Multibody simulation of railway vehicles with contact lookup tables

https://doi.org/10.1016/j.ijmecsci.2018.01.020Get rights and content

Highlights

  • An efficient formulation of the equations of motion of railway vehicles is presented.

  • Symbolic computation and the use of moving reference frames is shown and discussed.

  • Wheel-rail tread contacts and flange contacts are treated with pre-calculated lookup tables which can take into account the track irregularities.

  • Numerical results when applying this formulation to a railroad vehicle show that it could simulate the motion of a railway vehicle in real time.

Abstract

The use of contact lookup tables is widely used in multibody railway simulations to increase the computational efficiency. However, due to simplifying assumptions the use of contact lookup tables decreases the accuracy of the simulation results. This paper analyses the increase of computational efficiency and loss of accuracy for a particular multibody simulation. To this end the results based on contact lookup tables are compared with the results of the online solution of the wheel-rail contact constraints. The formulation used to compute the equations of motion of railway vehicles has the following features: (1) the equations of motion are obtained using a systematic procedure based on multibody dynamics, (2) generalized forces included in the equations of motion are obtained using symbolic computations when possible, (3) generalized coordinates are referred to a non-inertial track frame, (4) the equations of motion are obtained using a velocity transformation of the Newton–Euler equations of the vehicle bodies, which are assumed to be rigid and (5) wheel-rail tread contact and flange contact are treated with pre-calculated lookup tables which can take into account the track irregularities. The comparative study presented in this paper shows that this formulation can be used to simulate the dynamics of a railway vehicle in real-time.

Introduction

The interaction between railway vehicles and track has been of interest in many investigations through the literature generally by the use of multibody dynamics techniques [1]. It is a complex dynamical system in engineering because it can be defined by many bodies, include many degrees of freedom (DOFs) and involve a high degree of complexity. This is one of the reasons why dealing with efficient but accurate models that account for the dynamics of railroad vehicles is of great interest for the research community.

When modeling the dynamics of railroad vehicles, one has to consider the high computational cost that solving the non linearities of the multibody system entails. It is important to find an adequate formulation of the equations of motion and a proper integration method for its computer implementation [2]. In addition, special attention must be paid to the solution of the wheel-rail contact scenario since the analysis of the complex geometry of bodies in contact requires an important computational effort.

In multibody dynamics, the wheel-rail contact problem is one of the most fundamental issues for evaluating vehicle stability, ride comfort or curve negotiations [3]. It is assumed to occur at a single point where the normal and tangential contact forces are applied. In this context, two well-known and widely explained procedures can be used [4]; the Constraint Method, where the contact point is assumed to occur in a single point that occupies the same position in space for both bodies [5], and the Elastic Method, where the contact point in each body can occupy different position in space, allowing to occur indentation [6]. Both approaches are greatly time-consuming and as a consequence, many research publications focused on reliable efficient methods to evaluate the contact points locations can be found in the literature.

In general, the search of the contact points can be done in two ways: the online and the offline approach. In the online one, the location of the contact points is derived throughout the numeric simulation by solving the geometric equations that govern both the rigid or the elastic method. In [6], an optimized approach for searching all possible contact points in which each candidate point is grouped into a region of penetration called batch, is presented. In addition, Malvezzi et al. [7], presented two semi-analytic procedures for the contact point detection between wheel and rail and based on the known analytic expressions of the surfaces in contact. One of these efficient semi-analytic methods is implemented in the work of Auciello et al. [8], where its computational efficiency is analyzed. Another interesting work related to the location of the wheel/rail contact points is presented by Sugiyama et al. [9], and it is based on the constraint formulation where tread, flange and back-of-flange contacts are allowed to be evaluated. Moreover, Recuero et al. [10], uses an accurate online elastic method that updates the geometry of the wheel-rail interface due to the rail flexibility.

With regard to the offline approach, which is used in this work, the concept of contact lookup table arises. A contact lookup table has solutions for specific wheelset positions relative to the track and can have their spatial derivatives and other geometric data associated with the contact points that are needed for the numerical simulations. Schupp et al. [11] proposed a quasi-elastic contact model that accounts qualitatively for the elastic deformation of the contact interface and approximates by two-dimensional splines the contact solution, whose coefficients are stored as a table in a pre-processing step. This method has been implemented in the commercial simulation package Simpack MBS [12]. Other works using the lookup table approach are those presented in [9], [13], [14]. In Meli et al. [13], the compass search together with the simplex numerical algorithm are used for the point detection, which is based on an analytical procedure that runs offline. Santamaría et al. [14] developed a procedure that accounts for the elastic contact in a lookup table approach. Here, four initial DOFs define the relative position of the wheelset with respect to the rail that are later reduced to three DOFs with the assumption that the contact point lies on radial sections at the wheel. Using an optimal number of discretized positions and the symmetric property of the problem, a reduced-size with 3-DOFs lookup table is obtained. Additionally, Sugiyama et al. [9] developed a hybrid procedure to account for the contact points both online and offline, in which a lookup table is used for evaluating the candidate points in the wheel tread while an iterative search is employed for predicting flange contact.

When evaluating the wheel-rail contact points, it is of major importance to include the effect of track irregularities, since its locations are highly influenced by these defects. However, the consideration of track irregularities when using contact lookup tables, as it is presented in further sections of this work, is a difficult task. Despite it allows a great reduction in the computational cost, it also augments the number of independent variables or entries to the table. Therefore, an efficient and simple solution to account for track irregularities when using contact lookup tables is proposed.

