Research Article Vertical Random Vibration Analysis of...
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Research Article Vertical Random Vibration Analysis of Track-Subgrade Coupled System in High Speed Railway with Pseudoexcitation Method Xinwen Yang, Xiaoshan Liu, Shunhua Zhou, Xiaoyun Ma, Jiangang Shen, and Jingchao Zang Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 201804, China Correspondence should be addressed to Shunhua Zhou; [email protected] Received 25 February 2016; Revised 16 August 2016; Accepted 20 September 2016 Academic Editor: Carlo Trigona Copyright © 2016 Xinwen Yang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In order to reduce the ground-borne vibration caused by wheel/rail interaction in the ballastless track of high speed railways, viscoelastic asphalt concrete materials are filled between the track and the subgrade to attenuate wheel/rail force. A high speed train-track-subgrade vertical coupled dynamic model is developed in the frequency domain. In this model, coupling effects between the vehicle and the track and between the track and the subgrade are considered. e full vehicle is represented by some rigid body models of one body, two bogies, and four wheelsets connected to each other with springs and dampers. e track and subgrade system is considered as a multilayer beam model in which layers are connected to each other with springs and damping elements. e vertical receptance of the rail is discussed and the receptance contribution of the wheel/rail interaction is investigated. Combined with the pseudoexcitation method, a solution of the random dynamic response is presented. e random vibration responses and transfer characteristics of the ballastless track and subgrade system are obtained under track random irregularity when a high speed vehicle runs through. e influences of asphalt concrete layer’s stiffness and vehicle speed on track and subgrade coupling vibration are analyzed. 1. Introduction Unballasted track has been extensively used in high speed railway construction in China for its high regularity, high stability, and high reliability. But when a high speed train runs on large-rigid unballasted tracks, more noise and vibration are produced than ballasted track, which cause serious impact on the environment, buildings, and residents’ living. Many scholars have released a series of studies in view of the vehicle and track coupling vibration and ground vibration. Back in 1995, Knothe and Grassie [1, 2] said that one of the major problems of railway engineering dynamics research is to give full consideration to the subgrade system, which should be included into the effective dynamic analysis model of structures. Using the Green function method, Krylov [3] obtained an analytical expression that considers considering the impact of track-subgrade system on ground vibration. Matsuura [4, 5] studied the track-foundation dynamics responses under the simple moving harmonic loads with the green function method in Japanese Shinkansen railway, in addition, considering the layered soil influence on wave propagation. Kaynia et al. [6–8] treated the foundation as a layered viscoelastic half-space body and the rail as Euler- Bernoulli beam and established a calculation model of the foundation-rail coupling system under moving loads and derived the corresponding dynamic equations using green function. e dynamic behavior of the subgrade under the train load was studied based on the vehicle-track coupling dynamics theory and the finite element method [9–13]. Xu and Cai [14] established a vehicle-track coupling model to study the dynamic behavior of track and subgrade system with modal method. Based on finite element method, the dynamic response of the roadbed under high speed train loads was analyzed, and the effect of the train speed, complex modulus of the subgrade, and vibrating condition of the vehi- cles were discussed [15]. Based on the theory of Timoshenko beam on elastic foundation and its dynamic solution under a moving load [16], the train-induced reaction forces between Hindawi Publishing Corporation Shock and Vibration Volume 2016, Article ID 7064297, 12 pages http://dx.doi.org/10.1155/2016/7064297
Transcript of Research Article Vertical Random Vibration Analysis of...
Research ArticleVertical Random Vibration Analysis of Track-Subgrade CoupledSystem in High Speed Railway with Pseudoexcitation Method
Xinwen Yang Xiaoshan Liu Shunhua Zhou Xiaoyun MaJiangang Shen and Jingchao Zang
Key Laboratory of Road and Traffic Engineering of the Ministry of Education Tongji UniversityShanghai 201804 China
Correspondence should be addressed to Shunhua Zhou zhoushh2015163com
Received 25 February 2016 Revised 16 August 2016 Accepted 20 September 2016
Academic Editor Carlo Trigona
Copyright copy 2016 Xinwen Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In order to reduce the ground-borne vibration caused by wheelrail interaction in the ballastless track of high speed railwaysviscoelastic asphalt concrete materials are filled between the track and the subgrade to attenuate wheelrail force A high speedtrain-track-subgrade vertical coupled dynamicmodel is developed in the frequency domain In thismodel coupling effects betweenthe vehicle and the track and between the track and the subgrade are considered The full vehicle is represented by some rigidbody models of one body two bogies and four wheelsets connected to each other with springs and dampers The track andsubgrade system is considered as a multilayer beam model in which layers are connected to each other with springs and dampingelementsThe vertical receptance of the rail is discussed and the receptance contribution of the wheelrail interaction is investigatedCombined with the pseudoexcitation method a solution of the random dynamic response is presented The random vibrationresponses and transfer characteristics of the ballastless track and subgrade system are obtained under track random irregularitywhen a high speed vehicle runs throughThe influences of asphalt concrete layerrsquos stiffness and vehicle speed on track and subgradecoupling vibration are analyzed
1 Introduction
Unballasted track has been extensively used in high speedrailway construction in China for its high regularity highstability and high reliability But when a high speed train runson large-rigid unballasted tracks more noise and vibrationare produced than ballasted track which cause serious impacton the environment buildings and residentsrsquo living Manyscholars have released a series of studies in view of thevehicle and track coupling vibration and ground vibrationBack in 1995 Knothe and Grassie [1 2] said that one of themajor problems of railway engineering dynamics researchis to give full consideration to the subgrade system whichshould be included into the effective dynamic analysis modelof structures Using the Green function method Krylov [3]obtained an analytical expression that considers consideringthe impact of track-subgrade system on ground vibrationMatsuura [4 5] studied the track-foundation dynamicsresponses under the simple moving harmonic loads with
the green function method in Japanese Shinkansen railwayin addition considering the layered soil influence on wavepropagation Kaynia et al [6ndash8] treated the foundation asa layered viscoelastic half-space body and the rail as Euler-Bernoulli beam and established a calculation model of thefoundation-rail coupling system under moving loads andderived the corresponding dynamic equations using greenfunction The dynamic behavior of the subgrade under thetrain load was studied based on the vehicle-track couplingdynamics theory and the finite element method [9ndash13] Xuand Cai [14] established a vehicle-track coupling model tostudy the dynamic behavior of track and subgrade systemwith modal method Based on finite element method thedynamic response of the roadbed under high speed trainloads was analyzed and the effect of the train speed complexmodulus of the subgrade and vibrating condition of the vehi-cles were discussed [15] Based on the theory of Timoshenkobeam on elastic foundation and its dynamic solution under amoving load [16] the train-induced reaction forces between
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 7064297 12 pageshttpdxdoiorg10115520167064297
2 Shock and Vibration
V
McIcy 120573c
Zc
Csz Ksz
120573t1120573t2
Zt1Kpz
Zw2 Zw1
Fw1Fw2
Zt2
Zw3Zw4
Fw3Fw4
Mt Ity
Cpz
Kp
Kb
Kf
120578p
120578ss
120578ff
zr
zs
zf
Er Ar 120588r Ir 120578r
Es As 120588s Is 120578s
Ef Af 120588f If 120578f
Car body
Bogie
Wheelset
Rail
Sleeper and slab
Subgrade bottom layer
Foundation
Second suspension
Primary suspension
Pad
Asphalt top layer
Wheelrail contact
Figure 1 Model of vehicle-double block ballastless track-subgrade coupled system
ground surface and embankment were calculated Exertingthe reaction forces on the ground surface the correspondingsteady-state stresses in the ground were obtained by integrat-ing the basic solution of a moving point load on the surfaceof an elastic half-space
Themechanism of track and subgrade coupling vibrationcaused by trains is relatively complicated involving prob-lems like vehicles and track parameters matching wheelrailcontact coupling track irregularity and the time-frequencyanalysis of vibration and foundation coupling Environ-mental vibration caused by ballastless tracks of high speedrailways is more than the one caused by ballasted tracksso environmental vibration control of ballastless track isone of the problems in the design and construction of highspeed railway The asphalt concrete layer (ACL) has certainstrength and elasticity which can be used on the trackstructure to attenuate wheelrail vibration In the presentpaper a high speed train-track-subgrade vertical coupleddynamic model is developed in the frequency domain Inthe proposed model coupling effects between vehicle andtrack and between track and subgrade are considered Thefull vehicle is represented by a rigid body model of onebody two bogies and four wheelsets connected to each otherwith springs and dampers The track and subgrade system isconsidered as a multilayer beam model in which layers areconnected to each other with springs and damping elementsThe receptance of track and subgrade system is discussedand receptance contribution of the wheelrail system isinvestigated Combined with the pseudoexcitation methoda solution of the random dynamical response of the train-track-subgrade coupling system is presented The random
vibration response and transfer characteristics of ballastlesstrack and subgrade system are obtained under track ran-dom irregularity when a high speed vehicle runs throughThe influences of the asphalt concrete layerrsquos stiffness andvehicle speed on track and subgrade coupling vibration areanalyzed
2 The Vehicle-Track-Subgrade CouplingDynamic Model
21 Physical Model A vehicle-track-subgrade coupling dy-namic model is established based on vehicle-track coupleddynamic theory [17] as is illustrated in Figure 1 (with thefour-axle locomotive or vehicle as an example)
In the model 119872119888 119872119905 119872119908 are respectively the mass ofthe car body the frame and the wheel-set 119868119888119910 and 119868119905119910 arerespectively the pitch moment of inertia of the car bodyand the frame 119870119904119911 and 119862119904119911 are respectively the verticalstiffness and the damping of the secondary suspension at oneside of the frame 119870119901119911 and 119862119901119911 are respectively the verticalstiffness and the damping of each primary suspension 119881is the vehicle running speed The material constants of therail are 119864119903 Youngrsquos modulus 119866119903 the shear modulus 119860119903the cross-sectional area 119868119903 the second moment of area 120588119903the density 120581 the shearing factor and 120578119903 the loss factorThe material constants of the slab are 119864119904 Youngrsquos modulus119866119904 the shear modulus 119860 119904 the cross-sectional area 119868119904 thesecond moment of area 120588119904 the density and 120578119904 the lossfactor The material constants of the subgrade layer are119864119891 Youngrsquos modulus 119860119891 the cross-sectional area 119868119891 thesecond moment of area 120588119891 the density and 120578119891 the loss
Shock and Vibration 3
Table 1 Degrees of freedom of the system model
System component Type of motionVertical Pitch
Wheelset 1 1198851199081 1205731199081Wheelset 2 1198851199082 1205731199082Wheelset 3 1198851199083 1205731199083Wheelset 4 1198851199084 1205731199084Front bogie frame 1198851199051 1205731199051Real bogie frame 1198851199052 1205731199052Car body 119885119888 120573119888Rail 119911119903 mdashSlab 119911119904 mdashSubgrade 119911119891 mdash
factor 119870119901 119870119904 and 119870119891 are respectively the stiffness of therail pad elastic supporting of track slab and supporting ofsubgrade 120578119901 120578119904119904 and 120578119891119891 are respectively the loss factor ofthe rail pad elastic supporting of track slab and supporting ofsubgrade
In the vehicle-track-subgrade coupling dynamic modelthere are respectively the vehicle subsystem the ballastlesstrack subsystem and the subgrade subsystem from top tobottom as is shown in Figure 1 In the vehicle subsystem thecar body the frame and the wheelset are assumed as rigidbodies and both the car body and the frame have two DOFs(degrees of freedom) of nodding and vertical movement andthe wheelset has only verticalmovement so the whole vehiclesubsystem has 10 DOFs which are listed in Table 1 Theconnections between car body and two bogies are the secondsuspension and the connection between each bogie and twowheelsets is the primary suspension
Thedouble-block ballastless track systemon the subgradeis mainly composed of rails high elastic fasteners double-block sleepers improved with truss reinforcing steel barsslabs and the rigid supporting layers under slabs The rail ismodeled as an infinite Timoshenko beam with continuouselastic discrete supports which has one DOF of verticalmotion And the double-block sleepers slabs and supportinglayers are considered as a whole called slabmodeled as a free-free Euler beam which has a DOF of vertical motion Andthe rail is vertically connected with the sleepers by viscousdamping and linear springs
