Analytical dynamic and quasi-static model of railway vehicle transit to curved track

c AFM, EDP Sciences 2012 DOI:10.1051/meca/2012017 www.mechanics-industry.org

M echanics

& I ndustry

Analytical dynamic and quasi-static model of railway vehicle transit to curved track

Habib Bettaieb

^a

Ecole pr´´ eparatoire aux acad´emies militaires de Sousse, Tunisie

Laboratoire de g´enie m´ecanique, ´Ecole nationale d’ing´enieurs de Monastir, avenue Ibn-Eljazzar, 5000 Monastir, Tunisia Received 12 December 2011, Accepted 27 June 2012

Abstract – This paper shows that the dynamic study of a railway vehicle in curve can be reduced to the study of a bogie and two wheelsets in the quasi-static case. Every wheelset supports the load of the quarter of the vehicle. From this model, we can optimize the suspension to define the mechanical build characteristics of vehicle.

Key words: Simulation / curving behaviour / dynamic and quasi-static model of railway vehicle / Circulation in tight curve / lateral -creep force / longitudinal creep force

R´esum´e – Mod`ele analytique dynamique et quasi-statique d’un v´ehicule ferroviaire en cir- culation sur une voie en courbe. Cet article montre que l’´etude dynamique d’un v´ehicule ferroviaire en circulation sur une voie en courbe, peut ˆetre r´eduite `a l’´etude d’un bogie `a deux essieux dans le cas quasi-statique. Chaque essieu supporte la charge du quart du v´ehicule. De ce mod`ele, on peut optimiser la suspension afin de d´efinir les caract´eristiques m´ecaniques du v´ehicule d’une fa¸con pr´ecise.

Mots cl´es : Simulation / comportement en courbe / mod`ele dynamique / mod`ele quasi-statistique / circulation en courbe de faible rayon / force de pseudo-glissement lat´erale / force de pseudo-glissement longitudinale

1 Introduction

The research for a long life span of the railway ve- hicles and for safety obliged railway companies to plan big investments to build ways of high geometrical, rec- tilinear qualities and curves of big radius. The majority of the existing vehicles are optimized in stability in align- ment for high speed, but not for the traffic of short radius curve. Such a matter induces problems of maintenance of both wheel and rail. Among these problems, we notice that the wear of the wheel as well as that of the rail cost much money to railway companies and create the risk of derailment. A number of investigations have worked on improving the rail vehicle design between the stability and the curving performance.

Bettaieb [1] developed also a model using a rail in- clination. This model studies the variation of rail incli- nation on the motion of the vehicle. This will allow us to master the transverse displacement of the wheelset, to avoid flange contact in a curve, which causes high wear on the wheel and the rail while preserving a good stability in alignment. This will reduce the cost of maintenance

a Corresponding author:

[email protected]

and will improve the comfort of the passengers.

Bettaieb et al. [2] used the model of a complete vehi- cle for the study of the vehicle stability at high speed and the quasi-static model of bogie and two wheelsets for the study of the traffic in short radius curved tracks.

They used the genetic algorithms for simple objective and multi-optimization. Indeed, they showed that this method always allows for finding the optimal values of several vari- ables (parameters of construction of the vehicle). These values allow the vehicle to traffic with a very high speed with safety in alignment. These same optimal values allow the vehicle to traffic without sliding, and without flange contact between the wheel and the rail for the big and short radius curved tracks. Bettaieb et al. [3,4] used the model to study the quasi-static model of bogie and two wheelsets for the study of the traffic in short radius curved tracks. Indeed, they showed that the model is used to op- timize the mechanical characteristic vehicles design and analyses of its performance. Boocok et al. [5] considered the steady-state motion of bogie and two-wheelset vehi- cles. Special attention was given to the effect of primary and secondary suspension parameters and the results were evaluated with experiments. Dukkipati et al. [6] consid- ered the steady state motion of conventional bogie and

Article published by EDP Sciences

Nomenclature

2a Wheelbase Rc Curve radius

2A Distance between the

bogies centres

rj Rolling circle radius of the wheel at the points of contact

C₁₁, C₂₂, C₂₃andC₃₃ Creep coefficients of Kalker

Tkij Lateral creep forces

ej Distance between the centre

of inertia of the wheelset and the wheel and rail points of contact

V Vehicle velocity

e₀ Half distance between the

contact points in the central position

Xkij Longitudinal creep forces

Kx Longitudinal stiffness of

primary suspension

Z Height of wheelset centre

Kx Longitudinal stiffness of

secondary suspension

γe Wheel equivalent conicity

Ky Lateral stiffness of

secondary suspension

γnc Not compensated acceleration γ₀ Wheel conicity (new wheel)

