Dynamic parameters
evaluations of rail wheels with uncertain material parameters
T. Uhl (Department of Robotics and
Шадрина Н.Ю. (ПГУПС, Санкт – Петербург, РФ)
1 Introduction
The interaction between the wheel and the rail
is the physical basis of rolling stock traffic over railroads. The traffic
safety and main technical and economic characteristics of the track and rolling
stock service depend upon parameters of this interaction. So, in particular, the
energy loss caused by wear in the wheel-rail system equals to 10-30% to be
spent on fuel and energy for train traction [1]. In addition, expenses on
renovation and rails and wheel pairs are a mere part of the total expenditure
on maintenance sections and locomotive and railcar depots respectively. Locomotive
and railcar depots incur especially large costs due to such expenses, as the
mean life of a wheel pair has considerably decreased for the last half a
century.
The contact between a highly worn
rail and a new or worn wheel changes the shape of a pressure distribution
region. The contact area size essentially decreased, it gets shifted towards
the outer surface of the exterior rail leading to an increase of the contact
pressure values, which levels can reach the yield strength and result in
plastic deformation of the rail head.
High contact stress arises in the
event that the wheel profile is supported by the rail with its external edge or
the contact area does not reach the external wheel edge leading to the
appearance of a shoulder (false flange) in the area of outer part of the wheel
rolling surface.
The contact stress value and
distribution significantly depend upon the wheel and rail profiles and upon
what kind of contact occurs, namely, single-point or and double-point. With a
conformal profile, the contact area size increases leading to a decrease in the
contact stress level as compared with non-conformal profiles.
Many researchers in the
whole world have investigated the dynamic parameter problem for the wheel-rail
system, for example, A.M. Fridberg, Marek Sitarz, Aleksander
Sladkowski, Kevin Sawley, Huimin Wu. In solving this
problem by finite element method, four types of grids are used, namely: linear
tetrahedral, linear hexahedral, quadratic tetrahedral or quadratic hexahedral. As
the practice ha shown, the most accurate agreement between a full-scale
experiment and a numerical experiment occurs with the use of a quadratic
hexahedral grid in the finite element model. It should be noted that in studying
by finite element method it is necessary to determine whether the h-method or
the p-method will be used. In using the h-method, the accuracy of results is
directly proportional to the number of finite elements. The
more the number of elements, the more accurate the results. In using the
p-method, the accuracy of results depends upon the length of the polynomial of
the finite element shape function. The more accurately the shape function is
described, the more accurate the results.
At the present time, the
most popular finite element software packages being used for studying the
wheel-rail system are MSC.Patran, MSC.Nastran, FEMAP, ANSYS, CosmosWorks, Pro/Mechanica. In this paper we present a MSC.Patran [3] software package
as the model pre-processor and post-processor whereas a MSC.Nastran
software package is the solver.
We consider a problem of influence of the
changes in such finite element model parameters as Young modulus and material
density on natural frequencies and modes of the wheel-rail system oscillations.
And for evaluating the total stressed state in the wheel-rail pair it is
sufficient to solve the Hertz problem for a single-point contact between the
shroud and the rail. This problem has been solved within the MSC.Nastran software package [4].
2. Description of
used model
In constructing the wheel-rail model
and making calculations, we have considered an R65 rail type according to GOST
18267-82 and a shroud according to GOST 398-96. The mechanical properties of
the shroud and the rail being used for calculations are given in Table 1.
Table 1 - The mechanical properties
of the shroud and the rail
|
The mechanical properties |
the shroud GOST 398 – 96 |
the rail GOST 18267 – 82 |
|
density, kg/m3 |
7850 |
7850 |
|
Young's modulus, Pa |
|
|
|
Poisson's ratio |
0,3 |
0,3 |
Fig. 1 shows geometrical parameters of the
wheel with a profile according to GOST.
Fig.1. Wheel geometry with the GOST profile
The diagram
of forces loading the wheel pair is given in Fig. 2.
Fig.2. Diagram of forces loading the wheel pair
3 Dynamic analyses
In the global Cartesian coordinate system, the wheel pair has
been divided into quadratic finite elements [2]. Two identical solid-state models with the
same kinematic and boundary conditions but with
different grids have been built for studying the system behaviour under the
variation of material parameters. The number of finite elements is 11772 and 53470 for the first and second models respectively. The finite element grid for the first model
is given in Fig. 3.
|
Fig. 3. Finite element grid |
In doing
so, each finite element
model has four sub-models
having different Young modulus and material density values. In simulating in accordance with the vertex
method, the mechanical parameters of steel specified in GOST and given in
Table 1 have been used as well as those under which the Young modulus and material density values for the wheel
varied within a range of ±10%. |
The vertex method being used in
solving such problems has to a sufficient extent described in [5].
This method approximates
the range of the result of a numerical procedure by introducing all possible
combinations of the boundary values of the input intervals into the analysis.
For N input intervals, there are 
vertices for which the analysis
has to be performed. These vertices are denoted by 
.
Each of these represents one unique combination of lower and upper bounds on
the N input intervals. The approximate analysis range is deduced from the
extreme values of the set of results for these vertices:
(1.1.)
Despite its simplicity, this
method has some important disadvantages. It is clear from equation (1.1.) that
the computational cost increases exponentially with the number of input
intervals. This limits the applicability of the vertex method to rather small
systems, or systems with very few interval entries in the system matrices.
