INFLUENCE OF THE WHEEL-RAIL CONTACT ...

INFLUENCE OF THE WHEEL-RAIL CONTACT INSTATIONARY PROCESS ON CONTACT PARAMETERS L. BAEZA, F. J. FUENMAYOR, J. CARBALLEIRA, AND A. RODA Abstract. The rapid convergence of the tangential rolling contact parameters to their stationary values combined with the high computational cost associated with calculations using instationary models have meant that stationary models are usually employed in railway dynamics. However, the validity of stationary models when the applied contact conditions are subjected to rapid changes has not been sufficiently investigated. With the objective of deducing the effects of the evolution of the instationary process on the contact parameters, the tangential contact problem is solved for a set of reference conditions. For this purpose a calculation model is adapted, from which it is possible to analyse the evolution of the contact parameters when the forces exerted between rail and wheel are subjected to rapid changes. From the calculations done, situations impossible to simulate by means of stationary theories are obtained according to the frequency of variation of the forces, such as slip zones in the leading edge of the contact area and reverse contact (locally, the traction field is opposite to the direction of the external force transmitted to the contact).

1. INTRODUCTION Most rolling tangential contact models used in railway dynamics belong to stationary theories. In such models the parameters defined with respect to the contact area (e.g. traction distribution) do not depend on time. Creepage therefore only depends on the tangential forces applied, and not on the previously existing contact conditions. The application of stationary theories is an approximation to the real problem since, if the external conditions outside the contact change, the parameters associated with the contact will be affected by a process of evolution in time to adapt to the new conditions. The results of a more accurate model that includes this evolution or instationary theory will converge to those of the stationary theory if the applied external conditions remain constant for a sufficient period of time.

Key words and phrases. steady-state rolling contact, instationary rolling contact. Address correspondence to: Luis Baeza. Dpto de Ingenier´ıa Mec´ anica. Universidad Polit´ ecnica de Valencia. Camino de Vera s.n., E46022 Valencia, Spain. Tel.: +34 – 96 3877621; E-mail: [email protected]. 1

2

L. BAEZA, F. J. FUENMAYOR, J. CARBALLEIRA, AND A. RODA

The instationary model is not a dynamic model (in which inertia forces are involved), and only follows the variations of the static contact parameters due to the change in time of the normal or longitudinal external forces. The dynamic effects, which only have a noticeable effect for train speeds higher than 500 km/h (see Ref. [1, 2]), have not been considered in this work. The instationary process of the contact converges to the stationary configuration very rapidly. If the applied external conditions are kept constant, the wheel must roll over a distance ranging from two to four times the contact patch size in order to obtain the same parameters as those predicted by the stationary contact theories. This fact enables us to simulate railway dynamics through stationary theories adequately in most cases. However, there are some railway problems for which the contact conditions vary rapidly and the instationary contact process may influence the results. This is the case of shortwavelength rail corrugation and certain rolling noise and rolling contact fatigue problems, in which the railhead or the wheel tread has an irregular profile due to head/tread checks, shelling, etc. Only a few researchers have contributed with instationary rolling contact models. Kalker (see Ref. [3]) developed a three dimensional contact model based on an exact theory that models the elastic behaviour of the bodies as halfspaces. The high computational cost associated with this method motivated the development of simplified ones in which two-dimensional contact models were adopted. The solution was found using a polynomial approach [4], the Boundary Element Method [5] or a set of analytical functions [6]. An additional way to reduce computational requirements was the adoption of linearised hypotheses. Groß-Thebing and Knothe in [7, 8] considered the linearised relationship between external forces and creepages through the creep coefficients when the longitudinal external forces vary harmonically. The problem was solved in the frequency domain and the creep coefficients derived from this approach were complex values (their argument represented the phase angle between force and creepage, whereas force and creepage were in phase in the steady state case). The aim of this paper is to demonstrate the effects related to the rolling contact process that only can be simulated through a non-linear three-dimensional instationary contact model. In order to analyse such effects, the results corresponding to three different load cases are presented in Section 3. Sections 3.2 and 3.3 show the traction evolution in the contact patch and the creepages when normal and longitudinal forces vary harmonically. These results correspond to the stationary relationship between external harmonic forces and contact parameters after a transitory associated with the choice of a set of arbitrary initial

WHEEL-RAIL CONTACT INSTATIONARY PROCESS

' (

!

