SCIENCE CHINA Physics, Mechanics & Astronomy

Science China Press and Springer-Verlag Berlin Heidelberg 2010 phys.scichina.com www.springerlink.com

*Corresponding author (email: zeng_qingyuan@hotmail.com)

Research Paper April 2010 Vol.53 No.4: 751757 doi: 10.1007/s11433-010-0158-2

Analysis theory of random energy of train derailment in wind

CHEN RuiLin1,2,3, ZENG QingYuan3*, HUANG YunQing1, XIANG Jun3, WEN Ying3, GUO XiaoGang2, YIN ChangJun2, DONG Hui2 & ZHAO Gang4

1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; 2 School of Civil Engineering & Mechanics, Xiangtan University, Xiangtan 411105, China;

3 School of Civil Architecture, Central South University, Changsha 410075, China; 4 AMEC, Toronto M5A5G7, Canada

Received December 1, 2009; accepted February 8, 2010

Based on the analysis theory of random energy of train derailment, an analysis theory of random energy of train derailment in wind is suggested. Two methods are proposed -the time domain method and the frequency domain method of analysis theory of random energy of train derailment in wind. The curves of pw-v under various wind speeds are obtained through the compu-tation. The original curve of p-v is expanded, which turns the analysis theory of random energy of train derailment into the all-weather theory. Train derailment condition has been established under wind action. The first and second criterions of train derailment have been proposed in light of wind action. The analysis of train derailment cases at home or abroad is made, in-cluding the first analysis of Xinjiang train derailment case encountered 13-level of gale, which explained the inevitability of train derailment. The analysis theory of random energy of train derailment in wind shows its validity and accuracy. The input energy pw of the transverse vibration of train-track(bridge)-wind system is linked to train speed. With the establishment of the analysis theory of random energy of train derailment in wind, It is likely to initiate an all-weather speed limit map for a train or any high-speed train.

train derailment, analysis theory of random energy of train derailment, analysis theory of random energy of train de-railment in wind, train derailment condition in wind, the criterion of train derailment in wind, train derailment case

PACS: 45.40.-f, 47.11.-j, 47.85.-g

In the early time, researchers mainly concentrated on aero-dynamic drag in train aerodynamics. As the enhancement of train velocity makes train lighter, more wind or air influence is pronounced against the train. Although the research on aerodynamics is more and more thorough, which makes it possible to design fighter aircraft and the ultra-large pas-senger plane with good manoeuvring performance, the safety research on train derailment under wind effects is far from success. For instance when the world first high-speed railroad was built in Japan, as a result of underestimating the aerodynamic effect on train operation safety, the multi-ple track spacing and the tunnel cross sectional area were

too small, which saved project investment. Japan has stud-ied train aerodynamics for decades, but train aerodynamics performance is still the major issue to prevent Japanese railroad train speed from enhancing. With higher train speeds, train movement resistance grows dramatically, so does energy consumption; Train-crossing at a high speed can produce air pressure pulse resulting in the large distor-tion of the side wall of a passenger train, accompanied by an intense air demolition sound, which can crush window panes of the vehicle. It can potentially overturn a dou-ble-decked container freight train. The important channel Lanchow Sinkiang line of Asian- European land bridge passes through the Xinjiang gale Gobi area where the natu-ral condition is extremely severe. In its hundred kilometers

752 CHEN RuiLin, et al. Sci China Phys Mech Astron April (2010) Vol. 53 No. 4

wind area (150 km), the instantaneous maximum wind speed can reach 64 m/s, which is twice the speed of 12-level of wind, topping the world at the railroads. The serious ac-cident of the whole train blown to overturn repeatedly oc-curs in the hundred kilometers wind area (HKWA). Trains are frequently forced to halt due to the extreme wind condi-tion in gale season. Large numbers of passengers and cargos are detained, restricting the economical development in western China. At the same time, a series of aerodynamics questions need to be solved when the high-speed train is developed in China. In order to guarantee the traffic safety, China Planning Commission, the Ministry of Science &Technology, the Ministry of Railway, the Ministry of Education and National Natural Science Foundation of China list safety development as the key science& technology, the National Basic Research Program of China (973 project), National High-Tech Industrial Production, and National Natural Science Foundation of China. A series of achieve-ments in scientific research [13] have been accomplished.