The subgrade subsystem consists of the surface layer ofthe bed the bottom layer of the bed and the subgrade layerThe surface layer of the bed which is asphalt concrete layerregarded as an elastic cushion ismodeled as linear springs andviscous damping elements And the bottom layer of the bed isregarded as a free-free Euler beam with one DOF of verticalmotion And the subgrade layer is simulated by linear springsand viscous damping elements Therefore the connectionsbetween the track bed and the bottom layer of the bed andthose between the bottom layer of the bed and the subgradeare linear springs and viscous damping elements
An interaction between the vehicle subsystem and theballastless track subsystem is the wheelrail forceThe verticalwheelrail interaction force is approximately calculated using
the Hertz contact theory The coupling effect between theballastless track subsystem and the subgrade subsystem canbe coordinated by the interaction force between the slab andthe subgrade surface
22 Equations of Motion of Track-Subgrade System
221 The Rail The rail is considered as an infinite Timo-shenko beam and its differential equations of motion in thefrequency domain are
minus 1205881199031198601199031205962119911119903 (119909) + 119866119903119860119903120581 (1205931015840 (119909) minus 11991110158401015840119903 (119909))= 119873119908sum119895=1
119865119908119895120575 (119909 minus 119909119908119895) minus119873119901sum119894=1
119891119901ℎ119894120575 (119909 minus 119909119901119894)minus 1205881199031198681199031205962120593 (119909) + 119866119903119860119903120581 (120593 (119909) minus 1199111015840119903 (119909)) minus 11986411990311986811990312059310158401015840 (119909)= 0
(1)
where 119911119903(119909) is the vertical displacement of the rail and 120593(119909) isthe rotate angular of the cross section of the rail 119865119908119895 denotesthe amplitude of the force applied on the rail by the 119895thwheel and 120596 is the angular frequency 119909119901119894 is the longitudinalcoordinate of the discretely distributed rubber pads under therail and 119894 is the ordinal number of the rubber pads 119873119901 isthe total number of the rubber pads 119909119908119895 is the longitudinalcoordinate of the wheels 119895 is the ordinal number of thewheels and 119873119908 is the total number of the wheels and 120575(sdot)is the Dirac-delta function 119891119901ℎ119894 denotes the reaction of thesupporting force applied on the rail by the 119894th rail pad as isgiven by
119891phi = 119870119901119894 (119911 (119909119901119894) minus 119911 (119909119901119894)) (2)
where119870119901119894 is the stiffness of the 119894th rail padThe vertical displacement of any point of the rail in 119909
direction is
119911119903 (119909) =119872119908sum119895=1
119865119908119895120572119903 (119909 119909119908119895) minus119873119901sum119894=1
119891phi120572119903 (119909 119909119901119894) (3)
where 120572119903(119909 119909119908119895) and 120572119903(119909 119909119901119894) are the dynamic flexibility ofthe rail at position 119909 due to a unit force acting at positions 119909119908119895and 119909119901119894 respectively and the dynamic flexibility of the rail isgenerally given by
120572119903 (1199091 1199092) = 1199061119890minus1198941198961|1199091minus1199092| + 1199062119890minus1198962|1199091minus1199092|
1198961 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
+ ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
4 Shock and Vibration
1198962 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
minus ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
1199061 = 119894119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962121198601199031198961 (11989621 + 11989622)
1199062 = 1119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962221198601199031198962 (11989621 + 11989622)
(4)
222 The Track Slab The track slab is considered as a finitefree-free Euler-Bernoulli beam and its differential equationof vibration in the frequency domain is
minus 1205962120588119904119860 119904119911119904 (119909) + 1198641199041198681199041199111015840101584010158401015840119904 (119909)= 119873119901sum119894=1
119891119901119894120575 (119909 minus 119909119901119894) minus119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) (5)
where the material constants of the slab are 119864119904 Youngrsquosmodulus 119866119904 the shear modulus 119860 119904 the cross-sectionalarea 119868119904 the second moment of area 120588119904 the density and 120578119904the loss factor 119911119904 119909119901119894 and 119909119904119895 are respectively the verticaldisplacement of the slabs the longitudinal coordinate at thefasteners and the longitudinal coordinate at the supportingpoints of the slabs 119891119901119894 is the interaction force between therail and the slab and 119891119904119895 is the interaction force between theslab and the subgrade 119873119901 and 119873119904 are respectively the totalnumbers of the supporting points of fasteners and slabs
119891119904119901 = 119870119904 (119906119904 (119909119904119901) minus 119906119887 (119909119904119901)) (6)
Because there is a rigid connection between the sleeperand the slab in the double-block ballastless track the effect ofthe inertia force produced by sleeper blocks vibration on thevibration of track slabs is important as is shown in Figure 2
Assuming that 119911119903(119909 119905) and 119911119904(119909 119905) denote respectivelythe vertical displacement of the rail and the track slab on therail pad in the time domain the inertia force produced by thesleeper block is
119891sleep = minus119872119904 12059721205971199052 119911119904 (119909 119905) (7)
Track slab
Sleeper
Rail
FastenerMs
Kp
Figure 2 Combination of sleeper and slab of twin-block ballastlesstrack structure
So 119891119901ℎ119894 is the force applied to the track slab by the rail onthe 119894th pad as is given by
119891119901ℎ119894 = 119870119901 (119911119903 (119909119901ℎ119894) minus 119911119904 (119909119901ℎ119894)) minus1198721199041205962119911119904 (119909119901ℎ119894) (8)
where119870119901 is the pad stiffness and119872119904 is the sleeper blockmassThe vertical displacement of any point of the track slab in119909 direction is
119911119904 (119909) =119873119901sum119894=1
119891119901119894120572119904 (119909 119909119901119894) minus119873119904sum119901=1
119891119904119901120572119904 (119909 119909119904119901) (9)
where 120572119904(119909 119909119901119894) and 120572119904(119909 119909119904119901) are respectively the dynamicflexibility of the track slab at position119909due to a unit harmonicforce acting at positions 119909119901119894 and 119909119904119901 and the dynamic flexi-bility of the track slab is generally [18]
120572119904 (1199091 1199092) = NMSsum119899=1
119882119899 (1199092)119882119899 (1199091)1205962119899 minus 1205962 (10)
where 120596119904119899 = 1205732119899radic119864119904119868119904120588119904119860 119904 is the 119899th mode frequency of theslab and 120573119899 is the wavenumber NMS is the modal numberof the track slabs in calculation 119882119899(119909) is the modal shapefunction of the free-free Euler beam
223 The Subgrade Layer The subgrade layer is consideredas an infinite Euler-Bernoulli beam and its differential equa-tion of vibration in the frequency domain is
minus 1205962120588119891119860119891119911119891 (119909) + 1198641198911198681198911199111015840101584010158401015840119891 (119909)= 119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) minus119873119891sum119902=1
119891119891119902120575 (119909 minus 119909119891119902) (11)
where 119911119891 119909119904ℎ119901 and 119909119891ℎ119902 are respectively the vertical dis-placement of the subgrade layer the longitudinal coordinateat the supporting points of the slabs and the subgrade layer119891119904ℎ is the interaction force between the slab and the subgradelayer and 119891119891119902 is the interaction force between the subgradelayer and the subgrade119873119891 and119873119904 are respectively the totalnumbers of the supporting points of the subgrade layer andthe slabs
119891119891119902 = 119870119891119911119891 (119909119891119902) (12)
The vertical displacement of any point of the subgradelayer in 119909 direction is
119911119891 (119909) =119873119904sum119901=1
119891119904119901120572119891 (119909 119909119904119901) minus119873119891sum119902=1
119891119891119902120572119891 (119909 119909119891119902) (13)
Shock and Vibration 5
where120572119891(119909 119909119904119901) and120572119891(119909 119909119891119902) are respectively the dynamicflexibility of the track slab denoting the displacementresponse at position 119909 due to a unit harmonic force acting atpositions 119909119904119901 and 119909119891119902 and the dynamic flexibility of the trackslab is generally
120572119891 (1199091 1199092) = 11986211198901205821|1199091minus1199092| + 11986221198901205822|1199091minus1199092| (14)
where 1198621 and 1198622 are complex constants [19]All the above equations except for the dynamic flexibility
of rail and slab do not have explicit damping But actuallythe bending stiffness of beam and the stiffness of spring arein complex stiffness form that is 119866119903(1 + 119894120578119903) 119864119903119868119903(1 + 119894120578119903)119864119904119868119904(1+119894120578119904)119864119891119868119891(1+119894120578119887)119870119901(1+119894120578119901)119870119904(1+119894120578119904119904) and119870119891(1+119894120578119891119891) where 120578119903 120578119904 120578119891 120578119901 120578119904119904 and 120578119891119891 are respectively the lossfactors of rail track slab subgrade rail pad elastic supportingof track slab and supporting of subgrade
Assuming that the amplitude of the dynamic wheelrailforce induced by track irregularity is 119875(Ω) the displace-ment amplitude of any point of track and subgrade system119911TS(Ω) can be obtained by
119911TS (Ω) = [120572TS] 119875 (Ω) (15)
where [120572TS] is the dynamic flexibility of track-subgradesystem at the position of the wheelset which is composed ofdynamic flexibility of track-subgrade system structure 120572119903 120572119904and 12057211989123 Equations of Motion of Vehicle The vehicle system iscomposed of car body two bogies and fourwheelsets for highspeed train The equation of movement of the vehicle systemis [19]
[119872119881] 119881 (119905) + [119862119881] 119881 (119905) + [119870119881] 119911119881 (119905)= 119876119881 (119905) (16)
where [119872119881] [119862119881] and [119870119881] are respectively the generalizedmass matrix generalized damping matrix and generalizedstiffness matrix of the vehicle system 119911119881(119905) 119881(119905) and119881(119905) are respectively the generalized displacement vec-tor generalized velocity vector and generalized accelerationvector of the vehicle system and 119876119881(119905) is the generalizeddynamic force vector of the vehicle system
[119872119881] = diag 119872119862 119869119862 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084
[119870119881] =
[[[[[[[[[[[[[[[[[[[[[[[[[
21198701199042 0 minus1198701199042 minus1198701199042 0 0 0 0 0 0211989721198881198701199042 minus1198971198881198701199042 1198971198881198701199042 0 0 0 0 0 0
21198701199041 + 1198701199042 0 0 0 minus1198701199041 minus1198701199041 0 021198701199041 + 1198701199042 0 0 0 0 minus1198701199041 minus1198701199041
211989721199051198701199041 0 minus1198701199041119897119905 1198701199041119897119905 0 0211989721199051198701199041 0 0 minus1198701199041119897119905 1198701199041119897119905
symmetry 1198701199041 0 0 01198701199041 0 0
1198701199041 01198701199041
]]]]]]]]]]]]]]]]]]]]]]]]]
[119862119881] =
[[[[[[[[[[[[[[[[[[[[[[[[
21198621199042 0 minus1198621199042 minus1198621199042 0 0 0 0 0 0211989721198881198621199042 minus1198971198881198621199042 1198971198881198621199042 0 0 0 0 0 0
21198621199041 + 1198621199042 0 0 0 minus1198621199041 minus1198621199041 0 021198621199041 + 1198621199042 0 0 0 0 minus1198621199041 minus1198621199041
211989721199051198621199041 0 minus1198621199041119897119905 1198621199041119897119905 0 0211989721199051198621199041 0 0 minus1198621199041119897119905 1198621199041119897119905
symmetry 1198621199041 0 0 01198621199041 0 0
1198621199041 01198621199041
]]]]]]]]]]]]]]]]]]]]]]]]
6 Shock and Vibration
119911119881 (119905) = 119911119888 (119905) 120593119888 (119905) 1199111199051 (119905) 1199111199052 (119905) 1205931199051 (119905) 1205931199052 (119905) 1199111199081 (119905) 1199111199082 (119905) 1199111199083 (119905) 1199111199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119876119881 (119905) = 0 0 0 0 0 0 minus1198751199081 (119905) minus1198751199082 (119905) minus1198751199083 (119905) minus1198751199084 (119905)119879
(17)
Assuming 119875(Ω) as the amplitude of dynamic wheelrailforce induced by track irregularity yields the amplitude ofwheel displacement 119911119908(Ω) is
119911119908 (Ω) = [120572119908] 119875 (Ω) (18)
where [120572119908] = [119867][120572119881][119867]119879 is the dynamic flexibility of thevehicle at the wheelrail contact point and Ω is spatial fre-quency (radm) [120572119881] is the dynamic flexibility of the vehicle
24 The WheelRail Interaction The wheelrail interactionforce is described by
119875 (Ω) = ([120572119882] + [120572TS] + [120572Δ])minus1 Δ119911 (Ω) (19)
where [120572Δ] is the dynamic flexibility of the wheelrail contactspring [120572Δ] = 1119896119867 Since the dynamic displacements aresmall 119896119867 is linearized wheelrail contact stiffness accordingto the preload between the wheel and the rail and theirradii of curvature Δ119911(Ω) is the relative displacement (trackrandom irregularity) between the wheel and the rail andcan be obtained by using pseudoexcitation method [20ndash23]119875(Ω) is the amplitude of the wheelrail force induced bytrack irregularity then the wheelrail force in time domaincan be expressed as
119875 (119905) = 119875 (Ω) 119890119894Ω119905 (20)
There are generally two kinds of excitation inputmethodsof analyzing the vehicle-track coupling vibration One isfixed-point excitation and the other is moving-point excita-tion For the former supposing that the vehicle and track arefixed the track irregularity excitation moves backward at acertain speed For the latter the vehicle runs on the track withrandom irregularity at a certain speed In the present paperthe fixed-point excitation method is used
When a train runs on an irregular track it can be assumedthat the different wheels on the same rail are excited bythe same track irregularity but there will be time delayamong different wheels 1199051 1199052 119905119873 Track random excita-tion Δ119911(119905) at wheelsets119873 is