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# $ % &

HJ9I:

) = - ?0 . @2 / A 1 B 3 C 4 D 5 E 6 F7 G8

jJZ\k[

3

K ^L ]+

M ` _O N a0 P b2 Q c R d S e2 T fW U g V hX iY

Figure 1. Normal tractions at contact for 3 different times (t1 < t2 < t3 ) in a twodimensional problem, obtained from supposing a potential contact area with 10 elements. It can be seen that, although the value of the resultant force is increasing, the largest tractions occur at time t2 .

conditions. With the objective of illustrating the transitory characteristics, a calculation using the model in [3], which corresponds to the convergence to the stationary solution, is performed in Section 3.1. The present paper contributes with the creepage evolution process to its stationary value. The tangential contact model used in this work is based on Prof. Kalker’s CONTACT method (see reference [3] for a detailed description). The technique is summarised in Section 2. Modifications have been made to the original method in order to be able to consider normal force variations. These modifications are aimed at eliminating the errors associated with methods based on an assumed potential contact area. When a variation occurs in the normal force value, the normal tractions in the elements may not vary in the same direction as the resulting variation if a new element enters into contact, as can be observed in the scheme in the Figure 1. Errors related to the discretization of the problem are reduced when more refined meshes are employed. Another possibility is to use discretizations of the contact area adapted to its size. In this case, the discretization by Paul and Hashemi [9] may be appropriate for reproducing the tangential traction field. As can be seen in Figure 2, this discretization divides the contact area in rectangles or strips of the same width oriented in the rolling direction with each strip divided into the same number of elements. The advantage of this discretization is that it reproduces similar discretization errors in all the strips. This is the case when longitudinal creep is predominant, since similar stress distributions in the rail direction are obtained independently of the longitudinal section of the contact area analysed. As an initial hypothesis, the response of the wheel and rail materials is considered to be elastic. Although the load is high enough to reach the yield point, residual stresses, strain hardening and deformed geometry may enable (after a few load cycles) the materials to shake down to a perfectly elastic behaviour (see [10]). The viscous damping of the materials is neglected.

4


Figure 2. Discretization of the contact area according to Paul and Hashemi’s model.

The contact is assumed to happen in the wheel tread. Taking into account the reduced dimensions of the contact patch in comparison with the curvature radii of the bodies in the contact point vicinity, the contact is considered to be non-conformal (see the classification of contact types by Nowell and Hills in [11]). According to this principle, the normal contact tractions do not depend on the tangential contact tractions (or on the resultant tangential contact force), so that the tangential stresses can be calculated a posteriori, when the normal tractions are known. Also, the small magnitude of the ratio contact patch size/curvature radii leads us to consider the elastic behaviour of the bodies as halfspaces. Zili Li investigated in [12] the wheel-rail conformal contact and stated that the non-conformal hypothesis and the halfspace approximation can be adopted if the contact occurs between railhead and wheel tread. The normal contact problem can be approached by means of a non-conformal contact model such as, for example, those proposed in References [13, 14]. Bearing in mind that the model for the normal contact will not affect the results of the present work, the Hertzian contact model will be used.

2. METHOD OF CALCULATION 2.1. Kinematic model. The methodology employed to model the kinetics of the contact is a general approach in continuum mechanics of the type proposed by Mase and Mase in Reference [15]. As a preliminary stage to the development of the model, a reference co-ordinate system is defined associated with the non-deformed wheel and rail surfaces Ox1 x2 x3 (see Figure 3). The origin O is located at the contact point of the undeformed surfaces, the x1 axis being in line with the rail and positive in the rolling direction. The x1 and x2 axes define the tangential contact plane and the x3 axis is perpendicular to the surfaces at the contact point.


5

Figure 3. System of coordinates. Considering the characteristics of the non-conformal contact, the magnitudes associated with the problem can be expressed as functions of the point x = {x1 , x2 , 0}. Thus, for example, the spatial domain in which all the particles of the rail and the wheel are located is flat. In the case of the wheel, it would correspond to the development of the wheel tread surface. As regards the rail, we make use of the inertial co-ordinate system Ox1 x2 at a time t and of a railfixed system Ox10 x20 which coincides with Ox1 x2 when t = 0 (see Figure 4). We define a Reference Configuration for the undeformed rail. By means of the material coordinates Xr0 we define the position of a particle Pr in the Reference Configuration. This particle will be displaced due to elastic deformations until it reaches the position given by the spatial coordinates xr0 and xr from the fixed system and the inertial system, respectively. Let ur be the vector that defines the displacement associated with the deformation. From these magnitudes the spatial position of the particle Pr is obtained in the form

(2.1)

xr0 = Xr0 + ur

A Reference Configuration for the wheel tread is defined in its non-deformed state and with no displacement with respect to the rail. The kinematic contact model must take the wheel displacement into consideration so that, from the Reference Configuration, any other configuration can be reached through w w a displacement and a rotation of the rigid solid plus a deformation. The vectors Xw and uw 0 , x0 , x

6


!"

