Research ArticleResearch on the Vibration Insulation of High-Speed TrainBogies in Mid and High Frequency

Jia Liu 12 and Xuesong Jin 1

1State Key Laboratory of Traction Power Southwest Jiaotong University Chengdu 610031 China2Science and Technology on Reactor System Design Technology Laboratory Nuclear Power Institute of China Chengdu 610213 China

Correspondence should be addressed to Xuesong Jin xsjinhomeswjtueducn

Received 24 September 2017 Accepted 24 December 2017 Published 13 February 2018

Academic Editor Evgeny Petrov

Copyright copy 2018 Jia Liu and Xuesong Jin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

According to a large amount of the test data themid and high frequency vibrations of high-speed bogies are very notable especiallyin the 565sim616Hz range which are just the passing frequencies corresponding to the 22nd to 24th polygonal wear of the wheel Inorder to investigate the main cause of wheel higher-order polygon formation a 3D flexible model of a Chinese high-speed trainbogie is developed using the explicit finite element method The results show that the couple vibration of bogie and wheelset maylead to the high-order wears of wheel In order to reduce the coupled resonance of thewheelset and the bogie frame the effects of thestiffness and damping of the primary suspensions wheelset axle radius and bogie frame strength on the vibration transmissibilityare discussed carefully The numerical results show that the resonance peaks in high frequency range can be reduced by reducingthe stiffness of axle box rotary arm joint reducing the wheelset axle radius or strengthening the bogie frame location The relatedresults may provide a reference for structure improvement of the existing bogies and structure design of the new high-speed bogies

1 Introduction

Nowadays more and more people consider high-speed trainsto be a comfortable safe low and clean energy consumptiontransportation tool However increasing the operation speedand mileage accelerated the polygonal wear of high-speedtrain wheels which leads to the fierce vibration of the vehicle-track system in a wide frequency range According to sitetests and experiences of the authors the high-order polygonalwear of wheel and the mid and high frequency vibrations ofhigh-speed bogies are very notable [1] These mid and highfrequency vibration behavior existing in bogie frames havebeen considered to accelerate high-order polygonal wear ofhigh-speed train wheels To slow down the development ofwheel polygon and offer a comfortable vibration environ-ment for the high-speed train it is necessary to investigatethe dynamic characteristics of primary suspension and thestructure parameters of bogie components on the vibrationtransmissibility of a high-speed bogie system This study can

provide a basis for high-speed vibration reduction inmid andhigh frequency

The multibody modeling and dynamic behavior of therailway vehicles are well understood in the low frequencyrange and the wheelset can be modeled well in the frequencyrange up to several thousand Hertz [2] However the workfocused on middle and high frequency vibration of railwaybogies and its influence on the system dynamics is relativelyinsufficient Alexander et al [3] established a FE model ofthe whole bogie this model covers a high number of beamsand bars andmost of the primary suspensions and secondarysuspensions are simplified in this model The model mainlyserves as a support to investigate the actuator performanceand evaluate several actuator concepts Ren et al [4] built aflexible vehicle system dynamics model based on multibodymodeling and FE method which was used to investigate thevibration and frequency transmission characteristics of thehigh-speed EMU The vibration transference from axle boxto car body was analyzed in detail

HindawiShock and VibrationVolume 2018 Article ID 6759105 13 pageshttpsdoiorg10115520186759105

2 Shock and Vibration

In order to reduce the vibrations of the high-speed bogiein in mid and high frequency a 3D flexible model of aChinese high-speed train bogie is developedusing the explicitfinite element (FE) method Based on the bogie FE modelthe vertical vibration responses of the axle boxes and thebogie frame are obtained in frequency domains The reasonswhich cause the high-order wear of wheel are analyzed andthe effects of the parameters of the primary suspensionswheelset axle radius and stiffened thin plates in bogie frameon the vibration transmissibility are discussed carefully Itis noted that the numerical methods and the results wouldbe helpful in understanding the mid and high frequencyvibration characteristics and the vibration transmissibility ofthe high-speed train bogie

2 Measurement of Wheel Polygon andVehicle Vibration

Both wheel polygon and vehicle vibration of the trailer bogiewere tested Wheel polygon measurements before and afterreprofiling were completed while the bogies were standing onthe rail Vibrations of the axle box and the bogie frame beforereprofiling weremonitored while the train was running at thespeed of 250 kmh

21 Characteristic of Wheel Polygon The wheel polygon cancause a series of vibration and noise problems of high-speedtrain according to a lot of previous test data Therefore thewheel roughness of thewheel circumferencewas tested firstlyTest of wheel polygon was carried out by OSD-RRM01 [15] Displacement sensors were installed vertically to wheeltreads to record the wheel diameter difference Rotationsensors were used to measure the circumference of wheel inorder to record the wheel polygon orders exactly

The wheel roughness is defined with logarithmic form 119871119871 = 20 log10 119903119899119903ref (1)

where 119903119899 is wheel roughness in different order and 119903ref isreference of the wheel diameter difference usually being1 120583m

The polygon order distributions before and after repro-filing are compared in Figure 1 The horizontal axis illus-trates the polygon orders and the vertical axis denotes theamplitudes of the wheel polygons The peaks mean that thecorresponding polygons have a large contribution to theunevenwear of thewheels Figure shows that the roughness ofthe wheel before reprofiling is very large especially the 22ndto 24th polygonal wear of the wheel After reprofiling theroughness level of the 5th to 24th polygon reduces evidentlybut peaks at the ordinates from 22 to 24 are still high In theprocessing of profiling higher-order polygons are reservedbecause the tool head does not contact with wave valleysegment of polygonal wear

22 Vibration of Vehicle Figure 2 gives the test photo ofbogie vibration Two accelerates were fixed on the axle boxand the bogie frame respectively to measure the vertical

2 4 6 8 10 12 14 16 18 20 22 24 26 28 300Polygon order

minus30

minus20

minus10

0

10

20

30

40

Roug

hnes

s lev

el (d

Bmiddotre

1

m)

After reprofilingBefore reprofiling

Figure 1 Wheel polygon order

Axle box

Bogie frame

Figure 2 Test photo of accelerations installed on the bogie frameand axle box

vibration of the bogie The vibration data was acquired byusing BampK Type 3560D hardware The sampling frequencyand the sampling numbers are 3200Hz and 800 respectivelyThe high-speed train was running on a straight track and itsspeed was 250 kmh

Figure 3 shows the vibration results of the axle boxand the bogie frame before reprofiling There is an obviousreduction in the vibration from the axle box to the bogieframe especially in the frequency band of 650sim800Hz Inthe 565sim616Hz range magnitudes of serious vibrations onthe axle box are close to that of the bogie frame Herefrequencies in the band 565sim616Hz are just the passingfrequencies corresponding to the 22nd to 24th polygonalwear of the wheel Hence measures should be taken toimprove the vibration isolation in this frequency band forslowing down the development of the wheel polygonal wearThe vibration isolation can be increased by optimizing eitherthe parameters of primary suspension or the structure of

Shock and Vibration 3

0 100 200 300 400 500 600 700 800Frequency (Hz)

minus30

minus20

minus10

0

10

20

Vert

ical

vib

ratio

n (d

B re

1m

Ｍ2)

Axle boxBogie frame

Figure 3 Vibrations of bogie frame and axle box

the bogie In this paper the material parameters that affectthe vibration transmissibility of the primary suspensioncomponents are analyzed

When wheel rotates a circle the rotating frequency 1198910 isgiven by

1198910 = V120587119863 = 250 km sdot hminus1

086m times 314 times 36 = 257Hz (2)

where the wheel diameter119863 is 086mThe passing frequency of the wheel polygon 119891119899 is calcu-

lated by

119891119899 = 1198991198910 = 257119899 (3)

where 119899 is the polygon order

3 Flexible Model of the Bogie andIts Validation

31 Flexible Bogie Model Figure 4 indicates the 3D transientfinite element (FE) model of the entire bogie system whichconsists of two wheelsets a bogie frame and a series ofthe primary suspension parts developed with ANSYS Theprimary suspension parts include eight coil springs (insidespring and outside spring) four rubber bearings under thecoil springs four axle box rotary arms four rotary arm rubberjoints four vehicle primary dampers and eight rubber jointsabove or below the dampers All the geometries of this bogiesystem are the same as that of the previously tested trailerThe brake disc on the wheelsets and the brake disc seat onthe bogie frame are also considered in this study

Figure 5 shows the FE model of some of the primarysuspension components The figures in the left- and right-hand sides are the 3D geometries and the FE model respec-tively From Figure 5 it can be observed that the nonlinear

Figure 4 FEM of the whole bogie system

influences of rubber materials are ignored The rotary armrubber joint the rubber bearing and the damper rubberjoint are simulated using spring-damper elements [6] Thevehicle damping is simulated using damper elements Thecar body is simulated as a lumped mass connected to thebogie frame through secondary suspension The secondarysuspension is simulated using the spring-damper elementsThe axle box rotary arm and the coil spring are meshedusing 3D solid elements taking their actual geometries intoconsideration [7] Table 1 lists the values of the stiffness anddamping involved in this paper [8]

To improve the calculating efficiency some 1D and 2Delements are applied to the model For example beamelements are used to simulate bogie frame horizontal beamsand axles mass elements are used to simulate the brakedisc seat on the bogie frame and shell elements are usedto simulate bogie side frames The total element and nodenumbers are 234165 and 184209 respectively

32Theory ofWheelTrack Force Remington [9] put forwardthe most comprehensive single model of wheel-rail forceand wheel-rail noise The model assumes that the small-scaleroughness on the running surfaces of the wheel and rail is theprimary mechanism for the force generation Included in themodel are such effects as the spatial filtering of the roughnessdue to the finite area of contact between wheel and rail andthe interaction between the wheel and rail including localcontact stiffness119903 is defined as the displacement-input vector Assumingthat there are 119873 coupling coordinates 1199111 119911119873 one canuse 119903119895 to represent the coupled displacements of the wheeland rail 119911119877119895 for the rail 119911119882119895 for wheel and 119911119862119895 for the relativemotion in the contact zone The latter can be separated intotwo parts 119911119862119882119895 for local deformations of the wheel and 119911119862119877119895for the local deformations of the rail Then

