Experimental and Numerical Analysis and Prediction...

Research ArticleExperimental and Numerical Analysis and Prediction ofGround Vibrations Due to Heavy Haul Railway Viaduct

Jingjing Hu 12 Yi Luo 1 and Jiayun Xu1

1Hubei Key Laboratory of Roadway Bridge amp Structure Wuhan University of Technology China2School of Civil Engineering and Architecture Wuhan University of Technology China

Correspondence should be addressed to Yi Luo yluowhuteducn

Received 12 September 2018 Revised 29 November 2018 Accepted 9 January 2019 Published 13 February 2019

Academic Editor Gisella Tomasini

Copyright copy 2019 Jingjing Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The ground vibration wave induced by the viaduct section of the heavy freight wagons is transformed into the ground vibrationproblem under the condition of point source excitation after it is transmitted to soil through pier When the speed and axle loadof wagon become larger and larger the impact on the surrounding buildings will also increase In this paper the ground vibrationaround Shenshan Village in section of Shuo-Huang railway line is monitored and numerical analyzed the 3D numerical model ofthe Bridge-Pier-Field-House system is established The relationship between peak vibration velocity (PPV) and the distance to thepier caused by heavy freight wagons at different speeds and different type of wagons is analyzed The power function relationshipbetween the two whenmeasuring line perpendicular to the centerlineof railway is verified Based on themodified Sodevrsquos equationthe relationship between PPV of ground points and wagon speed axle load and soil properties is proposed and the value of everyparameter in the formula is discussed in detail The concept of energy index is put forward for the first time in the formula andthe relationship between energy index and wagon speed and wagon weight is analyzed by regression analysis The accuracy of thecalculationmodel and prediction formula is verified by comparing field test results an analytical method is proposed to predict theground vibration induced by viaduct

1 Introduction

Railway is one of the most efficient transport systems inrecent years railway transportation develops very fast inChina and abroad With the development of transportationthe vibration caused by railways has an increasing impacton the environment The environmental vibration causedby traffic load has the characteristics of repeatability andlong-term When it exceeds a certain level it will affect thedaily life and work of people in nearby buildings When thevibration reaches a larger level it may also cause damageto the structure of the building thus affecting the normalfunction of the building [1ndash3]

On the other hand due to the influence of terrain moreand more sections are designed as railway viaducts to ensurethe smoothness of the line [4] For example the total lengthof the Seoul-Busan high-speed railway in Korea is 412kmthe highest running speed is 300kmh and the viaductproportion is 271 The Madrid-Seville high-speed railway

in Spain has 31 viaducts with a total length of 15 km thebridge ratio is 32 The Beijing-Shanghai railway line hasa total length of 1318 km with 244 main bridges and a totallength of 1060km The bridge length accounts for 807 ofthe total line However the study of railway viaduct mostlyfocuses on the deformation and stability of bridges which isquite complex due to the vehicle-coupled-guideway vibration[5ndash7] Due to the different ways of excitation the groundvibration caused by viaducts should also be concerned [8 9]

On the other hand with heavy load long marshallingand fast running speed the dynamic effect of heavy freightwagons is stronger than that of ordinary trains and thedynamic influence of wagons on the railway structure andits surrounding structures is more serious [10 11] The envi-ronmental vibration problem caused by the railway viaductsystem involves many complicated dynamic systems suchas wagons bridges piers and sites and buildings involvinga large number of nonlinear problems [1 12 13] Usuallythe system is decomposed into several subsystems to study

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2751815 15 pageshttpsdoiorg10115520192751815

2 Mathematical Problems in Engineering

separately The current theoretical research has not yetreached a high degree of precision or obtained a simplifiedanalytical solution Field measurements are the most com-mon means of studying such problems [14ndash16]

However railway line is usually long and the complex-ity of the problem is high given the dynamic interactionbetween distinct domains namely the train the viaductthe ground and the building It is not realistic to detecteach area nearby the line As response to these needsit is necessary to put forward the general law of groundvibration attenuation and to provide a method for evaluatingthe magnitude of train vibration in a viaduct area to bebuilt

Vibration assessment is a key element of the environ-mental impact assessment process for railway projects Forthe propagation and attenuation law of ground vibrationmany scholars have put forward different prediction for-mulas They can be broadly classified into two categoriesone is the amplitude representation and the other is thevibration level representation [17ndash19] Among them Bornitz[20] put forward the attenuation formula of ground vibrationamplitude when disturbance force acts on the surface ofhalf space of soil considering the geometric attenuationof the vibration propagation and the medium absorptionattenuation and the influence of propagation distance of thevibration G Volberg [21] based on tests at three differentsites regressed the relationship between the vibration leveland train speed Chen Jianguo et al [22] established an envi-ronmental vibration prediction formula based on the resultsof finite element vibration analysis The formula reflects thelogarithmic relationship between vibration acceleration leveland distance and also considers the vibration level correctionvalues of train speed bridge span and unit length andquality of the bridge by the additional function methodHowever most of the above formulas can only reflect theattenuation relationship of vibration with distance or wagonspeed and cannot clearly reflect the influence of geologicalconditions More accurate prediction formulas should be putforward

A large number of studies have shown that the vibrationvelocity and vibration frequency of the particle are closelyrelated to the damage of the building [23ndash25] It can directlyreflect the vibration energy and play an important role in theevaluation of the vibration of building Accurate predictionof the peak vibration velocity (PPV) of ground points underdifferent site conditions is of great significance In the fieldof traffic environment vibration especially for the groundvibration caused by the heavy freight wagons there are fewstudies using PPV as the evaluation index Regarding theorganization of the paper firstly a finite element model oftrack box girder pier field and building is established onthe basis of field test Numerical result of vibration responseof ground and structure is studied by analyzing vibrationvelocity and spectra induced by trains of different weight andspeed on a railway viaduct Based on modified Sodevrsquos for-mula the relationship between PPV andwagon speed wagonweight distance and site is established Finally regressionanalysis is used to discuss the procedures of determining eachparameter in the forecast formula

2 Field Test of Railway Viaduct inShenshan Village

Shuo-Huang railway is an important part of the second largestchannel of the ldquoWest-EastCoal Transportationrdquo inChinaThetotal length of the railway is nearly 600 km the transportationcapacity exceeds 255 million tons and the freight volumeincreases by tens of millions of tons every year It is locatedin the connecting area between mountainous area and plainThe relative height difference is 200m-700m The length ofthe bridge section accounts for more than 10 of the totallength In the northeast of Yuanping City Shanxi ProvinceShuo-Huang Heavy-Haul Line crosses Shenshan Village inthe form of viaducts The area is densely populated and thereare a large number of residential buildings within 30 metersof the railway line as shown in Figure 1

Shuo-Huang railway is a first-class electrified railwaywithdouble track ballast track jointless track and the rail of 60kgmThe commonly used types of trucks are C64k C70 andC80 The axle weights of heavy freight wagons are about 21t23t and 25t and the axle weights of empty wagons are about58t 60t and 50t The unspring mass of the wagon is about1500 kg per wheel set

Monitoring points in the field test can be divided into3 parts pier measuring points site measuring points andhouse measuring points as shown in Figure 2 Among themthe site measuring points are mainly arranged on two differ-ent routes perpendicular to the viaduct axis (Rout MR andRout PL) Limited by the conditions of the site there are fourmeasuring points on each of the two lines and the distancebetween adjacent points is not completely equal Based onmeasured data the basic law of site vibration attenuationon two measuring routs is put forward In order to furtheranalyze and verify the proposed rules a numerical model isset up below