Δ119911 (119905) =
Δ1199111 (119905)Δ1199112 (119905)Δ119911119873 (119905)
=
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)
Δ119911119873 (119905 minus 119905119873)
(21)
where Δ119911(119905) is the history value of vertical track randomirregularity 119905119895 (119895 = 1 2 119873) is the time delay of eachwheel 119905119895 = (119886119895 minus 1198861)V 119886119895 is the longitudinal coordinate ofthe 119895th wheelset at the time 119905 = 0 and 1198861 is the longitudinalcoordinate of the first wheelset Δ119911(119905) can be regarded as ageneralized single excitation If the power spectrum densityfunction of Δ119911(119905) is 119878V(Ω) the pseudoexcitation of verticaltrack random irregularity can be expressed as
Δ119911 (119905) = radic119878V (Ω)119890119894Ω119905 (22)
Then the pseudoexcitation corresponding to (21) is
Δ119911 (119905) =
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)Δ119911119873 (119905 minus 119905119873)
=
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω)119890119894Ω119905
= Δ119911 (Ω) 119890119894Ω119905
(23)
where Δ(Ω) is the harmonic irregularity amplitude of thepseudoexcitation of vertical track random irregularity
Δ119911 (Ω) =
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω) (24)
Replacing Δ119911(Ω) in (19) with Δ(Ω) in (24) we canhave that
(Ω) = Δ (Ω)([120572119882] + [120572TS] + [120572Δ]) (25)
where (Ω) is the amplitude of pseudodynamic wheelrailforce corresponding to pseudoexcitation Δ(Ω) By solv-ing (25) the amplitude of pseudodynamic wheelrail forceinduced by track irregularity can be obtained According tothe pseudoexcitation method the power spectral density ofthe dynamic wheelrail force 119878119875119875(Ω) is given as follows
119878119875119875 (Ω) = (Ω)lowast sdot (Ω)119879 (26)
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
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Shock and Vibration
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2 Shock and Vibration
V
McIcy 120573c
Zc
Csz Ksz
120573t1120573t2
Zt1Kpz
Zw2 Zw1
Fw1Fw2
Zt2
Zw3Zw4
Fw3Fw4
Mt Ity
Cpz
Kp
Kb
Kf
120578p
120578ss
120578ff
zr
zs
zf
Er Ar 120588r Ir 120578r
Es As 120588s Is 120578s
Ef Af 120588f If 120578f
Car body
Bogie
Wheelset
Rail
Sleeper and slab
Subgrade bottom layer
Foundation
Second suspension
Primary suspension
Pad
Asphalt top layer
Wheelrail contact
Figure 1 Model of vehicle-double block ballastless track-subgrade coupled system
ground surface and embankment were calculated Exertingthe reaction forces on the ground surface the correspondingsteady-state stresses in the ground were obtained by integrat-ing the basic solution of a moving point load on the surfaceof an elastic half-space
Themechanism of track and subgrade coupling vibrationcaused by trains is relatively complicated involving prob-lems like vehicles and track parameters matching wheelrailcontact coupling track irregularity and the time-frequencyanalysis of vibration and foundation coupling Environ-mental vibration caused by ballastless tracks of high speedrailways is more than the one caused by ballasted tracksso environmental vibration control of ballastless track isone of the problems in the design and construction of highspeed railway The asphalt concrete layer (ACL) has certainstrength and elasticity which can be used on the trackstructure to attenuate wheelrail vibration In the presentpaper a high speed train-track-subgrade vertical coupleddynamic model is developed in the frequency domain Inthe proposed model coupling effects between vehicle andtrack and between track and subgrade are considered Thefull vehicle is represented by a rigid body model of onebody two bogies and four wheelsets connected to each otherwith springs and dampers The track and subgrade system isconsidered as a multilayer beam model in which layers areconnected to each other with springs and damping elementsThe receptance of track and subgrade system is discussedand receptance contribution of the wheelrail system isinvestigated Combined with the pseudoexcitation methoda solution of the random dynamical response of the train-track-subgrade coupling system is presented The random
vibration response and transfer characteristics of ballastlesstrack and subgrade system are obtained under track ran-dom irregularity when a high speed vehicle runs throughThe influences of the asphalt concrete layerrsquos stiffness andvehicle speed on track and subgrade coupling vibration areanalyzed
2 The Vehicle-Track-Subgrade CouplingDynamic Model
21 Physical Model A vehicle-track-subgrade coupling dy-namic model is established based on vehicle-track coupleddynamic theory [17] as is illustrated in Figure 1 (with thefour-axle locomotive or vehicle as an example)
In the model 119872119888 119872119905 119872119908 are respectively the mass ofthe car body the frame and the wheel-set 119868119888119910 and 119868119905119910 arerespectively the pitch moment of inertia of the car bodyand the frame 119870119904119911 and 119862119904119911 are respectively the verticalstiffness and the damping of the secondary suspension at oneside of the frame 119870119901119911 and 119862119901119911 are respectively the verticalstiffness and the damping of each primary suspension 119881is the vehicle running speed The material constants of therail are 119864119903 Youngrsquos modulus 119866119903 the shear modulus 119860119903the cross-sectional area 119868119903 the second moment of area 120588119903the density 120581 the shearing factor and 120578119903 the loss factorThe material constants of the slab are 119864119904 Youngrsquos modulus119866119904 the shear modulus 119860 119904 the cross-sectional area 119868119904 thesecond moment of area 120588119904 the density and 120578119904 the lossfactor The material constants of the subgrade layer are119864119891 Youngrsquos modulus 119860119891 the cross-sectional area 119868119891 thesecond moment of area 120588119891 the density and 120578119891 the loss
Shock and Vibration 3
Table 1 Degrees of freedom of the system model
System component Type of motionVertical Pitch
Wheelset 1 1198851199081 1205731199081Wheelset 2 1198851199082 1205731199082Wheelset 3 1198851199083 1205731199083Wheelset 4 1198851199084 1205731199084Front bogie frame 1198851199051 1205731199051Real bogie frame 1198851199052 1205731199052Car body 119885119888 120573119888Rail 119911119903 mdashSlab 119911119904 mdashSubgrade 119911119891 mdash
factor 119870119901 119870119904 and 119870119891 are respectively the stiffness of therail pad elastic supporting of track slab and supporting ofsubgrade 120578119901 120578119904119904 and 120578119891119891 are respectively the loss factor ofthe rail pad elastic supporting of track slab and supporting ofsubgrade
In the vehicle-track-subgrade coupling dynamic modelthere are respectively the vehicle subsystem the ballastlesstrack subsystem and the subgrade subsystem from top tobottom as is shown in Figure 1 In the vehicle subsystem thecar body the frame and the wheelset are assumed as rigidbodies and both the car body and the frame have two DOFs(degrees of freedom) of nodding and vertical movement andthe wheelset has only verticalmovement so the whole vehiclesubsystem has 10 DOFs which are listed in Table 1 Theconnections between car body and two bogies are the secondsuspension and the connection between each bogie and twowheelsets is the primary suspension
Thedouble-block ballastless track systemon the subgradeis mainly composed of rails high elastic fasteners double-block sleepers improved with truss reinforcing steel barsslabs and the rigid supporting layers under slabs The rail ismodeled as an infinite Timoshenko beam with continuouselastic discrete supports which has one DOF of verticalmotion And the double-block sleepers slabs and supportinglayers are considered as a whole called slabmodeled as a free-free Euler beam which has a DOF of vertical motion Andthe rail is vertically connected with the sleepers by viscousdamping and linear springs
The subgrade subsystem consists of the surface layer ofthe bed the bottom layer of the bed and the subgrade layerThe surface layer of the bed which is asphalt concrete layerregarded as an elastic cushion ismodeled as linear springs andviscous damping elements And the bottom layer of the bed isregarded as a free-free Euler beam with one DOF of verticalmotion And the subgrade layer is simulated by linear springsand viscous damping elements Therefore the connectionsbetween the track bed and the bottom layer of the bed andthose between the bottom layer of the bed and the subgradeare linear springs and viscous damping elements
An interaction between the vehicle subsystem and theballastless track subsystem is the wheelrail forceThe verticalwheelrail interaction force is approximately calculated using
the Hertz contact theory The coupling effect between theballastless track subsystem and the subgrade subsystem canbe coordinated by the interaction force between the slab andthe subgrade surface
22 Equations of Motion of Track-Subgrade System
221 The Rail The rail is considered as an infinite Timo-shenko beam and its differential equations of motion in thefrequency domain are
minus 1205881199031198601199031205962119911119903 (119909) + 119866119903119860119903120581 (1205931015840 (119909) minus 11991110158401015840119903 (119909))= 119873119908sum119895=1
119865119908119895120575 (119909 minus 119909119908119895) minus119873119901sum119894=1
119891119901ℎ119894120575 (119909 minus 119909119901119894)minus 1205881199031198681199031205962120593 (119909) + 119866119903119860119903120581 (120593 (119909) minus 1199111015840119903 (119909)) minus 11986411990311986811990312059310158401015840 (119909)= 0
(1)
where 119911119903(119909) is the vertical displacement of the rail and 120593(119909) isthe rotate angular of the cross section of the rail 119865119908119895 denotesthe amplitude of the force applied on the rail by the 119895thwheel and 120596 is the angular frequency 119909119901119894 is the longitudinalcoordinate of the discretely distributed rubber pads under therail and 119894 is the ordinal number of the rubber pads 119873119901 isthe total number of the rubber pads 119909119908119895 is the longitudinalcoordinate of the wheels 119895 is the ordinal number of thewheels and 119873119908 is the total number of the wheels and 120575(sdot)is the Dirac-delta function 119891119901ℎ119894 denotes the reaction of thesupporting force applied on the rail by the 119894th rail pad as isgiven by
119891phi = 119870119901119894 (119911 (119909119901119894) minus 119911 (119909119901119894)) (2)
where119870119901119894 is the stiffness of the 119894th rail padThe vertical displacement of any point of the rail in 119909
direction is
119911119903 (119909) =119872119908sum119895=1
119865119908119895120572119903 (119909 119909119908119895) minus119873119901sum119894=1
119891phi120572119903 (119909 119909119901119894) (3)
where 120572119903(119909 119909119908119895) and 120572119903(119909 119909119901119894) are the dynamic flexibility ofthe rail at position 119909 due to a unit force acting at positions 119909119908119895and 119909119901119894 respectively and the dynamic flexibility of the rail isgenerally given by
120572119903 (1199091 1199092) = 1199061119890minus1198941198961|1199091minus1199092| + 1199062119890minus1198962|1199091minus1199092|
1198961 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
+ ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
4 Shock and Vibration
1198962 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
minus ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
1199061 = 119894119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962121198601199031198961 (11989621 + 11989622)
1199062 = 1119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962221198601199031198962 (11989621 + 11989622)
(4)
222 The Track Slab The track slab is considered as a finitefree-free Euler-Bernoulli beam and its differential equationof vibration in the frequency domain is
minus 1205962120588119904119860 119904119911119904 (119909) + 1198641199041198681199041199111015840101584010158401015840119904 (119909)= 119873119901sum119894=1
119891119901119894120575 (119909 minus 119909119901119894) minus119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) (5)
where the material constants of the slab are 119864119904 Youngrsquosmodulus 119866119904 the shear modulus 119860 119904 the cross-sectionalarea 119868119904 the second moment of area 120588119904 the density and 120578119904the loss factor 119911119904 119909119901119894 and 119909119904119895 are respectively the verticaldisplacement of the slabs the longitudinal coordinate at thefasteners and the longitudinal coordinate at the supportingpoints of the slabs 119891119901119894 is the interaction force between therail and the slab and 119891119904119895 is the interaction force between theslab and the subgrade 119873119901 and 119873119904 are respectively the totalnumbers of the supporting points of fasteners and slabs
119891119904119901 = 119870119904 (119906119904 (119909119904119901) minus 119906119887 (119909119904119901)) (6)
Because there is a rigid connection between the sleeperand the slab in the double-block ballastless track the effect ofthe inertia force produced by sleeper blocks vibration on thevibration of track slabs is important as is shown in Figure 2
Assuming that 119911119903(119909 119905) and 119911119904(119909 119905) denote respectivelythe vertical displacement of the rail and the track slab on therail pad in the time domain the inertia force produced by thesleeper block is
119891sleep = minus119872119904 12059721205971199052 119911119904 (119909 119905) (7)
Track slab
Sleeper
Rail
FastenerMs
Kp
Figure 2 Combination of sleeper and slab of twin-block ballastlesstrack structure
So 119891119901ℎ119894 is the force applied to the track slab by the rail onthe 119894th pad as is given by
119891119901ℎ119894 = 119870119901 (119911119903 (119909119901ℎ119894) minus 119911119904 (119909119901ℎ119894)) minus1198721199041205962119911119904 (119909119901ℎ119894) (8)
where119870119901 is the pad stiffness and119872119904 is the sleeper blockmassThe vertical displacement of any point of the track slab in119909 direction is
119911119904 (119909) =119873119901sum119894=1
119891119901119894120572119904 (119909 119909119901119894) minus119873119904sum119901=1
119891119904119901120572119904 (119909 119909119904119901) (9)