Figure 4. Kinematic model of the rail.

!

"$# %

Figure 5. Kinematic model of the wheel. are defined for the wheel in the same way as for the rail, with the necessary addition of the variable dw corresponding to the displacement of the undeformed configuration (see Fig. 5). For wheel particles the relation between displacements is (2.2)

w w w xw 0 = X0 + d + u

Up to this point, the fact that the rail or wheel particle may or may not be within the contact area has not been considered. There will be a set of rail and wheel points for each instant defined as contact area. Given that both rail and wheel have the same elastic behavior and that the tractions in the contact area are of equal magnitude but with opposite direction compared to those in the rail, it will be verified that (2.3)

uw (x) = −ur (x)


7

Consider two particles in the wheel and the rail at time t that have the same spatial coordinates. By prescribing the difference between the expressions (2.1) and (2.2) to be zero, we obtain (2.4)

r w w r 0 = xw 0 − x0 = X0 + d + 2 u − X0

where, since u = uw = −ur , the superindex has been omitted in the variable u. The partial derivative, with respect to time, of the expression (2.4) corresponds to the local slip is calculated as (2.5)

s=

r ∂ (xw ∂ dw ∂u 0 − x0 ) = +2 ∂t ∂t ∂t

where the partial derivative of dw corresponds to the velocity obtained when considering the contact of the undeformed bodies (local rigid slip). This magnitude can be calculated through the velocity field equation from the theoretical contact point velocity or rigid slip. That is (2.6)

dw = vG + ω w × rGO

where vG is the velocity of the wheelset mass centre, ω w is the wheelset angular velocity and rGO is the vector from the centre of mass to the theoretical contact point. If the rigid slip is divided by the vehicle velocity V , the creepages ξ1 and ξ2 are obtained (longitudinal and transversal, respectively). The spin angular velocity is defined as the projection of the wheelset angular velocity to the normal vector of the undeformed surfaces at the contact point, that is (2.7)

ωsp = ω w · n

where n is the normal unit vector. Similarly, the ξsp spin creepage can be calculated by dividing the spin angular velocity by the vehicle velocity. Expression (2.5) is then stated thus:

(2.8)

   ξ1 − x2 ξsp  ∂u s=V +2  ξ +x ξ  ∂t 2

1 sp

2.2. Numerical algorithm. The method of calculation proposes a double discretization of the problem, one in the spatial and one in the time domain. The former consists of considering, as it was proposed in the introduction, the discretization of the contact area into elements in which each magnitude is considered constant and equal to the value which it would reach in the centre of the element. Reserving the superindex in the notation to identify the element with which the magnitude is associated, equation

8


(2.8) corresponding to element J is      sJ   ξ1 − xJ ξsp  ∂ uJ 1 2 (2.9) =V +2  sJ   ξ + xJ ξ  ∂t 2 2 1 sp The temporal discretization establishes an approximation to the derivative of uJ by means of finite differences, i.e. uJ − uJo ∂ uJ ≈ ∂t ∆t

(2.10)

where uJo represents the displacements in element J associated with the deformation corresponding to the discretization carried out at time t. These displacements are due to the action of a distribution of tractions corresponding to a previous instant t − ∆t and applied on the discretization carried out for the previous instant t − ∆t. 2.3. Elastic model. Constant tractions are applied to each element J, pJ3 in the normal contact direction and pJ1 , pJ2 in the tangential directions . These variables refer to the magnitudes applied from rail to wheel and gives the resulting forces F1 , F2 and F3 , and the moment Msp , with its direction normal to the plane of contact. Supposing that the rectangular dimensions of element J are 2 bJ1 × 2 bJ2 , the resulting forces are obtained as (2.11)

Fj = 4

N X

bJ1 bJ2 pJj

j = 1, 2, 3

J=1

(2.12)

