a study on mechanical mechanism of train derailment and preventive measures for derailment

7/27/2019 A Study on Mechanical Mechanism of Train Derailment and Preventive Measures for Derailment

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A study on mechanical mechanismof train derailment and preventivemeasures for derailmentX Jun & Z Qingyuana Civil and Architectural Institute, Central South University,Changsha, Hunan, 410075, People's Republic of ChinaVersion of record first published: 06 Aug 2006.

To cite this article: X Jun & Z Qingyuan (2005): A study on mechanical mechanism of trainderailment and preventive measures for derailment, Vehicle System Dynamics: International

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A study on mechanical mechanism of train derailment andpreventive measures for derailment

X. JUN* and Z. QINGYUAN

Civil and Architectural Institute, Central South University, Changsha, Hunan 410075, PeoplesRepublic of China

The research status of train derailment is summarized. Major problems existing in currentderailment research are pointed out. By applying system dynamics stability concepts, themechanical mechanism of train derailment is described. The theory of random energy analysisfor train derailment is then further expounded and preventive measures for train derailmentand a calculation method for an anti-derailment safety coefficient (of train track time variantsystem) are introduced. Finally, some train derailment cases are analysed. Six train track timevariant system vibration cases are calculated, four of which derailed and two that did not. Theconclusion compares the results of the theoretical analysis with that which actually occurred.

1. Introduction

Since the inception of the railway in England in 1825, train derailment has been aproblem troubling railway scientists and technologists alike. Train derailment, whichoccurs frequently in China and other countries of the world, has become a centennialproblem. In the authors opinion, this is due to an insufficient understanding of themechanics of derailment, which means feasible precaution measures are not taken.

Since the launch of the speed-up campaign in the Chinese railways, the operation

speed of the freight trains has increased from 50 60km h7 1

up to 70 80 km h7 1

.Proportionately, more and more derailment accidents have taken place, particularly onstraight lines, constituting a grave threat to safety and a serious interference to thenormal order of railway transportation. As a result, the Ministry of Railways is underpressure to take measures such as speed limitation to secure the safety of trainoperation. However, speed limitation is in conict with the objectives specied in thetenth ve-year plan of railway development of science and technology, which clearlydenes that by 2005, rapid freight trains shall achieve a speed of 120 km h 7 1, a rapid

*Corresponding author. Email: [email protected]

Vehicle System DynamicsVol. 43, No. 2, February 2005, 121 147

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online 2005 Taylor & Francis Group Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/0042311041233132201

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freight transport network shall take shape; normal freight trains shall achieve90 km h 7 1 and track capacity will improve. Speed limits have acted as a bottleneckrestraining the railway transportation economy, severely hindering the tenth ve-yearplan, and longer term plans. Therefore, there is a pressing need to study the train

derailment mechanism and seek measures to prevent derailment of freight trains and toreduce, and ultimately even eradicate, derailment accidents.

2. Current status of studies on derailment

2.1 Study on critical derailment coefficient

In 1896, Nadal, a French engineer, put forward an equation to calculate the criticalderailment coefficient Q/P according to the relationship of normal force N andtangential friction T with transverse wheel-rail force Q and vertical wheel-rail force P ,as shown in gure 1 [1]. He then regarded the following equation as the basis of commencement of derailment:

QP

tga m

1 mtga1

where m is the dynamic friction coefficient of the wheel-rail contract and a is the angeangle [1]. Japanese scientists take standard wheel ange angle a = 60 8, frictioncoefficient m= 0.3, Q/P = 0.95 according to equation (1). The estimated safetycoefficient is 1.2, so the critical derailment coefficient (start of derailment) Q/P = 0.8.Building upon Nadals equation (1), scientists around the world have investigated thevalue of the critical derailment coefficient in terms of calculation and test (e.g. [2 5]).Japanese scholars use the single wheelset calculation model as shown in gure 2a, andvertical load W = 2 P 0, where P 0 is static wheel load. The wheel load is presumed staticand the waveform of the transverse force acting on single wheelset is as shown in gure2b. Without consideration of the rail action, the relationship between Q/P and wheellift value is as shown in gure 2c. We can see from gure 2c, when Q/P = 0.8, the wheellift value is very small, satisfying the conception of Nadals critical derailmentcoefficient and being very close to the calculated Q/P value from equation (1). InAmerica, scientists measure the vertical and transverse anti-action force of wheelset by

using track loading vehicle functioning on normal vertical force and gradually

Figure 1. Wheel-rail action at the start of derailment.

122 X. Jun and Z. Qingyuan

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increasing the transverse force on bogie. The critical derailment coefficient is measuredby tests on the wheel lift of the single wheelset. Additionally, taking the load to be thesame as the test load, the wheel critical derailment coefficient Q/P of the single wheelset

is calculated with the NUCARS software. The calculated result is close to the testedresult, Q/P 1.0 (new rail) and 1.4 (old rail) as shown in gures 3a and 3b. The fact thatQ/P of the new rail is different from that of the old rail presents the effect of the frictioncoefficient. The friction coefficient of the new rail is large Q/P is lower while thefriction coefficient of the old rail is small Q/P is higher.

Figure 2a. Single wheelset model for calculation of derailment in Japan. Figure 2b. Curve of lateral force onsingle wheelset supposed in Japan. Figure 2c. Calculation relationship of derailment coefficient and wheel liftvalue.