Mkij Spin creep moment νkij Creepages at the contact point

m Wheelset mass ρxki Radius of inertia of the

ρyki wheelset about axis

ρzki (G₀kixδpki, G₀kiyδpki, G₀kizδpki) respectively

M Bogie mass μ Friction coefficient

M Body mass ωN kij Angular velocity about normal in

Contact pointj

N Normal load for a wheel Ωx Radius of inertia of the

Ωy body about axis

Ωz (Gxδp,Gyδp, G zδp) respectively

R Lateral radius of rail profile Ωxk Radius of inertia of

Ωyk the bogie about axis

Ωzk (O₀kxδpk, O₀kyδpk, O₀kzδpk) respectively

R Transverse radius of wheel

profile

σk Angles between the longitudinal axis of the body and bogies.

r₀ Rolling circle radius for

a central position

σki Angles between the longitudinal axes of the bogies and middle of the wheelset.

δpki Superelevation to the

wheelset position

δpk Superelevation of the bogies position

δp Superelevation of the

body position

two-wheelset vehicle, the effects of unsymmetrical suspen- sions were also included. Many designs are considered to provide good dynamic performance and the ability of the vehicle to steer around a curve. Bettaieb [10] has devel- oped a thesis, which leads to the conclusion that in the design of the railway vehicle there is a conflict between dynamic stability at a high speed and the ability of the vehicle to steer around curves.

To approach most real systems, it is necessary to adopt a mathematical model, which describes well the system to be studied. The aim of this work, is to find

the mathematical model, which describes the behaviour of complete vehicle in curve.

2 Vehicle system model

We define with precision the parameters used in the mechanical model. A railway vehicle is taken as a multiple rigid body system composed of wheelsets, bogie frames, and car body (see Fig. 1). These rigid bodies are con- necting to each other by elements of suspension, primary Kx;Ky;Kzand secondary suspensionKx;Ky;Kz. The

I¹ 1

r Ψ

G

e

G

e

⁰

e

¹

u

(ψ+π/2)

z

0

R'

y

0 O'1 O1

A¹

Β^{1 δ0} δ1 0

r

u

(ψ)

Fig. 1.Wheelset parameters.

Table 1.Degrees of fredoom of the vehicle.

lateral displacement yaw roll

Bogie (Ck):k= 1,2 yk αk θk

Car body (C) Y α θ

Wheelsets (Ski) k, i= 1,2 yki αki ψki=Γ.yki

dampers are putting in parallel with the springs. The steps followed in this process begin with the geometri- cal study of the wheel and rail contact. This study takes linear wheel/rail geometry with two points of contact into account. The distancesej,rj between the centre of inertia of the wheelset and the wheel and rail points of contact are a function of the position parameters of the wheelset [10]

as shown in Figure 3. These expressions are given by Eqs. (A.1)–(A.6). The application of the Kalker’s the- ory [9] determines the actions of contact between the rail and the wheel. It should be mentioned that for the investi- gations of the curving behaviour, the values of the body’s centrifugal forces due to lateral curvature of the truck’s central line are introduced. It is assumed that the lateral and the vertical vibrations of the system are uncoupled, thus, only the lateral vibrations of the system are taken into consideration for this analysis. The wheelset is given with an indexi= 1 (leading wheelset) ori= 2 (trailing wheelset), a contact point on the wheel tread is indicated by the indexj = 1 (left wheel in the direction of travel) and j = 2 (right wheel). The bogie frame is given with an indexk = 1 (the first in the direction of travel), and k= 2 (the second). For all solids, the movements consid- ered are the lateral displacement, the yaw and the roll.

These movements are noted in Table1.

2.1 Used references and parameters

We must clearly define the notations and reference frame used for the passage from one basis to another.

The parameter setting of different solids in the vehicle moving from the alignment to the connecting curve is as follows:

O₀: mobile centre related to the car body, OOk: mobile centre related to the bogieCk, OOki: mobile centre related to the wheelsetsSki. O₀ki: moving along the axis track geometry.

The plans (Og, xg, yg) and (O₀, x₀, y₀) are parallel.

