dynamic interaction between a lumped mass vehicle ... - Science Direct

Computers & Yrucrures Vol. 63, No. 5, pp. 981497, 1991 1997 Ekvier Science Ltd. All ri@tts rcsmed Printed in Great Britain 004s7949/97 s17.00 + 0.00

0

Pergamon PII: s0045-7949p6po4o14

DYNAMIC INTERACTION BETWEEN A LUMPED VEHICLE AND A DISCRETELY SUPPORTED CONTINUOUS RAIL TRACK

MASS

W. Zhait and Z. Cai$$ tTr,tin and Track Research Institute, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China $Department of Civil Engineering, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5L0 (Received 31 March 1995) Abstract-This paper describes the formulation and application of a numerical model that simulates the vertical dynamic interaction between a train vehicle and the rail track. The considered vehicle model is supported on two double-axle bogies at each end and is described as a IO-degree-of-freedom lumped mass system comprising the vehicle body mass and its moment of inertia, the two bogie masses and their moments of inertia, and four wheelset unsprung masses. The bogie sideframe mass is linked with the wheel unsprung mass through the primary suspension springs and linked with the vehicle body mass through the secondary suspension springs. In the track model, the rail is treated as an infinitely long beam discretely supported at rail/sleeper junctions by a series of springs, dampers and masses representing the elasticity and damping effects of the rail pads, the ballast, and the subgrade, respectively. Shear springs and dampers are also introduced between the ballast masses to model the shear coupling effects in the ballast. The mass, stiffness and damping values of these track components and the sleeper spacings can be arbitrarily varied, so that vari,ations in track component properties and track geometric errors can be taken into account. The dynamic interaction between the wheelsets and the rail is accomplished by using the non-linear Hertzian theory. Example applications of the model are given. 0 1997 Elsevier Science Ltd.

I. INTRODU~ION

From a mechanical point of view, the train vehicle and the rail track: can be treated as two separate vibrating systems while the train is running on the track. However, they are dynamically coupled as an integral entity at the wheel/rail contacts where the common source of excitation to both systems takes place. Severe dynamic disturbances at the wheel/rail interface occur when geometric irregularities exist, either along the rail head surface-such as rail joints, imperfect rail welds and rail corrugations, or around the wheel circumference-such as wheel shells, flat spots, or a completely out-of-round wheel. The ultimate consequence of irregular wheel/rail interaction is the production of large wheel/rail impact forces, which are the principal cause of damage to the wheels, rails, the track foundation and other vehicle and track components. The formation and development of wheel/rail irregularities and the intensity of dynamic wheel/rail interaction are mutually responsible, each being the casualty of the other. The existence of a defective spot, either on the wheel circumference or on the rail 9 To whom correspondence should be addressed.

surface, gives rise to wheel impact force on the rail, which in turn, causes the existing defect to grow and may initiate other types of irregularities on the wheel or rail surfaces-resulting in a ‘vicious circle’. An adequate understanding of the mechanisms of these aspects of the wheel/rail and track interactive system is essential to formulate effective strategies to reduce or eliminate the adversities of the vehicle/track system. The dynamic interaction between the vehicle, wheelset, and the rail track systems has been the subject of extensive research by the world’s major railways and research institutions during the past 25 years. Since the early 7Os, numerous theoretical models for wheelset and track dynamic interaction have been developed and field experiments carried out by railway researchers and engineers in many countries. Representative studies include those in Australia [l-3], Britain [412], Canada [13-201, France [21,22], Germany [23-251, Japan [26-281, Sweden [29-311 and the United States [32-351. A state-of-art review on wheel/rail and track dynamics has been given by Knothe and Grassie (361. Recently, the increasing demand for high speed and heavy haul transportation in the Chinese railway 987

988

W. Zhai and Z. Cai

industry has accelerated the need for research into train and track dynamics in the Chinese academia [37-391. This paper describes a numerical vehicle and track interaction model used to investigate the dynamic effects at the wheel/rail interfaces on typical Chinese mainline tracks. 2. VEHICLE AND TRACK INTERACTION