Safety performance research of train derailment under the wind action has not made enough progress to meet train safety requirement in the wind area. The reasons [4,5] are summarized approximately as follows: (1) researchers tend to research the aerodynamics of aviation and astronautics and study hydrodynamics performance of ship, traditionally ignoring the aerodynamics performance of train. However the speed of modern high-speed train is approach that of the earlier airplane. To study aerodynamics performance of train for safe operation is urgent; (2) the train-track-wind time-variant system (hereafter refers to as the system with wind) is a highly complex system involving many fields and subjects, such as railway engineering, vehicles project, aerodynamics, analysis theory of train derailment, bridge and tunnel engineering, and soil and rock mechanics. the present system is limited by the advancement of these fields. For instance, the problem of train derailment has remained a puzzle for more than one hundred years. Academician Zeng and his team have made encouraging achievements [416], but the influence of wind has not been considered in the analysis of train derailment; (3) Experimental difficulties: Can we obtain the test data of the real system with wind? We can not gather all the data because of the high cost, dif-ficulty to meet all wind conditions and to do the tests of train derailment. So we attempt to use the CFD method to calculate aerodynamics performance of train for safe opera-tion. However, there are many difficulties at the present stage, especially for train-track-wind system [15]. If we apply the DNS method, the large-scale rapids requests high resolution, which will require huge computation that the most advanced computer will miss several magnitudes. If we have to consider the vibration of train-track system itself, this will be three dimensional Fluid-Structure Interaction (FSI) problem, which can not be resolved by the up-to-date computer. other methods, such as LES and the RANS

method must be applied to this system, besides the inherent choice problem of turbulence model. Still, the problem of separated flow will be a very big barrier. Other methods originated in airfoil theory, such as the panel method, the theory of lift surface can coincide with the experimental results in only few special situations.

Based on the analysis theory of random energy of train derailment, two methods- the time domain method and the frequency domain method of analysis theory of random energy of train derailment in wind are proposed in this pa-per. after the train buffeting response spectrum has been conducted thoroughly, the curves of pw-v under various wind speeds are obtained through the computation. The original curve of p-v has been l expanded, which turns the analysis theory of random energy of train derailment into an all-weather theory.

1 Proposal of the analysis theory of random energy of train derailment in light of wind action

Two methods-the time domain method and the frequency domain method of analysis theory of random energy of train derailment under wind action are proposed, based on the energy stochastic analysis theory of train derailment estab-lished by the academician ZENG QingYuan [615].

1.1 The method in time domain

We regard the train, track and wind as an integral system, and take into account the clearance between wheel and rail and the wheel-rail displacement connecting conditions, that is, the wheel displacement (transverse, vertical) is equal to rail displacement (transverse, vertical) plus rail irregularity (transverse, vertical) and plus wheel-rail relative displace-ment (transverse, vertical). According to the principle of total potential energy with the stationary value in elastic system dynamics and the rule of set-in -right-position for formulating system matrices [6,15], the spatial vibration equations of the system can be established. Because the equations satisfy the wheel-track displacement connecting conditions, this kind of equations of the system can reflect the wheel-track contact conditions and relative displacement, and conform to mathematics physics equation theory re-quirement.

1.1.1 Vibration equations analysis of the system without wind

The measured hunting waves of bogie frames can be treated as the exciting source of vibration in the deterministic analysis of the transverse vibration of the system. When wind load is not considered, the transverse vibration equa-tions of the system are as follows:

[ ]{ } [ ]{ } [ ]{ } 0,M u C u K u+ + = (1)

CHEN RuiLin, et al. Sci China Phys Mech Astron April (2010) Vol. 53 No. 4 753

where [M], [C] and [K] are the mass, damping and stiffness matrix of transverse vibration of the system, respectively, and { }u , { }u and { }u are the acceleration, velocity, and displacement vector of the system, respectively. If k vibra-tion responses are known, and n responses unknown, then through eq. (1), we obtain:

0.

kk kn k kk kn k

nk nn n nk nn n

kk kn k

nk nn n

M M u C C u

M M u C C u

K K u

K K u

+ + =

(2)

To expand eq. (2), we have

[ ]{ } [ ]{ } [ ]{ }

[ ]{ } [ ]{ } [ ]{ } ,nn n nn n nn n

nk k nk k nk k

M u C u K u

M u C u K u

+ +=

(3)