3 Numerical Analysis of Ground VibrationInduced by Viaduct

A 3D numerical model is established to analyze the vibrationattenuation rule of site points under various wagon speed andaxle load more deeply Due to the complexity of rail systemthe tracks and the ballast were considered to be contactdirectly the field was considered as homogeneous sand soil

31 Calculation Model and Parameter Setting According tothe actual site conditions of Shenshan railway viaduct anumerical model of Bridge-Pier-Field-House system is estab-lished using ANSYSLS-DYNA as can be seen in Figure 3which shows the finite elements (FE) mesh of the crosssection The size of the field model is 125m length perpen-dicular to the tracks direction and 1325m in the directionof the tracks including 5 piers with each span of 265mThe thickness of the soil layer is 80m under which there isbed rock the nodes at the bottom boundary are fixed in alldirections to simulate the bedrock Pier foundation adoptsindependent foundation The maximum element size is 225m the minimum element size is 02m satisfying the require-ments of an element dimension in finite element analysis The

Mathematical Problems in Engineering 3

Figure 1 The field site of Shenshan railway viaduct

75m 75m 75m 75m 19m

BP6 BP7 BP8 BP9MP11

MP10 (12)

1m 65m 75m 13m 05m

BP5BP4BP3

BP2

MP1(Foundation)

Route MR

Route PL

Figure 2 Layout of measuring points in the field test (unit m)

Pier1

2 34 5 12 pier

Test line

Non-reflectingboundary

y

(a) 3D view mode

Sym

met

ric b

ound

ary

225m

625m

16m

(b) Front view



15m

1325m(c) Right view

Figure 3 Numerical model of Bridge-Pier-Field-House system


Track

Cushion

Box girder

Pier

Soil

Figure 4 Schematic diagram of each part of the numerical model

minus10 minus5 0 5 104

6

8

10

12

Distance (m)

PPV

(ms

)

1Pier2Pier3Pier

minus4times 10

Figure 5 The PPV of points around piers 1 2 and 3

3D model mesh contains 226912 eight-node brick elementsThe type of elements which are being used is solid simulatingall the part in system As the structure of the symmetryin order to reduce the computation the establishment of ahalf model simulation A symmetric boundary condition isintroduced into the model at the center of the pier

The longitudinal direction of the rail and the ballast isrestricted to simulate their continuity As the vibration wavepasses through the pier to the soil it becomes a problem ofground vibration The whole viaduct system is simplified intotwo subsystems the vibration source system (track-cushion-box girder) and the vibration transmission system (pier-residence-soil) The schematic diagram of each part is shownin Figure 4 The elastic parameters of the system materialare listed in Table 1 Material damping is modeled using theRayleigh damping modelThe damping ratio is assumed to be5 in the frequency range of interest

The propagation of vibration in soil is infinite and thevibration simulated by FEmodel is propagated in the bound-ary of the model Nonreflecting boundaries are implementedto prevent wave reflections at the edges The basic principleis to apply viscous normal stress and shear stress on theboundary which are opposite to the direction of boundarystress to absorb the energy propagating to the boundary Astudy has been done in order to determine the accuracy ofnonreflective boundary comparing the PPV on the test linein Figure 3(a) parallel to the track Considering the influenceof different piers the center of each pier is taken as the originand 17 test points are evenly placed within 20 meters aroundthe pier The PPV of points around Piers 1 2 and 3 werecompared in Figure 5 and the variances of points on the threecurves are calculated respectively It can be found that thePPV variance of the particles around Pier 1 (which is closerto the boundary) is larger than that of pier 3 (which is fartheraway from the boundary) the PPV variance of the particlesaround Pier 3 is 3613e-9 ms and the value is very small Inthis paper the locations of the measured points are all nearPier 3 (Figure 7) which are less affected by the boundaryThe analysis results do not depend on the size of the model

32 Simplification of Train Load According to [26 27] sincethe straight railway lines account for a large proportion onlythe vertical bearing forces are applied to the Train-Track-Bridge system to solve the ground vibration For the purposeof comparing the vibration characteristics of computationalmodel and field model the wheelbase of four-axle wagonsand load distribution are stimulated as shown in Figure 6Four car bodies are considered while the actual numbercan reach more than 100 Using a series of moving loads torepresent the effects of the moving train is an alternative forTrain-Structure Interaction The load amplitude takes intoaccount the static wheel load and dynamic load coefficientand the track irregularity is also taken into account It hasbeen proved effective in many researches [28 29] The trainload which is represented by a series of axles load Pnlocated in the wheel-track contact points is formulated witha constant moving speed v The train load on the track is aone-way pulse stress wave

In Figure 6 119897c is the wheelbase of the wagon 119897w is thewheelbase of the bogie 119897b is thewheelbase between the bogiesand 119897v is the length of a single wagon

The C64 C70 and C80 track which are commonlyused in China for heavy haul transport are selected in thisnumerical model During the regular arrangement of axleload vertical loading frequency of wagon mainly depends onwagon speed V (kmh) and wheelbase d (m) When loadingfrequency is f = v(36sdotd) and train speed is 80-160 km h theloading frequency of the freight wagon is calculated as shownin Table 2 It can be seen that the vertical loading frequency ofthe train is within 26 Hz and the influence of the wheelbasebetween bogies andwheelbase inside the bogie on the loadingfrequency is greater In the model the rail is divided into auniform grid with a side length of 02 m In the calculationthe load time history curve is applied to each unit of the railThe load curve on each unit is the same but the action timeis different The running speed of the wagon is controlled by


Table 1 Elastic material parameters of the system

Position Thickness (m) Density(kgm3) Modulus of elasticity (Mpa) Poisson ratioTrack 7800 60E+05 022Cushion 10 2200 16E+04 020Steel beam 30 7800 20E+05 022Pier 210 2200 25E+04 020Layer 80 2000 4662 026Bedrock infin 2200 285E+04 020

Table 2 Vertical loading frequency of wagons

Type of wagon 119897119887m 119897119908m 119897119888m Wheelbase between wagonsmC64k 2988 175 695 1519C70 269 183 738 1556C80 197 183 637 1356Loading frequency range Hz 744-2256 1214-2539 301-698 146-328

the peak rise time and duration time of wheel-rail load curveand the different axle load is simulated by changing the loadamplitude

33 Monitoring Point Arrangement More measuring pointsare arranged in the numerical model compared with the fieldtest which are shown in Figure 7 BP1 BP2 and BP3 are themeasuring points of the top of pier the measuring point at 2m on the pier from the ground and the measuring point ofpier foundation respectively Two vertical lines are placed onthe field Route PL starts from a pier with eight MPs (No1to 8) in horizontal distance of 10m 75 m 15 m 225 m 30m 40m 50 m and 60m perpendicular to viaduct axis MR1-MR7 are located at the RouteMRMP1-MP3 are placed on thefoundation of the residential building the center of a roomonthe ground floor and the roof respectively

34 Ground Vibration Analysis In order to verify the validityof the numericalmodel Figure 8 compares the calculated andmeasured velocity time history and Fourier spectrum of PL3under C80 running at a speed of 80kmh The amplitudeand trend of velocity time histories are very close betweenthe calculated and measured value as demonstrated in thefigure Comparison of frequency spectrum shows that themain frequency range of vibration is the same there is a goodcorrespondence between the calculated andmeasured resultsso the results of numerical simulation are reliable