MODEL

Figure 1 illustrates the various components of the interactive train vehicle and rail track system model. The considered vehicle is supported on two double-axle bogies at each end and is modelled as a lo-degree-of-freedom lumped mass system comprising the vehicle body mass and its moment of inertia (M,, J,), the two bogie masses and the associated moments of inertia (M,, J1), and four wheelset unsprung masses (M,). The bogie sideframe mass is linked with the wheel unsprung mass through the primary suspension springs (KS,, C,,) and linked with the vehicle body mass through the secondary suspension springs (KS*,C,,). The denotations of the symbols in Fig. 1 are given in Table 1. The vehicle is assumed to run in the horizontal direction at a constant speed (v). In the track model, the rail is described as an

infinitely long beam discretely supported at rail/ sleeper junctions by a series of springs, dampers and masses. The three layers of discrete springs and dampers represent the elasticity and damping effects of the rail pads, the ballast, and the subgrade, respectively. The two layers of discrete masses below the rail represent the sleepers and the ballast, respectively. In order to account for the shear continuity of the interlocking ballast particles, shear springs and dampers are introduced between the ballast masses to model the shear coupling effects in the ballast. The mass, stiffness and damping values of these track components and the distances between sleepers can be arbitrarily varied so that variations in track component properties and track geometric errors can be taken into account. In many situations, such variations may be the primary cause of undesirable wheel/rail dynamic actions.

3. SYSTEM EQUATIONS

3.1. Equations of vehicle For the vehicle body, the equations governing the bounce and pitch motions of the lumped masses can be entirely described using second order ordinary

1-v

I

OF MOTION

I

I

Fig. 1. Vehicle and track vertical interaction

system model.

Interaction between a train vehicle and rail track

989

Table 1. Notations for vehicle and track interaction model (see Fig. I)

-

car body mass bogie mass wheelset mass car body mass inertia bogie mass inertia primary suspension stiffness primary suspension damping secondary suspension stiffness second suspension damping displacement of car body pitch rotation of car body displacement of bogie frame pitch rotation of bogie frame displacement of wheelset irregularity function wheel/rail contact force

differential equations in the time domain. They are formulated by applying the D’Alembert principle and are given below: Car body bounce motion:

bogie distance wheel distance rail mass per unit length sleeper mass ballast mass railpad stiffness railpad damping ballast stiffness ballast damping subgrade stiffness subgrade damping ballast shear stiffness ballast shear damping rail bending stiffness rail displacement sleeper displacement ballast displacement

Bogie 2 pitch motion:

- Kd(Z,,

M& + 2C&c -I- 2K,A - C&,, - K&t,

+ &)

- Zw,) = 0.

Equations of motion of the four wheels:

+ Zd = 0.

(1) (7)

Car body pitch motion: J& + 2C&$c -I- 2K&h

- C,J,(i,,

- K&Z,,

- 2~)

- Za) = 0.

(2)

Bogie 1 bounce motion: %%

(6)

- C~,ft$,, - K&u

+ (Cd -t. 2C,,)i,, + (Ksz+ 2K,,)Z,, - C&%

+ k)

- C&c

+ I,$,) - K&Z, + C$e) = 0.

M,&

+ C,,(& + C,,&

- K,,(Zw, + Zwz)

-

212)

+

+ P,(t) = 0,

(9)

Ks,(Zw4- Zu)

+ K,,l,IL_o+ P&) = 0.

(10)

3.2. Equationsof rail track (3)

Bogie 1 pitch motion: .&&I + 2Cd%

+ 2K,,th -

- C,,l&

- &)

K,,W%,I- Zwz) = 0. (4)

The equations of motion of the track system consist of that of the rail, which has continuity in space (horizontal direction) and is modelled as a simple beam, and those of the individual sleepers and ballast masses. They are described below. Equation of motion of the rail is given by E*a4zsx,

a2

Bogie 2 bounce motion:

+ L)

- C&c

‘- I&)

rar2

= - i F&)S(x - Xi) + i P,(t)& i-1

Mt.& + (Ct2 + 2C,l)& + (KS2+ 2Kdzt2 - C&L

t) + m a2qx, 1)

- &(Zws + ZVA) - Kdzc - I&)

= 0.