where 120572119904(119909 119909119901119894) and 120572119904(119909 119909119904119901) are respectively the dynamicflexibility of the track slab at position119909due to a unit harmonicforce acting at positions 119909119901119894 and 119909119904119901 and the dynamic flexi-bility of the track slab is generally [18]
120572119904 (1199091 1199092) = NMSsum119899=1
119882119899 (1199092)119882119899 (1199091)1205962119899 minus 1205962 (10)
where 120596119904119899 = 1205732119899radic119864119904119868119904120588119904119860 119904 is the 119899th mode frequency of theslab and 120573119899 is the wavenumber NMS is the modal numberof the track slabs in calculation 119882119899(119909) is the modal shapefunction of the free-free Euler beam
223 The Subgrade Layer The subgrade layer is consideredas an infinite Euler-Bernoulli beam and its differential equa-tion of vibration in the frequency domain is
minus 1205962120588119891119860119891119911119891 (119909) + 1198641198911198681198911199111015840101584010158401015840119891 (119909)= 119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) minus119873119891sum119902=1
119891119891119902120575 (119909 minus 119909119891119902) (11)
where 119911119891 119909119904ℎ119901 and 119909119891ℎ119902 are respectively the vertical dis-placement of the subgrade layer the longitudinal coordinateat the supporting points of the slabs and the subgrade layer119891119904ℎ is the interaction force between the slab and the subgradelayer and 119891119891119902 is the interaction force between the subgradelayer and the subgrade119873119891 and119873119904 are respectively the totalnumbers of the supporting points of the subgrade layer andthe slabs
119891119891119902 = 119870119891119911119891 (119909119891119902) (12)
The vertical displacement of any point of the subgradelayer in 119909 direction is
119911119891 (119909) =119873119904sum119901=1
119891119904119901120572119891 (119909 119909119904119901) minus119873119891sum119902=1
119891119891119902120572119891 (119909 119909119891119902) (13)
Shock and Vibration 5
where120572119891(119909 119909119904119901) and120572119891(119909 119909119891119902) are respectively the dynamicflexibility of the track slab denoting the displacementresponse at position 119909 due to a unit harmonic force acting atpositions 119909119904119901 and 119909119891119902 and the dynamic flexibility of the trackslab is generally
120572119891 (1199091 1199092) = 11986211198901205821|1199091minus1199092| + 11986221198901205822|1199091minus1199092| (14)
where 1198621 and 1198622 are complex constants [19]All the above equations except for the dynamic flexibility
of rail and slab do not have explicit damping But actuallythe bending stiffness of beam and the stiffness of spring arein complex stiffness form that is 119866119903(1 + 119894120578119903) 119864119903119868119903(1 + 119894120578119903)119864119904119868119904(1+119894120578119904)119864119891119868119891(1+119894120578119887)119870119901(1+119894120578119901)119870119904(1+119894120578119904119904) and119870119891(1+119894120578119891119891) where 120578119903 120578119904 120578119891 120578119901 120578119904119904 and 120578119891119891 are respectively the lossfactors of rail track slab subgrade rail pad elastic supportingof track slab and supporting of subgrade
Assuming that the amplitude of the dynamic wheelrailforce induced by track irregularity is 119875(Ω) the displace-ment amplitude of any point of track and subgrade system119911TS(Ω) can be obtained by
119911TS (Ω) = [120572TS] 119875 (Ω) (15)
where [120572TS] is the dynamic flexibility of track-subgradesystem at the position of the wheelset which is composed ofdynamic flexibility of track-subgrade system structure 120572119903 120572119904and 12057211989123 Equations of Motion of Vehicle The vehicle system iscomposed of car body two bogies and fourwheelsets for highspeed train The equation of movement of the vehicle systemis [19]
[119872119881] 119881 (119905) + [119862119881] 119881 (119905) + [119870119881] 119911119881 (119905)= 119876119881 (119905) (16)
where [119872119881] [119862119881] and [119870119881] are respectively the generalizedmass matrix generalized damping matrix and generalizedstiffness matrix of the vehicle system 119911119881(119905) 119881(119905) and119881(119905) are respectively the generalized displacement vec-tor generalized velocity vector and generalized accelerationvector of the vehicle system and 119876119881(119905) is the generalizeddynamic force vector of the vehicle system
[119872119881] = diag 119872119862 119869119862 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084
[119870119881] =
[[[[[[[[[[[[[[[[[[[[[[[[[
21198701199042 0 minus1198701199042 minus1198701199042 0 0 0 0 0 0211989721198881198701199042 minus1198971198881198701199042 1198971198881198701199042 0 0 0 0 0 0
21198701199041 + 1198701199042 0 0 0 minus1198701199041 minus1198701199041 0 021198701199041 + 1198701199042 0 0 0 0 minus1198701199041 minus1198701199041
211989721199051198701199041 0 minus1198701199041119897119905 1198701199041119897119905 0 0211989721199051198701199041 0 0 minus1198701199041119897119905 1198701199041119897119905
symmetry 1198701199041 0 0 01198701199041 0 0
1198701199041 01198701199041
]]]]]]]]]]]]]]]]]]]]]]]]]
[119862119881] =
[[[[[[[[[[[[[[[[[[[[[[[[
21198621199042 0 minus1198621199042 minus1198621199042 0 0 0 0 0 0211989721198881198621199042 minus1198971198881198621199042 1198971198881198621199042 0 0 0 0 0 0
21198621199041 + 1198621199042 0 0 0 minus1198621199041 minus1198621199041 0 021198621199041 + 1198621199042 0 0 0 0 minus1198621199041 minus1198621199041
211989721199051198621199041 0 minus1198621199041119897119905 1198621199041119897119905 0 0211989721199051198621199041 0 0 minus1198621199041119897119905 1198621199041119897119905
symmetry 1198621199041 0 0 01198621199041 0 0
1198621199041 01198621199041
]]]]]]]]]]]]]]]]]]]]]]]]
6 Shock and Vibration
119911119881 (119905) = 119911119888 (119905) 120593119888 (119905) 1199111199051 (119905) 1199111199052 (119905) 1205931199051 (119905) 1205931199052 (119905) 1199111199081 (119905) 1199111199082 (119905) 1199111199083 (119905) 1199111199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119876119881 (119905) = 0 0 0 0 0 0 minus1198751199081 (119905) minus1198751199082 (119905) minus1198751199083 (119905) minus1198751199084 (119905)119879
(17)
Assuming 119875(Ω) as the amplitude of dynamic wheelrailforce induced by track irregularity yields the amplitude ofwheel displacement 119911119908(Ω) is
119911119908 (Ω) = [120572119908] 119875 (Ω) (18)
where [120572119908] = [119867][120572119881][119867]119879 is the dynamic flexibility of thevehicle at the wheelrail contact point and Ω is spatial fre-quency (radm) [120572119881] is the dynamic flexibility of the vehicle
24 The WheelRail Interaction The wheelrail interactionforce is described by
119875 (Ω) = ([120572119882] + [120572TS] + [120572Δ])minus1 Δ119911 (Ω) (19)
where [120572Δ] is the dynamic flexibility of the wheelrail contactspring [120572Δ] = 1119896119867 Since the dynamic displacements aresmall 119896119867 is linearized wheelrail contact stiffness accordingto the preload between the wheel and the rail and theirradii of curvature Δ119911(Ω) is the relative displacement (trackrandom irregularity) between the wheel and the rail andcan be obtained by using pseudoexcitation method [20ndash23]119875(Ω) is the amplitude of the wheelrail force induced bytrack irregularity then the wheelrail force in time domaincan be expressed as
119875 (119905) = 119875 (Ω) 119890119894Ω119905 (20)
There are generally two kinds of excitation inputmethodsof analyzing the vehicle-track coupling vibration One isfixed-point excitation and the other is moving-point excita-tion For the former supposing that the vehicle and track arefixed the track irregularity excitation moves backward at acertain speed For the latter the vehicle runs on the track withrandom irregularity at a certain speed In the present paperthe fixed-point excitation method is used
When a train runs on an irregular track it can be assumedthat the different wheels on the same rail are excited bythe same track irregularity but there will be time delayamong different wheels 1199051 1199052 119905119873 Track random excita-tion Δ119911(119905) at wheelsets119873 is
Δ119911 (119905) =
Δ1199111 (119905)Δ1199112 (119905)Δ119911119873 (119905)
=
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)
Δ119911119873 (119905 minus 119905119873)
(21)
where Δ119911(119905) is the history value of vertical track randomirregularity 119905119895 (119895 = 1 2 119873) is the time delay of eachwheel 119905119895 = (119886119895 minus 1198861)V 119886119895 is the longitudinal coordinate ofthe 119895th wheelset at the time 119905 = 0 and 1198861 is the longitudinalcoordinate of the first wheelset Δ119911(119905) can be regarded as ageneralized single excitation If the power spectrum densityfunction of Δ119911(119905) is 119878V(Ω) the pseudoexcitation of verticaltrack random irregularity can be expressed as
Δ119911 (119905) = radic119878V (Ω)119890119894Ω119905 (22)
Then the pseudoexcitation corresponding to (21) is
Δ119911 (119905) =
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)Δ119911119873 (119905 minus 119905119873)
=
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω)119890119894Ω119905
= Δ119911 (Ω) 119890119894Ω119905
(23)
where Δ(Ω) is the harmonic irregularity amplitude of thepseudoexcitation of vertical track random irregularity
Δ119911 (Ω) =
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω) (24)
Replacing Δ119911(Ω) in (19) with Δ(Ω) in (24) we canhave that
(Ω) = Δ (Ω)([120572119882] + [120572TS] + [120572Δ]) (25)
where (Ω) is the amplitude of pseudodynamic wheelrailforce corresponding to pseudoexcitation Δ(Ω) By solv-ing (25) the amplitude of pseudodynamic wheelrail forceinduced by track irregularity can be obtained According tothe pseudoexcitation method the power spectral density ofthe dynamic wheelrail force 119878119875119875(Ω) is given as follows
119878119875119875 (Ω) = (Ω)lowast sdot (Ω)119879 (26)
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
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Shock and Vibration
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Shock and Vibration 3
Table 1 Degrees of freedom of the system model
System component Type of motionVertical Pitch
Wheelset 1 1198851199081 1205731199081Wheelset 2 1198851199082 1205731199082Wheelset 3 1198851199083 1205731199083Wheelset 4 1198851199084 1205731199084Front bogie frame 1198851199051 1205731199051Real bogie frame 1198851199052 1205731199052Car body 119885119888 120573119888Rail 119911119903 mdashSlab 119911119904 mdashSubgrade 119911119891 mdash
factor 119870119901 119870119904 and 119870119891 are respectively the stiffness of therail pad elastic supporting of track slab and supporting ofsubgrade 120578119901 120578119904119904 and 120578119891119891 are respectively the loss factor ofthe rail pad elastic supporting of track slab and supporting ofsubgrade
In the vehicle-track-subgrade coupling dynamic modelthere are respectively the vehicle subsystem the ballastlesstrack subsystem and the subgrade subsystem from top tobottom as is shown in Figure 1 In the vehicle subsystem thecar body the frame and the wheelset are assumed as rigidbodies and both the car body and the frame have two DOFs(degrees of freedom) of nodding and vertical movement andthe wheelset has only verticalmovement so the whole vehiclesubsystem has 10 DOFs which are listed in Table 1 Theconnections between car body and two bogies are the secondsuspension and the connection between each bogie and twowheelsets is the primary suspension
Thedouble-block ballastless track systemon the subgradeis mainly composed of rails high elastic fasteners double-block sleepers improved with truss reinforcing steel barsslabs and the rigid supporting layers under slabs The rail ismodeled as an infinite Timoshenko beam with continuouselastic discrete supports which has one DOF of verticalmotion And the double-block sleepers slabs and supportinglayers are considered as a whole called slabmodeled as a free-free Euler beam which has a DOF of vertical motion Andthe rail is vertically connected with the sleepers by viscousdamping and linear springs
The subgrade subsystem consists of the surface layer ofthe bed the bottom layer of the bed and the subgrade layerThe surface layer of the bed which is asphalt concrete layerregarded as an elastic cushion ismodeled as linear springs andviscous damping elements And the bottom layer of the bed isregarded as a free-free Euler beam with one DOF of verticalmotion And the subgrade layer is simulated by linear springsand viscous damping elements Therefore the connectionsbetween the track bed and the bottom layer of the bed andthose between the bottom layer of the bed and the subgradeare linear springs and viscous damping elements
An interaction between the vehicle subsystem and theballastless track subsystem is the wheelrail forceThe verticalwheelrail interaction force is approximately calculated using
the Hertz contact theory The coupling effect between theballastless track subsystem and the subgrade subsystem canbe coordinated by the interaction force between the slab andthe subgrade surface
22 Equations of Motion of Track-Subgrade System
221 The Rail The rail is considered as an infinite Timo-shenko beam and its differential equations of motion in thefrequency domain are
minus 1205881199031198601199031205962119911119903 (119909) + 119866119903119860119903120581 (1205931015840 (119909) minus 11991110158401015840119903 (119909))= 119873119908sum119895=1
119865119908119895120575 (119909 minus 119909119908119895) minus119873119901sum119894=1
119891119901ℎ119894120575 (119909 minus 119909119901119894)minus 1205881199031198681199031205962120593 (119909) + 119866119903119860119903120581 (120593 (119909) minus 1199111015840119903 (119909)) minus 11986411990311986811990312059310158401015840 (119909)= 0
(1)
where 119911119903(119909) is the vertical displacement of the rail and 120593(119909) isthe rotate angular of the cross section of the rail 119865119908119895 denotesthe amplitude of the force applied on the rail by the 119895thwheel and 120596 is the angular frequency 119909119901119894 is the longitudinalcoordinate of the discretely distributed rubber pads under therail and 119894 is the ordinal number of the rubber pads 119873119901 isthe total number of the rubber pads 119909119908119895 is the longitudinalcoordinate of the wheels 119895 is the ordinal number of thewheels and 119873119908 is the total number of the wheels and 120575(sdot)is the Dirac-delta function 119891119901ℎ119894 denotes the reaction of thesupporting force applied on the rail by the 119894th rail pad as isgiven by