119911119877119895 = 119911119882119895 minus 119911119862119882119895 minus 119911119862119877119895 + 119903119895 (4)

The wheel and rail can each then represented by an 119873 sdot119873 matrix of point receptances 120572119882119895 and 120572119877119895 The transferproperties of the contact zone (relative motion 119911119862119882119895 and 119911119862119877119895 per unit force) can similarly be represented by receptances

4 Shock and Vibration

C K

(a) Coil spring and the rubber bearing

C

K

KC

C

(b) Primary damper and the rubberjoint

K C

(c) Rotary arm rubber joints

Figure 5 3D geometry and FEM of the primary suspension

Table 1 Values of stiffness and damping involved in this study

Stiffness (Nmm) Damping (Nsdotsmm)Vertical Horizontal Longitudinal Vertical Horizontal Longitudinal

Primary suspensionRubber bearings 4000 400 4000 002 002 024Primary dampers - - - - - - - - 33 - -Damper rubber joints 70000 7500 70000 166 018 166Rotary arm rubber joints 160000 18000 76000 9 043 379

Secondary suspension 1150 174 174 37 294 245

120572119862119882119895119896 and 120572119862119877119895119896 An interaction force acting in the 119896th coor-dinate direction 119875119896 results from the coupling (the force istransmitted by the contact springs) and the same force actson both the wheel and the rail (in opposite directions) Theforces and displacements at a specific frequency 120596 can berelated by the receptances of the various subsystems

119911119877119895 =119873sum119896=1

120572119877119895119896119875119896

119911119882119895 = 119873sum119896=1

120572119882119895119896 (minus119875119896)

119911119862119877119895 = 119873sum119896=1

120572119862119877119895119896 119875119896

119911119862119882119895 = 119873sum119896=1

120572119862119882119895119896 119875119896(5)

Shock and Vibration 5

Contact filter

Wheel receptances

Rail receptances

Contact receptances

Wheel-rail interaction force

Wheel roughness

Rail roughness

Combined roughness

Figure 6 Framework of theoretical model of wheel-rail force

Or the upper matrix can be translated to other forms

119911119877 = [120572119877] 119875 119911119882 = minus [120572119882] 119875 119911119862119877 = [120572119862119877] 119875 119911119862119882 = [120572119862119882] 119875

(6)

Note the negative signs for the wheel since here the forceacts in the opposite direction to that of the displacementHence from equation (4)

119903 = [120572119877 + 120572119862119877 + 120572119862119882 + 120572119882] 119875 = [120572] 119875 (7)

where [120572] is defined as the matrix of combined wheel railand contact zone receptances

119875 = [120572]minus1 119903 (8)

Equation (8) is the wheel-rail force calculated modelwhich was put forward byThompson [10] based on Reming-tonrsquos model A wheel-rail interaction model is used to calcu-late wheel-rail dynamic force base on the wheel-rail com-bined roughness Figure 6 shows the flow chart of the progressto calculate the wheel-rail force The wheel roughness andrail roughness are used to calculate the wheel-rail combinedroughness and then combinedwith the receptances of wheeland rail the wheeltrack force is obtained

The local deformations of two bodies in contact actas nonlinear stiffness between them However for smalldisplacement linearized Hertzian contact stiffness 119870V invertical direction may be calculated as

119870V = ( 32120585)[(43

1198641 minus 1205832)

2 1198750 4119877119882119877119877119877119882 + 119877119877]13

(9)

where 119877119882 is the radius of wheel and 119877119877 is the radius ofcurvature of rail surface 119864 is the plain strain elastic and 120583 isPoissonrsquos ratio of wheel and rail 1198750 is the static component ofthe vertical load at the contact 120585 is a dimensionless quantitydependent on the radius of curvature of the two surfaces (119877119877and 119877119882) and hence the shape of the contact patch 120585 is relatedto 120579 which is defined as follows

120579 = 119886119888119903 cos 10038161003816100381610038161003816100381610038161003816119877119877 minus 119877119882119877119877 + 119877119882

10038161003816100381610038161003816100381610038161003816 (10)

The contact patch wavenumber filter is presented byRemington which is given by

|119867 (119896)|2 = 4120572 (119896119887)2 int

arctan120572

0[1198691 (119896119887 sec119909)]2 119889119909 (11)

where 119896 is the wavenumber along the length of the rail oraround the circumference of the wheel 120572 is a constant deter-mining the degree of correlation between parallel roughnessprofiles at a given wavenumber 1198691(119909) is the first-order Besselfunction 119909 is a variable and 119887 is circular contact patch ofradius

Prestress and wheelrail contact forces are applied to thepoints on the four wheel treads (Figure 5) The prestress is72000N computed based on the single wheel weight of thevehicle The wheelrail contact forces are calculated basedon the TWINS model Rail roughness and wheel roughnesswere obtained by the site test results and the measurementswere conducted by using CAT and BBM respectively Railreceptance is obtained by using hammer and the verticalacceleration frequency response of rail surface is obtainedby exiting the same position Acceleration is translated intodisplacement by using (12) in the frequency domain Wheelreceptance is obtained by using the upper 3D transient finiteelement (FE) model of the entire bogie system Force isapplied to the points on the four wheel treads the driving-point and transfer-pointmobility are calculated then Figure 7illustrates the wheel-track forces in the frequency domainfrom this figure it can be shown that the wheel-track exci-tation energies are mainly focused on the frequency bandwhich is under 200Hz

119886119894 = minus41205872 times 1198911198942 times 119889119894 (119894 = 1 2 119899) (12)

where 119886119894 is the acceleration in the frequency 119894 and 119889119894 is thedisplacement in the frequency 11989433 Model Validation Based on the above FE model thevibrations of the axle box and the bogie frame both in timedomain and frequency domain are calculated byANSYSTheFEmodel takeswheeltrack force in time domain as input thevibrations of the axle box and the bogie frame are calculatedby transient dynamic analysis method Figure 8 shows the

6 Shock and VibrationFo

rce (

N)

200 300 400 500 600 700 800100Frequency (Hz)

105

104

103

102

101

Figure 7 Wheelrail contact force in frequency domain

0

2

4

6

8

10

Axle box

Testing Simulation

Bogie frame

Vibr

atio

n (g

)

285 290 295 300 305 310 315 320280Time (s)

minus6

minus8

minus10

minus4

minus2

Figure 8 Vibration comparison of testing and simulation (in timedomain)

vibration responses in time domain Through FFT analysisthe vibrations are translated into the frequency domain below800Hz Figure 9 shows the vibration frequency spectrum Intime domain the vibration magnitudes of bogie frame andaxle box obtained by simulation are in good agreement withthemeasured results In frequency domain the spectrumdis-tributions and vibration magnitudes of the axle box obtainedby using the FEmethod are similar to themeasurement in thefrequency band of 0sim800Hz which validates that the modelof wheelset and axle box are accurate and the wheeltrackforce is close to that in practice As for the vibration of thebogie frame in spite of some differences in magnitudes inthe frequency band of 0sim800Hz the spectrum distributionsare in good agreement with the measured results Thus theFEmodel and the calculated method are validated effectivelyand a series of investigations are conducted using thismodel

4 Vibration Analysis of Bogie Flexible Model

Harmonic response analysis is used to calculate the steadyvibration responses of bogie key components under theswept frequency excitation Then the vibration peaks fromthe frequency response function are counted out In thesepeak frequencies vibration displacement color clouds of thewhole bogie are invested The car bodyrsquos 6 DOFs are allconstrained The harmonic forces are applied to the pointson the four wheel treads The magnitude of the force is50N and the analysis frequency band is from 0Hz to1000Hz and the frequency resolution is 4HzThe calculatedprogress is as follows First of all the modal parameters andextended models are calculated Based on the modal resultsthe vibrations of axle box and bogie frame are extractedby the full arithmetic The vibration results in one-thirdoctave of four bogie key part compartments which are bogieover the primary damper bogie over the spring rotary armsupport and axle box respective are shown as Figure 10From this figure it can be shown that as the frequencyincreased the magnitudes of every part components of bogieraise significantly which illustrates that as for the samewheeltrack force higher frequency is easier to excite theserious vibration In the centre bandof 500Hz themagnitudeof bogie frame is almost the same as axle box which is up to20 dB Vibrations of rotary arm support are the smallest Inthe centre band of 630Hz vibration energies of bogie frameover primary damper are bigger than axle box axle box androtary arm support are almost the same and bogie framesover spring are the smallest

Furthermore the FFT vibrations of these four componentparts in the frequency band of 400sim800Hz are calculatedResults are shown as Figure 11 from this picture in the fre-quency bandof 520sim560Hzmagnitudes of serious vibrationson the axle box are close to that of the bogie frame In thefrequency bandof 550sim620Hz there are no obvious vibrationpeaks on bogie frame and axle The vibration energies fromaxle box are translate into bogie framewithout reductionThevibrationmagnitudes of bogie frame are higher than axle boxin some specific frequencies Vibration of rotary arm supportis smaller than the bogie frame over the primary damperand over the spring respectively Therefore the vibrationreductions of axle box and bogie frame over the primarydamper are mainly discussed in Section 5

FromFigure 11 resonance peaks of bogie systemare in thefrequency band of 520sim560Hz Therefore the displacementcolor clouds of the whole bogie in this band are shown inFigure 12 In the 520sim560Hz the vibration deformations aremainly shown as the couple vibration of bogie side beam localmovements and the third bend ofwheelsetTherefore the twomodes may be the main reasons which lead to the resonancevibration in high frequency and then induce the high-orderwear of wheel

5 Vibration Isolation Design in Mid andHigh Frequency

The vibrations at three points on the bogie frame near thevertical damper coil spring and the rotary arm rubber joint

Shock and Vibration 7

200 300 400 500 600 700 800100Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

Axl

e box

vib

ratio

n (d

B re

1ＧＭ

2)

(a) Axle box

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

200 300 400 500 600 700 800100Frequency (Hz)

Bogi

e fra

me v

ibra

tion

(dB

re1ＧＭ

2)

(b) Bogie frame

Figure 9 Vibration comparison of testing and simulation (in frequency domain)

80 160 315 63040Frequency (Hz)