Because the horizontal distance between Route PL andthe center line is smallest and the horizontal distance betweenRoute MR and the center line is largest different points onRoute PL and Route MR with same distance perpendicularto viaduct axis are selected to observe the attenuation law ofvibration on the ground One case is studied with availabledata for C80 with a speed of 80kmh the change law of rootmean square (RMS) velocity vibration level on the two routesis compared The RMS of a signal is the average of its squaredamplitude and is typically calculated over one second interval[30] The average vibration level and maximum vibrationlevel of different points are shown in Table 3

As an amplitude descriptor velocity level 119871V is defined as

119871119881 = 20 lg( 1198810119881ref) (1)

where 1198810 is the amplitude of the velocity time history inms and 119881ref is the reference value amplitude 254times10minus8 ms

It can be seen from Table 3 obviously that the averagevibration level of the measuring point on Route PL is largerthan that on Route MR The attenuation rate of velocityvibration level on Route PL is faster than that on Route MRat the measuring point within 30 m (near source MPs) Withthe distance from the axle of the viaduct increases from 75m to 30 m the average vertical vibration level on Route PLdecreases from 8087 dB to 6956 dB which decreased 1131dB while the average vertical vibration level on Route MRdecreases from 7785 to 7311 dB which decreases by 474 dBThe vibration attenuation rate of the Route PL is rapidly firstand then slowly with time which is basically the same as themeasured law

The variation of vertical vibration velocity on either routeis analyzed by comparing the RMS velocity level whichcalculated from (1) as shown in Figure 9 As can be seenthe velocity vibration levels of the measuring points on eachroute decrease with the increase of distance regularly butsudden change (suddenly magnified or suddenly reduced)also occurs locally those regions called the local amplificationregion (LAR) For example the average value of verticalvelocity vibration levels at 40 m on Route PL is 257 dB largerthan that at 30 m This phenomenon had been observed inother field tests [31 32]The reason is uncertain one possibleview is that the presence of hard bedrock beneath the soillayer the wave velocity of the bedrock is usually larger thanthat of the surface layerWhen the incident wave generated bythe vibration source which is reflected and refracted on thesurface of bedrock and the surface wave propagating alongthe ground surface reaches a certain point on the ground atthe same time the superposition of thewaveswill form a localamplification


Table 3 Maximum and average velocity levels at different routes (dB)

Vertical vibration Horizontal vibrationRoute PL Route MR Route PL Route MR

Max Average Max Average Max Average Max Average

MPs

75 9499 8087 8593 7485 9352 8047 8749 759615 8829 7401 810 7131 8458 7378 8507 7285225 8332 7255 8448 7154 8506 7436 8293 701830 8062 6956 8298 7011 7398 6307 8449 717840 8217 7213 7862 6722 7799 6686 7472 646150 8009 6862 7287 6251 7861 6672 7431 628660 7481 6495 6903 5900 7617 6620 7349 6169

lＰ

l＝ l＝ l＝ l＝

lＱ lＱ lＱ lＱ lＱ lＱ lＱ lＱ

PＨ

l＜

P1P2P3P4

Figure 6 The layout and geometrical specification of wagons

MP2MP1

MP3

66

76 74

Foundation

BP1

BP2BP3 2

011

Figure 7 Layout of measuring points in the numerical model (unit m)

Figure 10 displays the horizontal and vertical velocity ofall the points on Rout PL introduced previously (Section 33)in the form of one-third center frequency band within thescope of 1-80Hz Due to the vibration-absorbing effect of soilthe vibration usually decreases with the increase of the dis-tance but there are exceptions in some frequency bands likevertical vibration at 75mwhich is obviously larger than that at10m in 25Hz to 35Hz frequency bands The measuring pointwithin 30m has an amplification phenomenon at the centerfrequency of 8sim16Hz and 20sim25Hz and the amplification

is only obvious near the center frequency of 10Hz beyond30m The frequency less than 8 Hz is rapidly attenuatedfrom 1 m to 75 m and then remains stable However thefrequency band above 20 Hz is uniformly attenuated withdistance It indicates that vibration in frequency between8 and 16 Hz where PPV lies attenuates at a lower rateIt is an interesting phenomenon that no much attenuationis observed for distances larger than 30 m the higherfrequencies of the ground vibration wave attenuate fast whilethe lower frequencies attenuate slowly Higher frequencies at


0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)

Measured valueCalculated value


minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)

Figure 8 Comparison of calculated and measured values of velocity time and spectrum of ground point

75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)

(a) Horizontal vibration on Route PL

75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)

(b) Vertical vibration on Route PL

0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)

(c) Horizontal vibration on Route MR

Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m

(d) Vertical vibration on Route MR

Figure 9 The attenuation law of vibration on different route


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m

(a) Horizontal vibration

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m

0 10 20Frequency (Hz)

30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5

(b) Vertical vibration

Figure 10 The effective value of 13 octave band velocity on Route PL

distances larger than 30 m have been attenuated more thanhalf (perhaps 80 percent) On the other hand Rayleigh wavewill play a leading role in the far field which accounts for 67of the total energy The attenuation speed of Rayleigh waveis much slower than that of body wave so the attenuation ofvibration is slower at distances larger than 30 m

4 Regression Analysis Using SodevrsquosModified Formula

Propagation characteristics of vibrations generated by variousvibration sources may be dependent on the type of thegenerated waves which can be assessed bymeasuring particlemotions According to the analysis of the literature themagnitude of the dynamic load transmitted to the groundby the pier is mainly affected by the train running speed andaxle weight After the vibration wave is transmitted to the soilthrough the pier it is converted into the ground vibrationproblem purely under the point source excitation conditionIt is similar to the vibration effect induced by blasting in themiddle and far region In fact the propagation medium ofblasting vibration and pier vibration caused by wagons areboth rock and soilmass and the vibration ofmediumnear theprotected object is elastic vibration The relationship between

the PPVand the distance to vibration source can be describedby the modified Sodevrsquos formula as shown in (2) to (4)

119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)

where 1198961015840 and1205721015840 are the coefficients related to local geolog-ical condition 1198761015840 is a comprehensive indicator consideringvarious factors which affect the source of energy (hereinafterreferred to as the energy index) and119863 is the scaled distanceFor vibration source two adjustments are applied includingthe train type and train speed

In this section the following steps are taken to obtain thePPVof the ground point which is d from the center of the pieraccording to (2) (1) Referring to the method of determiningsite coefficient in blasting 1198961015840 and 1205721015840 are obtained (2) Whenthe wagon weight is the same multiple sets of data are usedfor regression analysis based on the (3) and then 11989610158401015840 is gotand it is substituted in (4) to get Qrsquo at different wagon speeds


20 40 600Distance to the center line (m)

0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80


Figure 11 Power function fitting of PPV under different wagon weights

and the change law of Qrsquo with the wagon speed can be fittedby least square method (3) In the same way as step (2) therelationship between Qrsquo and wagon weight at the same speedcan be fitted (4) Repeat steps (2) and (3) multiple sets ofchange law of Qrsquo with wagon speed and Qrsquo with wagon weightare obtained so as to fit the calculation equation of Qrsquo underany wagon speed and wagon weight (5) After Qrsquo is obtainedsubstitute it into (2) to obtain the PPV at any point

41 PPV and Wagon Weight A total of four types of trains(between C50 and C80) were analyzed in the model andthe values of 11989610158401015840 and 1205721015840 in (3) are discussed when the trainspeed is 80kmh The fitting result between PPV and distanceto the center line (119863) of each MPs on Rout PL is shown inTable 4 Figure 11 shows the relationship curves between PPVand distance under different wagon weights