119891phi = 119870119901119894 (119911 (119909119901119894) minus 119911 (119909119901119894)) (2)
where119870119901119894 is the stiffness of the 119894th rail padThe vertical displacement of any point of the rail in 119909
direction is
119911119903 (119909) =119872119908sum119895=1
119865119908119895120572119903 (119909 119909119908119895) minus119873119901sum119894=1
119891phi120572119903 (119909 119909119901119894) (3)
where 120572119903(119909 119909119908119895) and 120572119903(119909 119909119901119894) are the dynamic flexibility ofthe rail at position 119909 due to a unit force acting at positions 119909119908119895and 119909119901119894 respectively and the dynamic flexibility of the rail isgenerally given by
120572119903 (1199091 1199092) = 1199061119890minus1198941198961|1199091minus1199092| + 1199062119890minus1198962|1199091minus1199092|
1198961 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
+ ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
4 Shock and Vibration
1198962 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
minus ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
1199061 = 119894119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962121198601199031198961 (11989621 + 11989622)
1199062 = 1119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962221198601199031198962 (11989621 + 11989622)
(4)
222 The Track Slab The track slab is considered as a finitefree-free Euler-Bernoulli beam and its differential equationof vibration in the frequency domain is
minus 1205962120588119904119860 119904119911119904 (119909) + 1198641199041198681199041199111015840101584010158401015840119904 (119909)= 119873119901sum119894=1
119891119901119894120575 (119909 minus 119909119901119894) minus119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) (5)
where the material constants of the slab are 119864119904 Youngrsquosmodulus 119866119904 the shear modulus 119860 119904 the cross-sectionalarea 119868119904 the second moment of area 120588119904 the density and 120578119904the loss factor 119911119904 119909119901119894 and 119909119904119895 are respectively the verticaldisplacement of the slabs the longitudinal coordinate at thefasteners and the longitudinal coordinate at the supportingpoints of the slabs 119891119901119894 is the interaction force between therail and the slab and 119891119904119895 is the interaction force between theslab and the subgrade 119873119901 and 119873119904 are respectively the totalnumbers of the supporting points of fasteners and slabs
119891119904119901 = 119870119904 (119906119904 (119909119904119901) minus 119906119887 (119909119904119901)) (6)
Because there is a rigid connection between the sleeperand the slab in the double-block ballastless track the effect ofthe inertia force produced by sleeper blocks vibration on thevibration of track slabs is important as is shown in Figure 2
Assuming that 119911119903(119909 119905) and 119911119904(119909 119905) denote respectivelythe vertical displacement of the rail and the track slab on therail pad in the time domain the inertia force produced by thesleeper block is
119891sleep = minus119872119904 12059721205971199052 119911119904 (119909 119905) (7)
Track slab
Sleeper
Rail
FastenerMs
Kp
Figure 2 Combination of sleeper and slab of twin-block ballastlesstrack structure
So 119891119901ℎ119894 is the force applied to the track slab by the rail onthe 119894th pad as is given by
119891119901ℎ119894 = 119870119901 (119911119903 (119909119901ℎ119894) minus 119911119904 (119909119901ℎ119894)) minus1198721199041205962119911119904 (119909119901ℎ119894) (8)
where119870119901 is the pad stiffness and119872119904 is the sleeper blockmassThe vertical displacement of any point of the track slab in119909 direction is
119911119904 (119909) =119873119901sum119894=1
119891119901119894120572119904 (119909 119909119901119894) minus119873119904sum119901=1
119891119904119901120572119904 (119909 119909119904119901) (9)
where 120572119904(119909 119909119901119894) and 120572119904(119909 119909119904119901) are respectively the dynamicflexibility of the track slab at position119909due to a unit harmonicforce acting at positions 119909119901119894 and 119909119904119901 and the dynamic flexi-bility of the track slab is generally [18]
120572119904 (1199091 1199092) = NMSsum119899=1
119882119899 (1199092)119882119899 (1199091)1205962119899 minus 1205962 (10)
where 120596119904119899 = 1205732119899radic119864119904119868119904120588119904119860 119904 is the 119899th mode frequency of theslab and 120573119899 is the wavenumber NMS is the modal numberof the track slabs in calculation 119882119899(119909) is the modal shapefunction of the free-free Euler beam
223 The Subgrade Layer The subgrade layer is consideredas an infinite Euler-Bernoulli beam and its differential equa-tion of vibration in the frequency domain is
minus 1205962120588119891119860119891119911119891 (119909) + 1198641198911198681198911199111015840101584010158401015840119891 (119909)= 119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) minus119873119891sum119902=1
119891119891119902120575 (119909 minus 119909119891119902) (11)
where 119911119891 119909119904ℎ119901 and 119909119891ℎ119902 are respectively the vertical dis-placement of the subgrade layer the longitudinal coordinateat the supporting points of the slabs and the subgrade layer119891119904ℎ is the interaction force between the slab and the subgradelayer and 119891119891119902 is the interaction force between the subgradelayer and the subgrade119873119891 and119873119904 are respectively the totalnumbers of the supporting points of the subgrade layer andthe slabs
119891119891119902 = 119870119891119911119891 (119909119891119902) (12)
The vertical displacement of any point of the subgradelayer in 119909 direction is
119911119891 (119909) =119873119904sum119901=1
119891119904119901120572119891 (119909 119909119904119901) minus119873119891sum119902=1
119891119891119902120572119891 (119909 119909119891119902) (13)
Shock and Vibration 5
where120572119891(119909 119909119904119901) and120572119891(119909 119909119891119902) are respectively the dynamicflexibility of the track slab denoting the displacementresponse at position 119909 due to a unit harmonic force acting atpositions 119909119904119901 and 119909119891119902 and the dynamic flexibility of the trackslab is generally
120572119891 (1199091 1199092) = 11986211198901205821|1199091minus1199092| + 11986221198901205822|1199091minus1199092| (14)
where 1198621 and 1198622 are complex constants [19]All the above equations except for the dynamic flexibility
of rail and slab do not have explicit damping But actuallythe bending stiffness of beam and the stiffness of spring arein complex stiffness form that is 119866119903(1 + 119894120578119903) 119864119903119868119903(1 + 119894120578119903)119864119904119868119904(1+119894120578119904)119864119891119868119891(1+119894120578119887)119870119901(1+119894120578119901)119870119904(1+119894120578119904119904) and119870119891(1+119894120578119891119891) where 120578119903 120578119904 120578119891 120578119901 120578119904119904 and 120578119891119891 are respectively the lossfactors of rail track slab subgrade rail pad elastic supportingof track slab and supporting of subgrade
Assuming that the amplitude of the dynamic wheelrailforce induced by track irregularity is 119875(Ω) the displace-ment amplitude of any point of track and subgrade system119911TS(Ω) can be obtained by
119911TS (Ω) = [120572TS] 119875 (Ω) (15)
where [120572TS] is the dynamic flexibility of track-subgradesystem at the position of the wheelset which is composed ofdynamic flexibility of track-subgrade system structure 120572119903 120572119904and 12057211989123 Equations of Motion of Vehicle The vehicle system iscomposed of car body two bogies and fourwheelsets for highspeed train The equation of movement of the vehicle systemis [19]
[119872119881] 119881 (119905) + [119862119881] 119881 (119905) + [119870119881] 119911119881 (119905)= 119876119881 (119905) (16)
where [119872119881] [119862119881] and [119870119881] are respectively the generalizedmass matrix generalized damping matrix and generalizedstiffness matrix of the vehicle system 119911119881(119905) 119881(119905) and119881(119905) are respectively the generalized displacement vec-tor generalized velocity vector and generalized accelerationvector of the vehicle system and 119876119881(119905) is the generalizeddynamic force vector of the vehicle system
[119872119881] = diag 119872119862 119869119862 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084
[119870119881] =
[[[[[[[[[[[[[[[[[[[[[[[[[
21198701199042 0 minus1198701199042 minus1198701199042 0 0 0 0 0 0211989721198881198701199042 minus1198971198881198701199042 1198971198881198701199042 0 0 0 0 0 0
21198701199041 + 1198701199042 0 0 0 minus1198701199041 minus1198701199041 0 021198701199041 + 1198701199042 0 0 0 0 minus1198701199041 minus1198701199041
211989721199051198701199041 0 minus1198701199041119897119905 1198701199041119897119905 0 0211989721199051198701199041 0 0 minus1198701199041119897119905 1198701199041119897119905
symmetry 1198701199041 0 0 01198701199041 0 0
1198701199041 01198701199041
]]]]]]]]]]]]]]]]]]]]]]]]]
[119862119881] =
[[[[[[[[[[[[[[[[[[[[[[[[
21198621199042 0 minus1198621199042 minus1198621199042 0 0 0 0 0 0211989721198881198621199042 minus1198971198881198621199042 1198971198881198621199042 0 0 0 0 0 0
21198621199041 + 1198621199042 0 0 0 minus1198621199041 minus1198621199041 0 021198621199041 + 1198621199042 0 0 0 0 minus1198621199041 minus1198621199041
211989721199051198621199041 0 minus1198621199041119897119905 1198621199041119897119905 0 0211989721199051198621199041 0 0 minus1198621199041119897119905 1198621199041119897119905
symmetry 1198621199041 0 0 01198621199041 0 0
1198621199041 01198621199041
]]]]]]]]]]]]]]]]]]]]]]]]
6 Shock and Vibration
119911119881 (119905) = 119911119888 (119905) 120593119888 (119905) 1199111199051 (119905) 1199111199052 (119905) 1205931199051 (119905) 1205931199052 (119905) 1199111199081 (119905) 1199111199082 (119905) 1199111199083 (119905) 1199111199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119876119881 (119905) = 0 0 0 0 0 0 minus1198751199081 (119905) minus1198751199082 (119905) minus1198751199083 (119905) minus1198751199084 (119905)119879
(17)
Assuming 119875(Ω) as the amplitude of dynamic wheelrailforce induced by track irregularity yields the amplitude ofwheel displacement 119911119908(Ω) is
119911119908 (Ω) = [120572119908] 119875 (Ω) (18)
where [120572119908] = [119867][120572119881][119867]119879 is the dynamic flexibility of thevehicle at the wheelrail contact point and Ω is spatial fre-quency (radm) [120572119881] is the dynamic flexibility of the vehicle
24 The WheelRail Interaction The wheelrail interactionforce is described by
119875 (Ω) = ([120572119882] + [120572TS] + [120572Δ])minus1 Δ119911 (Ω) (19)
where [120572Δ] is the dynamic flexibility of the wheelrail contactspring [120572Δ] = 1119896119867 Since the dynamic displacements aresmall 119896119867 is linearized wheelrail contact stiffness accordingto the preload between the wheel and the rail and theirradii of curvature Δ119911(Ω) is the relative displacement (trackrandom irregularity) between the wheel and the rail andcan be obtained by using pseudoexcitation method [20ndash23]119875(Ω) is the amplitude of the wheelrail force induced bytrack irregularity then the wheelrail force in time domaincan be expressed as
119875 (119905) = 119875 (Ω) 119890119894Ω119905 (20)
There are generally two kinds of excitation inputmethodsof analyzing the vehicle-track coupling vibration One isfixed-point excitation and the other is moving-point excita-tion For the former supposing that the vehicle and track arefixed the track irregularity excitation moves backward at acertain speed For the latter the vehicle runs on the track withrandom irregularity at a certain speed In the present paperthe fixed-point excitation method is used
When a train runs on an irregular track it can be assumedthat the different wheels on the same rail are excited bythe same track irregularity but there will be time delayamong different wheels 1199051 1199052 119905119873 Track random excita-tion Δ119911(119905) at wheelsets119873 is
Δ119911 (119905) =
Δ1199111 (119905)Δ1199112 (119905)Δ119911119873 (119905)
=
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)
Δ119911119873 (119905 minus 119905119873)
(21)
where Δ119911(119905) is the history value of vertical track randomirregularity 119905119895 (119895 = 1 2 119873) is the time delay of eachwheel 119905119895 = (119886119895 minus 1198861)V 119886119895 is the longitudinal coordinate ofthe 119895th wheelset at the time 119905 = 0 and 1198861 is the longitudinalcoordinate of the first wheelset Δ119911(119905) can be regarded as ageneralized single excitation If the power spectrum densityfunction of Δ119911(119905) is 119878V(Ω) the pseudoexcitation of verticaltrack random irregularity can be expressed as
Δ119911 (119905) = radic119878V (Ω)119890119894Ω119905 (22)
Then the pseudoexcitation corresponding to (21) is
Δ119911 (119905) =
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)Δ119911119873 (119905 minus 119905119873)
=
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω)119890119894Ω119905
= Δ119911 (Ω) 119890119894Ω119905
(23)
where Δ(Ω) is the harmonic irregularity amplitude of thepseudoexcitation of vertical track random irregularity
Δ119911 (Ω) =
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω) (24)
Replacing Δ119911(Ω) in (19) with Δ(Ω) in (24) we canhave that
(Ω) = Δ (Ω)([120572119882] + [120572TS] + [120572Δ]) (25)
where (Ω) is the amplitude of pseudodynamic wheelrailforce corresponding to pseudoexcitation Δ(Ω) By solv-ing (25) the amplitude of pseudodynamic wheelrail forceinduced by track irregularity can be obtained According tothe pseudoexcitation method the power spectral density ofthe dynamic wheelrail force 119878119875119875(Ω) is given as follows
119878119875119875 (Ω) = (Ω)lowast sdot (Ω)119879 (26)
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
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4 Shock and Vibration
1198962 = ( 120596radic2)[( 120588119903119864119903 (1 + 120578119903) minus
