2014 numerical analysis of soil vibrations due to trains moving at critical speed.pdf

8/12/2019 2014 Numerical analysis of soil vibrations due to trains moving at critical speed.pdf

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R E S E A R C H P A P E R

Numerical analysis of soil vibrations due to trains movingat critical speed

Xuecheng Bian Chong Cheng Jianqun Jiang

Renpeng Chen Yunmin Chen

Received: 17 July 2013 / Accepted: 30 March 2014

Springer-Verlag Berlin Heidelberg 2014

Abstract High-speed train induced vibrations of track

structure and underlying soils differ from that induced bylow-speed train. Determining the critical speed of train

operation remains difficult due to the complex properties of

the track, embankment and ground. A dynamic analysis

model comprising track, embankment and layered ground

was presented based on the two-and-half-dimensional

(2.5D) finite elements combining with thin-layer elements

to predict vibrations generated by train moving loads. The

track structure is modeled as an EulerBernoulli beam

resting on embankment. The train is treated as a series of

moving axle loads; the embankment and ground are mod-

eled by the 2.5D finite elements. The dynamic responses of

the track structure and the ground under constant and

vibrating moving loads at various speeds are presented.

The results show that the critical speed of a train moving onan embankment is higher than the Rayleigh wave velocity

of the underlying soil, attributed to the presence of the

track structure and the embankment. It is found that the

dynamic response of ground induced by moving constant

loads is mostly dominated by train speed. While for the

moving load with vibration frequency, the ground response

is mostly affected by the vibration frequency instead of

train speed. Mach effect appears when the train speed

exceeds the critical speed of the trackembankment

ground system.

Keywords Critical speed 2.5D finite element methodHigh-speed train Mach effect Trackembankment

ground dynamic interaction

1 Introduction

In recent decades, high-speed railways have developed

rapidly all over the world as a form of fast and energy-

efficient mass transportation. As train speed increases,

dynamic responses of railway track and ground along the

railway line become more substantial. For a high-speed

train running on soft soil, resonance may occur, and con-

sequently, the dynamic responses of the track and ground

are dramatically amplified. The speed at which extraordi-

narily large dynamic response occurs is named as the

critical speed [17]. At the critical speed, train moving

loads induce strong vibration in track structure and increase

the risk of train derailing and track structure damage. In

1998, an extensive measurement was undertaken by the

Swedish State Railways on soft soil ground in Ledsgard

[15]. The test was performed using an X-2000 passenger

Invited Paper from the International Symposium on Geotechnical

Engineering for High-speed Transportation Infrastructure

(IS-GeoTrans 2012), October 26 to 28 2012, Hangzhou, China.

Co-Editors Prof. Xiong (Bill) Yu, Case Western Reserve University,

USA and Prof. Renpeng Chen, Zhejiang University, China.

X. Bian R. Chen (&) Y. ChenDepartment of Civil Engineering, Key Laboratory of Soft Soils

and Geoenvironmental Engineering, MOE, Zhejiang University,

Hangzhou 310058, China

e-mail: [email protected]

X. Bian


Y. Chen


C. Cheng J. JiangInstitute of Hydraulic Structure and Water Environment,

Zhejiang University, Hangzhou 310058, China


J. Jiang


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Acta Geotechnica

DOI 10.1007/s11440-014-0323-2


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problems in the trackground system under train loads

moving at high speeds have been reported and studied in

recent years. Madshus and Kaynia [18] analyzed the test

results at a site in Sweden, where the resonance problem

was encountered, providing a unique opportunity to mea-

sure the response of the railembankmentground system.

Dimitrovova and Varandas [7] deal with excessive ground

and track vibrations induced by high-speed trains when

moving from one region to another with very different

vertical stiffness of the system trackground. Huang and

Chrismer [13] investigated the impact of high-speed trains

running on a track structure. They used a dynamic track

model together with an discrete element model (DEM) to

investigate the railway ballast settlement under freight and

trains moving at critical speed.

The majority of the aforementioned works used sim-

plified beam on ground models to analyze the ground

vibrations under train moving loads. Although these

solutions could be used to understand the vibration gen-

eration mechanism under train moving loads, these

methods are not able to deal with the complex track

structures and ground soils. Determining the critical speed

of train operation remains difficult due to the complex

properties of the track, embankment and ground. Three-

dimensional finite element models may be used to

account for details of the track structure and ground. But

its main disadvantage is very time-consuming [3, 11, 27].