As can be seen from Table 4 and Figure 11 the correlationcoefficients in the fitting equation (3) are all greater than 88and the mean square error (MSE) is less than 08 The fittingaccuracy of vertical vibration is higher than that of horizontalvibration It is verified that the relationship between PPVand distance under different wagon weights conforms topower function The coefficient 11989610158401015840 varies greatly with theincrease of the total weight of the wagon For the horizontalvibration from C50 to C80 11989610158401015840 gradually increases from1901 to 2887 and 1205721015840 maintains at around 05 For verticalvibration 11989610158401015840 gradually increases from 2612 to 3533 and1205721015840 maintains around 065 indicating that the weight of thewagon has a great influence on the PPV especially the near-source measuring point

42 PPV andWagon Speed Taking the fully loaded train C80as an example a total of five speeds (between 80 and 160kmh) were analyzed in the section Table 5 and Figure 12

present the fitting result of PPV and119863 on Rout PL at differentwagon speeds according to formula (3)

It can be seen from Table 5 and Figure 12 that thecorrelation coefficients in fitting equation (3) are all greaterthan 85 and the MSE are less than 09754 As wagon speedincreases the coefficient 11989610158401015840 increases and 11989610158401015840 in the verticalvibration is larger than that in horizontal vibration while 120572is relatively stable When the wagon speed increases from80kmh to 160kmh for horizontal vibration 11989610158401015840 graduallyincreases from 2887 to 3666 and for vertical vibration 11989610158401015840gradually increases from 3533 to 8896 With the increase ofwagon speed the increasing speed of PPV at the same pointslows down

43 Determination of Site Coefficient The blasting vibrationsimulation was carried out on the site of the model todetermine the value of site coefficient Piers in the originalmodel were removed and the equivalent blasting load wasapplied in hole of the pier A simplified triangular load curveis adopted as shown inFigure 13 Among them119875w is the peakpressure of blasting 1199051 is boost time and 1199052 is total actiontime

The boost time is set to 2 ms total action time is set to 7ms the total calculation time is 300ms and the peak pressureof the blasting equivalent load is 664 MPa when the chargeis 120 kg The velocity time histories of particles at differentdistances on the site are extracted by calculation and therelationship between the PPV of the measuring points andthe distance from blasting center is fitted by Sodevrsquos formula(see (5))

119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)


Table 4 Fitting of PPV versus distance under different wagons

Vibration Component Wagon 11989610158401015840 1205721015840 Correlation coefficient MSE

Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188

Table 5 Fitting of PPV versus distance under different speeds

Vibration Component Speed(kmh) 11989610158401015840 1205721015840 Correlation coefficient MSE

Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131

where 119896 and 120572 are site coefficients 119876 is the mass ofthe blasting single-shot explosive (kg) Formula (5) takeslogarithms on both sides simultaneously

ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)

where 119910 = ln119881119901119890119886119896 119909 = ln( 3radic119876119863) = (13) ln119876 minus ln119863and 119887 = ln 119896

Equation (6) becomes the following linear form

119910 = 120572119909 + 119887 (7)

By means of linear regression with the above methodwhen the MSE of calculated value and fitted value are thesmallest the coefficients 120572 and b are obtained Results areas follows in horizontal direction 120572=1447 b=4056 k=5774and in vertical direction 120572=1479=1479 b=3351 k=2853

The site coefficients are substituted into Equation (3) 11989610158401015840119909represents horizontal direction and 11989610158401015840119911 represents verticaldirection

11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)

44 Prediction Formula of PPV If to predict PPV by (2)after site coefficients are calculated the energy index Qrsquo isalso needed The following two regression methods are usedto analyze the relationship between energy index and wagonspeed and wagon weight

(a) Linear Regression According to the corresponding valueof 11989610158401015840 at the same speed and different weight in Table 4 Qrsquocorresponding to each weight can be obtained by substitutingthem in (8) results are shown in Table 6Making linear fittingbetween energy index and total weight of wagon the fittingformula is shown in (9)

1198761015840 (119892) = 1198861119892 + 1198871 (9)

where 119892 is total weight of wagon in t 1198861 and 1198871 are linearfitting coefficient

Fitting results show that fitting correlation coefficient infitting equation (9) is greater than 90and themax differencevalue (D-value) percentage between the fitting value and thecalculated value is 636 D-value in vertical direction areall less than 1 Among them the linear fitting coefficientsof horizontal direction are a1=547times10minus7 and b1=-2689times10minus5while those of vertical direction are a1=1768times10minus6 and b1=-5022times10minus5 According to (9) the119876rsquo values of different wagonweights can be estimated under the same site condition andwagon speed

According to similar method above using the valueof 11989610158401015840 at same weight and different speeds in Table 5 Qrsquocorresponding to each speed can be obtained in Table 7Making linear fitting between energy index and wagon speedthe fitting formula is shown in (10)

1198761015840 (V) = 1198881V + 1198891 (10)

where V is wagon speed in kmh 1198881 and1198891 are linear fittingcoefficient


0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh


Figure 12 Power function fitting of PPV under different speeds

O

０Ｑ

t1 t2

Figure 13 Applied blasting load

Table 6 Energy index values under different wagon weight

Type of wagon Total weight (t) Horizontal Vertical1198761015840 (10-5) D-value 1198761015840 (10-4) D-value

C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059

Table 7 Energy index values under different wagon speed

Type of wagon Speed (kmh) Horizontal Vertical1198761015840 (10-5) D-value 1198761015840 (10-4) D-value

C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306


The fitting results show that energy index increases lin-early with the train speed between 80 kmh and 160kmh theD-value percentage betweenfitting value and calculated valueis all less than 8 Among them the linear fitting coefficientsof horizontal direction are c1=272times10minus7 and d1=1146times10minus5while those of vertical direction are c1=1158times10minus6 and d1=-0981times10minus6 The Qrsquo value in vertical is larger than horizontalwhich indicates that the increase of train speed has a greaterimpact on vertical vibration According to (10) the Qrsquo valuesof different wagon speeds can be estimated

According to (9) and (10) Qrsquo is both the function ofwagon weight and wagon speed Therefore Qrsquo is assumed tohave the following forms for it has a linear relationship bothwith wagon weight and wagon speed

1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)

where 1198601 1198611 and 1198602 1198612 are fitting coefficients At aspecific speed the form of Qrsquo is as follows

1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)

The values of a1 and b1 have been solved above andthe same method can be used to find out the value of1198862 1198872 119886V119899 119887V119899 According to (11) regression coefficient1198861 1198862 sdot sdot sdot 119886V119899 and 1198871 1198872 sdot sdot sdot 119887V119899 have linear relationship withwagon speed respectively it is expressed in the form as

119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)

The coefficients of A1 B1 and A2 B2 were calculated byfitting 1198861 1198862 sdot sdot sdot 119886V119899 1198871 1198872 sdot sdot sdot 119887V119899 and speed respectively andthen the change law of 1198761015840 with wagon speed and wagonweight can be found by substituting them in (11)

(b) Nonlinear Regression Volbergrsquos [21] study has shown thelogarithmic relationship between ground vibration velocitylevel caused by trains and train speed in a certain frequencyrange in 1983