Vibr

atio

n (d

B re

1ＧＭ

2)

minus40

minus30

minus20

minus10

0

10

20

30

40

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

Figure 10 Vibration in one-third octave

are calculated The responses of the axle box are obtainedAll of them are used to evaluate the vibration isolationcharacteristics from the axle box to the bogie 119867 is definedas the singular evaluation index of the primary suspensionsvibration isolation expressed by

119867 = 10 log 3sum800119891=1 1198861198942 (119891)sum800119891=1 [11988611990012 (119891) + 11988611990022 (119891) + 11988611990032 (119891)] (13)

where 1198861199001 1198861199002 1198861199003 are vibration responses of bogie framenear the vertical damping coil spring and the rotary armrubber joint respectively and 119886119894 is the vibration of axle box

Vibr

atio

n (d

B re

1ＧＭ

2)

minus70

minus50

minus30

minus10

10

30

50

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

500 600 700 800400Frequency (Hz)

Figure 11 Vibration in FFT

The stiffness and damping of the primary suspensionhave important influence on the vibration isolation so threeparameters are investigated they are respectively stiffnessof the rubber bearing under coil spring the rotary armrubber joint and the damping of the vehicle damper Beyondthat the radius of wheelset axle and the number of bogiestiffened plates are also considered Tables 2 and 3 showthe vibration isolation singular evaluation index 119867 of thosetracks Table 2 shows that the stiffness of rubber bearing anddamping of vertical damper has little effect on the vibrationisolation As the stiffness of rotary arm rubber joint reducesfrom 220000Nmm to 80000Nmm the overall vibration

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

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Shock and Vibration 3

0 100 200 300 400 500 600 700 800Frequency (Hz)

minus30

minus20

minus10

0

10

20

Vert

ical

vib

ratio

n (d

B re

1m

Ｍ2)

Axle boxBogie frame

Figure 3 Vibrations of bogie frame and axle box

the bogie In this paper the material parameters that affectthe vibration transmissibility of the primary suspensioncomponents are analyzed

When wheel rotates a circle the rotating frequency 1198910 isgiven by

1198910 = V120587119863 = 250 km sdot hminus1

086m times 314 times 36 = 257Hz (2)

where the wheel diameter119863 is 086mThe passing frequency of the wheel polygon 119891119899 is calcu-

lated by

119891119899 = 1198991198910 = 257119899 (3)

where 119899 is the polygon order

3 Flexible Model of the Bogie andIts Validation

31 Flexible Bogie Model Figure 4 indicates the 3D transientfinite element (FE) model of the entire bogie system whichconsists of two wheelsets a bogie frame and a series ofthe primary suspension parts developed with ANSYS Theprimary suspension parts include eight coil springs (insidespring and outside spring) four rubber bearings under thecoil springs four axle box rotary arms four rotary arm rubberjoints four vehicle primary dampers and eight rubber jointsabove or below the dampers All the geometries of this bogiesystem are the same as that of the previously tested trailerThe brake disc on the wheelsets and the brake disc seat onthe bogie frame are also considered in this study

Figure 5 shows the FE model of some of the primarysuspension components The figures in the left- and right-hand sides are the 3D geometries and the FE model respec-tively From Figure 5 it can be observed that the nonlinear

Figure 4 FEM of the whole bogie system

influences of rubber materials are ignored The rotary armrubber joint the rubber bearing and the damper rubberjoint are simulated using spring-damper elements [6] Thevehicle damping is simulated using damper elements Thecar body is simulated as a lumped mass connected to thebogie frame through secondary suspension The secondarysuspension is simulated using the spring-damper elementsThe axle box rotary arm and the coil spring are meshedusing 3D solid elements taking their actual geometries intoconsideration [7] Table 1 lists the values of the stiffness anddamping involved in this paper [8]

To improve the calculating efficiency some 1D and 2Delements are applied to the model For example beamelements are used to simulate bogie frame horizontal beamsand axles mass elements are used to simulate the brakedisc seat on the bogie frame and shell elements are usedto simulate bogie side frames The total element and nodenumbers are 234165 and 184209 respectively

32Theory ofWheelTrack Force Remington [9] put forwardthe most comprehensive single model of wheel-rail forceand wheel-rail noise The model assumes that the small-scaleroughness on the running surfaces of the wheel and rail is theprimary mechanism for the force generation Included in themodel are such effects as the spatial filtering of the roughnessdue to the finite area of contact between wheel and rail andthe interaction between the wheel and rail including localcontact stiffness119903 is defined as the displacement-input vector Assumingthat there are 119873 coupling coordinates 1199111 119911119873 one canuse 119903119895 to represent the coupled displacements of the wheeland rail 119911119877119895 for the rail 119911119882119895 for wheel and 119911119862119895 for the relativemotion in the contact zone The latter can be separated intotwo parts 119911119862119882119895 for local deformations of the wheel and 119911119862119877119895for the local deformations of the rail Then

119911119877119895 = 119911119882119895 minus 119911119862119882119895 minus 119911119862119877119895 + 119903119895 (4)

The wheel and rail can each then represented by an 119873 sdot119873 matrix of point receptances 120572119882119895 and 120572119877119895 The transferproperties of the contact zone (relative motion 119911119862119882119895 and 119911119862119877119895 per unit force) can similarly be represented by receptances

4 Shock and Vibration

C K

(a) Coil spring and the rubber bearing

C

K

KC

C

(b) Primary damper and the rubberjoint

K C

(c) Rotary arm rubber joints

Figure 5 3D geometry and FEM of the primary suspension

Table 1 Values of stiffness and damping involved in this study

Stiffness (Nmm) Damping (Nsdotsmm)Vertical Horizontal Longitudinal Vertical Horizontal Longitudinal

Primary suspensionRubber bearings 4000 400 4000 002 002 024Primary dampers - - - - - - - - 33 - -Damper rubber joints 70000 7500 70000 166 018 166Rotary arm rubber joints 160000 18000 76000 9 043 379

Secondary suspension 1150 174 174 37 294 245

120572119862119882119895119896 and 120572119862119877119895119896 An interaction force acting in the 119896th coor-dinate direction 119875119896 results from the coupling (the force istransmitted by the contact springs) and the same force actson both the wheel and the rail (in opposite directions) Theforces and displacements at a specific frequency 120596 can berelated by the receptances of the various subsystems

119911119877119895 =119873sum119896=1

120572119877119895119896119875119896

119911119882119895 = 119873sum119896=1

120572119882119895119896 (minus119875119896)

119911119862119877119895 = 119873sum119896=1

120572119862119877119895119896 119875119896

119911119862119882119895 = 119873sum119896=1

120572119862119882119895119896 119875119896(5)

Shock and Vibration 5

Contact filter

Wheel receptances

Rail receptances

Contact receptances

Wheel-rail interaction force

Wheel roughness

Rail roughness

Combined roughness

Figure 6 Framework of theoretical model of wheel-rail force

Or the upper matrix can be translated to other forms

119911119877 = [120572119877] 119875 119911119882 = minus [120572119882] 119875 119911119862119877 = [120572119862119877] 119875 119911119862119882 = [120572119862119882] 119875

(6)

Note the negative signs for the wheel since here the forceacts in the opposite direction to that of the displacementHence from equation (4)

119903 = [120572119877 + 120572119862119877 + 120572119862119882 + 120572119882] 119875 = [120572] 119875 (7)

where [120572] is defined as the matrix of combined wheel railand contact zone receptances

119875 = [120572]minus1 119903 (8)

Equation (8) is the wheel-rail force calculated modelwhich was put forward byThompson [10] based on Reming-tonrsquos model A wheel-rail interaction model is used to calcu-late wheel-rail dynamic force base on the wheel-rail com-bined roughness Figure 6 shows the flow chart of the progressto calculate the wheel-rail force The wheel roughness andrail roughness are used to calculate the wheel-rail combinedroughness and then combinedwith the receptances of wheeland rail the wheeltrack force is obtained

The local deformations of two bodies in contact actas nonlinear stiffness between them However for smalldisplacement linearized Hertzian contact stiffness 119870V invertical direction may be calculated as

119870V = ( 32120585)[(43

1198641 minus 1205832)

2 1198750 4119877119882119877119877119877119882 + 119877119877]13

(9)

where 119877119882 is the radius of wheel and 119877119877 is the radius ofcurvature of rail surface 119864 is the plain strain elastic and 120583 isPoissonrsquos ratio of wheel and rail 1198750 is the static component ofthe vertical load at the contact 120585 is a dimensionless quantitydependent on the radius of curvature of the two surfaces (119877119877and 119877119882) and hence the shape of the contact patch 120585 is relatedto 120579 which is defined as follows

120579 = 119886119888119903 cos 10038161003816100381610038161003816100381610038161003816119877119877 minus 119877119882119877119877 + 119877119882

10038161003816100381610038161003816100381610038161003816 (10)

The contact patch wavenumber filter is presented byRemington which is given by

|119867 (119896)|2 = 4120572 (119896119887)2 int

arctan120572

0[1198691 (119896119887 sec119909)]2 119889119909 (11)

where 119896 is the wavenumber along the length of the rail oraround the circumference of the wheel 120572 is a constant deter-mining the degree of correlation between parallel roughnessprofiles at a given wavenumber 1198691(119909) is the first-order Besselfunction 119909 is a variable and 119887 is circular contact patch ofradius

Prestress and wheelrail contact forces are applied to thepoints on the four wheel treads (Figure 5) The prestress is72000N computed based on the single wheel weight of thevehicle The wheelrail contact forces are calculated basedon the TWINS model Rail roughness and wheel roughnesswere obtained by the site test results and the measurementswere conducted by using CAT and BBM respectively Railreceptance is obtained by using hammer and the verticalacceleration frequency response of rail surface is obtainedby exiting the same position Acceleration is translated intodisplacement by using (12) in the frequency domain Wheelreceptance is obtained by using the upper 3D transient finiteelement (FE) model of the entire bogie system Force isapplied to the points on the four wheel treads the driving-point and transfer-pointmobility are calculated then Figure 7illustrates the wheel-track forces in the frequency domainfrom this figure it can be shown that the wheel-track exci-tation energies are mainly focused on the frequency bandwhich is under 200Hz