[ ]{ } [ ]{ } [ ]{ } [ ]{ }

[ ]{ } [ ]{ } 0,kk k kk k kk k kn n

kn n kn n

M u C u K u M u

C u K u

+ + ++ + =

(4)

eq. (4) is a non-independent equation set which is to be crossed out. All the items on the right side of equation (3) are known. With this, n unknown, transverse vibration re-sponses can be obtained. Thus, k known, vibration re-sponses become the exciting source of the transverse vibra-tion of the system. The items on the right side of eq. (3) become the equivalent self-exciting force causing transverse vibration of the system and the items on the left side of eq. (3) become the resistant force of the transverse vibration of the system. From refs. [5,14] by studying the process of trial calculation for defining the work done by the resistant force of the system against the transverse vibration, we under-stand the work done by the resistant force of the system is decided by the known items on the right side of eq. (3). So the curve of c-v for the system without wind can be drawn. 1.1.2 The condition of train derailment of train-track- wind system

When wind load is considered, the transverse vibration equations of the train-track-wind system are as follows:

[ ]{ } [ ]{ } [ ]{ } ( )M u C u K u f t+ + = (5) where [M], [C] and [K] are defined as eq. (1). f(t) mainly is the vector of transverse component of wind load. It is inevi-tably the unsteady force because of the pulsation of the at-mospheric rapids and randomness of this system transverse vibration.

If k vibration responses are known, and n responses un-known, then through eq. (5), the following equations can be written

kk kn k kk kn k

nk nn n nk nn n

M M u C C u

M M u C C u

+

( )( ) ,kk kn k knk nn n n

K K u f t

K K u f t

+ = (6)

where both fk(t) and fn(t) are known. To expand eq. (6), we have:

[ ]{ } [ ]{ } [ ]{ }

( ) [ ]{ } [ ]{ } [ ]{ } ,nn n nn n nn n

n nk k nk k nk k

M u C u K u

f t M u C u K u

+ +=

(7)

[ ]{ } [ ]{ } [ ]{ } [ ]{ }

[ ]{ } [ ]{ } ( ) ,kk k kk k kk k kn n

kn n kn n k

M u C u K u M u

C u K u f t

+ + ++ + =

(8)

eq. (8) is a non-independent equation set which is to be crossed out. All the items on the right side of eq. (7) are known. With this, n unknown, the transverse vibration re-sponses can be obtained. Thus, k known, vibration re-sponses become the exciting source of the transverse vibra-tion of the system. The items on the right side of eq. (7) become the equivalent self-exciting force causing transverse vibration of the system and the items on the left side of eq. (7) become the resistant force of the transverse vibration of the system. From literatures [4] and [15] by studying the process of trial calculation for defining the work done by the resistant force of the system against the transverse vi-bration, we can know the work done by the resistant force of the system is decided by the known items on the right side of eq. (7). So the curve of c-v [5,8,9,14] for the system without wind can be drawn. c(v) of train-track-wind sys-tem is obviously different from that of the system without wind.

1.2 The method in frequency domain

From literatures [5,14] we know that, the curve of p-v [5,8,9,14] can be measured for different kinds of trains at different rails, and the artificial bogie frame hunting wave can be simulated and treated as the exciting source of transverse vibration of the system. If there is wind, espe-cially in strong wind environment, p(v) is obviously dif-ferent from the measured p, that is to say, the exciting source of transverse vibration of the system has changed. Of course we can not measure the curve of p-v at different wind speeds, because even if we have enough time and fund, it is impossible to suffer all different wind speeds.

we are able to acquire the curve of p-v at different wind speeds only by calculation. With the analysis of Chapters 2 and 3 in ref. [5] we can see that if buffeting responses of bogie frame can be calculated, we can modify figures 5-7 and 5-8(They are respectively the bottom curve of figures 1 and 2) in ref. [5], in order to get the curve of p-v at differ-ent wind speeds.

bogie,windpw p = + (9)

754 CHEN RuiLin, et al. Sci China Phys Mech Astron April (2010) Vol. 53 No. 4

where pw is p at various wind speeds; bogie,wind is p un- der the wind action. In literature [5], horizontal wind spec- trum refers to the empirical formula Simiu [17] gave, and the vertical wind spectrum applies to the empirical formula Panofsky-McCormick [18] proposed. Aerodynamic admit- tance function uses the arithmetic mean of approximate formula of the cylindrical cross-section and prism cross- section [19] presented. The buffeting response spectrum of vehicle body and bogie frame is calculated. The first-order buffeting response spectrum of heave, transverse and roll of bogie frame can be drawn, so we can find bogie,wind . The detailed derivation of bogie,wind is as follows.