Msp = 4

N X ©

bJ1 bJ2

¡

xJ1 pJ2 − xJ2 pJ1

¢ª

J=1

N being the number of elements. Assuming linear elastic behaviour of the bodies in contact, it is possible to establish a relationship between tractions and displacements associated with elastic deformation of the type (2.13)

uJτ =

2 X N X

K DτJK κ pκ

τ = 1, 2

κ=1 K=1

where DτJK κ is the displacement according to direction τ in the centre of the J-th element when a constant unitary traction in direction κ is applied to element K. The value DτJK κ has an analytic expression if it is supposed that the bodies in contact behave elastically as infinite half-spaces. These coefficients were calculated by Kalker in [3]. From the expressions


9

q 2 2 (x2 + b2 ) + (x1 + b1 ) q (2.14) L1 (x1 , x2 ) = (x2 + b2 ) ln + 2 2 x1 − b1 + (x2 + b2 ) + (x1 − b1 ) q 2 2 x1 − b1 + (x2 − b2 ) + (x1 − b1 ) q + (x2 − b2 ) ln 2 2 x1 + b1 + (x2 − b2 ) + (x1 + b1 ) x1 + b1 +

q

2

2

(x2 + b2 ) + (x1 + b1 ) q + 2 2 x2 − b2 + (x2 − b2 ) + (x1 + b1 ) q 2 2 x2 − b2 + (x2 − b2 ) + (x1 − b1 ) q + (x1 − b1 ) ln 2 2 x2 + b2 + (x2 + b2 ) + (x1 − b1 )

(2.15) L2 (x1 , x2 ) = (x1 + b1 ) ln

x2 + b2 +

q (2.16) L3 (x1 , x2 ) =

q 2

2

2

2

(x2 + b2 ) + (x1 − b1 ) − (x2 + b2 ) + (x1 + b1 ) − q q 2 2 2 2 − (x2 − b2 ) + (x1 − b1 ) + (x2 − b2 ) + (x1 + b1 )

the expression of DτJK κ can be deduced from the value of the displacement at a point (x1 , x2 ) when in a rectangle with dimensions 2b1 × 2b2 , located at the origin of the reference system, a unitary value uniform traction is applied in the direction of the x1 axis. The displacement in the direction of x1 is (2.17)

u1 (x1 , x2 ) =

1+ν [L1 (x1 , x2 ) + (1 − ν) L2 (x1 , x2 )] πE

and in the direction of x2 (2.18)

u2 (x1 , x2 ) =

1+ν ν L3 (x1 , x2 ) πE

where E and ν are, respectively, Young’s modulus and Poisson’s ratio.

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2.4. Method for the solution of the tangential contact problem. By substituing the relationships (2.10) and (2.13) into (2.9), we obtain the system

(2.19)

      N  2 X JK K   ξ1 − xJ ξsp   sJ  X D p 2 J 2 κ 1κ 2 1 =V + − u  DJK pK  ∆t o  ξ + xJ ξ  ∆t  sJ  κ=1 K=1 2 sp κ 2κ 1 2

The problem can be solved in two different ways: the creepages or the resulting forces are used as input data. When the resulting forces are known, the problem can be solved by the system formed by (2.11), (2.12) and (2.19). In all cases and for each element J, the expression of the equation (2.19) will be determined after the condition (in adhesion or not) of the element is known. In the elements where adhesion exists, the value of sJ is zero and the value of the tangential traction is unknown. Assuming Coulomb’s law with a friction coefficient µ between the surfaces, for the elements in which there is slip (sJ 6= 0), the value of the tangential traction can be calculated as (2.20)

pJτ = −µ pJ3

sJτ k sJ k

τ = 1, 2

In bi-dimensional problems (a cylinder on an infinite half-space) the system of equations associated with (2.19) is linear. The tri-dimensional problem requires much more computation, since it is non-linear due to equation (2.20). Besides, it is necessary to consider which elements are in conditions of adhesion and which are slipping. For this reason, we will introduce a regularization of the friction coefficient. This technique performs a softer approach of Coulomb’s law, after which the tangential traction distribution vanishes as independent variable. This is carried out by the following approach (see Figure 6) Ã° °! ° sJ ° µ pJ3 sJτ 2 J (2.21) pτ ≈ − atan π ε k sJ k This expression converges to the original Coulomb’s Law when ε tends to zero. The solution of the contact problem is given through the system

(2.22)

     sJ   ξ1 − xJ ξsp  1 2 =V −  sJ   ξ + xJ ξ  2 2 1 sp

µ K ¶   ks k  K p atan  DJK sK  ε  3 4µ κ 1κ  − 2 uJ  −  K  DJK sK  ∆t o π ∆t κ=1 ks k K=1 κ 2κ 

2 X N X

The system is solved via Newton-Raphson iteration using an initial estimation of the solution.


