A railway track dynamics model based on modal substructuring and a cyclic boundary condition
Introduction
The coupled dynamic response of a railway vehicle and the track is of great practical interest to the industry. It is associated with some very important railway engineering problems such as wheelflats or other wheel tread irregularities (out of round wheels) [1], rail corrugation [2], rail head defects [3] and rolling noise [4]. In order to analyse these and other problems, many researchers have developed dynamic vehicle–track models and these models basically differ in how the track is treated. The track is a large system that can be considered as an infinite structure and it is made up of the ballast and elastomeric materials which have nonlinear behaviour. Due to these complex characteristics, models for only finite nonlinear or infinite linear track have been established and can be found in the literature.
A first approach of an infinite track consisted of a homogeneous track that was modelled as a Winkler beam. The infinite Euler beam on discrete elastic supports was analysed in [5] by means of Fourier series. A sophisticated homogeneous track model was developed in [6] for modelling a slab track with embedded rails. The slab was modelled as a Kirchhoff plate on an elastic foundation; the rails were Euler beams and were continuously supported on the slab. The wheel–rail contact force was supposed to be harmonic and the steady-state response was calculated. The solution was obtained by using the Fourier transform in the frequency and the wavenumber domains. A similar procedure was presented in references [7], [8]. The first work [7] considered an infinite double beam system that modelled a rail on a continuous slab. The second article [8] led to a model of the track with discrete supports (sleepers). The rails (Euler beams) were periodically supported on rigid sleepers on an infinite elastic layer. The method presented in [9] studied vibration induced by traffic and modelled the infinite soil in the frequency-wavenumber domain through the boundary element method. Following a different approach, the periodicity condition and the time Fourier transformation were employed in order to study an infinite Euler beam on discrete elastic supports [10].
The U-transformation method was employed in [11] for modelling a beam on discrete rigid supports. The U-transformation uncoupled the equations of motion together with the boundary and continuity conditions for N periodic structures. The method led to an infinite track when N approaches infinity.
Another approach was based on Mead’s works (e.g. [12]) which implemented a technique for analysing wave propagation in infinite periodic structures in order to model a railway track. Some researchers extended this approach to other railway applications: rolling noise [13], [14]; corrugation problem [15]; the dynamic response of the vehicle to track irregularities [16].
Nonlinear models of wheel–rail contact or any other interfaces of the system have been studied with a finite track. Finite-length track models are characterised by the end-rail effects. Waves originating from the vehicle–track interaction get reflected at the end-track and interact with the vehicle if the distance from the vehicle to the rail-end is small. Therefore the distance between the vehicle and the rail-end must be sufficiently long to avoid wave reflection; and thus such track models need a large number of degrees of freedom.
The Finite Element Method (FEM) is one of the most versatile techniques and it has been implemented for modelling the complete track in [17], [18]. The FEM is known to provide a large number of coordinates and it can be reduced through a modal approach [19], [20] where a small number of vibration modes are retained in the dynamic analysis. The paper presented in [21] led to a reduction of the computational cost through a substructuring technique where the rail vibration was described by a modal approach. The mode shapes of the rail were used to describe the deformed rail and its dynamic response and they were obtained from the free–free boundary conditions. The high-frequency dynamics of the track (400 Hz to 1.5 kHz) can be modelled properly through a Winkler beam [4], in which case the mode shapes of the real rail are very similar to those of a free–free beam. Consequently, the number of modes of the rail (and modal coordinates) necessary for simulating the behaviour of the track below 1.5 kHz is reduced.
The present work presents a method for simulating the dynamic interaction of a railway vehicle and the track. It adopts an infinite cyclic track–train system on which an infinite set of identical vehicles are operating and the rail is represented by a modal approach. The rail is modelled as a cyclic Timoshenko beam that allows bending in the vertical and horizontal directions, and torsion. The technique can consider nonlinearities.
The main characteristics of the cyclic track–train system are developed in Section 2 of the present article. The modal formulation of a cyclic beam is presented in Section 3 of this article. The train–track model is described in Section 4. The present method facilitates the studies of a variety of dynamic problems. The results of some practical problems associated with the ballast characteristics are presented in Section 5.
Section snippets
The cyclic train–track model
The cyclic model provides some benefits with respect to the finite track model and classical infinite model. The finite track models need to keep the vehicle on the track and consequently, the time interval of integration must be shorter than the track length divided by the train velocity; moreover the wheelsets keep distance from the extreme ends of the track in order to avoid the interaction with the waves that reflect at the track end; the length of the cyclic model does not depend on the
The cyclic Timoshenko beam
Consider the free vibration of a Timoshenko beam of constant cross-section. Its equations of motion are:where w=w(x, t) is the transversal displacement and ψ=ψ(x, t) is the beam section rotation; E is the Young’s modulus, I is the second moment of area, A is the cross-sectional area, κ is the Timoshenko shear coefficient, m is the linear mass density and G is the shear modulus. The solution has the following form:
Proposed model
The dynamic model employs a substructuring technique. The whole system consists of the following substructures: rails, the vehicle and the under-rail substructures. In the present work the under-rail substructures are the sleepers. The equations of motion of each substructure are considered separately, and the whole system is brought together through the forces at the wheel/rail contact and the railpad. The vehicle equations of motion are expressed in a set of physical coordinates,
Application to hanging sleepers and ballast compaction force
This section presents results from the dynamic model applied to a number of situations. The calculations are aimed at investigating certain engineering problems associated with ballast conditions. They are the presence of a hanging or unsupported sleeper and the ballast compaction types. Both cases concern the maintenance operations through tamping. The dynamics in the presence of a hanging sleeper was studied in [22] through an FE model; different cases of ballast compaction under the sleeper
Conclusions
The present work developed a new methodology for modelling the coupled dynamics of a railway vehicle and the track. The technique considered an infinite cyclic train–track system, on which an infinite set of identical railway vehicles were running. This reasonable simplification led to fairly realistic results and at the same time could accommodate nonlinearities such as a gap at the sleeper support or in the wheel–rail contact force model. The modal approach permitted simulations through a
Acknowledgements
The first author gratefully acknowledges the support for this work provided by the Project TRA2010-15669 (Ministerio de Ciencia e Innovación).
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