120588119903119866119903120581119903)2
+ 4120588119903119860119903119864119903 (1 + 120578119903) 1198681199031205962]12
minus ( 120588119903119864119903 (1 + 120578119903)+ 120588119903119866119903120581119903)
12
1199061 = 119894119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962121198601199031198961 (11989621 + 11989622)
1199062 = 1119864119903 (1 + 120578119903) 119868119903119866119903120581119903sdot 1205881199031198681199031205962 minus 119866119903120581119903119860119903 minus 119864119903 (1 + 120578119903) 1198681199031198962221198601199031198962 (11989621 + 11989622)
(4)
222 The Track Slab The track slab is considered as a finitefree-free Euler-Bernoulli beam and its differential equationof vibration in the frequency domain is
minus 1205962120588119904119860 119904119911119904 (119909) + 1198641199041198681199041199111015840101584010158401015840119904 (119909)= 119873119901sum119894=1
119891119901119894120575 (119909 minus 119909119901119894) minus119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) (5)
where the material constants of the slab are 119864119904 Youngrsquosmodulus 119866119904 the shear modulus 119860 119904 the cross-sectionalarea 119868119904 the second moment of area 120588119904 the density and 120578119904the loss factor 119911119904 119909119901119894 and 119909119904119895 are respectively the verticaldisplacement of the slabs the longitudinal coordinate at thefasteners and the longitudinal coordinate at the supportingpoints of the slabs 119891119901119894 is the interaction force between therail and the slab and 119891119904119895 is the interaction force between theslab and the subgrade 119873119901 and 119873119904 are respectively the totalnumbers of the supporting points of fasteners and slabs
119891119904119901 = 119870119904 (119906119904 (119909119904119901) minus 119906119887 (119909119904119901)) (6)
Because there is a rigid connection between the sleeperand the slab in the double-block ballastless track the effect ofthe inertia force produced by sleeper blocks vibration on thevibration of track slabs is important as is shown in Figure 2
Assuming that 119911119903(119909 119905) and 119911119904(119909 119905) denote respectivelythe vertical displacement of the rail and the track slab on therail pad in the time domain the inertia force produced by thesleeper block is
119891sleep = minus119872119904 12059721205971199052 119911119904 (119909 119905) (7)
Track slab
Sleeper
Rail
FastenerMs
Kp
Figure 2 Combination of sleeper and slab of twin-block ballastlesstrack structure
So 119891119901ℎ119894 is the force applied to the track slab by the rail onthe 119894th pad as is given by
119891119901ℎ119894 = 119870119901 (119911119903 (119909119901ℎ119894) minus 119911119904 (119909119901ℎ119894)) minus1198721199041205962119911119904 (119909119901ℎ119894) (8)
where119870119901 is the pad stiffness and119872119904 is the sleeper blockmassThe vertical displacement of any point of the track slab in119909 direction is
119911119904 (119909) =119873119901sum119894=1
119891119901119894120572119904 (119909 119909119901119894) minus119873119904sum119901=1
119891119904119901120572119904 (119909 119909119904119901) (9)
where 120572119904(119909 119909119901119894) and 120572119904(119909 119909119904119901) are respectively the dynamicflexibility of the track slab at position119909due to a unit harmonicforce acting at positions 119909119901119894 and 119909119904119901 and the dynamic flexi-bility of the track slab is generally [18]
120572119904 (1199091 1199092) = NMSsum119899=1
119882119899 (1199092)119882119899 (1199091)1205962119899 minus 1205962 (10)
where 120596119904119899 = 1205732119899radic119864119904119868119904120588119904119860 119904 is the 119899th mode frequency of theslab and 120573119899 is the wavenumber NMS is the modal numberof the track slabs in calculation 119882119899(119909) is the modal shapefunction of the free-free Euler beam
223 The Subgrade Layer The subgrade layer is consideredas an infinite Euler-Bernoulli beam and its differential equa-tion of vibration in the frequency domain is
minus 1205962120588119891119860119891119911119891 (119909) + 1198641198911198681198911199111015840101584010158401015840119891 (119909)= 119873119904sum119901=1
119891119904119901120575 (119909 minus 119909119904119901) minus119873119891sum119902=1
119891119891119902120575 (119909 minus 119909119891119902) (11)
where 119911119891 119909119904ℎ119901 and 119909119891ℎ119902 are respectively the vertical dis-placement of the subgrade layer the longitudinal coordinateat the supporting points of the slabs and the subgrade layer119891119904ℎ is the interaction force between the slab and the subgradelayer and 119891119891119902 is the interaction force between the subgradelayer and the subgrade119873119891 and119873119904 are respectively the totalnumbers of the supporting points of the subgrade layer andthe slabs
119891119891119902 = 119870119891119911119891 (119909119891119902) (12)
The vertical displacement of any point of the subgradelayer in 119909 direction is
119911119891 (119909) =119873119904sum119901=1
119891119904119901120572119891 (119909 119909119904119901) minus119873119891sum119902=1
119891119891119902120572119891 (119909 119909119891119902) (13)
Shock and Vibration 5
where120572119891(119909 119909119904119901) and120572119891(119909 119909119891119902) are respectively the dynamicflexibility of the track slab denoting the displacementresponse at position 119909 due to a unit harmonic force acting atpositions 119909119904119901 and 119909119891119902 and the dynamic flexibility of the trackslab is generally
120572119891 (1199091 1199092) = 11986211198901205821|1199091minus1199092| + 11986221198901205822|1199091minus1199092| (14)
where 1198621 and 1198622 are complex constants [19]All the above equations except for the dynamic flexibility
of rail and slab do not have explicit damping But actuallythe bending stiffness of beam and the stiffness of spring arein complex stiffness form that is 119866119903(1 + 119894120578119903) 119864119903119868119903(1 + 119894120578119903)119864119904119868119904(1+119894120578119904)119864119891119868119891(1+119894120578119887)119870119901(1+119894120578119901)119870119904(1+119894120578119904119904) and119870119891(1+119894120578119891119891) where 120578119903 120578119904 120578119891 120578119901 120578119904119904 and 120578119891119891 are respectively the lossfactors of rail track slab subgrade rail pad elastic supportingof track slab and supporting of subgrade
Assuming that the amplitude of the dynamic wheelrailforce induced by track irregularity is 119875(Ω) the displace-ment amplitude of any point of track and subgrade system119911TS(Ω) can be obtained by
119911TS (Ω) = [120572TS] 119875 (Ω) (15)
where [120572TS] is the dynamic flexibility of track-subgradesystem at the position of the wheelset which is composed ofdynamic flexibility of track-subgrade system structure 120572119903 120572119904and 12057211989123 Equations of Motion of Vehicle The vehicle system iscomposed of car body two bogies and fourwheelsets for highspeed train The equation of movement of the vehicle systemis [19]
[119872119881] 119881 (119905) + [119862119881] 119881 (119905) + [119870119881] 119911119881 (119905)= 119876119881 (119905) (16)
where [119872119881] [119862119881] and [119870119881] are respectively the generalizedmass matrix generalized damping matrix and generalizedstiffness matrix of the vehicle system 119911119881(119905) 119881(119905) and119881(119905) are respectively the generalized displacement vec-tor generalized velocity vector and generalized accelerationvector of the vehicle system and 119876119881(119905) is the generalizeddynamic force vector of the vehicle system
[119872119881] = diag 119872119862 119869119862 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084
[119870119881] =
[[[[[[[[[[[[[[[[[[[[[[[[[
21198701199042 0 minus1198701199042 minus1198701199042 0 0 0 0 0 0211989721198881198701199042 minus1198971198881198701199042 1198971198881198701199042 0 0 0 0 0 0
21198701199041 + 1198701199042 0 0 0 minus1198701199041 minus1198701199041 0 021198701199041 + 1198701199042 0 0 0 0 minus1198701199041 minus1198701199041
211989721199051198701199041 0 minus1198701199041119897119905 1198701199041119897119905 0 0211989721199051198701199041 0 0 minus1198701199041119897119905 1198701199041119897119905
symmetry 1198701199041 0 0 01198701199041 0 0
1198701199041 01198701199041
]]]]]]]]]]]]]]]]]]]]]]]]]
[119862119881] =
[[[[[[[[[[[[[[[[[[[[[[[[
21198621199042 0 minus1198621199042 minus1198621199042 0 0 0 0 0 0211989721198881198621199042 minus1198971198881198621199042 1198971198881198621199042 0 0 0 0 0 0
21198621199041 + 1198621199042 0 0 0 minus1198621199041 minus1198621199041 0 021198621199041 + 1198621199042 0 0 0 0 minus1198621199041 minus1198621199041
211989721199051198621199041 0 minus1198621199041119897119905 1198621199041119897119905 0 0211989721199051198621199041 0 0 minus1198621199041119897119905 1198621199041119897119905
symmetry 1198621199041 0 0 01198621199041 0 0
1198621199041 01198621199041
]]]]]]]]]]]]]]]]]]]]]]]]
6 Shock and Vibration
119911119881 (119905) = 119911119888 (119905) 120593119888 (119905) 1199111199051 (119905) 1199111199052 (119905) 1205931199051 (119905) 1205931199052 (119905) 1199111199081 (119905) 1199111199082 (119905) 1199111199083 (119905) 1199111199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119876119881 (119905) = 0 0 0 0 0 0 minus1198751199081 (119905) minus1198751199082 (119905) minus1198751199083 (119905) minus1198751199084 (119905)119879
(17)
Assuming 119875(Ω) as the amplitude of dynamic wheelrailforce induced by track irregularity yields the amplitude ofwheel displacement 119911119908(Ω) is
119911119908 (Ω) = [120572119908] 119875 (Ω) (18)
where [120572119908] = [119867][120572119881][119867]119879 is the dynamic flexibility of thevehicle at the wheelrail contact point and Ω is spatial fre-quency (radm) [120572119881] is the dynamic flexibility of the vehicle
24 The WheelRail Interaction The wheelrail interactionforce is described by
119875 (Ω) = ([120572119882] + [120572TS] + [120572Δ])minus1 Δ119911 (Ω) (19)
where [120572Δ] is the dynamic flexibility of the wheelrail contactspring [120572Δ] = 1119896119867 Since the dynamic displacements aresmall 119896119867 is linearized wheelrail contact stiffness accordingto the preload between the wheel and the rail and theirradii of curvature Δ119911(Ω) is the relative displacement (trackrandom irregularity) between the wheel and the rail andcan be obtained by using pseudoexcitation method [20ndash23]119875(Ω) is the amplitude of the wheelrail force induced bytrack irregularity then the wheelrail force in time domaincan be expressed as
119875 (119905) = 119875 (Ω) 119890119894Ω119905 (20)
There are generally two kinds of excitation inputmethodsof analyzing the vehicle-track coupling vibration One isfixed-point excitation and the other is moving-point excita-tion For the former supposing that the vehicle and track arefixed the track irregularity excitation moves backward at acertain speed For the latter the vehicle runs on the track withrandom irregularity at a certain speed In the present paperthe fixed-point excitation method is used
When a train runs on an irregular track it can be assumedthat the different wheels on the same rail are excited bythe same track irregularity but there will be time delayamong different wheels 1199051 1199052 119905119873 Track random excita-tion Δ119911(119905) at wheelsets119873 is
Δ119911 (119905) =
Δ1199111 (119905)Δ1199112 (119905)Δ119911119873 (119905)
=
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)
Δ119911119873 (119905 minus 119905119873)
(21)
where Δ119911(119905) is the history value of vertical track randomirregularity 119905119895 (119895 = 1 2 119873) is the time delay of eachwheel 119905119895 = (119886119895 minus 1198861)V 119886119895 is the longitudinal coordinate ofthe 119895th wheelset at the time 119905 = 0 and 1198861 is the longitudinalcoordinate of the first wheelset Δ119911(119905) can be regarded as ageneralized single excitation If the power spectrum densityfunction of Δ119911(119905) is 119878V(Ω) the pseudoexcitation of verticaltrack random irregularity can be expressed as
Δ119911 (119905) = radic119878V (Ω)119890119894Ω119905 (22)
Then the pseudoexcitation corresponding to (21) is
Δ119911 (119905) =
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)Δ119911119873 (119905 minus 119905119873)
=
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω)119890119894Ω119905
= Δ119911 (Ω) 119890119894Ω119905
(23)
where Δ(Ω) is the harmonic irregularity amplitude of thepseudoexcitation of vertical track random irregularity
Δ119911 (Ω) =
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω) (24)
Replacing Δ119911(Ω) in (19) with Δ(Ω) in (24) we canhave that
(Ω) = Δ (Ω)([120572119882] + [120572TS] + [120572Δ]) (25)
where (Ω) is the amplitude of pseudodynamic wheelrailforce corresponding to pseudoexcitation Δ(Ω) By solv-ing (25) the amplitude of pseudodynamic wheelrail forceinduced by track irregularity can be obtained According tothe pseudoexcitation method the power spectral density ofthe dynamic wheelrail force 119878119875119875(Ω) is given as follows
119878119875119875 (Ω) = (Ω)lowast sdot (Ω)119879 (26)
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
Shock and Vibration 5
where120572119891(119909 119909119904119901) and120572119891(119909 119909119891119902) are respectively the dynamicflexibility of the track slab denoting the displacementresponse at position 119909 due to a unit harmonic force acting atpositions 119909119904119901 and 119909119891119902 and the dynamic flexibility of the trackslab is generally
120572119891 (1199091 1199092) = 11986211198901205821|1199091minus1199092| + 11986221198901205822|1199091minus1199092| (14)
where 1198621 and 1198622 are complex constants [19]All the above equations except for the dynamic flexibility
of rail and slab do not have explicit damping But actuallythe bending stiffness of beam and the stiffness of spring arein complex stiffness form that is 119866119903(1 + 119894120578119903) 119864119903119868119903(1 + 119894120578119903)119864119904119868119904(1+119894120578119904)119864119891119868119891(1+119894120578119887)119870119901(1+119894120578119901)119870119904(1+119894120578119904119904) and119870119891(1+119894120578119891119891) where 120578119903 120578119904 120578119891 120578119901 120578119904119904 and 120578119891119891 are respectively the lossfactors of rail track slab subgrade rail pad elastic supportingof track slab and supporting of subgrade