Train derailment and preventive measures 123

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2.2 Research on wheel load reduction rate D P/P 0

As shown in gure 4, the wheel load at the wheel-rail contact point is reduced due tothe action of transverse force F and the moment M . The reduction of wheel loadD P = P 0 P d over the static wheel load P 0 is the wheel load reduction rate D P /P 0, andP d is the measured wheel load value. According to physical concept, the bigger the D P /

Figure 3a. The relationship between the tested and calculated derailment coefficient Q/P with new rail and theimpact angle f w. Figure 3b. The relationship between the tested and calculated derailment coefficient Q/Pwith old rail and the impact angle f w.


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P 0 , the more likely the chance of derailment. Japanese scientists have conrmed thestandard value of static wheel load reduction rate is 0.6 according to the calculations of single wheelset derailment and the results of derailment tests. The explanation given byJapanese scientists for static wheel load reduction is that on a transition curve, theirregularity of rail face, the twist action of the vehicle, the unbalanced load of passengers and goods, and excessive cant curve will lead to a mitigatory change of thewheel load, i.e. the change as a result of the static load. Experience shows that thestandard value of dynamic wheel load reduction rate is 0.8.

2.3 Specication standards for derailment prevention in various countries

2.3.1 Japan. Q/P 0.8, continuous action time of Q/P is less than 0.015 s, which isdetermined by the relationship between tc (the continuous action time when the carmodel passes a curve, wheelset has a certain impact angle and horizontal action , Q/P5 0.8 ) and wheel lift value, as shown in gure 5. D P /P 0 = 0.6 (static), 0.8 (dynamic).

2.3.2 Western Europe. Q/P 5 0.8 (Q/P average movement distance 2m).

2.3.3 North America. Q/P 5 1.0, D P /P 0 5 0.9.

2.3.4 China. Q/P = 1.0 (allowable limit), 1.2 (danger limit) and D P /P 0 = 0.60(allowable limit), 0.65 (danger limit).

2.4 Geometric rules of derailment

The China Academy of Railway Sciences (CARS) has carried out a full-scale wheelderailment course simulation test on a single wheelset rolling rig and measured thecurve of derailment course of wheel ange [6]. The test results show that when thewheel lift value mmax (t) = 25mm, and the relative transverse displacement of thewheelset ymax (t) = 54 mm, a derailment occurs.

In Japan mmax (t)=30 mm and ymax (t) = 70 mm determined by wheel and rail top

prole.According to the mmax (t) and ymax (t) and ange and rail top prole in China, thefollowing derailment geometric rules can be obtained [7]: when wheel lift value m(t)= mmax (t) = 25 mm, and the transverse displacement of wheel to the rail y(t) = ymax (t)= 54 mm, a true derailment will occur.

Figure 4. Lateral force F and torque M on wheelset.


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2.5 Principle problems in criteria of evaluation for train derailment

Specied derailment coefficient Q/P and wheel load reduction rate D P /P 0 cannotcontrol the maximum Q/P and D P /P 0 that can possibly occur within the specied limit.Both the actual measurements and calculations prove that the values of actualmaximum Q/P and D P /P 0 are much larger than the specied limit value [7 11]. Thereason for this is both that the existing criteria are made according to the experimentsand calculations of a single wheelset under the assumed lateral force, and the criterialack statistical regularity. It is very difficult to measure and calculate the actual Q/Pand D P /P 0 when the wheels start oating, and it will be even more difficult to obtainstatistical regularity. Hence, the current criteria cannot assure that derailment will notoccur. However if the possible maximum Q/P and D P /P 0 is within the specied limit,derailment will not happen.

3. Mechanical mechanism of train derailment

3.1 Train derailment is the result of losing stability from transverse vibration status of train track system

If wheel anges move in a snake-like way between two rails from beginning to end,

the wheel anges will be laterally restricted by the rails and move in the wheel-railclearance in the lateral direction, and the anges will not climb to the rail top, thewheels will not drop off the rails and the train will not derail. Such snakingmovement status of the wheel anges is stable, and for the train track system, thismeans that the transverse vibration of the system is stable. However, once the wheelanges climb to the rail top, the transverse movement of wheels will be relieved fromthe restriction of the rail and the wheels will run out of the track and derail. As aconsequence of the transverse movement status of the wheels between rails losingstability, the transverse vibration status of the system loses stability. For example, themovement status of bicycle wheels running vertically is stable, but once the wheels

incline, movement status of the wheel on vertical plane loses stability. From conceptsof physics, the stable system status withstands interference whereas the unstable onecannot. The stable system status can generate resistant force increment uponinterference, which will exceed or balance the load increment generated byinterference to maintain the former system status. When the unstable system status

Figure 5. Relationship between Q/P continuous action time tc and wheel lift value Z .


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is interfered with, the resistant force increment is smaller than the load increment andthe former system status will be destroyed. Interference exists and is inevitable. Inorder to protect the system status the system must have stability instinct. Hence themechanical mechanism of train derailment is as follows: derailment is the result of

transverse vibration status of the system losing stability; the transverse vibrationstatus of the system must be stable in order to ensure that the train will not derail.The concept of the stability of system movement status comes from the simplest

movement, the static state stability (which can be considered as movement of inniteperiod).

Figure 6. Drawing of stability analysis of balanced status of a wooden board in water.

Figure 7. Drawing of stability analysis of balanced status of a pressed bar.