The basis of R₀ is deduced from the basis of Rg by a rotation σ (axis z₀ = zg). The basis of R₀k is deduced from the basis of R₀ by a rotation σk (axis z₀k). The basis ofFki is deduced from the basisR₀K by a rotation σki (axis b_ki). The basis of Rδpki is deduced from the basis of Fki by a rotation σki (axis zδpki). The basis of Rδp= (O₀, xδp, yδp; zδp) is deduced from the basis ofR₀ by a rotation δp (axis x₀ = xδp). The basis of Rδpk = (O₀k, xδpk, yδpk, zδpk) is deduced from the basis (R₀k) by a rotationδpk(axisx₀k=xδpk). The basis ofRδpki= (O₀ki, xδpki, yδpki, zδpki) is deduced from the basis ofFki

by a rotationδpki(axisxδpki=b_ki) (see Fig. 2).

2.2 Expressions of force and moment of centrifugal and Coriolis force

The expressions of force and moment of centrifu- gal for the car body, bogie, wheelset; of force and mo- ment inertia Coriolis wheelset’s are given by Appendix A Eqs. (A.7)−(A.10).

h1

A h0

H

h1

h1

z_δp

1

x_δp

x_δp 21

zδp

11

xδp

Fig. 2. Configuration of a 17 DOF rail vehicle model.

Table 2.Used references.

Rg= (Og, xg, yg, zg) Galilean reference attached to the curve centre.

R₀ = (O₀, x₀, y₀, z₀) moving reference attached to the car body.

R₀k= (O₀k, x₀k, y₀k, z₀k) moving reference fixed to the bogieCk. Fki= (O₀ki, tki, nki, bki) Frenet frame.

2.3 Kinetic energy of the vehicle

Having completely defined the coordinate system, we calculate the kinetic energy (see Appendix A (Eq. (A.11)).

2.4 Power developed by the springs

Equation (A.12) gives the power developed by the springs and dampers.

2.5 Power developed by the action of gravity

The power developed by the gravity is given by Ap- pendix A Eq. (A.13).

2.6 Wheel and rail interactive forces

The traffic of vehicle in a tight curve (Rc < 500 m), the reduced sliding of wheels become more important. In that case, the force and moment contact are obtained by applying the law of saturation proposed by Johnson and Vermeulen [8], based on the coefficients of Kalker. The calculus of the wheel and rail interactive forces is given by Appendix A (Eqs. (A.14)–(A.18)).

2.7 Dynamic model

The set of differential equations is obtained by apply- ing Lagrange’s equations, withqi: generalized coordinate vector,q_i: derived from the generalized coordinate vector, T^g(D): kinetic energy,Π_i^g

Sj →Sj

: coefficient ofqi’ in expression of power.

d dt

∂T^(λ)(D)

∂q_i −∂T^(λ)(D)

∂qi =

j=n

j=1

Π_i^(λ)

Sj→Sj

−

j=n

j=1

λ Sj, qi

× A^e_Sj|^gλ

+

A^c_Sj|^gλ

(1)

A^e_Sj|^g_λ

corresponds to inertia centrifugal force and mo- ment of (Sj) in movement ofλrelative to the fixed refer- ence (Rg);

A^c_Sj|^gλ

to Coriolis inertia force and moment of solid (Sj) in movement of λrelative to the fixed (Rg) and

λ Sj, qi

= _∂q^∂ i

λ Sj

; λ

Sj

define the rotation and speed of solid Sj. The use of Lagrange’s equation gives us the dynamic equations. These equations stimu- late the dynamic behaviour of a vehicle in traffic of tran- sition curve of radiusRc. The curves of transition have an important role in the quality of the comfort, particularly for the high-speed vehicles. The dynamic model devel- oped in this research can simulate the dynamic vehicle of transition curve.

δpki

Fig. 3. Curve parameters.

2.8 Motion on a curve

Some hypotheses were adopted for the dynamic study of the vehicle in traffic on a way in curve: the vehicle travels with a constant speed; the system is at equilib- rium; the curve radiusRC is constant. Then we can ne- glect the damping forces and the inertia forces compared to the elastic forces. The rolls of the car body and bogie in the motion on a curve are small and we can write:

θ=θk = 0;δpki=δpk=δp= 0.07 rd;

γnc=V²

Rc−g.δp−g.δpandW= m+m +M 2 +M¯

4

.g ρ₁₁=−ρ₁₂=−ρ₂₁=−ρ₂₂= 1

Rc;σ₁=−σ₂=−A¯ Rc; σ₁₁=−σ₁₂=σ₂₁=−σ₂₂= −a

Rc (2)