To overcome the disadvantages of traditional numerical

methods for simulating train-induced soil vibrations, Yang

and Hung [25] originally proposed a high-efficient 2.5D

finite element method for treating vibrations in half-space

induced by moving trains, and the infinity of the ground

was dealt with by an infinite element in wavenumber

domain. And later, they used the same method to inves-

tigate underground train-induced ground vibrations [12,26]. The 2.5D finite element method later has also been

applied in studying soil vibrations induced by ground

surface trains [1, 4, 21].

In this paper, we present a 2.5D finite element model to

study ground vibrations induced by trains moving at vari-

ous speeds, accounting for the dynamic trackembank-

mentground interaction. In the following sections, first,

the 2.5D finite element method is briefly introduced, and

then, an elaborated trackembankmentground computa-

tional model is used to study the dynamic response of track

structure under a series of train wheel axle loads and the

consequent wave propagation in ground. A parametricstudy is performed based on this model to determine the

critical speed of train operation. Finally, some conclusions

are presented.

1.1 2.5D Finite element method

In this paper, we use a 2.5D finite element model combined

with thin-layer element to study the dynamic response of

the three-dimensional ground under train moving loads.

First, the wavenumber transform with respect to the track

direction is applied on the equations governing wave

propagation. Therefore, the vibration along the track

direction is expressed by discrete wavenumber. The

embankment and ground in the transversal vertical section

are discretized and modeled by 4-node quadrilateral ele-

ments with specific discrete wavenumbers. Each node of

the element has three degrees of freedom allowing wave

propagation in three-dimensional space to be taken into

account faithfully. This approach greatly reduces the

computational time compared with conventional numerical

models while maintaining computation accuracy of the

wave propagation in three-dimensional space.

Vibration problems induced by the train arise mainly

from movement and dynamics of the axle loads, as the

material and geometry of the track structure and sup-

porting ground can be regarded as constants in the

direction of the train movement. In this paper, we assume

that the train runs in the x-direction. The Fourier trans-

form is performed with respect to the time t and the train

running direction x (where subscript x and t represent

components in the wavenumber and frequency domain,

respectively).

Node 3Uy

Uz

Ux

Node 4 Uy

Uz

Ux

Node 2Uy

Uz

Ux

Node 1Uy

Uz

Ux

Fig. 2 2.5D finite element node

Fig. 3 Trackembankmentground interaction model (not to scale)

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uxt

Z1

1

Z1

1

ux; t expinxx expixtdnxdt 1

The corresponding inverse transform is,

u

Z1

1

Z1

1

uxtx; t expinxx expixtdnxdt 2

wherenx is the wavenumber in thex-direction, andx is the

frequency in the x-direction. The finite elementthin-layer

element model of the trackground system model is shown

in Fig. 1. This paper assumes that the vertical interface of

the track direction is continuous, and the ground materials

making up the embankment and track structure are con-

stant, with the sleeper distribution quality included in thetrack.

A typical 2.5D quadrilateral finite element is shown in

Fig.2, each node has three degrees of freedom where nx is

the wavenumber in the track direction. After introducing

the shape function, the discretized form of the governing

equation in the frequency domain can be derived by the

conventional finite element method.

Kxt x2MUxt Fxt 3

whereM, Kxt and Fxtare the mass matrix, stiffness matrix

and equivalent nodal force vectors, respectively, and their

detailed expressions are

MXe

qeZ Z

NTNjJjdgdf 4

Kxt Xe

Z Z BNTDBNjJjdgdf 5

Fxt Xe

Z Z NTfjJjdgdf 6

whereJ is the Jacobi matrix. The detailed expressions ofJ,

B and D have already been given in Bian et al. [ 4].

2 Trackembankmentground interaction model

and computation parameters

A simplified illustration of the computation model is shownin Fig.3. The train is modeled by a series of vertical point

loads at track surface along the rail which move along the

track at constant speed. Four observation points are indi-

cated in Fig. 3 at point A (the track center), point B (the

bottom of the embankment), points C and D (ground sur-

face). The distances of these points to the track centerline

are 0, 4, 14 and 24 m, respectively.