(5)

- x0,),

(11)

1-I

where N is the number of rail support points (or number of sleepers), S(x) is the Dirac delta function, xi is the horizontal coordinate of the ith sleeper, x0, is the coordinate of thejth wheel and Fnc(t) is the ith

W. Zhai and Z. Cai

990

force:

rail/sleeper

Equations of motion of the sleeper and ballast masses are given by:

Ed(r) = K,i[ZdX,, t) - Z,i(l)] M,-%(t) + (C,, + C&%(t)

Cp,[Z(x,,t) - zi(q.

+

+ (Kp, + Kbi)Z$(f)

(12)

To facilitate the solution of the partial differential equation of motion of the rail (eqn 1l), the following rail mode shape function is assumed:

- C,,&(t)

- KbrZdf) - C,, f

Yk(x,)Qk(t)

k=l

-

Kpi

5

Yk(&)qk(t)

=

i=l,2

0,

,...,

N.

k=l

Y,(~) = J-

--$ sin 7, r

(17)

(13) M&#(f)

where I is the effective length of the rail included in the track model and k is the mode number. Thus, the solution of Z,(x, t) in eqn (11) can be expressed as:

+ (C,; + c, + 2C,i)&(f)

+ (KM+ Kfi + 2K,,)Zsi(t) - Ct,,Zs,(t) - Kb,Zs,(f) - C,vi&i+ l,(t) - Kd&,+ l,(t)

Yk(XMf),

Zr(X, t) = i

(14)

k=l

where qk(t) is the kth mode time coordinate and K is the total number of modes considered. Preliminary numerical trials have shown that good convergence of solution can be obtained if K 2 60. By applying Ritz’s method, the fourth order partial differential equation of motion of the rail is then simplified to a series of second order ordinary differential equations in terms of the time coordinate qk(t):

C(t)

+

$

y

4qk(f)

=

-

+ i

i

(15)

j=l

Substituting eqns (14) and (12) into eqn (15) yields the detailed form of the ordinary differential equations of the rail:

Q(t)

+

f i=

+

cpiyk(&)

+

5

7

‘q*(t)

0

N.

(18)

In solving the above equations, all the initial and boundary conditions are assumed to be zero. 3.3. General equations of motion All the above equations for the individual vehicle and track components are combined to formulate a coherent set of ordinary differential equations that are expressed in terms of the following standard matrix form:

where [Mj, [C] and [K’j are the generalized mass, damping and stiffness matrices, respectively; {X(t)}, {P(r)), {X(f)} are the generalized displacement, velocity, and acceleration vectors, respectively and; {F(t)} is the corresponding force vector containing the wheel/rail interface forces P,(t) (i = 1 - 4) and the generalized modal forces given by the right side of eqn (16). Since the number of degrees of freedom of the vehicle is a constant (lo), the size of eqn (19) depends on the number (K) of rail vibration modes truncated in eqn (14) and the number (N) of sleepers (hence, the rail length) included in the track model. 3.4. Non-linear wheel/rail interaction

yk(X!)qk(t)

The system interaction between the vehicle and the track is achieved at the wheel/rail interfaces through wheel/rail force compatibility. This is described by the non-linear Hertzian theory commonly used in wheel/rail interaction problems:

k=l

5

i=

Yk(%)4k(f)

k=l

2 KpiYdxi) 5

,-I

-

2

t

,...,

I

k = 1,2,. . . , K.