119871V = 64 + 20 lg ( V40) (14)

where V is the wagon speed in kmh 119871V is vibrationvelocity level in dB From (14) it can be seen that the velocityvibration level is logarithmic with the wagon speed whilefrom (1) it can be seen that the velocity vibration level islogarithmic with PPV then it can be deduced that there is alinear relationship between PPVandwagon speed Accordingto (2) the relationship between Qrsquo and PPV can be deduced

1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)

Assuming the linear relationship between PPV andwagon speed the relationship between Qrsquo and wagon speedcan be assumed as follows

1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)

where p1 and p2 are linear fitting coefficient 120572rsquo is coeffi-cient for site condition Wagon speed and corresponding Qrsquovalues in Table 7 are regression analyzed according to (16)The results are compared with linear regression as shown inFigure 14

The nonlinear fitting coefficients in horizontal direc-tion are obtained like p1=2463times10minus5 and p2=00049 whilethe nonlinear fitting coefficients in vertical direction arep1=5668times10minus5 and p2=00075 As can be seen fromFigure 14results of 119876rsquo under different wagon speed calculated by thetwo methods are very close the RMS in vertical direction ofthe two fitting methods are compared with numerical valuethe nonlinear fitting is 1206times10minus6 and the linear fitting is1232 times10minus6

A similarmethod is used to fit the relationship betweenQrsquoand wagon weight assuming the relationship between themis as follows

1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)

Thewagon weight and corresponding Qrsquo values in Table 6are regression analyzed according to (17) Results are com-pared with the linear regression as shown in Figure 15

The nonlinear fitting coefficients in horizontal direc-tion are obtained like q1=6414times10minus5 and q2=-164times10minus4while nonlinear fitting coefficient in vertical direction areq1=923times10minus5 and q2=00027 As shown in Figure 15 thenonlinear fitting is more accurate for the RMS in the verticaldirection by the nonlinear fitting is 6427times10minus6 and is smallerthat of the linear fitting which is 6515 times10minus6

When wagon speed and wagon weight are both variablesthe expression ofQrsquo which satisfied (16) and (17) is as follows

1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)

where P1 Q1 and P2 Q2 are fitting coefficients Whenwagon speed is the same the forms of Qrsquo under differentwagon weights are as follows

1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)

The value of 1199021 and 1199022 have been solved above andthe same method can be used to find out the value of11990221 11990222 1199021198991 1199021198992 The regression coefficients 1199021 11990221 sdot sdot sdot 1199021198991


minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo

Nonlinear fittingLinear fittingNumerical value


minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)



Figure 14 Regression analysis of energy index Qrsquo and wagon speed

minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)


Figure 15 Regression analysis of energy index 119876rsquo and wagon weight

and 1199022 11990222 sdot sdot sdot 1199021198992 both have linear relationship with wagonspeed respectively according to (18) it is expressed in theform as follows

1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)

The coefficients 1198751 1198752 and 1198761 1198762 were calculated byfitting 1199021 11990221 sdot sdot sdot 1199021198991 1199022 11990222 sdot sdot sdot 1199021198992 and wagon speed respec-tively and then we can obtain the functional expression ofQrsquo by substituting them in (18) Setting Qrsquo into (2) PPV ofany surface point at different speed and wagon weight can beobtained


Table 8 Comparison of PPV of measuring points (mms)

Location of MPs m Measured value Numerical value Linear fitting Value Nonlinear fitting valueHorizontal Vertical Horizontal Vertical Horizontal Vertical Horizontal Vertical

1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355

45 Comparison of Fitting Result and Monitoring ResultWhen full C64 is running at 100kmh on the viaduct themonitoring results numerical results and fitting results of thePPV of four MPs with the distance between MPs and centerline are 1 75 15 and 225m which are shown in Table 8

It is found that the PPV at 15m is less than 05mmsthe ground vibration caused by wagons running on viaductsdecreases rapidly with the increase of distance Comparedwith PPV of MPs the numerical value is close to the mea-sured value which verifies the correctness of the numericalmodel Comparing the numerical value with the predictedvalue by two regressionmethods it is obvious that the nonlin-ear fitting value is closer to the numerical value Although thefitting value at 75m is smaller and the value at 15m is slightlylarger the deviation is less than 8 which can predict theattenuation trend of ground point vibration basically

5 Conclusion

In this paper the ground vibration attenuation rule andpropagation law are systematically discussed by establishingthe numericalmodel of Shenshan viaduct and its surroundingsite Based on the modified Sadevrsquos formula the predictionformulas of PPV under different wagon speeds and axle loadsare proposed by linear regression and nonlinear regressionrespectively Method to determine each parameter in theprediction formula is discussed on the basis of the simulationresults the principal conclusions (may be confined to thesimilar configurations as this paper) are as follows

(1) The comparison between numerical and measuredresults revealed a good agreement which confirms theintegrated Bridge-Pier-Field-House model established in thispaper is consistent with the actual project and can reflect thevibration propagation chain caused by train passing throughviaduct

(2) In the site the attenuation of velocity vibration levelof the MPs on Route PL is faster than MPs on Route MRmoreover the velocity vibration level of the MPs on Route PLshows a power function attenuation trend with the increaseof distance However there will also have the amplificationregions (LAR) the positions and quantities of LAR aredifferent in different directions

(3) The prediction formula proposed in the paper canpredict the PPV of ground vibration under different trainspeeds axle loads and site conditions effectively Comparingthe fitting valuewith numerical calculation values except thatthe result at 75m is smaller other results are close whichverifies the correctness of the prediction method

(4) The concept of energy index Qrsquo is proposed inprediction formula firstly the relationship between energyindex axle load and speed is regressed by linear fitting andnonlinear fitting The results show that the values obtained bynonlinear fitting are more accurate

The prediction formula presented in this paper doesnot consider the mutation phenomenon in the attenuationprocess and the more accurate formula needs to be studiedfurther under more cases

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest in preparing thisarticle

Acknowledgments

This work supported by the National Natural Science Foun-dation of China (51779197 and 51378500)

References

[1] D P Connolly G Kouroussis P K Woodward P Alves CostaO Verlinden and M C Forde ldquoField testing and analysisof high speed rail vibrationsrdquo Soil Dynamics and EarthquakeEngineering vol 67 pp 102ndash118 2014

[2] D P Connolly P A Costa G Kouroussis P Galvin P KWood-ward and O Laghrouche ldquoLarge scale international testing ofrailway ground vibrations across Europerdquo Soil Dynamics andEarthquake Engineering vol 71 pp 1ndash12 2015

[3] W Zhai K Wei X Song and M Shao ldquoExperimental inves-tigation into ground vibrations induced by very high speedtrains on a non-ballasted trackrdquo Soil Dynamics and EarthquakeEngineering vol 72 pp 24ndash36 2015

[4] H Xia and N Zhang ldquoDynamic analysis of railway bridgeunder high-speed trainsrdquo Computers amp Structures vol 83 no23-24 pp 1891ndash1901 2005

[5] L Liu C Tao R Sun H Chen and Z Lin ldquoNon-stationarychannel characterization for high-speed railway under viaductscenariosrdquo Chinese Science Bulletin vol 59 no 35 pp 4988ndash4998 2014

[6] J M Olmos and M A Astiz ldquoAnalysis of the lateral dynamicresponse of high pier viaducts under high-speed train travelrdquoEngineering Structures vol 56 pp 1384ndash1401 2013


[7] S Kusuhara S Fukunaga and N Toyama ldquoReevaluation ofaerodynamic stability for the Tozaki viaductrdquo Kozo KogakuRonbunshu A vol 56 pp 608ndash615 2010

[8] I Nakaya KHayakawa TNishimura andK Tanaka ldquoThe longdistance propagation mechanism of environmental groundvibration due to the viaductrdquo Journal of Japan Society of CivilEngineers Ser C vol 65 pp 196ndash212 2009

[9] P Li X Ling F Zhang Y Li and Y Zhao ldquoField testingand analysis of embankment vibrations induced by heavy haultrainsrdquo Shock and Vibration vol 2017 Article ID 7410836 14pages 2017

[10] L Pugi D Fioravanti andA Rindi ldquoModelling the longitudinaldynamics of long freight trains during the braking phaserdquoin Proceedings of the 2th Iftomm World Congress pp 18ndash21Micropolis Congress Center Besancon France 2007

[11] L Pugi A Palazzolo and D Fioravanti ldquoSimulation of railwaybrake plants An application to SAADKMS freight wagonsrdquoProceedings of the Institution of Mechanical Engineers Part FJournal of Rail and Rapid Transit vol 222 no 4 pp 321ndash3292008

[12] N Correia dos Santos J Barbosa R Calcada and R DelgadoldquoTrack-ground vibrations induced by railway traffic experi-mental validation of a 3D numerical modelrdquo Soil Dynamics andEarthquake Engineering vol 97 pp 324ndash344 2017

[13] P Alves Costa R Calcada and A Silva Cardoso ldquoTrackndashground vibrations induced by railway traffic In-situ mea-surements and validation of a 25D FEM-BEM modelrdquo SoilDynamics amp Earthquake Engineering vol 32 no 1 pp 111ndash1282012

[14] A H Mohammad N A Yusoff A Madun et al ldquoGroundvibration attenuation measurement using triaxial and singleaxis accelerometersrdquo Journal of Physics Conference Series p012113 2018

[15] A M Kaynia F Loslashvholt and C Madshus ldquoEffects of a multi-layered poro-elastic ground on attenuation of acoustic wavesand ground vibrationrdquo Journal of Sound and Vibration vol 330no 7 pp 1403ndash1418 2011

[16] Q Liu X Zhang Z Zhang and X Li ldquoIn-situ measurement ofground vibration induced by inter-city express trainrdquo AppliedMechanics and Materials vol 204-208 pp 502ndash507 2012

[17] G Kouroussis G Alexandrou J Florentin and O VerlindenldquoPrediction of railway-induced ground vibrations The use ofminimal coordinate method for vehicle modellingrdquo Notes onNumerical Fluid Mechanics and Multidisciplinary Design vol126 pp 329ndash336 2015

[18] P Vanhonacker and H Masoumi ldquoMitigation of groundvibrations from freight trainsrdquo in Proceedings of the TransportResearch Arena (TRA) 5th Conference Transport Solutions fromResearch to Deployment The National Academies of SciencesEngineering and Medicine Paris France 2014

[19] I G Lombaert S Lombaert G Grande G Lombaert GDegrande and D Clouteau ldquoRoad traffic induced free fieldvibrations numerical modelling and in situ measurementsrdquo inProceedings of the International Workshop Wave 2000 Wavepropagation Moving load Vibration reduction pp 195ndash207 AA Balkema Rotterdam Netherlands 2000

[20] D I G Bornitz Uber die Ausbreitung der von Groszligkolbenmas-chinen Erzeugten Bodenschwingungen in die Tiefe SpringerBerlin Germany 1931

[21] G Volberg ldquoPropagation of ground vibrations near railwaytracksrdquo Journal of Sound and Vibration vol 87 no 2 pp 371ndash376 1983

[22] J-G Chen H Xia and Y-M Cao ldquoAnalysis and predictionon environmental vibration induced by elevated rail transitrdquoGongcheng LixueEngineering Mechanics 2012

[23] M Hajihassani D Jahed Armaghani A Marto and E Ton-nizam Mohamad ldquoGround vibration prediction in quarryblasting through an artificial neural network optimized byimperialist competitive algorithmrdquoBulletin of Engineering Geol-ogy and the Environment vol 74 no 3 pp 873ndash886 2015

[24] H Dehghani and M Ataee-pour ldquoDevelopment of a modelto predict peak particle velocity in a blasting operationrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 48 no 1 pp 51ndash58 2011

[25] T Mokfi A Shahnazar I Bakhshayeshi A M Derakhsh andO Tabrizi ldquoProposing of a new soft computing-based model topredict peak particle velocity induced by blastingrdquo Engineeringwith Computers vol 34 no 4 pp 881ndash888 2018

[26] Y-S Wu and Y-B Yang ldquoA semi-analytical approach foranalyzing ground vibrations caused by trains moving overelevated bridgesrdquo Soil Dynamics and Earthquake Engineeringvol 24 no 12 pp 949ndash962 2004

[27] X Li Z Zhang and X Zhang ldquoUsing elastic bridge bearings toreduce train-induced ground vibrations An experimental andnumerical studyrdquo Soil Dynamics and Earthquake Engineeringvol 85 pp 78ndash90 2016

[28] K Liu E Reynders G D Roeck and G Lombaert ldquoExperi-mental and numerical analysis of a composite bridge for high-speed trainsrdquo Journal of Sound and Vibration vol 320 no 1-2pp 201ndash220 2009

[29] S-J Feng X-L Zhang Q-T Zheng and L Wang ldquoSimulationand mitigation analysis of ground vibrations induced by high-speed train with three dimensional FEMrdquo Soil Dynamics andEarthquake Engineering vol 94 pp 204ndash214 2017

[30] I Duvnjak L Herceg andM Rak ldquoExperimental investigationof railway train-induced ground vibrationrdquo in Proceedings of the28th Danubia-Adria-Symposium on Advances in ExperimentalMechanics DAS 2011 Scientific Society for Mechanical Engi-neering Siofok Hungary October 2011

[31] T Fujikake ldquoA predictionmethod for the propagation of groundvibration from railway trains on level tracks with welded railsrdquoJournal of Sound andVibration vol 128 no 3 pp 524ndash527 1989

[32] M Schevenels ldquoThe impact of uncertain dynamic soil charac-teristics on the prediction of ground vibrationsrdquo 2007

Hindawiwwwhindawicom Volume 2018

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


separately The current theoretical research has not yetreached a high degree of precision or obtained a simplifiedanalytical solution Field measurements are the most com-mon means of studying such problems [14ndash16]

However railway line is usually long and the complex-ity of the problem is high given the dynamic interactionbetween distinct domains namely the train the viaductthe ground and the building It is not realistic to detecteach area nearby the line As response to these needsit is necessary to put forward the general law of groundvibration attenuation and to provide a method for evaluatingthe magnitude of train vibration in a viaduct area to bebuilt

Vibration assessment is a key element of the environ-mental impact assessment process for railway projects Forthe propagation and attenuation law of ground vibrationmany scholars have put forward different prediction for-mulas They can be broadly classified into two categoriesone is the amplitude representation and the other is thevibration level representation [17ndash19] Among them Bornitz[20] put forward the attenuation formula of ground vibrationamplitude when disturbance force acts on the surface ofhalf space of soil considering the geometric attenuationof the vibration propagation and the medium absorptionattenuation and the influence of propagation distance of thevibration G Volberg [21] based on tests at three differentsites regressed the relationship between the vibration leveland train speed Chen Jianguo et al [22] established an envi-ronmental vibration prediction formula based on the resultsof finite element vibration analysis The formula reflects thelogarithmic relationship between vibration acceleration leveland distance and also considers the vibration level correctionvalues of train speed bridge span and unit length andquality of the bridge by the additional function methodHowever most of the above formulas can only reflect theattenuation relationship of vibration with distance or wagonspeed and cannot clearly reflect the influence of geologicalconditions More accurate prediction formulas should be putforward

A large number of studies have shown that the vibrationvelocity and vibration frequency of the particle are closelyrelated to the damage of the building [23ndash25] It can directlyreflect the vibration energy and play an important role in theevaluation of the vibration of building Accurate predictionof the peak vibration velocity (PPV) of ground points underdifferent site conditions is of great significance In the fieldof traffic environment vibration especially for the groundvibration caused by the heavy freight wagons there are fewstudies using PPV as the evaluation index Regarding theorganization of the paper firstly a finite element model oftrack box girder pier field and building is established onthe basis of field test Numerical result of vibration responseof ground and structure is studied by analyzing vibrationvelocity and spectra induced by trains of different weight andspeed on a railway viaduct Based on modified Sodevrsquos for-mula the relationship between PPV andwagon speed wagonweight distance and site is established Finally regressionanalysis is used to discuss the procedures of determining eachparameter in the forecast formula

2 Field Test of Railway Viaduct inShenshan Village

Shuo-Huang railway is an important part of the second largestchannel of the ldquoWest-EastCoal Transportationrdquo inChinaThetotal length of the railway is nearly 600 km the transportationcapacity exceeds 255 million tons and the freight volumeincreases by tens of millions of tons every year It is locatedin the connecting area between mountainous area and plainThe relative height difference is 200m-700m The length ofthe bridge section accounts for more than 10 of the totallength In the northeast of Yuanping City Shanxi ProvinceShuo-Huang Heavy-Haul Line crosses Shenshan Village inthe form of viaducts The area is densely populated and thereare a large number of residential buildings within 30 metersof the railway line as shown in Figure 1

Shuo-Huang railway is a first-class electrified railwaywithdouble track ballast track jointless track and the rail of 60kgmThe commonly used types of trucks are C64k C70 andC80 The axle weights of heavy freight wagons are about 21t23t and 25t and the axle weights of empty wagons are about58t 60t and 50t The unspring mass of the wagon is about1500 kg per wheel set

Monitoring points in the field test can be divided into3 parts pier measuring points site measuring points andhouse measuring points as shown in Figure 2 Among themthe site measuring points are mainly arranged on two differ-ent routes perpendicular to the viaduct axis (Rout MR andRout PL) Limited by the conditions of the site there are fourmeasuring points on each of the two lines and the distancebetween adjacent points is not completely equal Based onmeasured data the basic law of site vibration attenuationon two measuring routs is put forward In order to furtheranalyze and verify the proposed rules a numerical model isset up below

3 Numerical Analysis of Ground VibrationInduced by Viaduct

A 3D numerical model is established to analyze the vibrationattenuation rule of site points under various wagon speed andaxle load more deeply Due to the complexity of rail systemthe tracks and the ballast were considered to be contactdirectly the field was considered as homogeneous sand soil

31 Calculation Model and Parameter Setting According tothe actual site conditions of Shenshan railway viaduct anumerical model of Bridge-Pier-Field-House system is estab-lished using ANSYSLS-DYNA as can be seen in Figure 3which shows the finite elements (FE) mesh of the crosssection The size of the field model is 125m length perpen-dicular to the tracks direction and 1325m in the directionof the tracks including 5 piers with each span of 265mThe thickness of the soil layer is 80m under which there isbed rock the nodes at the bottom boundary are fixed in alldirections to simulate the bedrock Pier foundation adoptsindependent foundation The maximum element size is 225m the minimum element size is 02m satisfying the require-ments of an element dimension in finite element analysis The



75m 75m 75m 75m 19m

BP6 BP7 BP8 BP9MP11

MP10 (12)

1m 65m 75m 13m 05m

BP5BP4BP3

BP2

MP1(Foundation)

Route MR

Route PL


Pier1

2 34 5 12 pier

Test line


y

(a) 3D view mode

Sym

met

ric b

ound

ary

225m

625m

16m

(b) Front view



15m

1325m(c) Right view



Track

Cushion

Box girder

Pier

Soil



6

8

10

12

Distance (m)

PPV

(ms

)

1Pier2Pier3Pier

minus4times 10


















119871119881 = 20 lg( 1198810119881ref) (1)








MPs

75 9499 8087 8593 7485 9352 8047 8749 759615 8829 7401 810 7131 8458 7378 8507 7285225 8332 7255 8448 7154 8506 7436 8293 701830 8062 6956 8298 7011 7398 6307 8449 717840 8217 7213 7862 6722 7799 6686 7472 646150 8009 6862 7287 6251 7861 6672 7431 628660 7481 6495 6903 5900 7617 6620 7349 6169

lＰ

l＝ l＝ l＝ l＝


PＨ

l＜

P1P2P3P4


MP2MP1

MP3

66

76 74

Foundation

BP1

BP2BP3 2

011





0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)



minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)


75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)


75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)


0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)


Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m




0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018










75m 75m 75m 75m 19m

BP6 BP7 BP8 BP9MP11

MP10 (12)

1m 65m 75m 13m 05m

BP5BP4BP3

BP2

MP1(Foundation)

Route MR

Route PL


Pier1

2 34 5 12 pier

Test line


y

(a) 3D view mode

Sym

met

ric b

ound

ary

225m

625m

16m

(b) Front view



15m

1325m(c) Right view



Track

Cushion

Box girder

Pier

Soil



6

8

10

12

Distance (m)

PPV

(ms

)

1Pier2Pier3Pier

minus4times 10


















119871119881 = 20 lg( 1198810119881ref) (1)








MPs

75 9499 8087 8593 7485 9352 8047 8749 759615 8829 7401 810 7131 8458 7378 8507 7285225 8332 7255 8448 7154 8506 7436 8293 701830 8062 6956 8298 7011 7398 6307 8449 717840 8217 7213 7862 6722 7799 6686 7472 646150 8009 6862 7287 6251 7861 6672 7431 628660 7481 6495 6903 5900 7617 6620 7349 6169

lＰ

l＝ l＝ l＝ l＝


PＨ

l＜

P1P2P3P4


MP2MP1

MP3

66

76 74

Foundation

BP1

BP2BP3 2

011





0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)



minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)


75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)


75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)


0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)


Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m




0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018









Track

Cushion

Box girder

Pier

Soil



6

8

10

12

Distance (m)

PPV

(ms

)

1Pier2Pier3Pier

minus4times 10


















119871119881 = 20 lg( 1198810119881ref) (1)








MPs

75 9499 8087 8593 7485 9352 8047 8749 759615 8829 7401 810 7131 8458 7378 8507 7285225 8332 7255 8448 7154 8506 7436 8293 701830 8062 6956 8298 7011 7398 6307 8449 717840 8217 7213 7862 6722 7799 6686 7472 646150 8009 6862 7287 6251 7861 6672 7431 628660 7481 6495 6903 5900 7617 6620 7349 6169

lＰ

l＝ l＝ l＝ l＝


PＨ

l＜

P1P2P3P4


MP2MP1

MP3

66

76 74

Foundation

BP1

BP2BP3 2

011





0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)



minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)


75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)


75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)


0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)


Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m




0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018


















119871119881 = 20 lg( 1198810119881ref) (1)








MPs

75 9499 8087 8593 7485 9352 8047 8749 759615 8829 7401 810 7131 8458 7378 8507 7285225 8332 7255 8448 7154 8506 7436 8293 701830 8062 6956 8298 7011 7398 6307 8449 717840 8217 7213 7862 6722 7799 6686 7472 646150 8009 6862 7287 6251 7861 6672 7431 628660 7481 6495 6903 5900 7617 6620 7349 6169

lＰ

l＝ l＝ l＝ l＝


PＨ

l＜

P1P2P3P4


MP2MP1

MP3

66

76 74

Foundation

BP1

BP2BP3 2

011





0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)



minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)


75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)


75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)


0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)


Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m




0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018












MPs

75 9499 8087 8593 7485 9352 8047 8749 759615 8829 7401 810 7131 8458 7378 8507 7285225 8332 7255 8448 7154 8506 7436 8293 701830 8062 6956 8298 7011 7398 6307 8449 717840 8217 7213 7862 6722 7799 6686 7472 646150 8009 6862 7287 6251 7861 6672 7431 628660 7481 6495 6903 5900 7617 6620 7349 6169

lＰ

l＝ l＝ l＝ l＝


PＨ

l＜

P1P2P3P4


MP2MP1

MP3

66

76 74

Foundation

BP1

BP2BP3 2

011





0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)



minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)


75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)


75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)


0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)


Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m




0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018









0 1 2 3 4 5 6 7 8 9Time (s)

0

001

002

003

004

005

006

Am

plitu

de (m

ms

Hz)

5 10 15 20 250Frequency (Hz)



minus02

minus015

minus01

minus005

0

005

01

015

02Ve

loci

ty (m

ms

)


75m15m

225m30m

2 4 6 8 100Time (s)

0

20

40

60

80

100

Velo

city

leve

l (dB

)


75m15m

225m30m

0

20

40

60

80

100

Velo

city

leve

l (dB

)

2 4 6 8 100Time (s)


0 2 4 6 8 1020

30

40

50

60

70

80

90

Time (s)

75m15m

225m30m

Velo

city

leve

l (dB

)


Time (s)0 2 4 6 8 10

20

30

40

50

60

70

80

90

Velo

city

leve

l (dB

)

75m15m

225m30m




0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018









0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

100

10minus1

10minus2

10minus3

10minus4

10minus5

101

100

10minus1

10minus2

10minus3

10minus4

1m75m

15m225m

30m40m

50m60m


0 10 20 30 40 50 60 70 80Frequency (Hz)

Velo

city

(mm

s)

102

100

10minus2

10minus4

10minus6

1m75m

15m225m

30m40m

50m60m


30 40 50 60 70 80

Velo

city

(mm

s)

10minus3

10minus2

10minus1

100

10minus4

10minus5







119881119901119890119886119896 = 1198961015840 ( 3radic1198761015840119863 )1205721015840

(2)

119881119901119890119886119896 = 11989610158401015840119863minus1205721015840 (3)

11989610158401015840 = 1198961015840 (119876101584012057210158403) (4)





0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018










0

05

1

15

2

25

3PP

V (m

ms

)

C50C64

C70C80



0

05

1

15

2

25

3

35

4

PPV

(mm

s)

C50C64

C70C80











119881119901119890119886119896 = 119896( 3radic119876119863 )120572 (5)




Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

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Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

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3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018











Horizontal

C50 1901 05219 0885 03154C64 2036 05914 09504 01512C70 248 05318 09046 04369C80 2887 05181 08783 07745

Vertical

C50 2612 06494 09708 01472C64 3021 06576 09743 01477C70 3267 06474 09736 02071C80 3533 06163 0965 03188



Horizontal

80 2887 05181 08783 07745100 3109 04915 08581 08791120 3434 04191 08742 09754140 3580 04452 08825 08615160 3666 04097 09252 05892

Vertical

80 3533 06163 09650 03188100 3644 05465 09745 02282120 4082 04987 09607 04268140 4275 04691 09856 01549160 4896 04642 09775 03131


ln119881119901119890119886119896 = ln 119896 + 120572 ln( 3radic119876119863 ) (6)



119910 = 120572119909 + 119887 (7)



11989610158401015840119909 = 5774 lowast (Q10158400482)11989610158401015840119911 = 2853 lowast (Q10158400493)

(8)



1198761015840 (119892) = 1198861119892 + 1198871 (9)




1198761015840 (V) = 1198881V + 1198891 (10)



0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

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4

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5

55

6Q

rsquo



minus4times 10

12

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2

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Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

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35

Qrsquo


minus4times 10


06

07

08

09

1

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12

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Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018









0

05

1

15

2

25

3

35

4PP

V (m

ms

)


80kmh100kmh120kmh

140kmh160kmh


0

1

2

3

4

5

PPV

(mm

s)


80kmh100kmh120kmh

140kmh160kmh



O

０Ｑ

t1 t2




C50 70 1363 636 0734 019C64 84 1572 623 0926 031C70 94 2366 367 1126 032C80 105 3242 578 1355 0059



C80 80 3242 247 1355 633C80 100 378 248 1443 36C80 120 4645 456 1817 169C80 140 5064 217 1995 514C80 160 532 335 2626 306




1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

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5

55

6Q

rsquo



minus4times 10

12

14

16

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2

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Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

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Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018











1198761015840 (V 119892) = (1198601V + 1198611) 119892 + (1198602V + 1198612)= (1198601119892 + 1198602) V + (1198611119892 + 1198612)

(11)


1198761015840 (80 119892) = 1198861119892 + 11988711198761015840 (100 119892) = 1198862119892 + 1198872

1198761015840 (V 119892) = 119886V119899119892 + 119887V119899

(12)


119886V119899 = 1198601V + 1198611119887V119899 = 1198602V + 1198612 (13)



119871V = 64 + 20 lg ( V40) (14)


1198761015840 = (1198811199011198901198861198961198961015840 )3120572

1015840

sdot 1198633 (15)


1198761015840 = (1199011 sdot V + 1199012)31205721015840 (16)




1198761015840 = (1199021 sdot 119892 + 1199022)31205721015840 (17)




1198761015840 (119892 V) = [(1198751119892 + 1198761) V + (1198752119892 + 1198762)]31205721015840

= [(1198751V + 1198752) 119892 + (1198761V + 1198762)]31205721015840(18)


1198761015840 (80 119892) = (1199021119892 + 1199022)120572101584031198761015840 (100 119892) = (11990221119892 + 11990222)12057210158403

1198761015840 (V119899 119892) = (1199021198991119892 + 1199021198992)12057210158403

(19)



minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018









minus5times 10

100 120 140 16080Wagon speed (kmh)

3

35

4

45

5

55

6Q

rsquo



minus4times 10

12

14

16

18

2

22

24

26

28

Qrsquo

100 120 140 16080Wagon speed (kmh)




minus5times 10


70 80 90 100 11060Wagon weight (t)

05

1

15

2

25

3

35

Qrsquo


minus4times 10


06

07

08

09

1

11

12

13

14

Qrsquo

70 80 90 100 11060Wagon weight (t)




1199021198991 = 1198751V + 11987521199021198992 = 1198761V + 1198762 (20)





1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018











1 1762 2732 1803 27296 201 301 196 29775 0869 1212 08269 12654 0798 1118 0789 10815 0304 0402 03287 04113 0377 0418 0352 0412225 02639 03326 0281 0368 0285 0355



5 Conclusion







Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018










































Journal of












Journal of







Volume 2018






Volume 2018








Experimental and Numerical Analysis and Prediction...

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