In this paper, the ground is treated as a layer soft soil

resting on a layer of stiff soil. The parameters of

embankment and ground are shown in Table1.

Figure4 shows the geometry of the wheel axle weight

distribution of the high-speed train used in this study.PandLrepresent the axle load and the train length; a and b are

the distance between adjacent axles. The train used in this

paper is China high-speed train CRH3. It includes 8 coa-

ches, the length of a coach, L is 25 m,b is 15 m,a is 2.5 m

and the axle load P is 1.4 9 105 N. The mass of the two

rails is 120 kg/m. The bending stiffness of rail is

1.26 9 107 N m, and its loss factor is 0.01.

To explore vibration attenuation at distances from the track

center, this paper adopted the formula suggested by Esveld

[9]. The vibrations of the ground will be presented as dis-

placement or velocity on a logarithmic scale indB defined as:

LdB 20 logp1p2

7

where P1 is the amplitude of velocity vibration, and P2 is

the reference value. The result can be expressed by

displacement, velocity or acceleration. This paper

addressed the displacement and velocity responses, and

adopted the reference displacement and velocity values as:

P2 108m=s velocity andP2 10

11m(displacement)

Table 1 Parameters of the embankment and ground

Layer Depth (m) Density (kg/m3

) S-wave speed (m/s) Rayleigh wave speed (m/s) Loss factor Poisson ratio

Embankment 1.0 2,100 200.0 188 0.05 0.40

Soft soil 6.0 1,800 85.0 80 0.05 0.35

Stiff soil 17.0 2,000 150.0 142 0.05 0.45

Fig. 4 Geometry of wheel axle load distribution of a train

cFig. 5 Vertical displacement response at the speed 50, 120 and

200 m/s at observation points a A, b B, c C and d D (left) distance

history of displacement, (right) RMS value of displacement

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3 Numerical results and discussion

3.1 Moving constant load results

In this section, the train load is simplified as a series of

constant loads. Three typical speeds are adopted in the

numerical computation, super-critical speed (200 m/s),

estimated critical speed (120 m/s) and subcritical speed(50 m/s). The timehistory and root-mean-square (RMS)

displacement responses of the four observation points are

shown in Fig. 5. To compare the dynamic responses for

different train speed cases, the value of speed multiplying

time is adopted for the abscissa of the dynamic response

diagram. The ground vibrations from a train load moving

at a speed of 50 m/s can be regarded as similar to the

pseudo-static deformation induced by the total weight ofFig. 6 Vertical displacement response under varied speeds

Fig. 7 Spectrum of vertical displacement at observation points (y = 0, 4, 14, 24 m) during the pass-by of a constant train load at varied speeds.

a Point A at track center (y = 0), b point B at y = 4 m, c point C at y = 14 m, d point D at y = 24 m

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the whole train geometry, and no significant wave prop-

agation phenomena at ground surface are observed. The

displacement at points C and D is close to zero. The

propagation of vibration phenomenon at all four obser-

vation points becomes more obvious for a train running at

estimated critical speed (120 m/s) and super-critical speed

(200 m/s). At the track surface structure (points A and B),

vibrations induced at a speed of 120 m/s are much larger

than those induced at either sub- or super-critical speed,

which indicates that ground response does not always

increase with speed. Away from the track, the responses

at points C and D have large amplitudes, which decrease

slowly when train runs at critical and super-critical

speeds.

Figure6 presents the relationship between vertical dis-

placement amplitude and train speed. It shows that the

amplitude of displacement response increases slowly with

speed at relatively low speed. The response increases

sharply once the train moves at speed over the Rayleigh

wave velocity of the upper soil layer below the embank-

ment (80 m/s). The amplitude maintains a higher value

when the speed of train is higher than the Rayleigh wave

velocity of the subsoil. Based on Ref. [7], the critical speed

of the trackground system is defined as the train speed

which induces the maximum amplitude response of thetrack structure. Figure6 shows that the critical speed for

model is 120 m/s, which is higher than the Rayleigh wave

velocity (80 m/s) of the topsoil, that is, because of the

existence of track and embankment structures.