P,(t)Yk(XG;)

i=l,2

l,(t) - Kv,iZ,,,- l,(t) = 0,

F,&)Yk(Xi)

i=

0

- C,,Zt+

CpiYk(Xi)&(f)

-

,t,

&Yk(Xi)Zsi(t)

Pj(t)Yk(Xo,),

k = 192,. . . , K.

I

1 312

= i

I-1

(16)

P(t) = ; sz(t) [

,

Interaction between a train vehicle and rail track where G is the Hertzian wheel/rail contact coefficient, &Z(t) is the elastic wheel deformation (compression) in the vertical direction, which is given by:

Sat) = Zd?) - Zr(x, t) - zo(t),

or a wheel flat. zo(t) can be any kind of deterministic function (either spatial or time function).

4. SOLUTION

(21)

in which z,,(t) denotes geometric irregularities along the surface of the rail or around the wheel circumference, such as a rail joint, a corrugated rail,

991

METHOD

Due to the large size of eqn (19), coupled with the non-linearity of the force compatibility equation (eqn 20), the application of conventional direct time integration methods, such as the

Rail vehicle and track parameter input stomized to Chinese rail car bmes and track structures

c

Calculation of lnitral track/vehtcle

1 Formulation of vehicle and track 1 1 system dynamic equations J (Input wheel/rail interface irregularity function 1

Fig. 2. Flow chart of VICT (vertical interaction between cars and tracks) simulation system.

W. Zhai and Z. Cai

992

Table 2. Vehicle and track model parameters

a,

30

Vehicle model parameters

o Measured l Simulated

25

Mc =77OOOkg M, = 1lOOkg M,= 1200kg J, = 1.2 x IO6kgm2 J, = 760 kg m2 K,, = 2.14 x IO6N m-’ C,, =4.9 x l(rNsm-’ KS2= 5.32 x lo6 N m-l Cr2=7 x lO”Nsm-’

20

15

10

Track model parameters Mu= 51.5 kg m-’ EI=4.2 x lO”Nm* Ls = 0.545 m (sleeper spacing) M, = 237 kg Mb = 683 kg Kp= 1.2 x 108Nm-’ C,= 1.24 x 105Nsm-’ Kb = 2.4 x IO8N m-’ Cb = 5.88 x 10’ N s m-’ Kr=6.5 x lO’Nm-’ Cr=3.12 x 104Nsm-I K, = 7.84 x IO’N m-’ C,=8.0 x 104Nsm-’

C62A type car has no primary suspension.

5

dipped railjoint (0.02 rad)

0 30

I

40

1

1

{@“+I= {?}n +(1 +

(22b)

1

50 80 70 speed (km/h)

Fig. 3. Comparison of wheelset acceleration.

the Rung+Kutta method, or the Wilson-0 method, requires a large amount of computer storage and a long solution time. Based on Newmark’s implicit method, an improved explicit integration scheme has been constructed to solve eqn. (19). This procedure is expressed as follows [40]:

1x1n+,

=

4){X},At - 4{x},_,A.t,

{X}, + {2},At + (f + $){X}AP - JI{@_,At*

where At is the integration time step-size; subscripts n + 1, n, and II - 1 denote the integration time at (n + l)At, nAt, and (n - l)At, respectively; + and C#J are free parameters which are used to control the stability and numerical dissipation of the algorithm. Usually a value of 0.5 could be assigned to $ and $J to achieve such goals. Use of the above numerical scheme eliminates the need to solve any algebraic equation group during each time step because of the diagonal mass matrix [m] in eqn (19). Thus the computational efficiency is greatly enhanced and the solution algorithm can be implemented on common

(22a)

0 Measured l Simulated 20 0 0

15

0

0

10 t

0

jagged rail joint (1 mm height) I Oo

I

I

20

40

I

80 speed (km/h)

I

80

Fig. 4. Comparison of ballast acceleration.

1

0

20

80 ss

1 0

(k%h)

Fig. 5. Comparison of wheel/rail forces due to dipped rail joint.