2.8.1 Variable of relative positions

We propose that the vehicle is rigid with nominal ge- ometry describing perfect track design. The track centre line, which is an element of nominal geometry, always stays in horizontal plan. In a curve of constant radius, a free motion wheelset in a railway takes a stable posi- tion, which corresponds to a motion of rolling without slip. This position has a lateral displacement [10]:

y₀= e₀.r₀

γe.Rc (3)

We take this position as the origin of the lateral displace- ment, and we write:

Y=Y₁+Y₂

2 ; α=Y₁−Y₂

2A ; Y₁=Y₁^∗+Y₁^∗+y₁₁^∗ +y₁₂^∗ 2 +y₀; αk =α^∗_k+y_k^∗₁−y_k^∗₂

2a ; Y₂=Y₂^∗+Y₂^∗+y^∗₂₁+y₂₂^∗ 2 +y₀; yk=y_k^∗+y_k^∗₁+y_k^∗₂

2 +y₀; yki=y^∗_ki+y₀; αki (4) Y₁ and Y₂ are the lateral displacements of two points of the car body. These points belong to the axis of the elastic connection between bogie and the car body. They coincide with the geometrical axis of two bogies in position at rest.

The dynamic equations of the vehicle in the quasi-static case is: M(14, 14)(X)*X=B (Eq. (A.19)).

2.8.2 Decomposition of the rigidity matrix

The value of the longitudinal rigidity of the secondary suspensionKxis low in regard toKxandKyvalues, this condition makes the termKx.d²/Abe neglected, because 2A,the distance between the bogies centres, is high. From Equation (A.19), the relations relative to the car body are:

2KY.A.Y₁^∗−2KY.A.Y₂^∗= 0 2KY.Y₁^∗+ 2KY.Y₂^∗=M γnc

⇒Y₁^∗=Y₂^∗= M .γnc

4KY

(5)

Fig. 4.Wheelset position in a curve.

We replace the expressions of Y₁^∗ & Y₂^∗ in the relations relative of bogie Equation (6) and we obtain:

4KY.Y₁^∗−.2KY.Y₁^∗=M γnc

4KY.Y₂^∗−.2KY.Y₂^∗=M γnc

⇒Y₁^∗=Y₂^∗=

M +^M₂

.γnc

4KY (6)

We find the expressions above of the lateral displacement of the car body and the two bogies. The decoupling of the dynamic equations Equation (A.19) gives two uncoupled systems Equation (A.20) which describe the motions of the two identical bogies and the four wheelsets.

[M(5,5)] 0

0 [M(5,5)]

X₁ X₂

= B

B

;

[M(5,5)] (X₁).{X1}={B} (7) Each bogie and two wheelsets have 5 degrees of freedom {X₁} = {α^∗₁, y₁^∗_i, α₁i}^T and {X₂} = {α^∗₂, y^∗_2i, α₂i}^T i = 1,2. We have showed that a bogie and two wheelsets Equation (A.20) can replace the semi-static behaviour of the complete vehicle in curve.

We define a vectorial sub-space (Y₁^∗;α^∗₁; y₁₁^∗ , α₁₁; y₁₂^∗ ;α₁₂) of the decoupled bogie and two wheelsets, the additional sub-space (Y₁^∗;Y₂^∗;Y₂^∗;α^∗₂;y^∗₂₁, α₂₁; y^∗₂₂;α₂₂) of the complete vehicle. We note MS(8, 8)(X) the addi- tional matrix (Eq. (A.21)).

2.8.3 Derailment force

For the traffic on a way in curve, we suppose the ve- hicle travels with a constant speed and the curve radius

Fig. 5.Lateral effortFki.

is constant. The play between rail and wheel is 0,01 m. If the wheelset lateral displacementyki=y^∗_ki+y₀is equal at the play between rail-wheel, the variable yki is constant and presents known value. At this moment, the wheelset touches the rail and exercises a lateral effort Fki [2]. To avoid the derailment [7], it is necessary that:

Fki

N < tgθ_max−μ

1 +μ.tgθ_max (8) We supervise the value of the lateral displacement yki. If the lateral displacement yki is equal or greater than

100 150 200 250 300 350 400 450 500 0.5

1 1.5 2 2.5 3 3.5 4 4.5

5x 10^-3

Variation of Rc (m)

Variation of y*11 (m)

Variation of y*11 function of Rc

M(14,14) M(5,5) MS(8,8)

Fig. 6.Leading wheelset lateral displacementy^∗₁₁as function ofRc.

play = 10 mm, there is an effortF ki, otherwiseF ki= 0.