Assuming that the amplitude of the dynamic wheelrailforce induced by track irregularity is 119875(Ω) the displace-ment amplitude of any point of track and subgrade system119911TS(Ω) can be obtained by
119911TS (Ω) = [120572TS] 119875 (Ω) (15)
where [120572TS] is the dynamic flexibility of track-subgradesystem at the position of the wheelset which is composed ofdynamic flexibility of track-subgrade system structure 120572119903 120572119904and 12057211989123 Equations of Motion of Vehicle The vehicle system iscomposed of car body two bogies and fourwheelsets for highspeed train The equation of movement of the vehicle systemis [19]
[119872119881] 119881 (119905) + [119862119881] 119881 (119905) + [119870119881] 119911119881 (119905)= 119876119881 (119905) (16)
where [119872119881] [119862119881] and [119870119881] are respectively the generalizedmass matrix generalized damping matrix and generalizedstiffness matrix of the vehicle system 119911119881(119905) 119881(119905) and119881(119905) are respectively the generalized displacement vec-tor generalized velocity vector and generalized accelerationvector of the vehicle system and 119876119881(119905) is the generalizeddynamic force vector of the vehicle system
[119872119881] = diag 119872119862 119869119862 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084
[119870119881] =
[[[[[[[[[[[[[[[[[[[[[[[[[
21198701199042 0 minus1198701199042 minus1198701199042 0 0 0 0 0 0211989721198881198701199042 minus1198971198881198701199042 1198971198881198701199042 0 0 0 0 0 0
21198701199041 + 1198701199042 0 0 0 minus1198701199041 minus1198701199041 0 021198701199041 + 1198701199042 0 0 0 0 minus1198701199041 minus1198701199041
211989721199051198701199041 0 minus1198701199041119897119905 1198701199041119897119905 0 0211989721199051198701199041 0 0 minus1198701199041119897119905 1198701199041119897119905
symmetry 1198701199041 0 0 01198701199041 0 0
1198701199041 01198701199041
]]]]]]]]]]]]]]]]]]]]]]]]]
[119862119881] =
[[[[[[[[[[[[[[[[[[[[[[[[
21198621199042 0 minus1198621199042 minus1198621199042 0 0 0 0 0 0211989721198881198621199042 minus1198971198881198621199042 1198971198881198621199042 0 0 0 0 0 0
21198621199041 + 1198621199042 0 0 0 minus1198621199041 minus1198621199041 0 021198621199041 + 1198621199042 0 0 0 0 minus1198621199041 minus1198621199041
211989721199051198621199041 0 minus1198621199041119897119905 1198621199041119897119905 0 0211989721199051198621199041 0 0 minus1198621199041119897119905 1198621199041119897119905
symmetry 1198621199041 0 0 01198621199041 0 0
1198621199041 01198621199041
]]]]]]]]]]]]]]]]]]]]]]]]
6 Shock and Vibration
119911119881 (119905) = 119911119888 (119905) 120593119888 (119905) 1199111199051 (119905) 1199111199052 (119905) 1205931199051 (119905) 1205931199052 (119905) 1199111199081 (119905) 1199111199082 (119905) 1199111199083 (119905) 1199111199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119876119881 (119905) = 0 0 0 0 0 0 minus1198751199081 (119905) minus1198751199082 (119905) minus1198751199083 (119905) minus1198751199084 (119905)119879
(17)
Assuming 119875(Ω) as the amplitude of dynamic wheelrailforce induced by track irregularity yields the amplitude ofwheel displacement 119911119908(Ω) is
119911119908 (Ω) = [120572119908] 119875 (Ω) (18)
where [120572119908] = [119867][120572119881][119867]119879 is the dynamic flexibility of thevehicle at the wheelrail contact point and Ω is spatial fre-quency (radm) [120572119881] is the dynamic flexibility of the vehicle
24 The WheelRail Interaction The wheelrail interactionforce is described by
119875 (Ω) = ([120572119882] + [120572TS] + [120572Δ])minus1 Δ119911 (Ω) (19)
where [120572Δ] is the dynamic flexibility of the wheelrail contactspring [120572Δ] = 1119896119867 Since the dynamic displacements aresmall 119896119867 is linearized wheelrail contact stiffness accordingto the preload between the wheel and the rail and theirradii of curvature Δ119911(Ω) is the relative displacement (trackrandom irregularity) between the wheel and the rail andcan be obtained by using pseudoexcitation method [20ndash23]119875(Ω) is the amplitude of the wheelrail force induced bytrack irregularity then the wheelrail force in time domaincan be expressed as
119875 (119905) = 119875 (Ω) 119890119894Ω119905 (20)
There are generally two kinds of excitation inputmethodsof analyzing the vehicle-track coupling vibration One isfixed-point excitation and the other is moving-point excita-tion For the former supposing that the vehicle and track arefixed the track irregularity excitation moves backward at acertain speed For the latter the vehicle runs on the track withrandom irregularity at a certain speed In the present paperthe fixed-point excitation method is used
When a train runs on an irregular track it can be assumedthat the different wheels on the same rail are excited bythe same track irregularity but there will be time delayamong different wheels 1199051 1199052 119905119873 Track random excita-tion Δ119911(119905) at wheelsets119873 is
Δ119911 (119905) =
Δ1199111 (119905)Δ1199112 (119905)Δ119911119873 (119905)
=
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)
Δ119911119873 (119905 minus 119905119873)
(21)
where Δ119911(119905) is the history value of vertical track randomirregularity 119905119895 (119895 = 1 2 119873) is the time delay of eachwheel 119905119895 = (119886119895 minus 1198861)V 119886119895 is the longitudinal coordinate ofthe 119895th wheelset at the time 119905 = 0 and 1198861 is the longitudinalcoordinate of the first wheelset Δ119911(119905) can be regarded as ageneralized single excitation If the power spectrum densityfunction of Δ119911(119905) is 119878V(Ω) the pseudoexcitation of verticaltrack random irregularity can be expressed as
Δ119911 (119905) = radic119878V (Ω)119890119894Ω119905 (22)
Then the pseudoexcitation corresponding to (21) is
Δ119911 (119905) =
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)Δ119911119873 (119905 minus 119905119873)
=
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω)119890119894Ω119905
= Δ119911 (Ω) 119890119894Ω119905
(23)
where Δ(Ω) is the harmonic irregularity amplitude of thepseudoexcitation of vertical track random irregularity
Δ119911 (Ω) =
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω) (24)
Replacing Δ119911(Ω) in (19) with Δ(Ω) in (24) we canhave that
(Ω) = Δ (Ω)([120572119882] + [120572TS] + [120572Δ]) (25)
where (Ω) is the amplitude of pseudodynamic wheelrailforce corresponding to pseudoexcitation Δ(Ω) By solv-ing (25) the amplitude of pseudodynamic wheelrail forceinduced by track irregularity can be obtained According tothe pseudoexcitation method the power spectral density ofthe dynamic wheelrail force 119878119875119875(Ω) is given as follows
119878119875119875 (Ω) = (Ω)lowast sdot (Ω)119879 (26)
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
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Shock and Vibration
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Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Shock and Vibration
119911119881 (119905) = 119911119888 (119905) 120593119888 (119905) 1199111199051 (119905) 1199111199052 (119905) 1205931199051 (119905) 1205931199052 (119905) 1199111199081 (119905) 1199111199082 (119905) 1199111199083 (119905) 1199111199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119881 (119905) = 119888 (119905) 119888 (119905) 1199051 (119905) 1199052 (119905) 1199051 (119905) 1199052 (119905) 1199081 (119905) 1199082 (119905) 1199083 (119905) 1199084 (119905)119879119876119881 (119905) = 0 0 0 0 0 0 minus1198751199081 (119905) minus1198751199082 (119905) minus1198751199083 (119905) minus1198751199084 (119905)119879
(17)
Assuming 119875(Ω) as the amplitude of dynamic wheelrailforce induced by track irregularity yields the amplitude ofwheel displacement 119911119908(Ω) is
119911119908 (Ω) = [120572119908] 119875 (Ω) (18)
where [120572119908] = [119867][120572119881][119867]119879 is the dynamic flexibility of thevehicle at the wheelrail contact point and Ω is spatial fre-quency (radm) [120572119881] is the dynamic flexibility of the vehicle
24 The WheelRail Interaction The wheelrail interactionforce is described by
119875 (Ω) = ([120572119882] + [120572TS] + [120572Δ])minus1 Δ119911 (Ω) (19)
where [120572Δ] is the dynamic flexibility of the wheelrail contactspring [120572Δ] = 1119896119867 Since the dynamic displacements aresmall 119896119867 is linearized wheelrail contact stiffness accordingto the preload between the wheel and the rail and theirradii of curvature Δ119911(Ω) is the relative displacement (trackrandom irregularity) between the wheel and the rail andcan be obtained by using pseudoexcitation method [20ndash23]119875(Ω) is the amplitude of the wheelrail force induced bytrack irregularity then the wheelrail force in time domaincan be expressed as
119875 (119905) = 119875 (Ω) 119890119894Ω119905 (20)
There are generally two kinds of excitation inputmethodsof analyzing the vehicle-track coupling vibration One isfixed-point excitation and the other is moving-point excita-tion For the former supposing that the vehicle and track arefixed the track irregularity excitation moves backward at acertain speed For the latter the vehicle runs on the track withrandom irregularity at a certain speed In the present paperthe fixed-point excitation method is used
When a train runs on an irregular track it can be assumedthat the different wheels on the same rail are excited bythe same track irregularity but there will be time delayamong different wheels 1199051 1199052 119905119873 Track random excita-tion Δ119911(119905) at wheelsets119873 is
Δ119911 (119905) =
Δ1199111 (119905)Δ1199112 (119905)Δ119911119873 (119905)
=
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)
Δ119911119873 (119905 minus 119905119873)
(21)
where Δ119911(119905) is the history value of vertical track randomirregularity 119905119895 (119895 = 1 2 119873) is the time delay of eachwheel 119905119895 = (119886119895 minus 1198861)V 119886119895 is the longitudinal coordinate ofthe 119895th wheelset at the time 119905 = 0 and 1198861 is the longitudinalcoordinate of the first wheelset Δ119911(119905) can be regarded as ageneralized single excitation If the power spectrum densityfunction of Δ119911(119905) is 119878V(Ω) the pseudoexcitation of verticaltrack random irregularity can be expressed as
Δ119911 (119905) = radic119878V (Ω)119890119894Ω119905 (22)
Then the pseudoexcitation corresponding to (21) is
Δ119911 (119905) =
Δ1199111 (119905 minus 1199051)Δ1199112 (119905 minus 1199052)Δ119911119873 (119905 minus 119905119873)
=
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω)119890119894Ω119905
= Δ119911 (Ω) 119890119894Ω119905
(23)
where Δ(Ω) is the harmonic irregularity amplitude of thepseudoexcitation of vertical track random irregularity
Δ119911 (Ω) =
119890minus119894Ω1199051119890minus119894Ω1199052
119890minus119894Ω119905119873
radic119878V (Ω) (24)
Replacing Δ119911(Ω) in (19) with Δ(Ω) in (24) we canhave that
(Ω) = Δ (Ω)([120572119882] + [120572TS] + [120572Δ]) (25)
where (Ω) is the amplitude of pseudodynamic wheelrailforce corresponding to pseudoexcitation Δ(Ω) By solv-ing (25) the amplitude of pseudodynamic wheelrail forceinduced by track irregularity can be obtained According tothe pseudoexcitation method the power spectral density ofthe dynamic wheelrail force 119878119875119875(Ω) is given as follows
119878119875119875 (Ω) = (Ω)lowast sdot (Ω)119879 (26)
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
Excitation at mid-spanExcitation above a sleeper
103102101
Frequency (Hz)
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
(a) The present model
Excitation at mid-spanExcitation above a sleeper
10minus11
10minus10
10minus9
10minus8
10minus7
Rail
rece
ptan
ce (m
N)
102 103101
Frequency (Hz)
(b) FEMmodel
Figure 3 Vertical receptance of the rail
where (Ω)lowast and (Ω)119879 are the conjugated matrix andtransfer matrix of (Ω) respectively
According to (19) and (26) the power spectral density ofthe dynamic wheelrail force can be obtained Through (15)and (18) the dynamic response of any point of vehicle-track-subgrade coupling system can be calculated
3 Results and Discussion
31TheRail Receptance Thereceptance of the structure is thedynamic response of each part of the structure under a unitforce also known as the dynamic flexibility of the structureIt is generally in complex form of which the amplitude-frequency and phase-frequency curves reflect the transfercharacteristics of the structure dynamic responseThe ampli-tude of themobility is usually named as the receptance whichcan be divided into displacement velocity and accelerationreceptance In order to validate the effectiveness of theproposed method a numerical simulation for calculatingthe mobility of double-block ballastless track and subgradecoupling system is set up using the finite element analysissoftware ANSYS in which there are two slabs with a lengthof 125m each a rail and a subgrade layer with a length of25m A unit harmonic force is applied to the rail both at mid-span and above sleepers and the comparisons of the verticalreceptance of the rail at the excitation point are shown inFigure 3
In Figure 3 the vertical receptance of the double-blockballastless track is depicted both at mid-span and abovesleeper There are three peaks in the vertical receptance Thefirst one corresponds to the vertical vibration mode of thepad at 157Hz the second one corresponds to the first verticalpined-pined resonance of the rail which is related to the raildiscrete support at 1059Hz the last one is the high frequency
vibration of the rail at 2740Hz It can also be seen fromFigures 3(a) and 3(b) that the peak frequencies of the resultsof the proposed model are basically consistent with those ofthe finite element method
The wheelrail contact point applied at the mid-span ofthe adjacent fasteners the total receptance and phase curvesare shown in Figure 4