11

Total tangential traction

Total local slip

|| s ||

Figure 6. Regularization of Coulomb’s law. In continuous trace, actual Coulomb’s law. In dashed trace, regularised law.

3. RESULTS In this section results corresponding to the solution of the tangential contact problem are presented. The results obtained from the instationary contact model will be compared with the stationary model FASTSIM. FASTSIM performs a simplification of the elastic problem by means of a Winkler model, according to which the relationship (2.13) is expressed as follow

(3.1)

J uJτ ≈ DτJJ τ pτ

τ = 1, 2

By adopting this simplifying hypothesis, the computational costs are reduced noticeably, but nonnegligible errors are introduced. In order to correct this problem, a modified version of the algorithm was developed by Shen et al. [16]. The elastic coefficients of the formulation were adjusted in such way that FASTSIM provided the same results than Kalker’s Linear Theory for small creepages. The Linear Theory implements an exact elastic model that considers no slip area in the contact patch. Consequently, if the normal hertzian contact is assumed, the right solution of the tangential problem will converge to Linear Theory’s results in the limit as the creepages approach zero. The elastic coefficients of FASTSIM are chosen in order to satisfy this condition. The FASTSIM version modified by Shen, Hedrick and Elkins was adopted in this article.

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The calculations presented below were intended to test the convergence to the stationary solution of the described model, and to evaluate the results corresponding to normal and longitudinal contact forces with high-frequency variation in magnitude.

3.1. Convergence to the stationary solution. The reason that could justify the use of stationary contact models is the rapid convergence of the instantionary contact process to the stationary solution. The results given in Figures 7 and 8 may give an understanding of how this evolution occurs. The calculations belong to the tri-dimensional version of the problem From Cattaneo to Carter, which consists in applying a constant longitudinal force just from the start. The simulation obtains the traction evolution between a cylinder and a halfspace in contact, from the case considered by Cattaneo (constant normal and tangential forces are transmitted through the non-rolling contact) to the case analysed by Carter (twodimensional stationary rolling contact). From a practical point of view, this study could be compared to the sudden application of a traction torque to a motored wheelset. The forces applied are F1 =2000 N, F3 =10000 N, and the rest of the data correspond to those given in Table 1. In Figure 7 the tractions in the rolling direction at a centred longitudinal strip of the contact area are shown. The abscissa corresponds to the longitudinal axis of the rail and shows the positions of the successive contact areas. The stationary distribution of tractions is reached when the wheel covers a distance between two and three times the longitudinal semi-axis of the elliptical area. The last calculation is compared to the traction distribution provided by FASTSIM. The traction field is different since it employs a simplified elastic model. Nevertheless, the slip and adherence zones are comparable in both models. In the upper section of the graph, the traction field applied on the contact area for three instants in time is shown. The result shown in Figure 8 corresponds to the evolution of the creepage at the center of the wheel obtained through the instationary model. This calculation is compared with the constant creepage obtained from FASTSIM. A singularity appears at the initial instant as a result of the occurrence of an instantaneous deformation which would give rise to an infinite creepage. Apart from this phenomenon, the creepage differs up to about 80% with respect to the instationary model result. The difference compared to FASTSIM is around of 3% with respect to the stationary response obtained from the exact theory at the last computed time.

3.2. Variation of the normal force. The influence of the non-steady process is more evident when there is a variation in time of the forces transmitted through the contact. In Figure 9 is given the result


13

140

2

Longitudinal traction p1(x1,0) (MN/m )

120

100

80 ← FASTSIM 60

40

20

0

−2

0

2

4 6 Longitudinal axis x1 (mm)

8

10

12

Figure 7. The lower section of the graph shows the evolution of the longitudinal traction in a centred longitudinal section of the contact area when constant normal and tangential forces are applied (F1 =2000 N, F3 =10000 N). The abscissa represents the longitudinal position of the contact area. The final calculation was compared to the FASTSIM results (dashed line). The solutions given in bold lines are given for the complete contact area in the upper section. of a calculation in which the longitudinal force is maintained constant (2000 N), and the normal force varies sinusoidally according to

