

+

1

119866

Δ119885

119895119908119903]

12

[

1

119866

]

1003816

1003816

1003816

1003816

1003816

1003816

1003816Δ119885119895119908119903=0

=

3

2

1

119866

119875

13

(8)


119882(119909) = 0036119909

minus315

(9)


log11987010

= 119896 times (log11989110

minus log119891010

) + log119870010

(10)







Case 119870

0

119870

0

(119878) 119896


119891



log11987010

= 119896 times (log11989110

minus log119891010

) + log1198700(119878)10

(11)











Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014


A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3




PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3


Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8












5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)











119888


119905


119908


119888















minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00D

efor

mat

ions

of r

ail p

ads (

mm

)








119903








119901



0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)






Mc

Jc

Ksz

Kpz

Cp

Csz

MtJt

Cpz

Mw

Kp

ErIr mr







119901 (119905) = [

1

119866

Δ119885 (119905)]

32

(4)


1198950(119905) and


Δ119885 (119905) = Δ119885

1198950(119905) + Δ119885

119895119908119903(119905) (5)

Δ119885

1198950(119905) = 119866119875

23

0

Δ119885

119895119908119903(119905) = 119885

119908119895(119905) minus 119885

119903(119909

119866119895 119905) minus 119885

1198950(119905)

(6)




119875

119895119908119903(119905) = 119875

119895(119905) minus 119875

0

= [

1

119866

(Δ119885

1198950+ Δ119885

119895119908119903(119905))]

32

minus 119875

0

= [119875

23

0

+

1

119866

Δ119885

119895119908119903(119905)]

32

minus 119875

0

(7)


in

119870

ℎ=

120597119875

119895119908119903

120597Δ119885

119895119908119903

=

3

2

[119875

23

+

1

119866

Δ119885

119895119908119903]

12

[

1

119866

]

1003816

1003816

1003816

1003816

1003816

1003816

1003816Δ119885119895119908119903=0

=

3

2

1

119866

119875

13

(8)


119882(119909) = 0036119909

minus315

(9)


log11987010

= 119896 times (log11989110

minus log119891010

) + log119870010

(10)







Case 119870

0

119870

0

(119878) 119896


119891



log11987010

= 119896 times (log11989110

minus log119891010

) + log1198700(119878)10

(11)











Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014


A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3




PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3


Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8












5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119903








119901



0

10

20

30

40

50

Stat

ic st

iffne

ss o

f rai

l pad

s (kN

mm

)






Mc

Jc

Ksz

Kpz

Cp

Csz

MtJt

Cpz

Mw

Kp

ErIr mr







119901 (119905) = [

1

119866

Δ119885 (119905)]

32

(4)


1198950(119905) and


Δ119885 (119905) = Δ119885

1198950(119905) + Δ119885

119895119908119903(119905) (5)

Δ119885

1198950(119905) = 119866119875

23

0

Δ119885

119895119908119903(119905) = 119885

119908119895(119905) minus 119885

119903(119909

119866119895 119905) minus 119885

1198950(119905)

(6)




119875

119895119908119903(119905) = 119875

119895(119905) minus 119875

0

= [

1

119866

(Δ119885

1198950+ Δ119885

119895119908119903(119905))]

32

minus 119875

0

= [119875

23

0

+

1

119866

Δ119885

119895119908119903(119905)]

32

minus 119875

0

(7)


in

119870

ℎ=

120597119875

119895119908119903

120597Δ119885

119895119908119903

=

3

2

[119875

23

+

1

119866

Δ119885

119895119908119903]

12

[

1

119866

]

1003816

1003816

1003816

1003816

1003816

1003816

1003816Δ119885119895119908119903=0

=

3

2

1

119866

119875

13

(8)


119882(119909) = 0036119909

minus315

(9)


log11987010

= 119896 times (log11989110

minus log119891010

) + log119870010

(10)







Case 119870

0

119870

0

(119878) 119896


119891



log11987010

= 119896 times (log11989110

minus log119891010

) + log1198700(119878)10

(11)











Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014


A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3




PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3


Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8












5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119875

119895119908119903(119905) = 119875

119895(119905) minus 119875

0

= [

1

119866

(Δ119885

1198950+ Δ119885

119895119908119903(119905))]

32

minus 119875

0

= [119875

23

0

+

1

119866

Δ119885

119895119908119903(119905)]

32

minus 119875

0

(7)


in

119870

ℎ=

120597119875

119895119908119903

120597Δ119885

119895119908119903

=

3

2

[119875

23

+

1

119866

Δ119885

119895119908119903]

12

[

1

119866

]

1003816

1003816

1003816

1003816

1003816

1003816

1003816Δ119885119895119908119903=0

=

3

2

1

119866

119875

13

(8)


119882(119909) = 0036119909

minus315

(9)


log11987010

= 119896 times (log11989110

minus log119891010

) + log119870010

(10)







Case 119870

0

119870

0

(119878) 119896


119891



log11987010

= 119896 times (log11989110

minus log119891010

) + log1198700(119878)10

(11)











Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014


A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3




PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3


Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8












5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Case 1Case 2Case 3

2 4 6 8 10 12 140Frequency (Hz)

PSD

(m2

s4 H

z)

000

002

004

006

008

010

012

014


A

B

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 2000Frequency (Hz)

000

002

004

006

008

010

012

014

016

Case 1Case 2Case 3




PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8

9

Case 1Case 2Case 3


Case 1Case 2Case 3

PSD

(m2

s4 H

z)

25 50 75 100 125 150 175 200 225 2500Frequency (Hz)

0

1

2

3

4

5

6

7

8












5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















5 Conclusions








Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Competing Interests


Funding



Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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