Supposed that h, a and p denote vertical bending, torsion and lateral bending movement respectively. The subscript 1 denotes the first-order vibration mode. Standard sec- ond-order movement differential equations (we just take the first-order vibration mode of different directions) are:

1 1

1 1

1 1

~ ~2

1 1 1 1

~ ~2

1 1 1 1

~ ~2

1 1 1 1

2 ,

2 ,

2 .

h h h h

a a a a

p p p p

h h h Q

a a a Q

p p p Q

+ + = + + = + + =

(10)

where 2 21 1

2 2 4 *1 1 3 1 1 1

2 21 1

2 *1 1 1 1 1 1 1

1

4 *1 1 1 2 1 1 1

1

2 *1 1 1 1 1 1 1

1

,

/ ,

,

1(2 / ),

2

1(2 / ),

2

1(2 / ),

2

h h

a a a a a

p p

h h h h h hh

a a a a a aa

p p p p p pp

B A G m

B H G m

B A G m

B P G m

= = = = = =

1 1 1 10

1 1 1 10

1 1 1 10

( ) ( )d ,

( ) ( )d ,

( ) ( )d ,

L

h h

L

a a

L

p p

G q x q x x

G r x r x x

G s x s x x

= = =

10

1

10

1

10

1

1( , ) ( )d ,

1( , ) ( )d ,

1( , ) ( )d ,

L

hh

L

a ba

L

p bp

Q x t q x xm

Q M x t r x xm

Q D x t s x xm

= = =

where ( , ),iL x t ( , )jM x t and ( , )kD x t represent aerody-

namic lift, aerodynamic moment and aerodynamic drag re-

spectively, and the subscript b denotes buffeting force.

him , ajm and pkm are generalized mass. hi , aj and pk are damping ratios. hi , aj and pk are the natural

frequencies. We carry out a series of mathematical transformations

(including Fourier transformation) for the left items of the first formula of eq. (10), and finally the total response spec- tral density of h can be written:

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

21 1 1

2 22 21 1 1

, ,

,

hh h h

h h L

S x q x S x

UB H J S q x

== (11)

where 1

2| ( ) |hH is aerodynamic transfer function, ( ) 21| |hJ is called joint acceptance function, ( )LS is

the lift spectrum, q1(x) is the first order vibration mode function of h direction, x is the longitudinal coordinates of train, is air density, U is wind speed, and B is the width of train. Similarly, the total response spectral density of a and p can be obtained.

After the functions of response spectral density have been taken, the response variance can be obtained:

( ) ( )( ) ( ) ( )

( ) ( )

2

0

22 2

1 10

2

1

, d

d ,

h hh

h

h L

x S x

UB q x H

J S

=

=

(12a)

( ) ( )( ) ( ) ( )

( ) ( )

2

0

222 2

1 10

2

1

, d

d ,

a aa

a

a M

x S x

UB r x H

J S

=

=

(12b)

( ) ( )( ) ( ) ( )

( ) ( )

2

0

22 2

1 10

2

1

, d

d ,

p pp

p

p D

x S x

UB s x H

J S

=

=

(12c)

where bogie,wind is response standard deviation. Figures 1 and 2 are the curves of pw-v under various

wind speeds. Under the same train speed, the bigger the wind speed is, the bigger pw is. 1.2.1 The condition of train derailment

The analysis of refs. [5,14] shows it is the mechanics condi-tion of derailment commencement that the work c done by limit resistant force is equal to the maximum input energy p.max of transverse vibration of the system. It is difficult to compute the maximum input energy p.max of transverse vibration of the system, so the mechanics condition cant

CHEN RuiLin, et al. Sci China Phys Mech Astron April (2010) Vol. 53 No. 4 755

Figure 1 The relationship curves between p, the standard deviation of the hunting wave of the bogie frame of the passenger train, and v, the train speed.