The main disadvantage of
this method, however, is that it cannot identify local optima of the analysis
function which are not on the vertex of the input space. It only results in the
smallest hypercube if the analysis function is monotonic over the considered
input range. This is a strong condition that is difficult to verify for FE
analysis because of the complicated relation of analysis output to physical
input uncertainties. The approximation obtained when monotonicity
is not guaranteed is not necessarily conservative. This fact reduces the
validity of this method for design validation purposes.
4 Results
Fig. 4 and Tables 2 and 3 describe
methods for deriving 10 natural oscillation frequencies under variations of the
Young
modulus and material density values.
Fig. 4. Normal oscillation
modes for one of the natural oscillation frequencies
Table 2- Model with 11772 elements and node 20948, for a case, when
|
|
|
|
|
|
|
7.187736E+01 |
7.964701E+01 |
6.501552E+01 |
7.204343E+01 |
7.208932E+01 |
|
7.897687E+01 |
8.751394E+01 |
7.143726E+01 |
7.915934E+01 |
1.211863E+02 |
|
7.943273E+01 |
8.801909E+01 |
7.184960E+01 |
7.961626E+01 |
1.239039E+02 |
|
1.221843E+02 |
1.353919E+02 |
1.105199E+02 |
1.224666E+02 |
1.344894E+02 |
|
1.236157E+02 |
1.369781E+02 |
1.118146E+02 |
1.239013E+02 |
1.377132E+02 |
|
2.436237E+02 |
2.699584E+02 |
2.203659E+02 |
2.441866E+02 |
1.967976E+02 |
|
2.579305E+02 |
2.858118E+02 |
2.333069E+02 |
2.585265E+02 |
2.443459E+02 |
|
2.598499E+02 |
2.879386E+02 |
2.350431E+02 |
2.604503E+02 |
3.177438E+02 |
|
3.168070E+02 |
3.510526E+02 |
2.865627E+02 |
3.175390E+02 |
3.179060E+02 |
|
3.169713E+02 |
3.512346E+02 |
2.867113E+02 |
3.177036E+02 |
3.179443E+02 |
|
3.170099E+02 |
3.512774E+02 |
2.867462E+02 |
3.177423E+02 |
3.182450E+02 |
|
3.173091E+02 |
3.516090E+02 |
2.870169E+02 |
3.180423E+02 |
4.664863E+02 |
|
3.651834E+02 |
4.046583E+02 |
3.303208E+02 |
3.660272E+02 |
4.929135E+02 |
|
5.513638E+02 |
6.109641E+02 |
4.987273E+02 |
5.526378E+02 |
5.544852E+02 |
Table 3 - Model with 53470 elements and node 86115, for a case, when
|
|
|
|
|
|
|
7.180093E+01 |
7.956232E+01 |
6.494637E+01 |
7.196683E+01 |
7.201265E+01 |
|
7.771822E+01 |
8.611926E+01 |
7.029877E+01 |
7.789780E+01 |
7.794740E+01 |
|
7.783279E+01 |
8.624620E+01 |
7.040240E+01 |
7.801263E+01 |
7.806229E+01 |
|
1.181703E+02 |
1.309441E+02 |
1.068891E+02 |
1.184434E+02 |
1.185188E+02 |
|
1.185660E+02 |
1.313826E+02 |
1.072470E+02 |
1.188400E+02 |
1.189157E+02 |
|
2.406582E+02 |
2.666724E+02 |
2.176835E+02 |
2.412142E+02 |
2.413678E+02 |
|
2.522516E+02 |
2.795190E+02 |
2.281702E+02 |
2.528345E+02 |
2.529955E+02 |
|
2.528631E+02 |
2.801966E+02 |
2.287233E+02 |
2.534473E+02 |
2.536087E+02 |
|
3.148809E+02 |
3.489183E+02 |
2.848205E+02 |
3.156085E+02 |
3.158094E+02 |
|
3.149330E+02 |
3.489760E+02 |
2.848676E+02 |
3.156607E+02 |
3.158617E+02 |
|
3.151175E+02 |
3.491805E+02 |
2.850345E+02 |
3.158456E+02 |
3.160467E+02 |
|
3.151764E+02 |
3.492457E+02 |
2.850878E+02 |
3.159047E+02 |
3.161058E+02 |
|
3.610077E+02 |
4.000312E+02 |
3.265438E+02 |
3.618419E+02 |
3.620722E+02 |
|
5.470135E+02 |
6.061436E+02 |
4.947924E+02 |
5.482774E+02 |
5.486265E+02 |
5 Conclusions
The paper
contains the description of some problems relating to simulation methods for
the wheel-rail system given the contact interaction. In our investigations, we
have used the vertex method for variable parameter simulation. The vertex
method provides reliable and most accurate results provided the functions
describe the input and output parameter relationship unambiguously and
continuously. The main inconvenience of the vertex method consists of the fact
that it is necessary to make a considerable number of various simulations in
order to gain the required results. In this case, the result can be normal
oscillation modes and natural frequencies under variable Young modulus and
material density values.
The results obtained are only the first step in analysing
the wheel-rail contact problem, and they can be used further as the basis for
future investigations in problems relating to the dependence of fault
occurrence due to variation of basic wheel material properties.
Acknowledgements
This work was made
possible under the MADUSE EU Research & Training Network MRTN-CT-2003-505164.
References
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