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The balanced status of the board, as shown in gure 6a, is stable since its resistantforce increment after interference is Fe and the increment of load is zero, as shown ingure 6b. The balanced status of the board as shown in gure 6c is unstable as itsresistant force increment of the board after interference is zero, and the increment of

load is Fe , as shown in gure 6d.Supposing the rigidity constant of the spring is b, gure 7b shows that upon

interference, the top point B of the pressed bar is deviated al laterally, the pressure P isdeviated al , and a torque P al is generated at point A. This is the load incrementgenerated when the balanced status of the pressed bar is interfered with. At the sametime, the torque ba l 2 of point A from the spring tension ba l is its resistant forceincrement.

If

ba l 24 P al 2

the resistant force increment is bigger than the load increment. When the interferencedisappears, the spring will draw the bar to its original balanced status as shown ingure 7a. Therefore, the balanced status of the bar in gure 7a is stable.

If

ba l 25 P al 3

the resistant force increment is less than the load increment, the pressed bar will incline.Therefore, the balance status of the bar in gure 7a is not stable.

According to the dAlembert principle, when a dynamic issue of a system changesinto a dynamic balance issue, the inertial force, resistant force, elastic force andinterfered force of the system at any moment are balanced. Thus, the dynamic status of the system can be regarded as balanced. The stability of the system at dynamic status isits stability at balanced status (viewing from the force it received). The above conceptof static balanced status stability is also applicable for judgment of dynamic balancedstatus stability. Take gure 8 as an example: An aeroplane is ying horizontally on aA A straight line. If the aeroplane is disturbed (as impacted by up-going turbulence),it will y along track B moving up a bit at the beginning, and gradually returning to itsoriginal horizontal line. In this situation, the dynamic status of the aeroplane yingalong the straight line horizontally is stable. When disturbed, the aeroplane will yalong track C, deviating from the straight line, meaning its ying status along thestraight line is not stable.

3.2 A standard rule judging the balanced status stability of a system

From the above ideas, we can have:(1) Standard rules for resistant force increment and load increment of a systemResistant force increment 4 load increment stable balanced statusResistant force increment 5 load increment unstable balanced status

Figure 8. Drawing of stability analysis of balanced status of an aeroplane ying horizontally on a A Astraight line.


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Resistant force increment = load increment balanced status is at the borderbetween the stable and unstablestatus, i.e. critical stability lossstatus.

(2) Criterion for the resistant force working increment and input energy increment of the system

When a/2 is multiplied by equations (2) and (3), we obtain:

1=2b al 24 1=2 Pl a2 4

1=2b al 25 1=2 Pl a2: 5

Obviously, 1/2 b( al) 2 is the strain energy increment after the system is disturbed.1/2 Pl a2 is the working increment of the pressure P . As shown in gure 7a, whenthe top point B of the pressed bar deviates al , the vertical displacement of point Bis l (1 7 cosa) & 1/2 l a2. Additionally, the strain energy of the system is equal tothe work done by internal force of the system. Thus, 1/2 b( al) 2 is also called theworking increment of the system resistant force. When the work done by externalforce is turned into strain energy of the system, 1/2 Pl a2 is the increment of thesystem input energy. Therefore, according to equations (4) and (5), criterion for theresistant force work increment and input energy increment is:

Resistant force work increment 4 input energy increment stable balanced status

6

Resistant force work increment 5 input energy increment unstable balanced status

7

Resistant force work increment input energy incrementthe system is at critical stability loss status

8

4. Introduction to the theory of random energy analysis for train derailment

The theory of random energy analysis for train derailment was proposed in [10]and [11] but it was not expounded enough. This theory is now furtherexpanded.

From the mechanical mechanism of train derailment, the key to calculatingwhether a train derails lies in the calculation of the resistant force work incrementand input energy increment of the train track system. The following work shouldbe done:

(1) Set up a spatial vibration equation set of the system, which can reect the actualcontact status between wheel and rail.

(2) Determine the transverse vibration exciting source of the system.

(3) Develop the calculation method of random transverse vibration of the system.(4) Formulate the criteria of energy increment for judging whether a train derails ornot.

The above four items make up the theory of random energy analysis of trainderailment, which will be detailed in the following sections.


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4.1 Establishment of vibration equation set of the train track system

We regard the train and track as an integral system, and take into account theclearance between wheel and rail and the wheel-rail displacement connecting

conditions, that is, the wheel displacement (transverse, vertical) is equal to raildisplacement (transverse, vertical) plus rail irregularity (transverse, vertical) plus wheel-rail relative displacement (transverse, vertical). According to the principle of totalpotential energy with the stationary value in elastic system dynamics and the rule of set-in-right-position for formulating system matrices [12,13], the spatial vibrationequation set of the system can be established.

4.2 Determining the exciting source of transverse vibration of the train track system

When wind load is not considered, the transverse vibration equation set of the system isas follows:

M fdg C > f _dg K fdg 0 9

where [ M ], [C ] and [ K ] are the mass, damping and stiffness matrix of transversevibration of the system, respectively, and f dg, f _dg and ( d) are the acceleration,velocity, and displacement vector of the system, respectively. If k vibrationresponses are known and, n responses unknown, then through equation (9), weobtain:

M kk M knM nk M nn !

dkdn& ' C kk C knC nk C nn !

_dk_dn& ' K kk K knK nk K nn !dkdn& ' 0: 10

To expand equation (10), we have:

M nn fdng C nn _dnn o K nn dnf g M nk dkn o C nk _dkn o K nk dkf g 11M kk fdkg C kk _dk

n o K kk dkf g M kn dn

n o C kn _dn

n o K kn dnf g 0: 12

Equation (12) is a non-independent equation set which is to be crossed out. Allthe items on the right side of equation (11) are known. With this, n unknowntransverse vibration responses can be obtained. Thus, k known vibrationresponses becomes the exciting source of the transverse vibration of the system.The items on the right side of equation (11) become the equivalent self-excitingforce causing transverse vibration of the system and the items on the left sideof equation (11) become the resistant force of the transverse vibration of thesystem.