Figure 2 The relationship curves between p, the standard deviation of the hunting wave of the bogie frame of the wagon train, and v, the train speed.

become applicable evaluation. However, when wind load is considered, the mechanics condition turns into the first evaluation criterion to judge train derailment. The reasons follow. Figures 1 and 2 are similar in shape with Figures 5-7 and 5-8 in ref. [5], but the curves of p-v in Figures 5-7 and 5-8 are from the test and stand for no derailment. The curves in Figures 1 and 2 are from the calculation described in the former part, except that the bottom curve is from Fig-ures 5-7 and 5-8 in ref. [5]. Therefore, we take the standard deviation pw simulate input of transverse vibration, which can exceed the work c done by limit resistant force of the system. Then the first evaluation criterion to judge train derailment under wind action can be drawn:

,c P < (13) where pw substitutes p, in order to be consistent with the analysis theory of random energy of train derailment. Thus

we expanded the connotation of p. p can denote p of the train-track system with wind, as well as p of the train-track system without wind, with a detailed explanation in context. pw only indicates p of the train-track system with wind. We can get p at various wind speeds and train speed by looking into figures 1 and 2.

1.2.2 Determination of the input energy p of the trans-verse vibration of the train-track (bridge) system

Refs. [5,8,9,14] discuss how to determine the input energy p and the maximum input energy of transverse vibration of the train-track (bridge) system, which are discussed in the unclear transverse vibration. In fact, even if the known fac-tor is also may contained, because p is from the measured hunting wave of the bogie frame and contains the known and the unknown factors. In view of the forecited discussion, Figures 1 and 2 include the situation which a train does not derail, and also include the situation where a train derails. Wind load is the actual action in the train-track (bridge) system. If we can test the hunting wave of the bogie frame of the system in strong wind, it will be different from the measured hunting wave of the bogie frame of the train-track (bridge) system. Thus the hunting wave of the bogie frame of the system in strong wind may be treated as the exciting source of the transverse vibration of the system.

1.2.3 The work c done by limit resistant force of the train-track (bridge) system

Refs. [5,8,9,14] proposed the geometry criterion of wheel derailment as well as applied the hunting wave of bogie frame and trail calculation to work out the work c done by limit resistant force of train-track (bridge) system. For a certain type of train and track line (bridge), we can calculate the work c at a given train speed. That is to say, c is the function of train speed. In fact, the curve of c-v is the in-trinsic attribute of the train-track (bridge) system, which wont change so long as the vehicle type, the line (bridge) and the grouping method do not change.

1.2.4 The energy increment criterion of train derailment under wind action

The evaluation criterion to judge train derailment can be drawn according to deductive process in literatures [5,8,9,14]:

,cr pr < (14) where the increment is on the basis of the work c0 done by limit resistant force of train-track(bridge) system and the maximum input energy p0.max of transverse vibration of train-track(bridge) system at a train speed v0. Therefore the first evaluation criterion to judge train derailment under wind action should be met at a train speed v0. Eq. (14) can be regarded as the second evaluation criterion to judge train derailment under wind action.

756 CHEN RuiLin, et al. Sci China Phys Mech Astron April (2010) Vol. 53 No. 4

2 Analysis on derailment case in wind

Case 1 (Figure 3) At 2:05 on February 28, 2007, a pas-senger train numbered 5807 from Urumqi to Aqsu travelled at 42 kilometers +300 meters between Zhenzhu Spring and Red Shan Quan on the south Xinjiang line and the 9th to the 19th compartments are derailed because of the gale. The instantaneous wind speed achieves 13-level (37.041.4 m/s). Train speed is 66.6 km/h.

We Calculate the work done by the resisting force of the system when the train derails at 56.6 km/h and 66.6 km/h, and obtain c56.6=115 cm/s2 and c66.6=127 cm/s2, respec-tively. Through actual tests and according to figure 1, the standard deviations of the hunting wave of the bogie frame of the empty wagon at speeds 56.6 km/h and 66.6 km/h are p56.6=220 cm/s2 and p66.6=268 cm/s2, respectively.

Under the wind speed, we have c56.6< p56.6, c66.6< p66.6.

Therefore, according to the first evaluation criterion to judge train derailment, we may know whether at speed 56.6 km/h or at 66.6 km/h, so long as train encounters 13-level of wind, the transverse vibration of the time-variant system is unstable, and the train shall derail. The calculation coincides

Figure 3 Derailment of the passenger train numbered 58071).