Interaction between a train

I

train speed 40 km/h

I

t 8 $ 8

150 t 100 F

0 Measured 0 Simulated

50

I

t 01, 0

0.3

0.6 0.9 1.2 wavelength (mm)

1.5

Fig. 6. Comparison of wheel/rail forces due to corrugation. personal computers (such as 386 or 486 PCs) with fast solution speed. (For example, to simulate the entire process of wheel/rail shock and vibration occurring at a rail joint, the solution time for a 500 degrees of freedom system is about 50 min on a COMPAQ 386/25e type PC, and is under 10 min on an IBM 486/66 PC). The flow chart of the complete simulation process, which is dubbed VICT (vertical interaction between cars and tracks), is illustrated in Fig. 2. 5. EXPERIMENTAL

VERIFICATION

993

variety of vehicle and track conditions representative of Chinese freight trains and mainline tracks. Some typical results are described below. Figure 3 shows comparisons between simulated and measured peak wheelset acceleration data due to a dipped joint with a dip angle of 0.02 rad. and Fig. 4 shows comparisons between simulated and measured peak ballast acceleration data due to a jagged rail joint with a differential height of 1 mm, respectively. Both the vehicle and track parameters used in the simulations are given in Table 2. The vehicle model used is the C62A type freight train car commonly operated on Chinese mainline tracks. The P2 forces (see next section) due to the C62A train car running across the dipped joint is given in Fig. 5. Also given in Fig. 5 are the simulated and measured data for the Chinese C75 type train car, which is a heavy haul freight vehicle. The comparison of peak wheel/rail impact forces due to the C62A train car running on corrugated rails having various wavelengths is shown in Fig. 6. In all the simulations given above, the computer model gave close results to the measured values of the selected quantities. This provides confidence in the application of the model to simulate vehicle and track dynamic interaction problems. The following section describes some typical application results.

260 200

vehicle and rail track

6. EXAMPLE APPLICATION

6.1. PI and P2 forces due to rail joint The application of the VICT model in investigating the dynamic responses of vehicle/track systems can be demonstrated by the simulation of a train vehicle passing a rail joint. Figure 7 shows the predicted wheel/rail impact force time histories when the train vehicle passes a typical rail joint on

OF VICT MODEL

The validity of the VICT simulation model has been verified by field experiments conducted with a

train speed (km/h)

IA Pl

,’-. .

P,

v=250 v=200 v=160 v=120 v=60 , v=40

/ . F&

I

OJ

0

OF VICI’ MODEL

1

2

3

4

5

6

7

6

9

time (ms) Fig. 7. Simulated wheel/rail impact force history at a rail joint.

10

994

W. Zhai and Z. Cai

500

150025003500450055006500750085009SOO unsprung mass (kg) per axle

Fig. 8. Simulated effect of unsprung mass on PI and A forces.

Chinese mainline tracks at various train speeds. The data shows that both the PI and the P2 force, so designated by British Rail researchers [4] to denote the two peak forces of the wheel/rail impact at a rail joint, increase rapidly with increasing speed. This is consistent with the well-known fact that wheel/rail dynamic interaction due to wheel or rail irregularities will dramatically intensify as the train speed increases. Intensified dynamic activities at rail joint locations will inevitably accelerate deterioration of the joint assembly and degradation of the adjacent sleepers and ballast. Thus, it is of great importance that the use of rail joints on railway tracks be minimized by the increasing use of

10”

loo

continuously welded rails. (In China, the vast majority of existing rail lines were built with jointed rails.) Figure 8 illustrates the simulated theoretical relationship between the PI and P2 forces and the unsprung mass of the vehicle at a running speed of 80 km h-l. This figure indicates that the increase in the P2 force is more pronounced than in the PI force and the value of the Pz force exceeds that of the PI force when the unsprung mass is greater than 5000 kg. It has been demonstrated [4,7] that a significant portion of the P2 force transmits to the track formation and is a primary damaging source to the adjacent sleepers and the underlying ballast.