From Figure 4 it can be seen that in the frequency rangeof 1sim32Hz the receptance of the wheel is much larger thanthat of the rail and wheelrail contact and the dynamicflexibility amplitude and phase of wheelrail system arealmost equivalent to the flexibility of the wheel indicatingthat in the range of 1sim32Hz the receptance of the wheelplays the main role in the whole flexibility of the wheelrailsystem in the range of 32sim60Hz because the receptance ofthe wheel is in the same level with that of the rail the dynamicflexibility of wheelrail system is mainly decided both by railand by wheel and at 45Hz the total receptance has a localminimum in the range of 60sim277Hz the receptance of therail is larger than that of the wheel and wheelrail contactso the dynamic flexibility amplitude and phase of wheelrailsystem are mostly decided by the rail and at 200Hz thedynamic flexibility amplitude of wheelrail system has a peakvalue in the range of 277sim1000Hz both the rail and thewheel play a main role in the dynamic flexibility of wheelrailsystem above 100Hz the wheelrail contact receptance playsa main role in the dynamic flexibility of wheelrail system
32 Random Vibration Analysis of the System When thetrain is running at the speed of 300 kmh the rack randomirregularitywill lead to the randomvibration of vehicle-track-subgrade coupling system The track random irregularityspectrum curve is shown in Figure 5 which is the super-position of the results between track random irregularity
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
10minus14
10minus12
10minus10
10minus8
10minus6Re
cept
ance
(mN
)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
(a) Receptance
minus200
minus150
minus100
50
100
Phas
e (de
gree
s)
101 103102
Frequency (Hz)
SumRail
WheelWheelrail contact
minus50
0
(b) Phase
Figure 4 Displacement receptance contribution of the wheelrail system
103102101100
Frequency (Hz)
10minus5
100
Trac
k irr
egul
arity
spec
trum
(mm
2 Hz)
Figure 5 Track random irregularity
and rail surface roughness from Chinarsquos high speed railwaymeasurement The wheelrail force is shown in Figure 6Figure 7 demonstrates the vertical vibration accelerationpower spectrum curve of the rail at the excitation point andthe vertical vibration acceleration power spectrum curves ofthe slab and the subgrade at the cross section of excitation arerespectively shown in Figures 8 and 9
From Figure 6 it can be seen that the wheelrail interac-tion forces caused by track random irregularity are concen-trated in the range of 10sim1000Hz and the amplitude energy
10minus2
10minus1
100
101
102
103
Whe
elra
il fo
rce s
pect
rum
((kN
)H
z)
103102101
Frequency (Hz)
Figure 6 The amplitude spectrum of wheelrail force
of wheelrail force reaches maximum at 54Hz and secondlarge at 380Hz Above 1000Hz the wheelrail force hassome peaks but the corresponding vibration energy is smallFrom Figure 7 it can be seen that the vibration accelerationpower spectrum density of the rail caused by track ran-dom irregularity has three obvious peaks in the range of30sim1000Hz respectively at 43Hz 3412Hz and 5023Hzwhere the corresponding wavelength is respectively 2m
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 9
X 43Y 3923
X 3412Y 5123
X 5023Y 1529
103102101
Frequency (Hz)
0
100
200
300
400
500
600Ac
cele
ratio
n PS
D ((
ms
2 )2H
z)
Rail
Figure 7 The acceleration power spectrum of the rail
Track slab
0
2
4
6
8
10
12
Acce
lera
tion
PSD
((m
s2 )
2H
z)
103102101
Frequency (Hz)
X 4268Y 1002
X 5321Y 08907
Figure 8 The acceleration power spectrum of the track slab
024m and 017m under the train running at the speed of300 kmh From Figure 8 it can be seen that the vibrationacceleration power spectrum density of the slab caused bytrack random irregularity has two obvious peaks in the rangeof 30sim100Hz respectively at 426Hz and 532Hz wherethe corresponding wavelength is respectively 2m and 16munder the train running at the speed of 300 kmh FromFigure 9 it can be seen that very similar to that of the slab inFigure 11 the vibration acceleration power spectrum densityof the subgrade caused by track random irregularity has twoobvious peaks in the range of 30sim100Hz respectively at
SubgradeX 4268Y 4229
X 5321Y 03773X 2816
Y 01645
103102101
Frequency (Hz)
0
1
2
3
4
5
Acce
lera
tion
PSD
((m
s2 )
2H
z)Figure 9 The acceleration power spectrum of the subgrade
426Hz and 532Hz where the corresponding wavelength isrespectively 2m and 16m under the train running at thespeed of 300 kmh From Figures 7ndash9 it can also be seen thatall of the vibration acceleration power spectrum density ofrail slab and subgrade have a main peak value around 43Hzmainly due to the fact that the track irregularity has excitationwavelength in this band as is shown in Figure 5 anothermainfrequency of rail is induced by the shortwave irregularitywavelength below 1m in the rail surface in addition thefrequency band of the vibration energy spectrum of the rail iswider than that of the slab and subgrade and the frequencyband of the vibration energy spectrums of slab and subgradeis mainly concentrated in the low frequency around 43Hz
33 The Influence of ACLrsquos Stiffness on the Vibration Power ofTrack and Subgrade System In order to reflect the vibrationtransfer characteristics of track-subgrade coupling systemthe vibration power flow theory is used to analyze thevibration transfer characteristics of ballastless track-subgradecoupling systemThe average value of track-subgrade systemvibration power during a period is given by
119875 = int1198790119865 (119905) 119881 (119905) 119889119905
119879 (27)
where119875 is the average power119865(119905) is the external force appliedat the track-subgrade system and119881(119905) is the vibration veloc-ity in time domain And the vibration power in frequencydomain can be written as
119875 (Ω) = 12Re 119865lowast (Ω)119881 (Ω) (28)
where 119865lowast(Ω) is the conjugated matrix of the force ampli-tude matrix and 119881(Ω) is the vibration velocity in fre-quency domain Vibration power of track-subgrade system is
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Shock and Vibration
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
(b) Slab
18 times 107 Nm18 times 108 Nm18 times 109 Nm
101 102 103100
Frequency (Hz)
minus50
0
50
100
150
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
(c) Subgrade
Figure 10 The effect of ACLrsquos stiffness on the vibration power of track and subgrade system
expressed by the power level and reference vibration poweris 10minus12W
ACL has an important influence on vibration-reductionof track-subgrade coupling system In order to reflect theeffect of ACLrsquos stiffness on subgrade vibration-reductionstiffness parameters of three ACL are 18 times 107Nm 18 times108Nm and 18 times 109Nm respectively Figure 10 showsthe impact of ACLrsquos stiffness on the vibration power of trackand subgrade coupling system The other parameters areunchangeable
It can be seen from Figure 10 that for the rail theinfluence of ACLrsquos stiffness on rail vibration at 100sim150Hzis more than that at the other frequencies For the trackslab the larger the stiffness of ACL the lower the vibrationpower of the track bed at 1sim1800Hz When the stiffness
of ACL increases by ten times the vibration power of theslab will decrease by about 20 dB For the subgrade thelarger the stiffness of ACL the lower the vibration power ofthe subgrade bottom and the lower the peak frequency ofsubgrade below 100Hz The larger the stiffness of ACL thehigher the peak frequency of subgrade at 100ndash700Hz Thevibration power of the subgrade increases with the stiffnessof ACL above 1000Hz It can be concluded that some peaksin Figure 10 relate to nature frequencies of track-subgradesystem and track random irregularities
34 The Influence of Train Speed on the Track and SubgradeSystem Vibration Power To investigate the impact of trainspeed on the track and subgrade system vibration power thetrain speeds are respectively 100 kmh 200 kmh 300 kmh
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 11
101 102 103100
Frequency (Hz)
100 kmh200 kmh
300 kmh400 kmh
40
60
80
100
120
140
160Ra
ilrsquos v
ibra
tion
pow
er (d
B)
(a) Rail
101 102 103100
Frequency (Hz)
100 kmh400 kmh
minus20
0
20
40
60
80
100
120
140
Trac
k sla
brsquos v
ibra
tion
pow
er (d
B)
300 kmh200 kmh
(b) Slab
101 102 103100
Frequency (Hz)
minus40
0
40
80
120
160
Subg
rade
rsquos vi
brat
ion
pow
er (d
B)
100 kmh400 kmh300 kmh200 kmh
(c) Subgrade
Figure 11 The effect of vehicle speed on the vibration power of track and subgrade system
and 400 kmh Figure 11 shows the influence of train speed onthe vibration power of the track and subgrade system
It can be seen from Figure 11 that when the train speedis 400 kmh under excitation of track random irregularitiesthe vibration power of track-subgrade coupling system hasmaximum peak at about 40Hz and the maximum of vibra-tion power of the rail bed slab and subgrade reaches 150 dB130 dB and 125 dB respectively Figure 11 also indicates thatthe higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration power but thelower the frequency of resonance peak of them
4 Conclusions
In order to reduce the ground-borne vibration caused bywheelrail interaction in the ballastless track of high speedrailways viscoelastic asphalt concrete materials are filled
between the track and the subgrade to attenuate wheelrailforce In combination with double-block ballastless trackand subgrade structure characteristics considering the finitelength and discontinuity of the monolithic slab discretenessof the pads under rail and the continuity of subgrade ahigh speed train-track-subgrade vertical coupling dynamicmodel is developed and the analytic expressions of vibrationresponse are also derived Combined with the pseudoexci-tation method a solution of the dynamic response is putforward to calculate the receptance of ballastless track andsubgrade systemThe randomvibration response and transfercharacteristics of ballastless track and subgrade system areobtained under track random irregularity when a high speedvehicle runs through We can conclude the following
(1) The peak frequencies of the results of the proposedmodel are basically consistent with those of finiteelement method
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 Shock and Vibration
(2) The frequency band of the vibration energy spectrumof rail is wider than that of the slab and subgrade andthe frequency band of the vibration energy spectrumsof slab and subgrade ismainly concentrated in the lowfrequency around 43Hz
(3) The stiffness of ACL has relatively less influence onrail vibration than on the ballast bed and subgradevibration
(4) The higher the train speed the larger the amplitude ofresonance peak of the track-subgrade vibration powerbut the lower the frequency of resonance peak ofthem
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by National Natural ScienceFoundation of China (nos 51165017 51268030 and 51378395)and Open Fund of Traction Power State Key LaboratorySouthwest Jiaotong University (Grant no TPL1602)
References
[1] K Knothe ldquoPast and future of vehicletrack interactionrdquoVehicleSystem Dynamics (Supplement) vol 45 no 24 pp 1ndash3 1995
[2] S L Grassie ldquoReview of workshop aims and open questionsrdquoVehicle System Dynamics (Supplement) vol 45 no 24 pp 380ndash386 1995
[3] V V Krylov ldquoGeneration of ground vibrations by superfasttrainsrdquo Applied Acoustics vol 44 no 2 pp 149ndash164 1995
[4] A Matsuura ldquoImpulsive response of an elastic layered mediumin the anti-plane wave field based on a thin-layered elementand discrete wave number methodrdquo Structural EngineeringEarthquake Engineering vol 9 no 22 pp 119ndash128 1993
[5] A Matsuura ldquoSimulation for analyzing direct derailment limitof running vehicle on oscillating tracksrdquo Structural Engineer-ingEarthquake Engineering vol 15 no 1 pp 63ndash72 1998
[6] A M Kaynia C Madshus and P Zackrisson ldquoGround vibra-tion from high-speed trains prediction and countermeasurerdquoJournal of Geotechnical andGeo-environmental Engineering vol126 no 6 pp 531ndash537 2000
[7] H Takemiya ldquoSimulation of trackground vibrations due tohigh-speed trainsrdquo in Proceedings of the 8th International Con-gress on Sound and Vibration pp 2875ndash2882 Hong Kong 2001
[8] CMadshus andAMKaynia ldquoHigh-speed railway lines on softground dynamic behaviour at critical train speedrdquo Journal ofSound and Vibration vol 231 no 3 pp 689ndash701 2000
[9] Q Su and Y Cai ldquoA spatial time-varying coupling model fordynamic analysis of high speed railway subgraderdquo Journal ofSouthwest Jiaotong University vol 36 no 5 pp 509ndash513 2001
[10] L Dong C-G Zhao D-G Cai Q-L Zhang and Y-S YeldquoMethod for dynamic response of subgrade subjected to high-spped moving loadrdquo Engineering Mechanics vol 25 no 11 pp231ndash240 2008 (Chinese)
[11] B Liang H Luo and C-X Sun ldquoSimulated study on vibrationload of high speed railwayrdquo Journal of the China Railway Societyvol 28 no 4 pp 89ndash94 2006 (Chinese)
[12] X-N Ma B Liang and F Gao ldquoStudy on the dynamic prop-erties of slab ballastless track and subgrade structure on high-speed railwayrdquo Journal of the China Railway Society vol 33 no2 pp 72ndash78 2011 (Chinese)
[13] Q-Y Xu Y-F Cao and X-L Zhou ldquoInfluence of short-waverandom irregularity on vibration characteristic of train-slabtrack-subgrade systemrdquo Journal of Central South University vol42 no 4 pp 1105ndash1110 2011 (Chinese)
[14] P Xu and C-B Cai ldquoSpatial dynamic model of train-ballasttrack-subgrade coupled systemrdquo EngineeringMechanics vol 28no 3 pp 191ndash197 2011 (Chinese)
[15] J Li and K Li ldquoFinite element analysis for dynamic responseof roadbed of high-speed railwayrdquo Journal of the China RailwaySociety vol 17 no 1 pp 66ndash75 1995
[16] C-J Wang and Y-M Chen ldquoAnalysis of stresses in train-induced groundrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 24 no 7 pp 1178ndash1187 2005
[17] W M Zhai Vehicle-Track Coupled Dynamics Science PressBeijing China 2007
[18] H Liu A Study on Modelling Prediction and Its Control ofWheelRail Rolling Noises in High Speed Railway ShanghaiJiaotong University Shanghai China 2010
[19] H Wei Elevated Track Structure Vibration and TransmittingCharacteristics of UrbanMass Transit Tongji University Shang-hai China 2012
[20] J H Lin and Y H Zhang Pseudo ExcitationMethod in RandomVibration Science Press Beijing China 2004
[21] F Lu Q Gao J H Lin and F W Williams ldquoNon-stationaryrandom ground vibration due to loads moving along a railwaytrackrdquo Journal of Sound and Vibration vol 298 no 1-2 pp 30ndash42 2006
[22] J H Lin Y H Zhang and Y Zhao ldquoPseudo excitation methodand some recent developmentsrdquo Procedia Engineering vol 14pp 2453ndash2458 2011
[23] C C Caprani ldquoApplication of the pseudo-excitation methodto assessment of walking variability on footbridge vibrationrdquoComputers amp Structures vol 132 pp 43ndash54 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