The calculation of the lateral forces F₁i applied by each wheelset (i= 1, 2) on the rail is obtained from:

– Line (1 or 3) of matrix M(14, 14)(X) Equation (A.19);

F₁i= (m+m).γ nc−W.φ.(y0+y₁^∗_i)+2.χ.C₂₂.V J X₁i.α₁i

+ 2.Ky.Y^∗₁+ (−1)ⁱ⁺¹.2.Ky.a.α^∗₁ (9) – Line (1 or 3) of matrix M(5,5)(X₁) (Eq. (A.20)).

F₁i=W.γnc−W.φ(y₀+y₁^∗_i) + 2.χ.C₂₂.V J X₁i.α₁i

+ (−1)ⁱ⁺¹.2.Ky.a.α^∗₁ (10)

2.9 Method of resolution

The obtained equations form a non-linear system which write [K(X)]{X} = {B}. To resolve the system, we look for a vector, which makes the residue{R(X)}= [K(X)].{X}−{B}as close as possible to zero. The exact solution is equal to zero. The iterative process converges and stops whenn ≤εwithεa positive constant value (ε= 10⁻¹⁴).

3 Simulation results

We maintain the values below constant, except in case where there is a variation of the constants (Rc,Kx,γe)

Figures6and7show the transverse displacementy₁₁^∗ , y₁₂^∗ as function of Rc. For the large curves the leading lateral displacement decreases and it becomes small and constant, and it increases for tight curves.y^∗₁₁ is always positive,y₁₂^∗ is negative. The leading wheelset moves to- wards the outside of the curve; the trailing wheelset takes a centred position in the railway.

100 150 200 250 300 350 400 450 500

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5x 10^-3

Variation of Rc (m)

Variation of y*12 (m)

Variation of y*12 function of Rc

M(14,14) M(5,5) MS(8,8)

Fig. 7.Trailing wheelset lateral displacementy∗12as function ofRc(curve radius).

0 5 10 15

x 10⁵ 0

1 2 3 4 5 6x 10^-3

Variation of Kx (N/m)

Variation of y*11 (m)

Variation of y*11 function of Kx

M(14,14) M(5,5) MS(8,8)

Fig. 8. Wheelset transverse displacement y^∗₁₁ as function of primary suspensionKx.

Figure 8 shows the variations of the transverse dis- placementy^∗₁₁, as function of the longitudinal stiffness of primary suspensionKx. For a constant radius curveRc, the transverse displacement increases for high values of longitudinal stiffness of primary suspensionKx. It shows that stiffness of primary suspension Kx have a great in- fluence on the transverse displacement of wheelset. A soft stiffness of primary suspension on leading wheelset re- duces the transverse displacement in the traffic on a curve.

In this case, the wheel rail wear decreases. The railway company confirms these results by experiment [5,11].

Figure10shows the transverse displacementy₁₁^∗ as a function of the wheel conicity γe. For a constant radius curve Rc = 100 m, the transverse displacement of the leading wheelset decreases when γe increases. For small

Table 3.Design parameters.

γe= 0.2 γ₀= 0.025 e₀= 0.75 m Kx= 0 N.m⁻¹ Ky= 3.96×10⁶ N.m⁻¹ R1 =∞ d= 1 m 2a= 1.5 m Kx= 1,6×10⁶N.m⁻¹ δp= 0.07 rd R= 0.3 m r₀= 0.45 m Rc= 100 m Ky= 5×106 N.m⁻¹ m+m = 1750 kg 2A= 18.135 m M= 3020 kg d= 1 m M = 43 200 kg R=f(γ₀;γe; e₀;r₀)

100 150 200 250 300 350 400 450 500

0 0.5 1 1.5 2 2.5x 10^-3

Variation of Rc (m)

Variation of Alfa11 (rd)

Variation of Alfa11 function of Rc

M(14,14) M(5,5) MS(8,8)

Fig. 9. Leading wheelset yaw angleα₁₁ as function ofRc.

wheel conicity the lateral displacement of leading wheelset increases fastly, there is an evident flange contact between wheel and rail (see Fig.11), which causes more interaction efforts and yields a high wear of the wheels and rail.