10’ ld 103 rail pad Mfness (MN/m)

Fig. 9. Simulated effect of rail pad stiffness on wheel/rail forces.

lo’

Interaction between a train vehicle and rail track

995

ballast density (kg/ma) Fig. 10. Simulated effect of ballast density on ballast vibration level and P2 force. Therefore, the above simulation results demonstrate that the unsprung mass of the bogie should in principle be kept a minimum in any new design of train vehicles. 6.2. Effect of selected track parameters The effect of track system parameters on the dynamic wheel/rail vertical interaction has also been studied with the VICT simulation model. Figure 9 shows the simulate’d effect of the rail pad stiffness on the magnitude of the PI and P2 forces across the rail joint. The figure indicates that for the particular track configuration used., there appears to be an optimum rail pad stiffness value (approximately 50 MN m-l), above which both peak forces increase rapidly with increasing pad stifFness (most commercial rail pads ranging from 20 to have stiffness values 300 MN m-l). Figure 10 shows the simulated effect of the ballast density, which affects the ballast stiffness, on the peak vibration acceleration in the ballast and the magnitude of the P2 force. It is seen that while the effect of the ballast density on the PZ force is insignificant, the vibration level in the ballast is largely reduced as the ballast density increases. This illustrates the importance of maintaining a firm ballast layer below the track in order to reduce the vibration level in the ballast, and ultimately in the subgrade, thus keeping ballast and subgrade maintenance to a minimum. 6.3. Effect of damaged rail fastening When there are damaged components in the track structure, severe vibrations will occur as the wheels traverse the damaged site. Figure 11 demonstrates the influence of disfunctional rail fastenings on the rail acceleration and tlhe rail deflection response. The disfunctional fastenings are simulated by assigning

small stiffness values to the corresponding spring elements. The simulation results indicate that there is nearly a 30% increase in the rail acceleration and 6&70% increase in the rail displacement due to the disfunctioning of the rail fastening. Thus, any damaged track components should be promptly repaired or replaced to prevent further damages to the track due to the intensified dynamic track response. 7. CONCLUSIONS

The paper has presented a numerical model for the simulation of dynamic train and track interactions at the wheel/rail interface. Wheel/rail irregularities are the common source that creates dynamic disturbances to both the vehicle and the track system. It has been demonstrated that the model proposed in the paper is an efficient research tool to investigate the dynamic effects caused by any type of wheel and rail defects, which are damaging to the vehicle and track components. In particular, the dynamic response of the track at rail joints has been shown to dramatically increase with an increase in the train speed. The relationship between the vehicle components, namely the unsprung mass and the dynamic wheel/rail load, can be quantified using the established model. Further research using the model is necessary to explore its application in analyzing track/train dynamics and, hence, to investigate effective means of reducing the intensity of wheel/rail dynamic forces for the safety and economy of train operations. Acknowledgments-The research work presented in the paper was financially supported through a research grant awarded to the first author by National Natural Science Foundation of China. The authors would like to thank Professors Z. Shen and X. Sun for their valuable assistance.

W. Zhai and Z. Cai

996

150-

(a) rail acceleration A

damaged fastening

1

2

3

4

5

6

7

6

9

10

time (ms)

.. .

1.2

I

(b) rail deflection

E= l.OE.

.E 0.6B 3 0.6= E 0.4-

0

1

2

3

4

5 6 time (ms)

7

6

9

Fig. Il. Simulated effect of damaged fastening on rail vibration.

REFERENCES

1. R. I. Mair, Aspects of railroad track dynamics, part I-vertical response. BHP Melb. Res. Lab. Rep. h4RL 81/3, BHP, Melbourne, Australia (1974). 2. R. I. Mair, Natural frequency of rail track and its relationship to rail corrugation. The Ciuil Engineer Transactions. Institute of Engineers, Australia, pp. 6-l 1 (1977). 3. P. J. Mutton and T. Jeffs, Modelling the deformation behavior of welds. Proc. 4th Int. Conf. Contact Mechanics and Wear of Wheel/Rail Systems, Vancouver (1994). 4. H. H. Jenkins, J. E. Stephenson, G. A. Clayton, J. W. Morland and D. Lyon, The effect of track and vehicle parameters on wheel/rail vertical dynamic forces. Rai/way Engng J. Jan. issue, 2-16 (1974). 5. S. G. Newton and R. A. Clark, An investigation into

6.