Which displacement parameter should be set as a known number? The most directapproach is to measure out the transverse vibration displacement oscillogram of thewheelset with no distribution of sensors on it.

In 1984, Professor Zhentao P from the Shanghai Railway AdministrationBureau rst tested the transverse vibration acceleration oscillogram of the


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midpoint in longitudinal position of bogie frames of passenger and freight cars (weusually call this a hunting wave). We take the actually tested hunting waves of thecar bogie frames as the exciting source for conrmative analysis of transversevibration of the system. The calculated responses are properly close to the actually

tested responses [10,11]. Therefore, we come to realize that the hunting wave of the car bogie frame can be treated as the exciting source of transverse vibration of the system.

The following points should also be mentioned: (1) the measured huntingwaves of car bogie frames accurately reect the inuence of all factors which giverise to transverse vibration of the system. This is impracticable when thetransverse track irregularity is taken as the exciting source of vibration. Beingdeterministic itself, the measured hunting waves of car bogie frames can only beused in deterministic analysis of the transverse vibration of the system. (2) Thetransverse vibration of the system is caused by many factors and all these factorsare at random. So random analysis for transverse vibration of the system mustbe made.

4.3 Random energy analysis method for transverse vibration of the train track system

According to the principle of energy conservation and conversion, the input energy of the train track system generates a vibration response; the greater the input energy, thegreater the vibration response; the volume of the input energy corresponds to the sizeof the vibration response. In this way, the randomness of the response of the systemcan be regarded as that of its input energy. Then a random analysis of the multi-factorsystem response can be conversed to a random analysis of the single-factor inputenergy.

Nigeme [14] says if random variant x(t) is the displacement of a spring, x2(t) will bedirectly proportional to the deformation energy of the spring; if x(t) is the speed of a

Figure 9. Relationship curve between s p , the standard deviation of the hunting acceleration wave of the coachbogie frame, and v, the train speed (based on calculation and statistics of the measured data of a railinspection car on the Guangzhou Shenzhen line and the measured data of the transverse wheelset force of the German high-speed locomotives) [12,13].


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rigid body, x 2(t) will be proportional to the kinetic energy of the rigid body. Sinkoson[15] also says the mean-square deviation value is the mean work done in unit time. Itis known from these theories that the mean-square deviation of the measured huntingwave of the car bogie frame is the mean work done in unit time, which is the energyinput to the transverse vibration of the system. The root value of the mean-squaredeviation is the standard deviation s p. The following random simulation of x(t), thearticial hunting wave of the car bogie frame, uses only s p. For the purpose of

simplication, s p expresses the energy of transverse vibration input to the system. It isnecessary to start with random analysis of s p before we carry out random analysis of the transverse vibration of the system. Therefore, from a number of hunting waves of bogie frames of various cars under different speeds on mainlines, we take anoscillogram per kilometer as a sample and make statistics on the oscillograms of

Figure 10. Relationship curve between s p , the standard deviation of the hunting acceleration wave of thebogie frame of the loaded wagon, and v, the train speed (based on calculation and statistics of the datameasured on the Changsha Guiyang railway line) (Qingyuan and Xiangrong 1999, Qingyuan 2000).

Figure 11. Relationship between s p , the standard deviation of the hunting acceleration wave of the bogieframe of the empty wagon, and v, the train speed (based on calculation and statistics of the data measured on

the Beijing Tonghua railway line) [16].


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Figure 12. Measured hunting wave of bogie frame of locomotive on straight section with good trackcondition.

Figure 13. Measured hunting wave of bogie frame of empty wagon on straight section with good trackcondition.

Figure 14. Measured hunting wave of bogie frame of loaded wagon on straight section with good trackcondition.

Figure 15. Measured hunting wave of bogie frame of locomotive on straight section with poor trackcondition.

Figure 16. Measured hunting wave of bogie frame of empty wagon on straight section with poor trackcondition.


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tangent lines and curve lines respectively. According to the engineering probabilityanalysis method [12], we obtain relationship curves between the standard deviation s pof 99% probability and the train speed v as shown in gures 9 11. Further, by theMonte-Carlo random simulation method, we obtain the hunting wave of the car bogieframe with 99% probability, which follows the articial earthquake wave and is calledthe articial hunting wave of the car bogie frame.

The base frequency of the articial hunting wave of the car bogie frame is takenfrom the measured data at random. Each measured base frequency is taken to

simulate an articial hunting wave of the car bogie frame. By taking it as anexciting source, a sample of each response of the system is calculated. A number of samples can be calculated in this way. Finally, according to the engineeringprobability analysis method [12], the transverse vibration responses of the systemwith required probability are obtained. Such a calculation requires tremendousworkload. Calculation has proved that the measured base frequency of the huntingwave of the car bogie frame has quite an effect on the transverse vibrationresponses of the system [12]. Through trial calculation, we can obtain the articialhunting wave of the car bogie frame that could generate the maximum responses.With this wave, we can obtain the maximum transverse vibration responses of the

system in one calculation. The maximum responses of bridge calculated with thisrandom analysis method are properly close to the maximum responses measuredmany times [12,13].