Figure 11shows that for γe = 0.075 there is appear- ance of the effortF₁₁for the models M(14, 14), MS (8, 8) and M (5, 5). This effort decreases according to γe. It means that we have the advantage of using worn wheels in traffic in curve. The Equation (8) is used to avoid the derailment. For the model M(5, 5) of Figure11, we ob- tain:

F₁₁(γe= 0.075)/N= 8.6173×10⁴/6.8964×10⁴= 1.2495>1.06

F₁₁(γe= 0,137)/N= 7.0507×10⁴/6.8964×10⁴

= 1.024<1.06 For the equivalent conicityγebetween the values 0.075→ 0.137 the effort between the wheel and the rail cannot be avoided (see Fig. 11), as well as the derailment of the vehicle. For the values between 0.137→ 0.25, the effort between the wheel and the rail cannot avoid, but the de- railment of the vehicle is avoided.

Lateral forceF₁₁ applied by wheelset 1 on the rail as functions of the longitudinal rigidity and the equivalent conicityγeare represented in the Figures11and12.

Figure8givesy₁₁^∗ = 0.0003 m for all the models for a longitudinal rigidityKx = 0.5×10⁵ N.m⁻¹. The radius curve is 100 m, Equation (18) givesy₀ = 0.0169 m. So

0.060 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.005

0.01 0.015 0.02 0.025

Variation of GAMMAe

Variation of y*11 (m)

Variation of y*11 function of GAMMAe

M(14,14) M(5,5) MS(8,8)

Fig. 10.Trailing wheelset lateral displacementy₁₁^∗ as function of equivalent conicityγe.

0.062 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 3

4 5 6 7 8 9x 10⁴

Variation of GAMMAE

Variation of F1 (N)

Variation of F1 function of GAMMAE

M(14,14) M(5,5) MS(8,8)

Fig. 11. Lateral force F₁₁ applied by wheelset 1 on the rail as function ofγe.

Kx= 0.5×10⁵N.m⁻¹, there is appearance of the effort F1 = 2.7×10⁴ N, for the models M (14, 14), MS (8, 8) and M (5, 5) (see Fig.12). This effort is proportional to Kx.

Figure 13 shows the lateral displacement of leading wheelset function of the speed of the vehicle. For curves Rc= 150 m, the displacement is more than 0.01 m and

0 5 10 15 x 10⁵ 2.6

2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6x 10⁴

Variation of Kx (N/m)

Variation of F1 (N)

Variation of F1 function of Kx

M(14,14) M(5,5) MS(8,8)

Fig. 12. Lateral force F₁₁ applied by wheelset 1 on the rail as function ofKx.

30 40 50 60 70 80 90 100 110 120

0.013 0.0135 0.014 0.0145 0.015 0.0155 0.016

Variation of speed (Km/h)

Variation of y11 (m)

Variation of y11 function of speed of Rc=150m

M(14,14) M(5,5) MS(8,8)

Fig. 13.Trailing wheelset lateral displacementy₁₁as function of the speed of the vehicle.

there is always appearance of the effortF₁₁of the wheel on the rail for the models M(14, 14), MS (8, 8) and M (5, 5).

The model of bogie with two wheelsets represented by the matrix M (5, 5) described perfectly the complete ve- hicle (see Figs. 6 to 13). This model is simple and does not need much calculation time compared to the model M(14, 14). The model represented by the additional ma- trix MS (8, 8) follows the model M (5, 5). The variation of the lateral displacementy₁₁^∗ ,y^∗₁₂and yaw angleα₁₁func- tion of the radiusRc curve and the longitudinal rigidity Kxshows that the models M(14, 14), MS (8, 8), and M(5, 5) stay parallel and follow the same look. All these graphs of the lateral displacements and the yaw angles are very close. Most bibliographical studies [2,5,6,11–13] adapt the model bogie with two wheelsets.

4 Conclusion

This study showed that from the fundamental princi- ple of the dynamics of a railway vehicle in relative move- ment, we obtained the general dynamic equations, which describe the behaviour of the complete vehicle in transit to the curve. From reasonable hypotheses, we obtain the representative quasi-static mathematical model of a rail- way vehicle in traffic in curve. A bogie and two wheelsets represent this model; every wheelset supports the load of the quarter of the vehicle. The simulation shows that this model describes perfectly the complete vehicle. The ad- vantage of this model is simple and does not need much calculation time compared to a complete vehicle.