7. 8. 9.

10.

the dynamic effects on the track of wheelflats on railway vehicles. J. Mech. Engng Sci. 21, 287-297 (1979). R. A. Clark, P. A. Dean, J. A. Elkins and S. G. Newton, An investigation into the dynamic effects of railway vehicles running on corrugated rails. J. Me&. Engng Sci. 24, 6%76 (1982). C. 0. Frederick and D. J. Round, Vertical track loading. In Track Technology, pp. 135-149. Thomas Telford, London (1984). J. M. Tunna, Wheel-rail forces due to wheel irregularities. Proc. 9th Int. Wheefset Congress, Montreal, Canada, paper 6-2 (1988). S. L. Grassie, R. W. Gregory, D. Harrison and K. L. Johnson, The dynamic response of railway track to high frequency vertical excitation. J. Mec!i. Engng Sci. 24(2), 77-90 (1982). S. L. Grassie, R. W. Gregory and K. L. Johnson, The behavior of railway wheels&a and track at high

Interaction between a train vehicle and rail track

11.

12.

13.

14.

frequencies of excitation. J. Mech. Engng Sci. 24(2), 103-I 11 (1982). S. L. Grassie and S. J. Cox, The dynamic response of railway track with flexible sleepers to high frequency vertical excitation. Proc. Inst. Mech. Engrs 7(198D), 117-124 (1984). S. L. Grassie, A contribution to dynamic design of railway track. D:vnamics of Vehicles on Roads and on Tracks, Proc. 12th IAVSD Symposium, Lyon, France, pp. 195-209 (1991). Z. Cai and G. P. Raymond, Theoretical model for dynamic wheel/rail and track interaction. Proc. 10th Int. Wheelset Congress, Sydney, Australia, pp. 127-131 (1992). Z. Cai and G. P. Raymond, Use of a generalized beam/spring element to analyze natural vibration of rail track and its application. Int. J. Mech. Sci. 36(9),

Struct. Engng Mech. 2, 95112

dynamic interaction of wheelset and track. Railway Gaz. Int. 118(9h . ,_ 591-595 11992). 26. Y. Sato, T. Odaka and H.‘Tak&, Theoretical analyses on vibration of ballasted track. Quart. Rep. JNR 29(l), 30-32 (1988). 27. K. Ono and M. Yamada, Analysis of railway track vibration. J. Sound Vib. 130, 269-297 (1989). 28. M. Ishida and S. Miura, Relationship between rail surface irregularity and dynamic wheel load. Proc. 10th Int. Wheelset Congress, Sydney, Australia, pp. 175-179

(1992). 29. J. C. 0. Nielsen and T. J. S. Abrahamsson, Coupling of

physical and modal components for analysis of moving non-linear dynamic systems on general beam structures. Int. J. Numer. Meth. Engng 33, 1843-1859 (1992).

30. T. Dahlberg, B. Akesson and S. Sestberg, Modelling the dynamic interaction between train and track. Railway Gaz. Int. 149(6), 407412

863-874 (1994).

15. Z. Cai and G. F’. Raymond, Modelling the dynamic response of railway track to wheel/rail impact loading. (1994).

16. Z. Cai and G. P. Raymond and J. 0. Igwemezie, Contact loads from vertical dynamic wheel/rail and track interaction. Proc. 4th Int. Conf. Contact Mechanics and Wear of Wheel/Rail Systems, Vancouver (1994). 17. G. P. Raymond and Z. Cai, Dynamic track support loading from heavier/faster train sets. Transport. Res. Rec., TRB 1381, 53-59 (1993). 18. Z. Cai and G. F’. Raymond, Fundamentals of track vibration and dynamic track loading. Transport, Proc. Inst. Ciu. Engr (1995) (in press). 19. R. G. Dong and S. Sankar, The characteristics of impact loads due to wheel tread defects, Rail Transport., ASME 8, 23-30 111994).