119886119894 = minus41205872 times 1198911198942 times 119889119894 (119894 = 1 2 119899) (12)

where 119886119894 is the acceleration in the frequency 119894 and 119889119894 is thedisplacement in the frequency 11989433 Model Validation Based on the above FE model thevibrations of the axle box and the bogie frame both in timedomain and frequency domain are calculated byANSYSTheFEmodel takeswheeltrack force in time domain as input thevibrations of the axle box and the bogie frame are calculatedby transient dynamic analysis method Figure 8 shows the

6 Shock and VibrationFo

rce (

N)

200 300 400 500 600 700 800100Frequency (Hz)

105

104

103

102

101

Figure 7 Wheelrail contact force in frequency domain

0

2

4

6

8

10

Axle box

Testing Simulation

Bogie frame

Vibr

atio

n (g

)

285 290 295 300 305 310 315 320280Time (s)

minus6

minus8

minus10

minus4

minus2

Figure 8 Vibration comparison of testing and simulation (in timedomain)

vibration responses in time domain Through FFT analysisthe vibrations are translated into the frequency domain below800Hz Figure 9 shows the vibration frequency spectrum Intime domain the vibration magnitudes of bogie frame andaxle box obtained by simulation are in good agreement withthemeasured results In frequency domain the spectrumdis-tributions and vibration magnitudes of the axle box obtainedby using the FEmethod are similar to themeasurement in thefrequency band of 0sim800Hz which validates that the modelof wheelset and axle box are accurate and the wheeltrackforce is close to that in practice As for the vibration of thebogie frame in spite of some differences in magnitudes inthe frequency band of 0sim800Hz the spectrum distributionsare in good agreement with the measured results Thus theFEmodel and the calculated method are validated effectivelyand a series of investigations are conducted using thismodel

4 Vibration Analysis of Bogie Flexible Model

Harmonic response analysis is used to calculate the steadyvibration responses of bogie key components under theswept frequency excitation Then the vibration peaks fromthe frequency response function are counted out In thesepeak frequencies vibration displacement color clouds of thewhole bogie are invested The car bodyrsquos 6 DOFs are allconstrained The harmonic forces are applied to the pointson the four wheel treads The magnitude of the force is50N and the analysis frequency band is from 0Hz to1000Hz and the frequency resolution is 4HzThe calculatedprogress is as follows First of all the modal parameters andextended models are calculated Based on the modal resultsthe vibrations of axle box and bogie frame are extractedby the full arithmetic The vibration results in one-thirdoctave of four bogie key part compartments which are bogieover the primary damper bogie over the spring rotary armsupport and axle box respective are shown as Figure 10From this figure it can be shown that as the frequencyincreased the magnitudes of every part components of bogieraise significantly which illustrates that as for the samewheeltrack force higher frequency is easier to excite theserious vibration In the centre bandof 500Hz themagnitudeof bogie frame is almost the same as axle box which is up to20 dB Vibrations of rotary arm support are the smallest Inthe centre band of 630Hz vibration energies of bogie frameover primary damper are bigger than axle box axle box androtary arm support are almost the same and bogie framesover spring are the smallest

Furthermore the FFT vibrations of these four componentparts in the frequency band of 400sim800Hz are calculatedResults are shown as Figure 11 from this picture in the fre-quency bandof 520sim560Hzmagnitudes of serious vibrationson the axle box are close to that of the bogie frame In thefrequency bandof 550sim620Hz there are no obvious vibrationpeaks on bogie frame and axle The vibration energies fromaxle box are translate into bogie framewithout reductionThevibrationmagnitudes of bogie frame are higher than axle boxin some specific frequencies Vibration of rotary arm supportis smaller than the bogie frame over the primary damperand over the spring respectively Therefore the vibrationreductions of axle box and bogie frame over the primarydamper are mainly discussed in Section 5

FromFigure 11 resonance peaks of bogie systemare in thefrequency band of 520sim560Hz Therefore the displacementcolor clouds of the whole bogie in this band are shown inFigure 12 In the 520sim560Hz the vibration deformations aremainly shown as the couple vibration of bogie side beam localmovements and the third bend ofwheelsetTherefore the twomodes may be the main reasons which lead to the resonancevibration in high frequency and then induce the high-orderwear of wheel

5 Vibration Isolation Design in Mid andHigh Frequency

The vibrations at three points on the bogie frame near thevertical damper coil spring and the rotary arm rubber joint

Shock and Vibration 7

200 300 400 500 600 700 800100Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

Axl

e box

vib

ratio

n (d

B re

1ＧＭ

2)

(a) Axle box

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

200 300 400 500 600 700 800100Frequency (Hz)

Bogi

e fra

me v

ibra

tion

(dB

re1ＧＭ

2)

(b) Bogie frame

Figure 9 Vibration comparison of testing and simulation (in frequency domain)

80 160 315 63040Frequency (Hz)

Vibr

atio

n (d

B re

1ＧＭ

2)

minus40

minus30

minus20

minus10

0

10

20

30

40

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

Figure 10 Vibration in one-third octave

are calculated The responses of the axle box are obtainedAll of them are used to evaluate the vibration isolationcharacteristics from the axle box to the bogie 119867 is definedas the singular evaluation index of the primary suspensionsvibration isolation expressed by

119867 = 10 log 3sum800119891=1 1198861198942 (119891)sum800119891=1 [11988611990012 (119891) + 11988611990022 (119891) + 11988611990032 (119891)] (13)

where 1198861199001 1198861199002 1198861199003 are vibration responses of bogie framenear the vertical damping coil spring and the rotary armrubber joint respectively and 119886119894 is the vibration of axle box

Vibr

atio

n (d

B re

1ＧＭ

2)

minus70

minus50

minus30

minus10

10

30

50

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

500 600 700 800400Frequency (Hz)

Figure 11 Vibration in FFT

The stiffness and damping of the primary suspensionhave important influence on the vibration isolation so threeparameters are investigated they are respectively stiffnessof the rubber bearing under coil spring the rotary armrubber joint and the damping of the vehicle damper Beyondthat the radius of wheelset axle and the number of bogiestiffened plates are also considered Tables 2 and 3 showthe vibration isolation singular evaluation index 119867 of thosetracks Table 2 shows that the stiffness of rubber bearing anddamping of vertical damper has little effect on the vibrationisolation As the stiffness of rotary arm rubber joint reducesfrom 220000Nmm to 80000Nmm the overall vibration

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

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4 Shock and Vibration

C K

(a) Coil spring and the rubber bearing

C

K

KC

C

(b) Primary damper and the rubberjoint

K C

(c) Rotary arm rubber joints

Figure 5 3D geometry and FEM of the primary suspension

Table 1 Values of stiffness and damping involved in this study

Stiffness (Nmm) Damping (Nsdotsmm)Vertical Horizontal Longitudinal Vertical Horizontal Longitudinal

Primary suspensionRubber bearings 4000 400 4000 002 002 024Primary dampers - - - - - - - - 33 - -Damper rubber joints 70000 7500 70000 166 018 166Rotary arm rubber joints 160000 18000 76000 9 043 379

Secondary suspension 1150 174 174 37 294 245

120572119862119882119895119896 and 120572119862119877119895119896 An interaction force acting in the 119896th coor-dinate direction 119875119896 results from the coupling (the force istransmitted by the contact springs) and the same force actson both the wheel and the rail (in opposite directions) Theforces and displacements at a specific frequency 120596 can berelated by the receptances of the various subsystems

119911119877119895 =119873sum119896=1

120572119877119895119896119875119896

119911119882119895 = 119873sum119896=1

120572119882119895119896 (minus119875119896)

119911119862119877119895 = 119873sum119896=1

120572119862119877119895119896 119875119896

119911119862119882119895 = 119873sum119896=1

120572119862119882119895119896 119875119896(5)

Shock and Vibration 5

Contact filter

Wheel receptances

Rail receptances

Contact receptances

Wheel-rail interaction force

Wheel roughness

Rail roughness

Combined roughness

Figure 6 Framework of theoretical model of wheel-rail force

Or the upper matrix can be translated to other forms

119911119877 = [120572119877] 119875 119911119882 = minus [120572119882] 119875 119911119862119877 = [120572119862119877] 119875 119911119862119882 = [120572119862119882] 119875

(6)

Note the negative signs for the wheel since here the forceacts in the opposite direction to that of the displacementHence from equation (4)

119903 = [120572119877 + 120572119862119877 + 120572119862119882 + 120572119882] 119875 = [120572] 119875 (7)

where [120572] is defined as the matrix of combined wheel railand contact zone receptances

119875 = [120572]minus1 119903 (8)

Equation (8) is the wheel-rail force calculated modelwhich was put forward byThompson [10] based on Reming-tonrsquos model A wheel-rail interaction model is used to calcu-late wheel-rail dynamic force base on the wheel-rail com-bined roughness Figure 6 shows the flow chart of the progressto calculate the wheel-rail force The wheel roughness andrail roughness are used to calculate the wheel-rail combinedroughness and then combinedwith the receptances of wheeland rail the wheeltrack force is obtained

The local deformations of two bodies in contact actas nonlinear stiffness between them However for smalldisplacement linearized Hertzian contact stiffness 119870V invertical direction may be calculated as

119870V = ( 32120585)[(43

1198641 minus 1205832)

2 1198750 4119877119882119877119877119877119882 + 119877119877]13

(9)

where 119877119882 is the radius of wheel and 119877119877 is the radius ofcurvature of rail surface 119864 is the plain strain elastic and 120583 isPoissonrsquos ratio of wheel and rail 1198750 is the static component ofthe vertical load at the contact 120585 is a dimensionless quantitydependent on the radius of curvature of the two surfaces (119877119877and 119877119882) and hence the shape of the contact patch 120585 is relatedto 120579 which is defined as follows

120579 = 119886119888119903 cos 10038161003816100381610038161003816100381610038161003816119877119877 minus 119877119882119877119877 + 119877119882

10038161003816100381610038161003816100381610038161003816 (10)

The contact patch wavenumber filter is presented byRemington which is given by

|119867 (119896)|2 = 4120572 (119896119887)2 int

arctan120572

0[1198691 (119896119887 sec119909)]2 119889119909 (11)

where 119896 is the wavenumber along the length of the rail oraround the circumference of the wheel 120572 is a constant deter-mining the degree of correlation between parallel roughnessprofiles at a given wavenumber 1198691(119909) is the first-order Besselfunction 119909 is a variable and 119887 is circular contact patch ofradius