Figure7 presents the corresponding frequency spectra

of the displacement responses given in Fig. 5. There are

three peak responses indicated in the left of four diagrams,

which are excitation frequencies related to the main train

wheel axle load distribution: 2, 4.8 and 8 Hz. This indicates

that when the train is simplified as a series of constant

loads, the distance of the train wheel axle loads is an

important parameter in deciding the peak response of thetrack. The vibrations at the center of the track (point A)

induced by the three speeds are relatively large and

decrease slowly at higher frequencies. In the low-frequency

region, there are no clear differences in the spectral values

at three speeds. However, in the high-frequency region, the

spectral values for the train moving at 200 m/s are higher

than those of the two lower speeds, which indicate that

more high-frequency dynamic response is generated close

to a speed of 200 m/s. At observation points B, C and D,

the response induced at a speed of 50 m/s decreases shar-

ply at vibration frequencies above 2 Hz. At point D, the

response is almost 0. In the high-frequency region, the

amplitude of the response induced at speeds of 120 and

200 m/s remains high for most observation points. This

Fig. 8 Dispersion diagram for the SV modes of the layered ground

with load speed lines overlaid for constant load

Fig. 9 Variation in vertical intensity with distance from track center for train speed c = 50, 120, 200 m/s a displacement vibration level,

b velocity vibration level

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shows that at high speed, high-frequency modes in the

nearby ground are both induced and decline more slowly.

To analyze the dynamic characteristics of the layered

ground and the resonance phenomenon, we investigated the

relationship between the dispersion diagram and the speed

lines of the constant loads. This calculation was based upon

a dispersion diagram of the first eight modes of SV-wave

(Rayleigh wave) and the spectrum from 0 to 100 Hz,shown in Fig.8. From the load speed line equa-

tion:b X2pfj j=c, where c is the constant load speedand X is the load frequency, which is 0 for constant load.

Figure8shows the SV-wave modes overlaid with the lines

b X2pfj j=c for c = 50, 120 and 200 m/s. All thedispersion curves are located in between the upper and

lower lines, which are the Rayleigh wave velocity line of

the upper soil and shear wave speed line of the lower

ground. The cut-off frequencies of the layered ground are

5.88, 13.52, 22.30, 30.57, 39.56, 47.46 and 56.84 Hz for

the first eight modes. Figure7 shows that when the loads

travel at speed of 120 m/s, a peak is observed at 4.8 Hz,

which is close to the intersection of the load speed line and

the dispersion curve of the first SV-wave mode (Fig. 8).

This indicates that the first mode of the layered ground

plays an important role in the dynamic response of theground and the response of the ground and track peak at

frequencies near this intersection. No intersection occurs

for the load moving at a speed of 50 m/s (lower than the

minimum Rayleigh wave velocity of the layered ground,

80 m/s), which shows that the vibrational frequencies

induced by a constant load moving at less than 50 m/s are

below the resonant frequency of the ground. As the train is

simplified as a series of moving loads in this paper, the

Fig. 10 Vertical displacement responses at ground surface under different train velocities ( left) XY plane, (right) YZ plane a c = 50 m/s,

b c = 120 m/s, c c = 200 m/s

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Fig. 11 Vertical displacement responses for conditions of speed 50, 120 and 200 m/s. (left) distancehistory of displacement, (right) RMS value

of displacement a point A at track center (y = 0), b point B at y = 4 m, c point C at y = 14 m, d point D at y = 24 m

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length of the load interval may also induce peak responses

(Fig.7); however, these are weaker than that response of

the first mode SV-wave and load line crossover frequency.

The response of the ground is close to that of quasi-static

load conditions, especially when the speed is much lower

than the speed of the Rayleigh wave. The vibrations

induced by the loads moving at 50 and 200 m/s are lower

than those induced at 120 m/s, which could explain theconclusions drawn from Fig.5.