Appendix A

The distancesej,rj,zkibetween the centre of inertia of the wheelset and the wheel and rail points of contact are given by:

zki=φy²_ki

2 −ε₀γ₀α²_ki

2 (A.1)

eij =e₀+ (−1)^j.y_kiγe (A.2)

rij =r₀+ (−1)^jyki.γe (A.3)

γe= Rγ₀ R−R

e₀+Rγ₀ e₀−r₀.γ₀

(A.4) ε₀= ((R+ 2r₀)γ₀−e₀) (A.5)

ϕ= 1 R−R

e₀+Rγ₀ e₀−r₀γ₀

₂

(A.6) The expressions of force and moment of centrifugal for the car body, bogie, wheelset; of force and moment inertia Coriolis wheelset’s, are given by:

Car body:

A^e_c^gRδp

=

⎧⎪

⎨

⎪⎩ M

V²

4 (ρ₁₁+ρ₁₂+ρ₂₁+ρ₂₂)−h₀δ_p −→yδp

M(Ωx2

.δ_p−→ X+Ωz2

σ−→ Z

⎫⎪

⎬

⎪⎭

G

(A.7) Bogie:

A^e_Ck^gRδp

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ Mk

V²

4 (ρ₁₁+ρ₁₂+ρ₂₁+ρ₂₂)

−A(−1)¯ ^kσ+ (H+hh¯₁−h₀)δ_p−→yδp

Mk(Ωxk2

.δ_p−→

Xk+Ωzk2

σ−→ Zk

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

Gk

(A.8)

2T^Rδp(C+Ck+Ski) =M

y²+ (Ωx2

+h₀²)θ²+Ωz2

α²−2.h₀yθ

+ 2 k=1

Mk

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

y²_k + (Ω²_xk+h²₀)θ²_k + (H+h₁−h₀)².(δpk−δp)²

−2.h₀.y_kθk−2.h₀(H+h₁−h₀).(δpk−δp).θk

+2yk(H+h₁−h₀).(δpk−δp) +Ω_zk² .αk2

⎫⎪

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎭

+ 2 k=1

2 i=

⎧⎪

⎪⎨

⎪⎪

⎩

Γ².y_ki²(m.ρ²_x+md ²) +m.ρ²_y(φ²_ki+ 2.φ_ki.α_ki.yki.Γ) +α²ki(m.ρ²z+md ²)

⎫⎪

⎪⎬

⎪⎪

⎭

+ 2 k=1

2 i=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

+(m+m)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

y²_ki+ (H+h₁+h₁+l)².(δpk−δp)²+a²σ²_k+ (H+h₁)².(δpk−δp)²−2y_ki (H+h₁)(δpk−δp)+

2y_ki(H+h₁+h₁+l)(δpk−δp)−2a(−1)⁻¹σ_ki .y_ki

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎭

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎭

(A.11)

Wheelset:

A^e_Ski^gRδp

=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

(m+m) V²

4 (ρ₁₁+ρ₁₂+ρ₂₁+ρ₂₂

−(A(−1)^k+a(−1)ⁱ) σ+ (H+h₁+h₁+l).δ_p)−→yδp

+(mρ²_x+md ²)δ_p.−−→

Xki

(mρ²_z+m.d ²).σ.−→

Zki

⎫⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎭

G_ki

(A.9)

Expressions of force and moment inertia Coriolis of the wheelset:

A^c_Ski^gRδp

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−

→0

σϕ_kim(ρ²_z−ρ²_y−ρ²_x).−−→

Xki+ δ_p.ϕ_ki..m(ρ²_y+ρ²_z−ρ²_x).−→

Zki

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

G_ki

(A.10) The expression of the kinetic energy of the vehicle estab- lished the car body, the bogie and four-wheelsets is:

See Equation (A.11) above.

The expression of elastic power is:

See Equation (A.12) next page.

The power developed by the gravity is:

See Equation (A.13) next page.

Wheel and rail interactive forces

For traffic of vehicle in a tight curve (Rc < 500 m), the reduced sliding of wheels becomes more important. In

that case, the force and moment contact are obtained by applying the law of saturation proposed by Johnson and Vermeulen [8], based on the coefficients of Kalker.

(xδpki.u(γj)) defines the tangent plan to the point of contactIkij.C₁₁,C₂₂andC₂₃are the creep Kalker coeffi- cients, which depend on elasticity modulus, the Poisson’s ratio, and the ratio between the semi axes ellipse con- tact. The creepages in the contact plans (Eq. (A.15)) are derived following the development given by Bettaieb [10].