The goal of this paper is to develop a detailed, systematic, efficient and real-time capable procedure based on multibody dynamics to obtain the equations of motion of railway vehicles using pre-calculated contact lookup tables that account for track irregularities. To this end, this paper is described as follows: Section 2 summarizes the coordinates and reference frames used in the formulation of railway vehicles. Section 3 details the adopted notation and the kinematic description of vehicle bodies and moving frames employed. In addition, symbolic computations of the kinematic terms are derived when possible. Section 4 contains the Newton—Euler equations of motion for railroad vehicles while Section 5 presents a method to use relative coordinates associated with the kinematic joints to model the relative motion between group of bodies avoiding the use of constraints. This method is applied to a wheelset—two axleboxes group of bodies and to a carbody—two bogie frame. In Section 6, the innovative contact lookup table based on the constraint approach that accounts for track irregularities is presented. Finally, Section 7 provides the numerical solution when the proposed formulation is applied to a railway vehicle. A comparison of the efficient computational cost when using contact lookup tables is also developed and analyzed with respect to its loss in accuracy when the online constraint approach is used.

Section snippets

Coordinates and frames in multibody railway dynamics

In multibody dynamics the coordinates that are used to describe the motion of the bodies are used to obtain the position and orientation of the body frame (BF) with respect to an inertial and global frame (GF). Two families of coordinates can be used: reference coordinates, that include a fixed set of position and orientation coordinates for each of the bodies with respect to the inertial frame, and relative or joint coordinates that describe the position and orientation of the bodies with

Kinematics

This section explains in detail kinematic description used in the multibody Track Frame Formulation used in this work for the modeling and simulation of the dynamics of railway vehicles.

Newton–Euler equations for vehicle bodies

The equations of motion of the railroad vehicles obtained in this work are based on the Newton–Euler equations of the rigid bodies that comprise the vehicle. Newton equations are projected to the TF, as follows:miR¯¨Gi=F¯iwhere R¯¨Gi is the acceleration of the center of gravity of body i as given in Eq. (14) and F¯i is the sum of all forces applied to the body projected to the TF. This force vector includes applied forces (gravity force, suspension forces, aerodynamic forces and contact forces)

Models for railway vehicles with joint constraints

As said in Section 4, the relative motion of some railroad vehicle bodies can be constrained due to the existence of different kinematic joints such as revolute, spherical, prismatic or cylindrical joints. The coordinates selection and kinematics proposed in Section 3 is not efficient in this case because of the appearance of non-linear constraints that can be avoided. The method proposed in this work uses relative coordinates associated with the kinematic joints to model the relative motion

Wheel-rail contact with lookup tables

In the method proposed in this work wheel-rail contact does require the consideration of non-linear constraints. However, these constraints are treated in a very efficient manner using lookup tables. Lookup tables account for the permanent wheel-rail tread contact that is considered as perfectly rigid and also for the intermittent wheel-rail flange contact that is considered as an elastic contact, thus allowing interpenetration. This wheel-rail contact method is called hybrid.

In the following,

Simulation results and discussion: efficiency and accuracy of contact lookup tables

The goal of this section is two-fold:

  • 1

    To show that the general formulation presented for the simulation of railway vehicles works, and

  • 2

    To measure the loss in accuracy and gain in computational efficiency due to the use of contact lookup tables.

The loss of accuracy due to the use of contact lookup tables is due to two main reasons:

  • 1

    Because of the errors introduced in the geometry due to numerical interpolation, and

  • 2

    Because the influence of the wheelset-track relative yaw (angle of attack) in the

Conclusion

This paper starts with a discussion of the benefits and drawbacks of the use of absolute coordinates referred to a global frame or relative coordinates referred to a non-inertial track frame in multibody railway simulations. This work supports the use of relative coordinates and adopts a solution that requires a unique track frame to describe the motion of the whole vehicle. Up to five different types of frame are needed to analyze the motion of the vehicle bodies.

Kinematic analysis with

Acknowledgment

This research was supported by the Spanish Ministry of Economy, Industry and Competitiveness (MINECO) under the project TRA2017-86355-C2-1-R. This support is gratefully acknowledged.

References (20)

  • A.A. Shabana et al.

    Development of elastic force model for wheel/rail contact problems

    J Sound Vib

    (2004)
  • J.L. Escalona et al.

    Railroad vehicle dynamics—A roadmap to high speed trains

    J Comput Nonlinear Dyn

    (2012)
  • J. Cuadrado et al.

    Penalty, semi-recursive and hybrid methods for MBS real-time dynamics in the context of structural integrators

    Multibody Syst Dyn

    (2004)
  • A.A. Shabana et al.

    Railroad vehicle dynamics: a computational approach

    (2007)
  • A.A. Shabana et al.

    On the computer formulations of the wheel/rail contact problem

    Nonlinear Dyn

    (2005)
  • A.A. Shabana et al.

    An augmented formulation for mechanical systems with non-generalized coordinates: application to rigid body contact problems

    Nonlinear Dyn

    (2001)
  • M. Malvezzi et al.

    Determination of wheel–rail contact points with semianalytic methods

    Multibody Syst Dyn

    (2008)
  • J. Auciello et al.

    Dynamic simulation of railway vehicles: wheel/rail contact analysis

    Veh Syst Dyn

    (2009)
  • H. Sugiyama et al.

    On-line and off-line wheel/rail contact algorithm in the analysis of multibody railroad vehicle systems

    J Mech Sci Technol

    (2009)
  • A.M. Recuero et al.

    A nonlinear approach for modeling rail flexibility using the absolute nodal coordinate formulation

    Nonlinear Dyn

    (2016)
There are more references available in the full text version of this article.

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