Figure9 presents the vibration level of vertical velocity

and displacement at the ground surface from the track

center to the nearby ground. In the same figures, the

geometry of the track and embankment is plotted. First, all

the curves at the position of 0.7 m appear a small peak, as

that is the rail position. With increasing distance from the

track center, all the vibration level curves show anFig. 12 Vertical displacement response under varied speeds

Fig. 13 Spectrum of vertical displacement at observation points (y = 0, 4, 14, 24 m) during the pass-by of a constant train load at various

speeds.a Point A at track center (y = 0), b point B at y = 4 m, c point C at y = 14 m, d point D at y = 24 m

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attenuate trend. The decline of vibrations induced at a

speed of 50 m/s is faster than at other speed. On the dis-

placement vibration level diagram, the curve representing

the train moving at 50 m/s appears as a broken line in the

displacement vibration level figure, which shows that the

vibration decreases rapidly in the region 12 m from the

track center. The vibrations decrease more slowly beyond

12 m. Comparison of the curves at 120 and 200 m/s inFig.9shows that the level of vibrations induced at speed of

120 m/s is higher than that induced at 200 m/s close to the

track; however, farther from the track, the vibration level

induced at 200 m/s is greater than that at 120 m/s.

Figure10 shows the vertical displacements response of

the ground surface in XY plane and YZ plane for a train

moving at various speeds. At a speed of 50 m/s, vibration

is mainly induced near the axle positions, and there is little

propagation of vibration to the surrounding ground. When

the train travels at speeds of 120 and 200 m/s, which

exceed the Rayleigh wave velocity of the ground, vibra-

tional propagation occurs. The vertical deformation of the

ground in front of the load is small but more substantial

behind the load. Figure10b, c also presents the reverse

wave propagation lines from the load position, showing a

shock wave in the ground responses also known as a Machcone. This phenomenon is similar to the case of an aircraft

breaking the sound barrier. Figure 10 shows that the wave

propagation angle is h arcsin VRC

. As this paper is con-

cerned with analyzing the vertical response of the ground

surface, the observed Mach lines are mainly induced by

Rayleigh waves.

3.2 Harmonic moving loads results

Studies have shown that the excitation frequencies of

vertical wheelrail forces are mainly in three frequency

ranges: low-frequency range (0.510 Hz); mid-frequency

range (3060 Hz); and high-frequency range (100

400 Hz). We now consider the effect of a train simplified

as a series of loads with vibrational frequencies of 30 Hz.

Figure11 shows the vertical displacement distancehistory

curves and RMS value curves at the four observation points

at various speeds. In analysis of the impact of the vibra-

tional load, the differences of vibration induced at various

speeds are not clear from distancehistory diagrams.

Instead, the RMS value diagrams show these differences

more clearly: The greatest amplitude is induced at a speed

of 200 m/s and the lowest at 50 m/s. There appear to be

fluctuations of the amplitude below of 50 m/s, because of

the vibrating loads. Compared with Fig.5, the amplitude of

the response from the vibrating loads is much smaller thanFig. 14 Dispersion diagram for the SV modes of the layered ground

with load speed lines overlaid for a 30 Hz load

Fig. 15 Variation in vertical intensity with distance from track center for train velocities 50, 120 and 200 m/s.a Displacement vibration level,

b velocity vibration level

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constant loads, and the propagating waves are induced at

the lower speed.

Figure12 presents the relationship between the ampli-

tude of displacement response and train speed. The

amplitude of the displacement response is 0.71.0 mm, and

several response peaks are apparent. Compared with the

displacement results induced by the moving constant loads

(Fig.6), the amplitude of responses induced by the

vibrating loads is relatively small. This indicates that the

speed of harmonic loads has little impact on the amplitude

of vertical displacement response.

Figure13 presents the corresponding frequency spectra

of the displacement response given in Fig. 11. A peak is

observed at a frequency close to 30 Hz that is the

resonance frequency induced by the harmonic loads, and

this generates a larger dynamic response compared with

other frequencies. At observation point A, the amplitudes

of responses at the three speeds are similar in the low-

frequency region. The amplitudes of responses at 120 and

200 m/s are larger than that at 50 m/s in the high-frequency

region. At observation point B, the spectral amplitude

increases gradually at frequencies lower than 30 Hz. At the

distant observation points C and D, the dynamic responses

in the low-frequency region are almost 0. At higher fre-

quencies, there is a sudden jump in the spectrum. The

frequency of this jump coincided with the intersection

frequency of the moving load harmonic speed line and the

first SV mode dispersion according to the analysis of

Fig. 16 Vertical displacement responses at ground surface under varied train speeds (left) XY plane, (right) YZ plane. a c = 50 m/s,

b c = 120 m/s, c c = 200 m/s

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Fig.14 (spectrum of vertical displacement). At higher

speed, the region of the peak frequency becomes broader,

and large vibrations are induced over a wide range of

frequencies, which may explain the phenomenon described

in Fig. 11. The responses induced at speed below 50 m/s at

frequencies higher than 30 Hz decrease rapidly. However,

the responses induced at speeds of 120 and 200 m/s

decrease more slowly, and a substantial amplitude remainsin the high-frequency region. In general, the amplitude of

the higher frequency vibrations declines less rapidly at

higher speed.