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Xkij=C₁₁(νkij.xδpki)− C₁₁²

3μ.N(νkij.xδpki)² + C₁₁³

27μ.N(νkij.xδpki)³ Tkij=C₂₂(νkij.u(γj))− C₂₂²

3μ.N.(νkij.u(γj))² + C₂₂³

27μ.N.(νkij.u(γj))³ (A.14)

vkij =

⎧⎪

⎨

⎪⎩

(−1)^je₀

Vαki−(−1)^j e₀

Rcki(−1)^jγe

r₀yki

xδpki

χ

Vyki−αki

u(γj)

(A.15) The longitudinal, lateral creepages in the contact plan are derived following the development [3,4].

Xki1= C₁₁.γe

r₀y_ki^∗.

1 − C₁₁ 3μ.N

γe

r₀|yki^∗|+ C₁₁² 27μ².N²

γe

r₀y^∗_{ki 2}

(A.16)

P_SPRINGS=

⎧⎪

⎪⎨

⎪⎪

⎩−²

k=1

2 j=1

⎧⎪

⎪⎨

⎪⎪

⎩ Kxk

−d(−1)^j+1(αk−αki−σk) −d(−1)^j+1(α_k−α_ki−σ_k)

+Kzk

d(−1)^j+1(θ−θk+δpk−δpki) d(−1)^j+1(θ−θ_k +δ_pk −δ_pki )

⎫⎪

⎪⎬

⎪⎪

⎭

⎫⎪

⎪⎬

⎪⎪

⎭

+

⎧⎪

⎨

⎪⎩−²

k=1

2 j=1

⎧⎪

⎨

⎪⎩ +Kyk

y−yk−h₁.θ+H.θk+A(−1)^j+1α+h₁(δpk−δpki) y−y_k −h₁.θ+H.θ_k +A(−1)^j+1α

+h₁(δ_pk−δ_pki)

⎫⎪

⎬

⎪⎭

⎫⎪

⎬

⎪⎭

+

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

−²

k=1

2 i=1

2 j=1

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ +Kxk

−d(−1)^j+1(αk−αki−σki)

⎛

⎜⎝

−d(−1)^j+1. (α_k−α_ki−σ_ki)

⎞

⎟⎠

+Kyk

⎛

⎜⎝

yk−yki(1−lΓ)−h₁θk+a(−1)^j+1αk

+h₁(δpk−δpki)

⎞

⎟⎠

⎫⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎭

⎫⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎭

+

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

−²

k=1

2 i=1

2 j=1

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

y_k −y_ki (1−lΓ)−h₁θ_k+a(−1)^j+1α_k+h₁(δ_pk −δ_pki)

+Kzk

d(−1)^j+1(θk−Γ yki+δpk−δpki) d(−1)^j+1(θk−Γ yki+δpk−δpki )

⎫⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎭

⎫⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎭

(A.12)

Π^Rδp=

y(−M gδp) +θ

M gh₀(δp+θ)

⎧⎪

⎪⎨

⎪⎪

⎩ +

2 k=1

⎛

⎜⎜

⎝

+θ_k(Mk.g.h₀(δpk+θk) +y_k(−Mkgδpk)

−Mkg

δpk(H+h₁−h₀)(δ_pk −δ_p)−h₀sinθ(δ_pk−δ_p)

⎞

⎟⎟

⎠

⎫⎪

⎪⎬

⎪⎪

⎭

+ 2 k=1

2 i=1

yki −(mki+mki).g.δpki− mki+mki+Mk

2 +M 4

.g.φyki.yki

+α_ki mki+mki+Mk

2 +M

4 ).g.sin.ϕ₀.αki

(A.13)

Tki1=C₂₂.αki

1− C₂₂

3μ.N |αki|+ C₂₂² 27μ².N²α²_ki

Mki=−2.C₂₃αki; (A.17) We put

VJXki=

1− C₂₂

3.μ.N.|αki|+ C₂₂² 27.μ².N².α²_ki

;

VJTki=

1− C₁₁ 3.μ.N

γe

r₀ .|y_ki^∗|+ C₁₁² 27.μ².N²

γe

r_{0 2}

.y_ki^∗2 (A.18)

The dynamic equations of the vehicle in the quasi-static case is: M(14, 14)(X)*X = B.

See equation next page.

See Equation (A.19) next page.

M(5, 5): Matrix which describes the motions of the two identical bogies and the four wheelsets.

See Equation (A.20) in page243.

We note by MS(8, 8)(X) the additional matrix.

See Equation (A.21) in page243.