Prestress and wheelrail contact forces are applied to thepoints on the four wheel treads (Figure 5) The prestress is72000N computed based on the single wheel weight of thevehicle The wheelrail contact forces are calculated basedon the TWINS model Rail roughness and wheel roughnesswere obtained by the site test results and the measurementswere conducted by using CAT and BBM respectively Railreceptance is obtained by using hammer and the verticalacceleration frequency response of rail surface is obtainedby exiting the same position Acceleration is translated intodisplacement by using (12) in the frequency domain Wheelreceptance is obtained by using the upper 3D transient finiteelement (FE) model of the entire bogie system Force isapplied to the points on the four wheel treads the driving-point and transfer-pointmobility are calculated then Figure 7illustrates the wheel-track forces in the frequency domainfrom this figure it can be shown that the wheel-track exci-tation energies are mainly focused on the frequency bandwhich is under 200Hz

119886119894 = minus41205872 times 1198911198942 times 119889119894 (119894 = 1 2 119899) (12)

where 119886119894 is the acceleration in the frequency 119894 and 119889119894 is thedisplacement in the frequency 11989433 Model Validation Based on the above FE model thevibrations of the axle box and the bogie frame both in timedomain and frequency domain are calculated byANSYSTheFEmodel takeswheeltrack force in time domain as input thevibrations of the axle box and the bogie frame are calculatedby transient dynamic analysis method Figure 8 shows the

6 Shock and VibrationFo

rce (

N)

200 300 400 500 600 700 800100Frequency (Hz)

105

104

103

102

101

Figure 7 Wheelrail contact force in frequency domain

0

2

4

6

8

10

Axle box

Testing Simulation

Bogie frame

Vibr

atio

n (g

)

285 290 295 300 305 310 315 320280Time (s)

minus6

minus8

minus10

minus4

minus2

Figure 8 Vibration comparison of testing and simulation (in timedomain)

vibration responses in time domain Through FFT analysisthe vibrations are translated into the frequency domain below800Hz Figure 9 shows the vibration frequency spectrum Intime domain the vibration magnitudes of bogie frame andaxle box obtained by simulation are in good agreement withthemeasured results In frequency domain the spectrumdis-tributions and vibration magnitudes of the axle box obtainedby using the FEmethod are similar to themeasurement in thefrequency band of 0sim800Hz which validates that the modelof wheelset and axle box are accurate and the wheeltrackforce is close to that in practice As for the vibration of thebogie frame in spite of some differences in magnitudes inthe frequency band of 0sim800Hz the spectrum distributionsare in good agreement with the measured results Thus theFEmodel and the calculated method are validated effectivelyand a series of investigations are conducted using thismodel

4 Vibration Analysis of Bogie Flexible Model

Harmonic response analysis is used to calculate the steadyvibration responses of bogie key components under theswept frequency excitation Then the vibration peaks fromthe frequency response function are counted out In thesepeak frequencies vibration displacement color clouds of thewhole bogie are invested The car bodyrsquos 6 DOFs are allconstrained The harmonic forces are applied to the pointson the four wheel treads The magnitude of the force is50N and the analysis frequency band is from 0Hz to1000Hz and the frequency resolution is 4HzThe calculatedprogress is as follows First of all the modal parameters andextended models are calculated Based on the modal resultsthe vibrations of axle box and bogie frame are extractedby the full arithmetic The vibration results in one-thirdoctave of four bogie key part compartments which are bogieover the primary damper bogie over the spring rotary armsupport and axle box respective are shown as Figure 10From this figure it can be shown that as the frequencyincreased the magnitudes of every part components of bogieraise significantly which illustrates that as for the samewheeltrack force higher frequency is easier to excite theserious vibration In the centre bandof 500Hz themagnitudeof bogie frame is almost the same as axle box which is up to20 dB Vibrations of rotary arm support are the smallest Inthe centre band of 630Hz vibration energies of bogie frameover primary damper are bigger than axle box axle box androtary arm support are almost the same and bogie framesover spring are the smallest

Furthermore the FFT vibrations of these four componentparts in the frequency band of 400sim800Hz are calculatedResults are shown as Figure 11 from this picture in the fre-quency bandof 520sim560Hzmagnitudes of serious vibrationson the axle box are close to that of the bogie frame In thefrequency bandof 550sim620Hz there are no obvious vibrationpeaks on bogie frame and axle The vibration energies fromaxle box are translate into bogie framewithout reductionThevibrationmagnitudes of bogie frame are higher than axle boxin some specific frequencies Vibration of rotary arm supportis smaller than the bogie frame over the primary damperand over the spring respectively Therefore the vibrationreductions of axle box and bogie frame over the primarydamper are mainly discussed in Section 5

FromFigure 11 resonance peaks of bogie systemare in thefrequency band of 520sim560Hz Therefore the displacementcolor clouds of the whole bogie in this band are shown inFigure 12 In the 520sim560Hz the vibration deformations aremainly shown as the couple vibration of bogie side beam localmovements and the third bend ofwheelsetTherefore the twomodes may be the main reasons which lead to the resonancevibration in high frequency and then induce the high-orderwear of wheel

5 Vibration Isolation Design in Mid andHigh Frequency

The vibrations at three points on the bogie frame near thevertical damper coil spring and the rotary arm rubber joint

Shock and Vibration 7

200 300 400 500 600 700 800100Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

Axl

e box

vib

ratio

n (d

B re

1ＧＭ

2)

(a) Axle box

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

200 300 400 500 600 700 800100Frequency (Hz)

Bogi

e fra

me v

ibra

tion

(dB

re1ＧＭ

2)

(b) Bogie frame

Figure 9 Vibration comparison of testing and simulation (in frequency domain)

80 160 315 63040Frequency (Hz)

Vibr

atio

n (d

B re

1ＧＭ

2)

minus40

minus30

minus20

minus10

0

10

20

30

40

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

Figure 10 Vibration in one-third octave

are calculated The responses of the axle box are obtainedAll of them are used to evaluate the vibration isolationcharacteristics from the axle box to the bogie 119867 is definedas the singular evaluation index of the primary suspensionsvibration isolation expressed by

119867 = 10 log 3sum800119891=1 1198861198942 (119891)sum800119891=1 [11988611990012 (119891) + 11988611990022 (119891) + 11988611990032 (119891)] (13)

where 1198861199001 1198861199002 1198861199003 are vibration responses of bogie framenear the vertical damping coil spring and the rotary armrubber joint respectively and 119886119894 is the vibration of axle box

Vibr

atio

n (d

B re

1ＧＭ

2)

minus70

minus50

minus30

minus10

10

30

50

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

500 600 700 800400Frequency (Hz)

Figure 11 Vibration in FFT

The stiffness and damping of the primary suspensionhave important influence on the vibration isolation so threeparameters are investigated they are respectively stiffnessof the rubber bearing under coil spring the rotary armrubber joint and the damping of the vehicle damper Beyondthat the radius of wheelset axle and the number of bogiestiffened plates are also considered Tables 2 and 3 showthe vibration isolation singular evaluation index 119867 of thosetracks Table 2 shows that the stiffness of rubber bearing anddamping of vertical damper has little effect on the vibrationisolation As the stiffness of rotary arm rubber joint reducesfrom 220000Nmm to 80000Nmm the overall vibration

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

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Shock and Vibration 5

Contact filter

Wheel receptances

Rail receptances

Contact receptances

Wheel-rail interaction force

Wheel roughness

Rail roughness

Combined roughness

Figure 6 Framework of theoretical model of wheel-rail force

Or the upper matrix can be translated to other forms

119911119877 = [120572119877] 119875 119911119882 = minus [120572119882] 119875 119911119862119877 = [120572119862119877] 119875 119911119862119882 = [120572119862119882] 119875

(6)

Note the negative signs for the wheel since here the forceacts in the opposite direction to that of the displacementHence from equation (4)

119903 = [120572119877 + 120572119862119877 + 120572119862119882 + 120572119882] 119875 = [120572] 119875 (7)

where [120572] is defined as the matrix of combined wheel railand contact zone receptances

119875 = [120572]minus1 119903 (8)

Equation (8) is the wheel-rail force calculated modelwhich was put forward byThompson [10] based on Reming-tonrsquos model A wheel-rail interaction model is used to calcu-late wheel-rail dynamic force base on the wheel-rail com-bined roughness Figure 6 shows the flow chart of the progressto calculate the wheel-rail force The wheel roughness andrail roughness are used to calculate the wheel-rail combinedroughness and then combinedwith the receptances of wheeland rail the wheeltrack force is obtained

The local deformations of two bodies in contact actas nonlinear stiffness between them However for smalldisplacement linearized Hertzian contact stiffness 119870V invertical direction may be calculated as

119870V = ( 32120585)[(43

1198641 minus 1205832)

2 1198750 4119877119882119877119877119877119882 + 119877119877]13

(9)

where 119877119882 is the radius of wheel and 119877119877 is the radius ofcurvature of rail surface 119864 is the plain strain elastic and 120583 isPoissonrsquos ratio of wheel and rail 1198750 is the static component ofthe vertical load at the contact 120585 is a dimensionless quantitydependent on the radius of curvature of the two surfaces (119877119877and 119877119882) and hence the shape of the contact patch 120585 is relatedto 120579 which is defined as follows

120579 = 119886119888119903 cos 10038161003816100381610038161003816100381610038161003816119877119877 minus 119877119882119877119877 + 119877119882

10038161003816100381610038161003816100381610038161003816 (10)

The contact patch wavenumber filter is presented byRemington which is given by

|119867 (119896)|2 = 4120572 (119896119887)2 int

arctan120572

0[1198691 (119896119887 sec119909)]2 119889119909 (11)

where 119896 is the wavenumber along the length of the rail oraround the circumference of the wheel 120572 is a constant deter-mining the degree of correlation between parallel roughnessprofiles at a given wavenumber 1198691(119909) is the first-order Besselfunction 119909 is a variable and 119887 is circular contact patch ofradius