Figure 4 Derailment of a freight train in the county Griggs [20].

with the derailment accident. Case 2 (Figure 4) At 19:25 of American central time

zone on August 9, 2006, in county Griggs, more than 1800 meters long freight train is blown down on the Luverne bridge by the wind. The Luverne bridge is 49 meters high. The instantaneous maximum wind speed achieves 37.9 m/s approximately. On that evening between 7:15 and 7:25, the gust wind speed was between 26.8 m/s and 37.9 m/s. Train speed was 60 km/h.

Because the work done by the limit resisting force of the time-variant system cant be obtained temporarily by calcu-lation, we may take the similar type of bridge according to the front analysis, for instance Yinghe bridge (Figure 5).

Thus the work done by the resisting force of the system when the train derails at 50 km/h and 60 km/h, can be ob-tained c50=120 cm/s2 (from extrapolation) and c60=160 cm/s2, respectively. According to figure 2, the standard deviations of the hunting wave of the bogie frame of the empty wagon at train speeds 50 km/h and 60 km/h and at a wind speed 26.8 m/s are p50=115 cm/s2 and p60=150 cm/s2, respectively.

Under the wind speed 26.8 m/s, we have c50> p50.

Therefore, according to the first evaluation criterion to judge train derailment, we may know when the freight train is at the speed of 50 km/h and 60 km/h and at a wind speed of 26.8m/s, the transverse vibration of the time-variant sys-tem is stable, and the train shall not derail.

Under the wind speed of 26.8 m/s, we have c50=c60c50=40 cm/s2,

p50=p60p50=35 cm/s2. Therefore, c50> p50. According to the second evalua-tion criterion to judge train derailment, we may know when the freight train is at the speed 60 km/h and at a wind speed 26.8 m/s, the transverse vibration of the time-variant system

Figure 5 The relationship curve between c(p) and v of Yinghe bridge [14].

1) From Cusphoto http://www.chinanews.com.cn

CHEN RuiLin, et al. Sci China Phys Mech Astron April (2010) Vol. 53 No. 4 757

is stable, and the train shall not derail. In other words, when the wind speed is 26.8 m/s, whether the vehicle speed is 50 km/h or 60 km/h, the train shall not derail.

According to Figure 2, the standard deviations of the hunting wave of the bogie frame of the empty wagon at train speeds of 50 km/h and 60 km/h and at a wind speed of 37.9 m/s are p50=170 cm/s2 and p60=221 cm/s2, respec-tively.

Under the wind speed 37.9 m/s, we have c50< p50, c60< p60.

Therefore, according to the first evaluation criterion to judge train derailment, we may know whether at a speed of 50 km/h or 60 km/h, so long as the train encounters a wind speed of 37.9 m/s of gale, the transverse vibration of the time-variant system is unstable, and the train shall derail. The calculation coincides with the freight train derailment accident in the county Griggs.

3 Conclusions

Based on the analysis theory of random energy of train de-railment, the analysis theory of random energy of train de-railment in wind is suggested in this paper. Thus, the analy-sis theory of random energy of train derailment is extended to all weather conditions, i.e. common atmospheric wind field. The main contents of and contributions to train de-railment of the paper follow.

(1) Two kinds of methods are put forward, namely, time-domain method and frequency-domain method for the theory of random energy of train derailment, involving wind loads;

(2) With calculated results from former studies, curve p-v is extended to different wind speeds, which makes up the all-weather theory of random energy of train derailment;

(3) Train derailment condition involving wind is set up in this paper;

(4) First and second discriminative rules of train derail-ment in wind are proposed in this paper;

(5) Two cases of train derailment in strong wind, which occur frequently in China and other countries of the world, are analyzed in this paper. All the calculated results are consistent with the actual status, so the feasibility of the theory of energy random analysis involving strong wind for train derailment is verified.

(6) The key to success in analyzing the theory of random energy of train derailment lies in linking the input energy p of transverse vibration or the work c0 done by limit resis-tant force of train-track(bridge) system to the controlled train speed. In accordance with the view, we link the input energy pw of transverse vibration of train-track (bridge)- wind system to train speed. the establishment of the analysis theory of random energy of train derailment in wind is

likely to draw an all-weather speed limit map for a train or any high-speed train.

This work is Supported by the National Basic Research Program of China (Grant No. 2007CB714706), the National Natural Science Foundation of China(Grant Nos. 50078006 and 50678176) and Dr start-up fund research of Xiangtan university (09QDZ14)

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