The standard deviation of the hunting wave of the car bogie frame s p is used toreect the input energy of the system. This is supported by the measured results. In July

Table 1. Comparison of standard deviation of hunting wave of bogie frame of locomotive, empty wagon andloaded wagon on straight line sections with good or poor track status.

Sections in good track condition Sections in poor track condition

K252+100 * K253 + 550 K433 + 234 * K434+ 684Number Maximum/(mm) Number Maximum/(mm)

Prole 16 12 24 17Gauge 28 15 12 27Alignment 6 11 11 27Cross level 7 10 25 14Twist 4 11 10 12Locomotive s p/(cm s

7 2) 28 39Empty wagon s p/(cm s

7 2) 49 91Loaded wagon s p/(cm s

7 2) 41 68

Figure 17. Measured hunting wave of bogie frame of loaded wagon on straight section with poor trackcondition.


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2001, in cooperation with Shenyang Railway Administration, we carried out actualtests on the hunting wave of the bogie frame (gure 12 to gure 17) of a 60 km h 7 1

locomotive, an empty wagon and a loaded wagon on track sections in good and poorsmoothness on the Beijing-Tongliao railway line. Table 1 shows their standard

deviations [16].Figures 12 17 and table 1 show:

(1) When the track is in good condition, s p is small and the hunting wave hump of the car bogie frame is small. This means the input energy is small and so is thesystem vibration response.

(2) When the track is in poor condition, s p is big, the hunting wave hump is big.This means the input energy is big and so is the system vibration response.

(3) By improving the track condition, the standard deviation of the hunting waveof the car bogie frame can be lowered. This is benecial for derailmentprevention.

Train derailment is the result of drastic transverse vibration of the system. Since thetransverse vibration of the system is strongly random, so is train derailment.Furthermore, the fact of derailment also proves this. So the above energy analysismethod of transverse vibration of the system is also applicable to derailment randomanalysis. But in random analysis of train derailment, the calculated standard deviationof the articial hunting wave of the car bogie frame when wheel ange crawls on themid-point of the rail top, is the standard deviation of the hunting wave of the car bogieframe when the train derails, which is recorded as s e. The calculation is carried out bytrial calculation according to derailment geometrical rules mentioned above, since thehunting wave of the car bogie frame is very difficult to measure when the train derails[7].

4.4 Criteria of energy increment for judging train derailment

The above-mentioned equation (11) has shown the equivalent self-exciting force andthe resistant force of the transverse vibration of the system, but it cannot describe theirincrement. So the above-mentioned criteria of resistant force and load incrementcannot be used for judging the derailment of wheels. Only the criteria of the resistant

force work increment and input energy increment of the system (hereinafter refered toas energy increment criteria) shown in equations (6), (7) and (8), can be used for judging train derailment.

Equation (11) has conrmed that the measured hunting wave of the car bogie frameis the exciting source of the conrmative analysis of the transverse vibration of thesystem. Equation (11) and the random energy analysis method for transverse vibrationof the system has conrmed that the articial hunting wave of the car bogie frame is theexciting source of the random analysis of the transverse vibration of the system. Thestandard deviation s p of the articial hunting wave of the car bogie frame is the inputenergy of random transverse vibration of the system. It is the function of the train

speed v as shown in gures 9 11. When v changes, the input energy increment isD

s p.So we can calculate the input energy increment of transverse vibration of the system asfollows. Let us assume that the values of corresponding input energies of the system of two adjacent speeds v0 and vr are s po and s pr respectively, and we get the input energyincrement at the speed of vr is D s pr = s pr 7 pv. In many derailment cases, we set


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vr7 v0 =10 km h7 1, that is, if vr =70 km h

7 1, we get v0 =60 km h7 1. All the

results conrm the reality [7,11,17]. There is one important concept that needs to beexplained. Figures 9 11 are based on the measured hunting wave of the car bogieframe with limited times. They show the relationship between s p and v when the train

has not derailed. And s p is not the input energy when the train derails. Because themeasured times are limited, we cannot get the data when the train derails. Imagine thatwe carry out many tests on train oscillation, enough to make derailment happen, so wecan get the hunting wave of the car bogie frame with a very large input energy when thetrain derails. And moreover we can get the maximum input energy s p, max and therelationship between s p,max and v. Therefore, the input energy increment of the systemat the speed of vr is D s p,max = s p,max 7 s po ,max . The difference of generating conditionsbetween s po and s po ,max and that between s pr and s pr ,max are the test times (the formeris based on statistics of tests with limited times and the latter is based on statistics of alarge number of tests). The other conditions are the same. The difference of test timeshas an effect on the input energy s

p, which lies in the difference between the statistics

probability of s po and s pr , and that between s po, max and s pr ,max . (s pr 7 s po ) and(s pr ,max 7 s po ,max ) reect the effect of speed change D v = vr7 v0 on the ( s pr 7 s po ) and(s pr ,max 7 s po ,max ). According to the physical concept, D v should have the same effect onthem, so we get:

D s pr;max s pr;max s po;max s pr s po D s pr : 13

The standard deviation s c of the articial hunting wave of the car bogie frame whena train derails is the standard deviation s p that moves the wheel ange onto the railtop. If s p 5 s c, the articial hunting wave of the car bogie frame is impossible tomove the wheel ange on to the rail top. Since s p is the energy input to the transversevibration of the system, from equations (6), (7) and (8), s c is the work done by thesystem when it is resisting against the transverse vibration. This is similar to the caseof a pressed bar: when pressure P on the pressed bar is smaller than Euler criticalforce P cr , the stability status of the pressed bar will not be destroyed. The effect of s pand s c on the system is similar to that of P and P cr on the pressed bar. Since s c is theonly function of train speed v, if we subtract s co from s cr , we get the increment of work done by the resistant force of the system, that is, D s cr = s cr 7 s co . Fromequations (6), (7) and (8) we can get the criteria of energy increment to judge whethera train derails as below:

D s cr ! D s pr no derailment 14

D s cr 5D s pr derailment : 15

5. Preventive measures for derailment and calculation of the anti-derailment safetycoefficient K of the system

5.1 Prevention measures

(1) Lowering the transverse rigidity of the primary suspension can increase s c, thework done by the resistant force of the system (to be approved by calculation).