Prestress and wheelrail contact forces are applied to thepoints on the four wheel treads (Figure 5) The prestress is72000N computed based on the single wheel weight of thevehicle The wheelrail contact forces are calculated basedon the TWINS model Rail roughness and wheel roughnesswere obtained by the site test results and the measurementswere conducted by using CAT and BBM respectively Railreceptance is obtained by using hammer and the verticalacceleration frequency response of rail surface is obtainedby exiting the same position Acceleration is translated intodisplacement by using (12) in the frequency domain Wheelreceptance is obtained by using the upper 3D transient finiteelement (FE) model of the entire bogie system Force isapplied to the points on the four wheel treads the driving-point and transfer-pointmobility are calculated then Figure 7illustrates the wheel-track forces in the frequency domainfrom this figure it can be shown that the wheel-track exci-tation energies are mainly focused on the frequency bandwhich is under 200Hz

119886119894 = minus41205872 times 1198911198942 times 119889119894 (119894 = 1 2 119899) (12)

where 119886119894 is the acceleration in the frequency 119894 and 119889119894 is thedisplacement in the frequency 11989433 Model Validation Based on the above FE model thevibrations of the axle box and the bogie frame both in timedomain and frequency domain are calculated byANSYSTheFEmodel takeswheeltrack force in time domain as input thevibrations of the axle box and the bogie frame are calculatedby transient dynamic analysis method Figure 8 shows the

6 Shock and VibrationFo

rce (

N)

200 300 400 500 600 700 800100Frequency (Hz)

105

104

103

102

101

Figure 7 Wheelrail contact force in frequency domain

0

2

4

6

8

10

Axle box

Testing Simulation

Bogie frame

Vibr

atio

n (g

)

285 290 295 300 305 310 315 320280Time (s)

minus6

minus8

minus10

minus4

minus2

Figure 8 Vibration comparison of testing and simulation (in timedomain)

vibration responses in time domain Through FFT analysisthe vibrations are translated into the frequency domain below800Hz Figure 9 shows the vibration frequency spectrum Intime domain the vibration magnitudes of bogie frame andaxle box obtained by simulation are in good agreement withthemeasured results In frequency domain the spectrumdis-tributions and vibration magnitudes of the axle box obtainedby using the FEmethod are similar to themeasurement in thefrequency band of 0sim800Hz which validates that the modelof wheelset and axle box are accurate and the wheeltrackforce is close to that in practice As for the vibration of thebogie frame in spite of some differences in magnitudes inthe frequency band of 0sim800Hz the spectrum distributionsare in good agreement with the measured results Thus theFEmodel and the calculated method are validated effectivelyand a series of investigations are conducted using thismodel

4 Vibration Analysis of Bogie Flexible Model

Harmonic response analysis is used to calculate the steadyvibration responses of bogie key components under theswept frequency excitation Then the vibration peaks fromthe frequency response function are counted out In thesepeak frequencies vibration displacement color clouds of thewhole bogie are invested The car bodyrsquos 6 DOFs are allconstrained The harmonic forces are applied to the pointson the four wheel treads The magnitude of the force is50N and the analysis frequency band is from 0Hz to1000Hz and the frequency resolution is 4HzThe calculatedprogress is as follows First of all the modal parameters andextended models are calculated Based on the modal resultsthe vibrations of axle box and bogie frame are extractedby the full arithmetic The vibration results in one-thirdoctave of four bogie key part compartments which are bogieover the primary damper bogie over the spring rotary armsupport and axle box respective are shown as Figure 10From this figure it can be shown that as the frequencyincreased the magnitudes of every part components of bogieraise significantly which illustrates that as for the samewheeltrack force higher frequency is easier to excite theserious vibration In the centre bandof 500Hz themagnitudeof bogie frame is almost the same as axle box which is up to20 dB Vibrations of rotary arm support are the smallest Inthe centre band of 630Hz vibration energies of bogie frameover primary damper are bigger than axle box axle box androtary arm support are almost the same and bogie framesover spring are the smallest

Furthermore the FFT vibrations of these four componentparts in the frequency band of 400sim800Hz are calculatedResults are shown as Figure 11 from this picture in the fre-quency bandof 520sim560Hzmagnitudes of serious vibrationson the axle box are close to that of the bogie frame In thefrequency bandof 550sim620Hz there are no obvious vibrationpeaks on bogie frame and axle The vibration energies fromaxle box are translate into bogie framewithout reductionThevibrationmagnitudes of bogie frame are higher than axle boxin some specific frequencies Vibration of rotary arm supportis smaller than the bogie frame over the primary damperand over the spring respectively Therefore the vibrationreductions of axle box and bogie frame over the primarydamper are mainly discussed in Section 5

FromFigure 11 resonance peaks of bogie systemare in thefrequency band of 520sim560Hz Therefore the displacementcolor clouds of the whole bogie in this band are shown inFigure 12 In the 520sim560Hz the vibration deformations aremainly shown as the couple vibration of bogie side beam localmovements and the third bend ofwheelsetTherefore the twomodes may be the main reasons which lead to the resonancevibration in high frequency and then induce the high-orderwear of wheel

5 Vibration Isolation Design in Mid andHigh Frequency

The vibrations at three points on the bogie frame near thevertical damper coil spring and the rotary arm rubber joint

Shock and Vibration 7

200 300 400 500 600 700 800100Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

Axl

e box

vib

ratio

n (d

B re

1ＧＭ

2)

(a) Axle box

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

200 300 400 500 600 700 800100Frequency (Hz)

Bogi

e fra

me v

ibra

tion

(dB

re1ＧＭ

2)

(b) Bogie frame

Figure 9 Vibration comparison of testing and simulation (in frequency domain)

80 160 315 63040Frequency (Hz)

Vibr

atio

n (d

B re

1ＧＭ

2)

minus40

minus30

minus20

minus10

0

10

20

30

40

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

Figure 10 Vibration in one-third octave

are calculated The responses of the axle box are obtainedAll of them are used to evaluate the vibration isolationcharacteristics from the axle box to the bogie 119867 is definedas the singular evaluation index of the primary suspensionsvibration isolation expressed by

119867 = 10 log 3sum800119891=1 1198861198942 (119891)sum800119891=1 [11988611990012 (119891) + 11988611990022 (119891) + 11988611990032 (119891)] (13)

where 1198861199001 1198861199002 1198861199003 are vibration responses of bogie framenear the vertical damping coil spring and the rotary armrubber joint respectively and 119886119894 is the vibration of axle box

Vibr

atio

n (d

B re

1ＧＭ

2)

minus70

minus50

minus30

minus10

10

30

50

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

500 600 700 800400Frequency (Hz)

Figure 11 Vibration in FFT

The stiffness and damping of the primary suspensionhave important influence on the vibration isolation so threeparameters are investigated they are respectively stiffnessof the rubber bearing under coil spring the rotary armrubber joint and the damping of the vehicle damper Beyondthat the radius of wheelset axle and the number of bogiestiffened plates are also considered Tables 2 and 3 showthe vibration isolation singular evaluation index 119867 of thosetracks Table 2 shows that the stiffness of rubber bearing anddamping of vertical damper has little effect on the vibrationisolation As the stiffness of rotary arm rubber joint reducesfrom 220000Nmm to 80000Nmm the overall vibration

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

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6 Shock and VibrationFo

rce (

N)

200 300 400 500 600 700 800100Frequency (Hz)

105

104

103

102

101

Figure 7 Wheelrail contact force in frequency domain

0

2

4

6

8

10

Axle box

Testing Simulation

Bogie frame

Vibr

atio

n (g

)

285 290 295 300 305 310 315 320280Time (s)

minus6

minus8

minus10

minus4

minus2

Figure 8 Vibration comparison of testing and simulation (in timedomain)

vibration responses in time domain Through FFT analysisthe vibrations are translated into the frequency domain below800Hz Figure 9 shows the vibration frequency spectrum Intime domain the vibration magnitudes of bogie frame andaxle box obtained by simulation are in good agreement withthemeasured results In frequency domain the spectrumdis-tributions and vibration magnitudes of the axle box obtainedby using the FEmethod are similar to themeasurement in thefrequency band of 0sim800Hz which validates that the modelof wheelset and axle box are accurate and the wheeltrackforce is close to that in practice As for the vibration of thebogie frame in spite of some differences in magnitudes inthe frequency band of 0sim800Hz the spectrum distributionsare in good agreement with the measured results Thus theFEmodel and the calculated method are validated effectivelyand a series of investigations are conducted using thismodel

4 Vibration Analysis of Bogie Flexible Model

Harmonic response analysis is used to calculate the steadyvibration responses of bogie key components under theswept frequency excitation Then the vibration peaks fromthe frequency response function are counted out In thesepeak frequencies vibration displacement color clouds of thewhole bogie are invested The car bodyrsquos 6 DOFs are allconstrained The harmonic forces are applied to the pointson the four wheel treads The magnitude of the force is50N and the analysis frequency band is from 0Hz to1000Hz and the frequency resolution is 4HzThe calculatedprogress is as follows First of all the modal parameters andextended models are calculated Based on the modal resultsthe vibrations of axle box and bogie frame are extractedby the full arithmetic The vibration results in one-thirdoctave of four bogie key part compartments which are bogieover the primary damper bogie over the spring rotary armsupport and axle box respective are shown as Figure 10From this figure it can be shown that as the frequencyincreased the magnitudes of every part components of bogieraise significantly which illustrates that as for the samewheeltrack force higher frequency is easier to excite theserious vibration In the centre bandof 500Hz themagnitudeof bogie frame is almost the same as axle box which is up to20 dB Vibrations of rotary arm support are the smallest Inthe centre band of 630Hz vibration energies of bogie frameover primary damper are bigger than axle box axle box androtary arm support are almost the same and bogie framesover spring are the smallest

Furthermore the FFT vibrations of these four componentparts in the frequency band of 400sim800Hz are calculatedResults are shown as Figure 11 from this picture in the fre-quency bandof 520sim560Hzmagnitudes of serious vibrationson the axle box are close to that of the bogie frame In thefrequency bandof 550sim620Hz there are no obvious vibrationpeaks on bogie frame and axle The vibration energies fromaxle box are translate into bogie framewithout reductionThevibrationmagnitudes of bogie frame are higher than axle boxin some specific frequencies Vibration of rotary arm supportis smaller than the bogie frame over the primary damperand over the spring respectively Therefore the vibrationreductions of axle box and bogie frame over the primarydamper are mainly discussed in Section 5