20. R. G. Dong, S. Sankar and R. V. Dukkipati, A finite element model of railway track and its application to the wheel flat prolblem. J: Rail and Rapid Transit, Proc. Inst. Mech. Engr 208, 61-72 (1994).

21. J. Fortin, Dynamic track deformation. French Railway Rev., Int. edition 1, 3-12 (1983). 22. L. Girardi and I’. Recchia, Use of a comuutational model for assessing dynamical behavior of-a railway structure. Dynarnrcs of Vehicles on Roads and on Tracks, Proc. 12th IAVSD Symposium, Lyon, France, pp. 185-194 (1991). 23. H. Bias and S. Muller, A discrete-continuous track model for wheelsets rolling over short wavelength sinusoidal rail irregularities. Dynamics of Vehicles on Roads and on Tracks, Proc. 13th IAVSD

Symposium,

Chengdu, China, pp. 221-233 (1993). 24. K. Knothe and B Ripke, The effects of the parameters of wheelset, track and running conditions on the growth rate of rail corrugation. Dynamics of Vehicles on Roads and on Tracks, Proc. 11th IAVSD Symposium, Kingston, Ontario, Canada, pp. 345356 (1989). 25. K. Hempelmann, B. Ripke and S. Dietz, Modelling the

997

(1993).

31. M. Fermer and J. C. 0. Nielsen, Wheel/rail contact forces for flexible versus solid wheels due to tread irregularities. Dynamics of Vehicles on Roads and on Tracks, Proc. 13th IAVSD Symposium, Chengdu, China, pp. 142-157 (1993). 32. D. R. Ahlbcck, An investigation of impact loads due to wheel flats and rail joints. ASME Paper No. 80- WA/RT-1

(1980).

33. D. R. Ahlbeck and J. A. Hadden, Measurement and prediction of impact loads from worn railroad wheel gnd rail surface-profiles. J. Engng Ind. ASME 107, 197-205 (1985). and H. D. Harrison, The effect of 34. D. R. A&e& wheel/rail impact loading due to wheel tread run-out profiles. Proc. 9th Int. Wheelset Congress, Montreal, Canada, Paper 61, (1988). 35. H. D. Harrison, A. E. Bethune and D. R. Ahlbeck, Comparison of measured and predicted impact loads on loo-ton coal Gondolas. Proc. 4th Int. Heavy Haul Railway Conf., Brisbane, Australia, pp. 409-41?(1989). 36. K. Knothe and S. L. Grassie. Modellina of railwav track and vehicle/track interaction at high-frequencies. Vehicle Sys. Dyn. 22, 209-262 (1993). 37. Z. Shen, On principles and methods to reduce the wheel/rail forces for rail freight vehicles. Dynamics of Vehicles on Roa& and on Tracks, Proc. lith IAVSD Svmoosium. Lvon, France. DD. 584595 (1991).

38. %‘. Thai, k. bang, M. l?; -and J. Yan, Minimizing dynamic interaction between track and heavy haul freight cars. Proc. 5th Int. Heavy Haul Railway Conf., Beijing, China, pp. 157-162 (1993). 39. D. Chen, W. Wang, D. Hu, Y. Lu, G. Wei, L. Zhang and W. Li, Non-stationary random irregularities on heavy haul railway track in China. Proc. 5th Int. Heavy Haul Railway Conf., Beijing, China, pp. 489496 (1993). 40. W. Zhai, The explicit scheme of Newmark’s integration method for structural dynamic-analysis. Proc. Int. Conf. on Vibration Problems in Engineering, Vol. 1, pp. 157-162. International Academic, London (1990).