Figure14 shows load speed lines of the 30 Hz loads

under the speed of 50, 120 and 200 m/s plotted on the

dispersion diagram. The speed line is defined as

b X2pfj j=c, whereX = 30 Hz, the intersection of the

harmonic load speed line and layered ground first SV mode

is: 18.3 Hz (50 m/s), 11.9 Hz (120 m/s) and 8.9 Hz

(200 m/s). Combined with Fig.13, the minimum fre-

quency of the peak response is the point of intersection of

the vibrating load speed line and the first SV curve. Whenthe frequency of the ground response induced at different

speeds is close to the intersection frequency, the ground

will exhibit strong vibrations. Compared with the results

presented in Fig. 8, the main difference for the vibrating

loads is that at the speeds lower than the Rayleigh wave

speed of the ground surface or higher than the shear wave

speed of the subsoil, a corresponding resonance frequency

in the layered ground exists.

Figure15 presents the vertical displacement and

velocity vibration levels of the ground over a region from

the center of the track to a point nearby. It is found that all

curves in the two figures show a slow declining trend with

distance from the track. But there exists a rebounce in the

vibration transmission at the embankment toe. The differ-

ence in the vibrational level near the track structure is not

obvious, as the load moving under at all three speeds

induces resonance phenomenon. As the distance increases,

the response induced in the surrounding ground at a speed

of 200 m/s is slightly larger than those at 120 and 50 m/s.

Compared with the vibrational levels of constant loads

(Fig.9), the vibration induced by the vibrating loads

decreases slowly and spreads more widely, which indicates

that a wider range of vibration frequency modes is induced

by the moving harmonic loads.

Figure16 shows the vertical displacements response of

the ground surface in the XY and YZ plane for a train

moving at various speeds. For the vibrating loads, the

amplitude of vibrations induced is much smaller, but the

range is increased compared with the constant load even

when train is running at subcritical speed (50 m/s). The

Mach effect phenomenon under the moderate- and high-

speed conditions and from the vibration propagation is also

clearly obvious.

4 Conclusions

This paper applied a 2.5D finite element combined with

thin-layer element model to analyze the vibrations induced

by moving trains at subcritical speed, critical speed and

super-critical speed, accounting for both moving constant

load and moving load with vibration frequency. From the

computational results, the following conclusions can bemade:

1. There exists a critical speed in the trackembankment

ground system for the operation of a high-speed train.

As the embankment and track structure have greater

stiffness than that of the underlying ground soils, the

critical speed is higher than the Rayleigh wave

velocity of the upper layer soil.

2. For non-vibrating loads moving with a speed below the

critical speed, only a quasi-static response pattern at the

ground surface is produced. At train speed close to or

beyond the critical speed of the trackembankmentground system, a Mach effect appears at the ground

surface, and strong vibrations are generated in the ground.

3. Track response does not always increase with the train

speed. There exists a speed range (with inclusion of the

critical speed) in which the train induces peak

vibration intensity in the track. For train speeds outside

of this range, the track vibration remains small.

4. The response of the ground is dominated by the first

mode of the layered ground. The vibration of the track

structure reaches maximum amplitude near frequen-

cies equal to the intersection of the loads speed line

and the first mode of the SV-wave. The frequencydetermined by the intersection of the vibrating loads

speed line and first SV-wave intersection point corre-

sponds to the minimum frequency that induces peak

vibrational intensity.

5. When the load has a vibrational frequency, a resonance

may exist even if the speed of the moving load is lower

than the critical speed of the trackground system.

Acknowledgments Financial support from the Natural Science

Foundation of China (Grant Nos. 51178418 and 51222803) is grate-

fully acknowledged.

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