FromFigure 11 resonance peaks of bogie systemare in thefrequency band of 520sim560Hz Therefore the displacementcolor clouds of the whole bogie in this band are shown inFigure 12 In the 520sim560Hz the vibration deformations aremainly shown as the couple vibration of bogie side beam localmovements and the third bend ofwheelsetTherefore the twomodes may be the main reasons which lead to the resonancevibration in high frequency and then induce the high-orderwear of wheel

5 Vibration Isolation Design in Mid andHigh Frequency

The vibrations at three points on the bogie frame near thevertical damper coil spring and the rotary arm rubber joint

Shock and Vibration 7

200 300 400 500 600 700 800100Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

Axl

e box

vib

ratio

n (d

B re

1ＧＭ

2)

(a) Axle box

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

200 300 400 500 600 700 800100Frequency (Hz)

Bogi

e fra

me v

ibra

tion

(dB

re1ＧＭ

2)

(b) Bogie frame

Figure 9 Vibration comparison of testing and simulation (in frequency domain)

80 160 315 63040Frequency (Hz)

Vibr

atio

n (d

B re

1ＧＭ

2)

minus40

minus30

minus20

minus10

0

10

20

30

40

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

Figure 10 Vibration in one-third octave

are calculated The responses of the axle box are obtainedAll of them are used to evaluate the vibration isolationcharacteristics from the axle box to the bogie 119867 is definedas the singular evaluation index of the primary suspensionsvibration isolation expressed by

119867 = 10 log 3sum800119891=1 1198861198942 (119891)sum800119891=1 [11988611990012 (119891) + 11988611990022 (119891) + 11988611990032 (119891)] (13)

where 1198861199001 1198861199002 1198861199003 are vibration responses of bogie framenear the vertical damping coil spring and the rotary armrubber joint respectively and 119886119894 is the vibration of axle box

Vibr

atio

n (d

B re

1ＧＭ

2)

minus70

minus50

minus30

minus10

10

30

50

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

500 600 700 800400Frequency (Hz)

Figure 11 Vibration in FFT

The stiffness and damping of the primary suspensionhave important influence on the vibration isolation so threeparameters are investigated they are respectively stiffnessof the rubber bearing under coil spring the rotary armrubber joint and the damping of the vehicle damper Beyondthat the radius of wheelset axle and the number of bogiestiffened plates are also considered Tables 2 and 3 showthe vibration isolation singular evaluation index 119867 of thosetracks Table 2 shows that the stiffness of rubber bearing anddamping of vertical damper has little effect on the vibrationisolation As the stiffness of rotary arm rubber joint reducesfrom 220000Nmm to 80000Nmm the overall vibration

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

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Shock and Vibration 7

200 300 400 500 600 700 800100Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

Axl

e box

vib

ratio

n (d

B re

1ＧＭ

2)

(a) Axle box

minus40

minus30

minus20

minus10

0

10

20

Testing Simulation

200 300 400 500 600 700 800100Frequency (Hz)

Bogi

e fra

me v

ibra

tion

(dB

re1ＧＭ

2)

(b) Bogie frame

Figure 9 Vibration comparison of testing and simulation (in frequency domain)

80 160 315 63040Frequency (Hz)

Vibr

atio

n (d

B re

1ＧＭ

2)

minus40

minus30

minus20

minus10

0

10

20

30

40

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

Figure 10 Vibration in one-third octave

are calculated The responses of the axle box are obtainedAll of them are used to evaluate the vibration isolationcharacteristics from the axle box to the bogie 119867 is definedas the singular evaluation index of the primary suspensionsvibration isolation expressed by

119867 = 10 log 3sum800119891=1 1198861198942 (119891)sum800119891=1 [11988611990012 (119891) + 11988611990022 (119891) + 11988611990032 (119891)] (13)

where 1198861199001 1198861199002 1198861199003 are vibration responses of bogie framenear the vertical damping coil spring and the rotary armrubber joint respectively and 119886119894 is the vibration of axle box

Vibr

atio

n (d

B re

1ＧＭ

2)

minus70

minus50

minus30

minus10

10

30

50

Axle boxBogie over the primary damperBogie over spring

Rotary arm support

500 600 700 800400Frequency (Hz)

Figure 11 Vibration in FFT

The stiffness and damping of the primary suspensionhave important influence on the vibration isolation so threeparameters are investigated they are respectively stiffnessof the rubber bearing under coil spring the rotary armrubber joint and the damping of the vehicle damper Beyondthat the radius of wheelset axle and the number of bogiestiffened plates are also considered Tables 2 and 3 showthe vibration isolation singular evaluation index 119867 of thosetracks Table 2 shows that the stiffness of rubber bearing anddamping of vertical damper has little effect on the vibrationisolation As the stiffness of rotary arm rubber joint reducesfrom 220000Nmm to 80000Nmm the overall vibration

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

8 Shock and Vibration

0 788E minus 04394E minus 04 118E minus 03

158E minus 03197E minus 03

236E minus 03276E minus 03

315E minus 03355E minus 03

(a) The displacement color clouds of bogie frame in 528Hz

0 372E minus 04186E minus 04 558E minus 04

745E minus 04931E minus 04

112E minus 03130E minus 03

149E minus 03168E minus 03

(b) The displacement color clouds of bogie frame in 544Hz

Figure 12 Displacement color clouds of bogie frame in 520sim560Hz

Table 2 The overall vibration isolation (changing parameters of the primary suspension)

Stiffness of rubber bearing 119867 Stiffness of rotary arm rubber joint 119867 Damping of vertical damper 119867119870 = 6000 069 119870 = 220000 356 119862 = 66 063119870 = 4000 065 119870 = 160000 065 119862 = 33 065119870 = 2000 062 119870 = 80000 297 119862 = 10 067

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 9

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 160000K = 80000

K = 220000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 13 Different stiffness of the rotary arm rubber joint

Table 3The overall vibration isolation (changing geometries of thebogie)

Radius 119867 Stiffened plates 119867Original 065 119873 = 0 065Add 10mm 264 119873 = 1 168Add 20mm 339 119873 = 3 170Sub 10mm minus008 119873 = 5 170

isolation decreases by 291 dB at first and then increases by232 dB Table 3 shows that adding the radius of wheelset axleor numbers of bogie stiffened plates can increase the vibrationisolation

51 Effect of Primary Suspension

511 Stiffness of the Rotary Arm Rubber Compared to thestiffness of the rubber bearing the stiffness of the rotaryarm rubber joint is higher Therefore it has an obviouseffect on the overall vibration isolation performance as shownin Table 1 As the stiffness increased from 80000Nmm to160000Nmm the overall vibration isolation decreased by232 dB As current stiffness increased from 160000Nmmto 220000Nmm continuously the overall vibration isolationincreased by 291 dB

Figure 13(a) shows the spectrums of vibration isolation119867(120596) for the three caseswith stiffness 80000Nmm 16000Nmm and 220000Nmm A stiffness of 160000Nmm isapplied to the actual vehicle As shown in Figure 13(a) thestiffness of rotary arm rubber joint has an obvious effecton the primary suspension vibration isolation under thefrequency domain The peak positions are almost unchange-able but the magnitude varies a little under the stiffnessof 80000Nmm 160000Nmm and 220000Nmm In thecase of lower stiffness the vibration isolation of the primary

suspension isworse below 100Hz and tends to be the oppositeabove 100Hz

Figure 13(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness80000Nmm 16000Nmm and 220000NmmAs shown inFigure 13(b) as the stiffness decreased vibration magnitudeof axle box is almost unchangeable However the peakfrequencies decrease in the frequency ranges of 500sim520Hzand 750sim780Hz Vibration peak positions of bogie frame arealmost unchangeable while magnitude decreases graduallywith the stiffness decreasingThe vibration isolation betweenaxle box and bogie frame at stiffness of 80000Nmm isbetter than that of 220000Nmm The maximum vibrationisolation between axle box and bogie frame becomes upto approximately 21 dB in 500sim600Hz The bogie framersquosresonance frequency in 520Hz can escape from axle box of532Hz or 504Hz

512 Stiffness of the Rubber Bearing The rubber bearingunder coil spring is an important vibration absorbing ele-ment which connects the coil spring and the axle box Itsstiffness is lower than other rubber materials of the primarysuspension From Table 2 the stiffness of the rubber bearinghas little effect on the vibration isolation with the stiffnessincreasing from 2000Nmm to 6000Nmm the overallvibration isolation decreases by 007 dB

Figure 14(a) shows the spectrums of the vibration isola-tion for the three cases with stiffness 6000Nmm 4000Nmm and 2000Nmm Stiffness of 4000Nmm is applied tothe actual vehicle As shown in Figure 14(a) the stiffness of therubber bearing has almost no effect on the vibration isolationperformance above 200Hz Below 200Hz the peak positionsand magnitudes vary a little for different stiffness

Figure 14(b) shows the vibration spectrums of bogieframe and axle box for the three cases with stiffness2000Nmm 4000Nmm and 6000Nmm As shown in

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Shock and Vibration

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

K = 4000K = 2000

K = 6000(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 14 Different stiffness of the rubber bearing under coil spring

Figure 14(b) as the stiffness decreased vibrations of axle boxand bogie frame are almost unchangeable Stiffness of therubber bearing has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

513 Damping of the Primary Suspension As shown abovethe stiffness of some components has effects on the vibrationisolation Furthermore the influences of damping of theprimary suspension can be investedTheprimary damper canimprove the stability and comfortability of railway vehiclesfurthermore in this section the effect of the damping onisolation in high frequency is studied The damping of theprimary suspension parts mainly depends on the verticaldamper and the rubbermaterial damping As for the dampingof the rubber material it is hard to get so this paper mainlyinvests the influence of the vertical damper In Table 2the damping of the vertical damper has little effect onthe vibration isolation with the damping increasing from10Nsdotsmm to 66Nsdotsmm and the overall vibration isolationdecreasing by 004 dB