(2) improving the irregularity of the track (to prevent derailment on tracks).


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(3) improving the transverse and vertical rigidity of the bridge (to preventderailment on bridges).

5.2 Improvement parameter criteria

When the parameters are improved, the objective that trains operate safely andnormally (there is no wheel lift and the train has the stability specied in thespecications) can be realized.

5.3 Calculation of the safety coefficient K against derailment

Before the parameters are improved, when the train derails at speed vr , the minimumwork done by the resistant force of the system is s

cr, min, the maximum input energy is

s pr, max . When the train is at critical derailment status, s pr, max = s cr ,min . After theparameter is improved, s pr, max is lowered by l times so that the train runs normally.Then the maximum input energy under the normal train operation is:

s prs s pr;max ls pr;max s cr ;min ls cr ;min s cr ;min 1 l :

In addition, after the parameters are improved, the minimum work done by theresistant force of the system increases and becomes s crs . According to the criticalderailment status, after the parameters are improved the work done by the resistantforce s crs of the system should be equal to the maximum input energy when the trainruns under normal, safe conditions, and multiplied by the anti-derailment safetycoefficient K , i.e.

s crs K s prs K s cr ;min 1 l

So

K s crss prs

s crs

s cr ;min 1 l : 16

To prevent derailment on tracks, l in equation (16) may be determined by the s p,

obtained from statistics on the measured hunting wave of the car bogie frame on tracks

Figure 18. Time history curve of the wheel lift value of the derailment wheel.


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in good or poor irregularity; to prevent derailment on bridges, l in equation (16)may be determined by trial calculations. Let us suppose s prs , and simulate thearticial hunting wave of the car bogie frame, and calculate the vibration responsesof the train-bridge system and the Sperling comfort index W sp . When W sp = 3.0,s prs is the required s prs .

Figure 19. Time history curve of lateral force of the derailment wheelset.

Figure 20. Time history curve of derailment coefficient of the derailment wheelset.

Figure 21. Time history curve of the reduction rate of wheel load of the derailment wheel.


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6. Analysis on derailment case

6.1 Case 1

On July 8, 1997, train number 2422 with a full formation of empty wagons derailed atK730 + 56 in a straight section south of Xuzhou on the Tianjin Pukou line. The trainconsisted of 61 empty wagons with 8A bogies and ran at a speed of 70 km h 7 1 . Thetrack conditions of the section of the line conformed to the CARS (1997a)





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specications. It is required to calculate the possibility of derailment at 70 km h 7 1

under the above conditions.Calculate the work done by the resisting force of the system when the train derails at

60 km h 7 1 and 70 km h 7 1, and obtain s c60 =105 cm s7 2 and s c70 = 115 cm s

7 2 ,

respectively. Through actual tests and according to gure 11, the standard deviationsof the hunting wave of the bogie frame of the empty wagon at speed 60 km h 7 1 and70 km h 7 1 are s p60 = 68.5 cm s

7 2 and s p70 = 89.4 cm s7 2, respectively. Thus,

D s c =115 7 105=10 cm s7 2 , D s p = 89.4 7 68.5 = 20.9 cm s

7 2, so D s c5D s p . Ac-cording to equation (15), we can judge the freight train will derail at 70 km h 7 1 . Thecalculation coincides with the derailment accident. Figures 18 21 show the timehistory curves of calculation vibration response of derailment wheel.

6.2 Case 2

On August 6, 1997, train number 2344 with full formation of empty wagons derailed atK692 + 408 in a straight section south of Xuzhou on the Tianjin Pukou line. Thetrain consisted of 69 wagons with 8A bogies and ran at a speed of 62 km h 7 1. Thetrack conditions of the section of the line conformed to the CARS specications [8]. Itis required to calculate the possibility of derailment at 62 km h 7 1 under the aboveconditions.

Calculate the work done by the resisting force of the system when the train derails at52 km h 7 1 and 62 km h 7 1 , and obtain s c52 =98 cm s

7 2 and s c62 =105 cm s7 2 ,




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respectively. According to gure 11, the standard deviations of the hunting wave of thebogie frame of the empty wagon at speed 52 km h 7 1 and 62 km h 7 1

are s p52 = 52.5 cm s7 2 and s p62 = 69.5 cm s

7 2, respectively. Thus,D s c =105 7 98 = 7 cm s

7 2, D s p = 69.5 7 52.5 = 17 cm s7 2, so D s c 5 D s p. According





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to equation (15), we can judge the freight train will derail at 62 km h 7 1. Thecalculation coincides with the derailment accident. Figures 22 25 show the timehistory curves of calculation vibration response of derailment wheel.





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6.3 Case 3

On October 22, 1998, train number 2908 with full formation of empty wagons derailedat K711 + 220 in a straight section on the Beijing Kowloon line. The train consistedof 49 empty wagons with 8A bogies and ran at a speed of 73 km h 7 1. The trackconditions of the section of the line conformed to the CARS specications [8]. It isrequired to calculate the possibility of derailment at 73 km h 7 1 under the aboveconditions.

Calculate the work done by the resisting force of the system when the train derails at63 km h 7 1 and 73 km h 7 1 , and obtain s c63 = 114 cm s

7 2 and s c73 =130 cm s7 2 ,

respectively. According to gure 11, the standard deviations of the hunting wave of thebogie frame of the empty wagon at speed 63 km h 7 1 and 73 km h 7 1 ares p63 = 70.2 cm s

7 2 and s p73 = 99.4 cm s7 2, respectively. Thus, D s c =130

114 = 16 cm s 7 2 , D s p = 99.4 70.2 = 29.2 cm s7 2, so D s c 5 D s p. According to

equation (15), we can judge the freight train will derail at 73 km h 7 1 . The calculationcoincides with the derailment accident. Figures 26 29 show the time history curves of calculation vibration response of derailment wheel.

6.4 Case 4

On June 14, 1997, train number 2422 with a full formation of empty wagons derailed atK799 + 882 in a straight section south of Xuzhou on the Tianjin Pukou line. Thetrain consisted of 54 empty wagons with 8A bogies and ran at a speed of 71 km h 7 1 .The track conditions of the section of the line conformed to the CARS specications[8]. It is required to calculate the possibility of derailment at 71 km h 7 1 under theabove conditions.

Calculate the work done by the resisting force of the system when the train derails at61 km h 7 1 and 71 km h 7 1, and obtain s c61 = 103 cm s

7 2 and s c71 =113 cm s7 2 ,

respectively. According to gure 11, the standard deviations of the hunting wave of thebogie frame of the empty wagon at speed 61 km h 7 1 and 71 km h 7 1 are

s p61 = 66.5 cms

7 2

and s p71 = 92.6 cms

7 2

, respectively. Thus,D

s c =113 103 = 10 cm s 7 2 , D s p = 92.6 66.5 = 26.1 cm s7 2, so D s c 5 D s p. According to

equation (15), we can judge the freight train will derail at 71 km h 7 1 . The calculationcoincides with the derailment accident. Figures 30 33 show the time history curves of calculation vibration response of derailment wheel.



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6.5 Case 5

In order to further verify the judgment of the derailment by equations (14) and (15), aforecast was made to ascertain whether the freight trains on straight lines derail at

50 km h7 1

. We have calculated the work done by the derailment resisting force of thesystem at speed 40 km h 7 1 and 50 km h 7 1 and obtained s c40 =85 cm s

7 2 ands c50 =98 cm s

7 2, respectively.The standard deviations of the hunting wave of the bogieframe of the empty wagon at speed 40 km h 7 1 and 50 km h 7 1 are s p40 = 44.85 cm s

7 2

and s p50 = 51.45 cm s7 2 respectively according to gure 11. Then, D s c =98

85 = 13 cm s 7 2, CD s p = 51.45 44.85= 6.6 cm s7 2 , so D s c 4 D s p. According to

equation (14), we can judge that the freight trains will not derail at 50 km h 7 1 . On thebasis of investigation, we know that a number of freight train derailment accidents haveoccurred on major mainlines in China since the speed-up drive launched. Commonfeatures of derailment accidents appear to be: taking place on straight lines, on emptywagons, at the speed of 62 77 km h 7 1, etc. [8,9]. We can see that freight trains runningat 50 km h 7 1 have never derailed. This coincides with our forecast.

6.6 Case 6

Forecasts were made for the maximum responses in derailment site tests in order tofurther verify the theory of random energy analysis of train derailment. From May 9 toAugust 6, 1997, seven successive derailment accidents took place in the straight sectionbetween Shilibao and Taoshan south of Xuzhou on the Tianjin Pukou railway line. Inorder to nd out the causes of derailment on the line, CARS, the Jinan RailwayAdministration and the Xuzhou Railway Sub-Administration conducted derailmenttests from October 28 to November 11 in the section between Chuzhuangji andGaojiuing on the Tianjin Pukou railway line. No train derailed in the tests. Avibration response forecast was made for a train with full formation of empty wagonsat a speed of 80 km h 7 1 in view that empty wagons are of the maximum probability of derailment. Track conditions at the site of tests were: 60 kg m 7 1 rail, type 60reinforced concrete sleeper, crushed stone ballast, CWR track, long stretches of straight lines; small longitudinal slopes; being of good condition. The results of trackstatus inspection were: maximum prole 11.5 mm, maximum alignment 11 mm,maximum gauge error 7.5 mm, maximum cross level 6.5 mm, and maximum twist

8 mm [8].Calculate the work done by the resistant force of the system at 70 km h 7 1 and

80 km h 7 1 , and obtain s c70 =15 cm s7 2 and s c80 = 165 cm s

7 2, respectively.

Table 2. Comparison between the maximum calculated values and maximum measured values of the safetyindices against derailment of empty wagon.

Source of results

Item Calculated value Measured value [8]

Wheel lift (mm) 16 17Lateral wheelset force (kN) 50 56.11Derailment coefficient 5.9 4.98Wheel load reduction rate 1.0 1.0


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Table 3. Summary of derailment calculation for freight train running on straight line.

Case number Time Train number Derailment site Speed (km h 7 1) Formation D s c(cm s7 2)

1 July 8, 1997 2422 Jinpu lineK730+56

70 Full formation of empty wagon

10

2 Aug. 8, 1997 2344 Jinpu line

K692+ 408

62 Full formation of

empty wagon

7

3 Oct. 22, 1998 2908 Beijing Kowloon

K711+ 220

73 Mixed formationof loaded andempty wagon

16

4 June 14, 1997 2422 Jinpu lineK799+ 882

71 Mixed formationof loaded andempty wagon

10

5 Oct. 28, 1997 Test train No derailment 80 Full formation of empty wagon

50

6 Since speed-up No derailment 50 Full formation of empty wagon

13

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According to Figure 11, at speed 70 km h 7 1 and 80 km h 7 1, the standard deviationsof the hunting wave of the empty wagon bogie frame are s p70 = 89.4 cm s

7 2

and s p80 =126 cm s7 2, respectively. Then D s c =165 7 115=50 cm s

7 2 ,D s p =126 7 89.4 = 36.6 cm s

7 2, so D s c 4 D s p. This is in line with the criterion of

no derailment in equation (14). Thus, the train with full formation of empty wagonswill not derail at 80 km h 7 1. This coincides with the test result. According tos p80 =126 cm s

7 2 at 80 km h 7 1 as shown in gure 11, simulate the articial huntingwave of the empty wagon bogie frame at 80 km h 7 1 and then calculate the conditionsof the freight train formed with 29 empty wagons hauled by a locomotive running at80 km h 7 1 on the 500 m long, straight track. The calculation shows that the left wheelwith the rst axle of the third wagon behind the locomotive is of the maximum wheellift value, up to 16 mm. Comparison of the maximum calculated values and maximummeasured values of the wheel vibration response is shown in table 2. The calculatedresult is close to the measured result.

Table 3 shows the summary of derailment calculation for freight trains running onthe straight line, four of which are derailment cases and two are not. The calculationmatches with the practical case.

It can be seen from the six cases displayed in table 3 that the common features of train derailment are as follows:

(1) The derailment coefficient, wheel load reduction rate and transverse force of wheelset are very high when a train derails. These values are far beyond thespecied ones.

(2) The derailment time is very short, within 0.2 seconds or so.(3) Those derailed trains are mainly trains with full formation of empty wagons or

trains with mixed formation of loaded and empty wagons. Wagons derailed areall empty wagons.

7. Conclusions

(1) On the basis of the concept of the system dynamic stability and a thoroughunderstanding of the mechanical mechanism of train derailment, the loss of thetransverse movement stability of the train track time variant system isregarded as the cause of train derailment. The core of train derailment analysis

is to nd out the criteria for judging the stability of the system. According to therandom energy analysis theory of train derailment, calculations are made for sixvibration cases of train track time variant system, four of which are derailed,while two did not derail. The calculated results coincide with the practical case.This has proved that the energy increment criteria for judging train derailmentare correct and reliable.

(2) The preventive measures for train derailment and calculation method for theanti-derailment safety coefficient K are theoretically analysed for the purpose of ensuring the stability, normality and safety of train operation. They can serve asa reference for the departments of railway transportation and permanent way.

(3) The theory of train derailment analysis is a very complicated issue. So far aninitial survey has been made. In regard to putting the theory and methodsdescribed in this paper into practise, there is still a lot of work to be doneespecially in respect to tests on the hunting waves of bogie frames of locomotives and cars.


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Acknowledgements

The authors gratefully acknowledge that this research work has been supported by theNational Natural Science Foundation of China (No50078006) and Foundations of the

Science and Technology Section of the Railway Bureau in China (2001G029,2003G043) and Doctoral Research Foundation of the Education Ministry of China(20010533004).

References

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[3] Weinstock, H., 1984, Wheel climb derailment criteria for evaluation of rail vehicle safety. In Proceedingsof the ASME Winter Annual Meeting , 1 7.

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[5] Elkins, J. and Huimin, W. 2000, Angle of attack and distance-based criteria for ange climb derailment.Vehicle System Dynamics , 33, 293 305.

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[9] CARS, 1997b, Report on the Freight Train Derailment Test on the Datong Qinghuangdao Railway Line(Beijing: China Academy of Railway Sciences).

[10] Jun, X. and Qingyuan, Z., 2002, Simulation of the derailment courses of freight train on tangent track.Railway Journal , 24, 104 108.

[11] Jun, X., Qingyuan, Z. and Ping, L., 2003, Theory of random energy analysis for train derailment. Journal of Central South University Technology , 10 , 134139.

[12] Qingyuan, Z. and Xiangrong, G., 1999, Theory of Analysis of Vibration of the Train Bridge Time-Varying System and Its Application (Beijing: China Railway Press).

[13] Qingyuan, Z., 2000, The principle of the total potential energy with the stationary value in the elasticsystems dynamics. Academy Journal of Central China Polytechnic University , 28, 1 3.

[14] Nigeme, R.C. 1985, An Introduction to Random Vibration , translated by Chenhui, H. (Shanghai:Shanghai Communication University Press).

[15] Sinkoson, 1970, Analysis of Random Vibration , translated by Baoqi, C. (Beijing: Seismology Press).[16] Jun, X., Qinqyuan, Z. and Ping, L., 2001, Test Report on the Vibration of Freight Train on the Beijing

Tonghua Railway Line (Central South University).

[17] Jun, X., Qingyuan, Z. and Zhihui, Z., 2004, Mechanical mechanism and random energy analysis theoryof train derailment on the bridge and its application. Railway Journal , 26, 97 104.


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a study on mechanical mechanism of train derailment and preventive measures for derailment

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