µ (3.2)

F3 = 10000 + 4500 sin

2 π xwheel λ

¶ (N)

14

L. BAEZA, F. J. FUENMAYOR, J. CARBALLEIRA, AND A. RODA −4

−3

x 10

Longitudinal creepage ξ1

−3.5

−4

−4.5

−5

−5.5

−6

−6.5

0

10

20 30 40 50 60 70 Longitudinal wheel displacement xwheel (mm)

80

90

Figure 8. Evolution of longitudinal creepage as a function of wheel displacement, when constant normal and tangential forces are applied (F1 =2000 N, F3 =10000 N) . The continuous line represents the instationary model results and is compared to the FASTSIM results, shown by the dashed line. where xwheel is the position of the wheel on the rail. The calculation is performed for three wavelengths λ and compared to the FASTSIM results. The abscissa (xwheel /λ ratio) allows the different force frequencies to be compared. Significant effects of the contact process can be observed in the value of the longitudinal creepage. If we analyse the value of the longitudinal traction for the case in which the normal force varies more rapidly (λ=5 mm) we can observe a slip zone appearing in the leading edge of the contact patch. This cannot happen in a stationary model with pure longitudinal creepage. Figure 10 shows the traction field for a single application cycle of the normal force. The calculation is performed after applying various cycles of normal force with the aim of obtaining the permanent response. 3.3. Variation of the longitudinal force. Another interesting study aims at analysing the variation in the longitudinal force. This involves the application of a variable force µ (3.3)

F1 = 2000 + 1000 sin

2 π xwheel λ

¶ (N)

while a constant normal force F3 =10000 N is applied. As before, the contact problem is investigated for three wavelengths λ and compared with the FASTSIM results. When the creepage is analysed in


15

−3

0

x 10

−0.2


−0.4 −0.6 −0.8 −1 −1.2 −1.4

λ=12.5 mm λ=7.1 mm λ=5 mm FASTSIM

−1.6 −1.8 −2

0

0.5

1 1.5 2 2.5 3 Wheel position−Force wavelength Ratio xwheel /λ

3.5

4

Figure 9. Evolution of longitudinal creepage with the application of a constant longitudinal force of 2000 N and a variable normal force whose mean and alternating values are 10000 N and 4500 N, respectively. Figure 11, we observe, besides the discrepancies with FASTSIM, that ξ1 and F1 sometimes have the same direction, something that cannot happen with results from stationary models. The explanation of this phenomenon can be deduced from Figure 12, which shows the longitudinal traction field. This result corresponds to the case in which the longitudinal force varies more rapidly (λ=5 mm) after the application of several longitudinal force variation cycles in order to obtain the permanent response or limit cycle (contact parameter evolution after applying several load cycles). It can be seen that in these conditions a zone appears where the tractions are in the opposite direction to the resultant force. This effect, known as reverse slip, occurs in fretting problems [17]. 4. CONCLUSIONS In this work the evolution of the contact parameters is analysed using an instationary contact model. From the results obtained it can be seen that the instationary process has very little influence on the contact parameters. Significant differences that cannot be considered by means of stationary models only occur when the forces applied experience rapid variations. These differences may give rise to slip areas in the leading edge of the contact patch and/or reverse contact (or zones with tractions in the opposite direction to the resultant force). This could have consequences for problems in estimating wear and rolling contact fatigue. The application of this technique to vehicle dynamic simulation models may not

16


2


150

100

50

0

−2

−1

0

1

2 3 Longitudinal axis x1 (mm)

4

5

6

7

Figure 10. The lower graph shows the evolution of longitudinal traction with the application of a constant longitudinal force of 2000 N and a variable normal force with mean value 10000 N and amplitude 4500 N. The results correspond to the tractions applied on the centre longitudinal axis of the contact ellipse. The upper section shows the tractions applied on the complete contact area for the instants given in bold lines. be viable at present, due to the high computational cost involved. However, in calculations in which the contact problem is approached a posteriori (for example, as occurs in certain calculations for estimating corrugation patterns) this type of model could be of use. ACKNOWLEDGEMENTS This Project has been financed through the talgo–mec–feder plan tra2004-01828/tren.


17

−3

1.5

x 10

1


0.5 0 −0.5 −1 −1.5 λ=12.5 mm λ=7.1 mm λ=5 mm FASTSIM

−2 −2.5 −3

0

0.5

1 1.5 2 2.5 3 Wheel position−Force wavelength Ratio xwheel /λ

3.5

4

Figure 11. Evolution of longitudinal creepage with the application of a constant normal force of 10000 N and a variable longitudinal force with mean value and amplitude of 2000 and 1000 N, respectively. Table 1. Parameters of the model

Radius of railhead curvature

300 mm (convex)

Radius of wheel profile

409 mm (concave)

Wheel radius

500 mm

Vehicle speed

100 km/h

Friction coefficient

0.4 2.1 1011 N/m2

Young’s modulus Poisson’s ratio

0.3

Number of strips in spatial discretization

30

Number of elements per strip in spatial discretization

30

Train speed × time step (V ∆t)

0.05 mm

References [1] Kalker, J.J., 1979: Survey of wheel-rail rolling contact theory. Vehicle System Dynamics, 8, pp. 317-358. [2] Wang, G., Knothe, K., 1989: Influence of inertia forces on steady-state rolling contact. Acta Mechanica, 79 (3-4), pp. 221-232. [3] Kalker, J. J., 1990: Three dimensional elastic bodies in rolling contact. Kluwer Academic Publishers. ISBN 0-79230712-7.

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140

2


120

100

80

60

40

20

0

−20

−40 −3

−2

−1

0

1

2 3 Longitudinal axis x1

4

5

6

7

8

(mm)

Figure 12. The lower graph shows the evolution of the longitudinal tractions with the application of a constant normal force of 10000 N and a variable longitudinal force whose mean value is 2000 N and whose amplitude is 1000 N. The results correspond to the traction applied on the longitudinal axis of the contact ellipse. For the instants given in bold lines, the upper section shows the tractions applied on the complete contact area. [4] Nielsen, J. B. 1998: New developments in the theory of wheel/rail contact mechanics, Ph.D. Thesis, Department of Mathematical Modelling, Technical University of Denmark, Denmark. alez, J.A., Abascal, R., 2001: Solving 2D transient rolling contact problems using the BEM and mathematical [5] Gonz´ programming techniques. International Journal for Numerical Methods in Engineering, 53 (4), pp. 843-874. [6] Johnson, K.L., 1985: Contact Mechanics. Cambridge University Press, ISBN 0-521-34796-3. [7] Knothe, K., Gross-Thebing, A., 1986: Derivation of frequency dependent creep coefficients based on an elastic half-space model. Vehicle System Dynamics, 15 (3), pp. 133-153.


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[8] Knothe, K., Gross-Thebing, A., 1984: High-frequency contact mechanics: the derivation of frequency-dependent creep coefficient. Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science, 198 (12), pp. 167-173. [9] Paul, B., Hashemi, J., 1981: Contact pressures in closely conforming elastic bodies. Journal of Applied Mechanics, 48, pp 543–548. [10] Kapoor, A. and Johnson, K.L., 1992: Effect of changes in contact geometry on shakedown of surfaces in rolling/sliding contact. International Journal of Mechanical Sciences, 34(3), pp. 223-239. [11] Hills, D. A., Nowell, D., 1994: Mechanics of fretting fatigue. Kluwer Academic Publishers. ISBN 0-7923-2866-3. [12] Li, Z., 2002: Wheel-rail rolling contact and its application to wear simulation. PhD Thesis. Delft University Press, ISBN 90-407-2281-1. [13] Kik, W., Piotrowski, J., 1996: A fast approximate method to calculate normal load at contact between wheel and rail and creep forces during rolling. Proceedings of the 2nd mini conference on contact mechanics and wear of rail/wheel systems, pp 52–61. [14] Alonso, A., Gim´ enez, J. G., 2005: A new method for the solution of the normal contact problem in the dynamic simulation of railway vehicles. Vehicle System Dynamics, 43, pp 149 – 160. [15] Mase, G. T., Mase, G. E., 1999: Continuum mechanics for engineers. CRC Press, 1999. ISBN 0-8493-1855-6. [16] Shen, Z. Y., Hedrick, J. K., Elkins, J. A., 1984: A comparison of alternative creep-force models for rail vehicle dynamic analysis. Proceedings of the 8th IAVSD Symposium. Swets & Zeitlinger. [17] Tur, M., S´ aez, R. C., Fuenmayor, F. J., 2004: Analytic approach to obtain shear traction in a cylindrical contact with reverse slip. Journal of Strain Analysis for Engineering Design, 39 (6), pp 717 – 727.

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