Figure 15(a) shows the spectrums of the vibration isola-tion for the three cases with damping 66Nsdotsmm 33Nsdotsmmand 10Nsdotsmm respectively Damping of 33Nsdotsmm isapplied to the actual vehicle As shown in Figure 15(a)damping has a little effect on the vibration isolation As for thedifferent damping peak positions and magnitudes of peaksare almost unchanged above 300Hz Below 300Hz lowerdamping the vibration isolation of the primary suspensionis worse Therefore it is not a good way to apply the lowdamping to decrease vibration of high-speed vehicle systemin mid and high frequency

Figure 15(b) shows the vibration spectrums of bogieframe and axle box for the three cases with damping10Nsdotsmm 33Nsdotsmm and 66Nsdotsmm As shown in Fig-ure 15(b) as the damping decreased vibrations of axle box

and bogie frame are almost unchangeable Damping of theprimary damper has no effect on the resonance frequency ofbogie frame and axle box at 520Hz

52 Effect of Bogiersquos and Wheelsetrsquos Geometries The coupledresonance of wheelset and bogie frame at 520Hz may be themain reason which leads to the high-order wheel polygon sothe vibration in high frequencymay be reduced by improvingthe structure of these two components

521 Radius of theWheelset Axle Because of the high weighwheelset can generate large vibration energy once its modelis excitated which lead to the serious wheel-rail force Inthe frequency band of 520sim550Hz the model deformationof wheelset is shown as the third bend of axle So this modecan be improved by changing the radius of the wheelset axleFrom Table 2 the radius of the wheelset axle has obviouseffect on the vibration isolation As the radius increases by10mmand 20mm the overall vibration isolation increases by199 dB and 274 dB respectively while as the radius decreasesby 10mm the overall vibration isolation decreases by 073 dB

Figure 16(a) shows the spectrums of the vibration iso-lation for the four cases with original radius and improvedradius which include adding 10mm adding 20mm andreducing 10mm respectively As shown in Figure 16(a) theradius of wheelset has almost no effect on the vibrationisolation under 500Hz while vibration isolation magnitudeshows a big fluctuation above 500Hz

Figure 16(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different radiuseswhich include adding 10mm adding 20mm decreasing10mm and the original radius respectively As shown inFigure 16(b) as the radius increases vibration magnitudesof axle box and bogie frame decrease in the frequency bandof 400sim530Hz and 570sim800Hz and tend to be opposite in

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 11

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(a) Vibration isolation of the primary suspension (0sim800Hz)

axle box

bogie frame

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

C = 33

C = 10

C = 66

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 15 Different dampings of the vertical damper

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Original

Add 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(a) Vibration isolation of the primary suspension (0sim800Hz)

bogie frame

axle box

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30

Vibr

atio

n (d

B re

1ＧＭ

2)

OriginalAdd 10ＧＧ

Add 20ＧＧ

Sub 10ＧＧ

(b) Vibrations of bogie frame and axle box (400sim800Hz)

Figure 16 Different radiuses of the wheelset axle

the range of 530sim570Hz With the radius adding 10mmpeak frequency at 520Hz of axle box increases by 40Hz butbogie frame generates peak at the same frequency which is at560HzWhile radius proceeds to add 20mm peak frequencyat 560Hz of axle box increases by 10Hz and bogie framegenerates peaks at 570Hz Based on the original radius theradius deceases by 10mm and peak frequency at 520Hz ofaxle box diminishes but bogie frame has not generate anypeak at the same frequency and the magnitude at 520Hzdecreases obviously In a whole adding the radius of wheelsetaxle is not an effective way to avoid the high resonanceBecause it will make the resonance peak of axle box and bogieframe move at the same time and bring in new problemsof resonances at 560Hz and 570Hz However decreasing

radius may be a good way to reduce high frequency vibrationbecause it can reduce the magnitudes of axle and bogieframe at the same time and improve the vibration isolationMeanwhile new vibration peak cannot be generated

522 Bogie with Stiffened Thin Plates Because of the highweigh bogie frame can generate large vibration energy onceits model is excitated Then the wheel-rail force and thevibration increase dramatically Modal shapes of bogie framein low orders are almost shown as the global deformationsas the modal frequencies increase side beamsrsquo and horizonbeamsrsquo local deformations become dominant In the fre-quency range of 520sim550Hz the modal shapes are alwaysshown as the local movements of the end of side beamsThus

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Shock and Vibration

Figure 17 Stiffened plates of bogie

100 200 300 400 500 600 700 8000Frequency (Hz)

minus30

minus20

minus10

0

10

20

30

Vibr

atio

n iso

latio

n (d

B re

1ＧＭ

2)

Three stiffened platesOne stiffened plate Without stiffened plates

Five stiffened plates

(a) Vibration isolation of the primary suspension (0sim800Hz)

500 600 700 800400Frequency (Hz)

minus30

minus10

10

30

minus30

minus10

10

30Vi

brat

ion

(dB

re1ＧＭ

2)

Three stiffened plates

Without stiffened platesOne stiffened plate

Five stiffened plates

(b) Vibration of bogie frame and axle box (400sim800Hz)

Figure 18 Different numbers of the stiffened thin plates

adding stiffened thin plates in the side beams can improve thelocal stiffness of bogie frame and then change these modalfrequenciesThe stiffened thin plates are located in themiddleof the upper covers lower covers outside covers and insidecovers and numbers are one three and five respectivelyFigure 17 shows the positions of stiffened thin plates

From Table 2 after adding stiffened plates the overallvibration isolation increases by 103 dB but as the numbersincrease the overall vibration isolation almost varies a little

Figure 18(a) shows the spectrums of the vibration isola-tion for the four cases with original bogie frame and afteradding stiffened thin plates which include one plate threeplates and five plates respectively As shown in Figure 18(a)the vibration isolation increases in 520sim550Hz and 672Hzafter adding stiffened plates while the vibration isolationdecreases in 550sim570Hz Other frequencies are almostunchangeable Different numbers of stiffened plates almosthave no effect on the vibration isolation in the frequencyband

Figure 18(b) shows the vibration spectrums of bogieframe and axle box for the four cases with different numbersof stiffened plates As shown in Figure 18(b) the vibration

peaks of axle box at 520Hz increase by 30Hz after adding thestiffened thin plates Bogie frame vibration peaks at 520Hzhave not changed in which magnitudes decrease by 1233 dBAnd the magnitudes of bogie frame at 550Hz vary a littleDifferent numbers of stiffened plates have no effect on thevibration of bogie frame and axle box so adding stiffened thinplates is an effective way to improve the vibration isolationof the whole bogie at 520Hz and only one number can besatisfied

6 Conclusions

To reduce the vibration of the bogie in the mid and highfrequency a 3D flexible model of a whole bogie is developedusing the explicit finite element (FE) method Then theinfluences of material and structure parameters on bogieare analyzed carefully Based on the obtained results thefollowing conclusions are drawn

(1) The field tests show that the mid and high frequencyvibrations of high-speed bogies are very notable especiallyin the 560sim590Hz range which are very close to the passingfrequencies corresponding to the 22nd to 23rd polygonalwear of the wheel

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 13

(2) Based on the roughness and receptance by site testthe wheelrail contact forces are calculated as the FE modelrsquosinput Then vibrations of the axle box and the bogie frameobtained by simulation are in good agreement with themeasured results Thus the FE model and the calculatedmethod are validated effectively

(3) The vibration responses of the bogie FE model arecalculated by the harmonic response analysis The numericalresults show that magnitudes of serious vibrations on the axlebox are close to that of the bogie frame in the frequencyband of 520sim560HzThe vibration deformations of the wholebogie in the frequency band of 520sim560Hz aremainly shownas the couple vibration of bogie side beam local movementsand the third bend of wheelsetThe couple vibrations of bogieframe and wheelset may be the main reasons which lead tothe resonance vibration in high frequency and then inducethe high-order wear of wheel

(4)The numerical results show that stiffness of the rubberbearing and damping of the primary damper have little effecton the vibrations of the bogie frame and axle box while theresonance peaks in high frequency range can be reduced byreducing the stiffness of axle box rotary arm joint reducingthe wheelset axle radius or strengthening the bogie framelocation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The present work was supported by the National KeyRampD Program of China (nos 2016YFB1200503-022016YFB1200506-08 2016YFE0205200) the National Nat-ural Science Foundation of China (nos U1434201 U173420151475390) and the Scientific Research Foundation of StateKey Laboratory of TractionPower China (no 2015TPL T08)

References

[1] G Han J Zhang X Xiao D Cui and X Jin ldquoStudy on high-speed train abnormal interior vibration and noise related towheel roughnessrdquo Jixie Gongcheng XuebaoJournal of Mechan-ical Engineering vol 50 no 22 pp 113ndash121 2014

[2] K Popp H Kruse and I Kaiser ldquoVehicle-track dynamics in themid-frequency rangerdquoVehicle SystemDynamics vol 31 no 5-6pp 423ndash464 1999

[3] P Alexander S Stefan A Roder et al ldquoActive vibration controlfor high speed train bogiesrdquo Smart Mater Structure vol 14 pp1ndash18 2005

[4] Z-S Ren G Yang S-S Wang and S-G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMU withflexible modelrdquo Acta Mechanica Sinica vol 30 no 6 pp 876ndash883 2014

[5] J ZhangGHanX andXXiaoB ldquoInfluence ofwheel polygonalwear on interior noise of high-speedtrainsrdquoZhejiangUniversity-Science (Applied physics amp Engineering) vol 15 no 12 pp 1002ndash1018 2014

[6] M Sjoberg On Dynamic Properties of Rubber Isolators [PhDthesis] The Royal Institute of Technology Sweden 2002

[7] S Bruni J Vinolas M Berg O Polach and S Stichel ldquoMod-elling of suspension components in a rail vehicle dynamicscontextrdquo Vehicle System Dynamics vol 49 no 7 pp 1021ndash10722011

[8] Y X Chen An experimental and theoretical investigation of thedynamic properties of rubber [PhD thesis] Southwest JiaotongUniversity Chengdu China 2008

[9] P J Remington ldquoWheelrail noise part I characterization of thewheelrail dynamic systemrdquo Journal of Sound andVibration vol46 no 3 pp 359ndash379 1976

[10] D J Thompson ldquoWheel-rail noise generation Part I Introduc-tion and interactionmodelrdquo Journal of Sound andVibration vol161 no 3 pp 387ndash400 1993

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom