Keywords: multibody systems, trajectory coordinate system, railroad track flexibility.

This paper describes a particular formulation using trajectory coordinates. The use of the trajectory coordinate system (TCS) for describing the dynamics of ground vehicles is well-known in railroad dynamics [1]. However, as a specific feature, a single TCS is used as a coordinate system for all the bodies coordinates in this work. The kinematic description of a point in an arbitrary body serves as a reference to obtain the global translational acceleration of the center of mass and local angular acceleration to set up the Newton-Euler equations of motion. Subsections are devoted to presenting the way the spring-damper and gravity forces are calculated. A velocity transformation matrix that involves the kinematics of the TCS as well as the bodies' coordinates are used to convert global forces and local moments into generalized forces. This formulation, though more involved than that in the global reference coordinates, allows for a straightforward calculation of the steady motion in a curved track since the coordinates, except for the pitch motion and the arc length, remain constant. Furthermore, eigenvalue analysis gives meaningful results while negotiating a curve if the trajectory coordinates are used.

Advantages and limitations of the TCS formulation are discussed after presenting the formulation. To show the features of the formulation, numerical results of two vehicle models are presented. The unstable hunting motion of a wheelset on a straight rigid track and the time-domain response of a wagon below the critical speed when two sets of initial conditions are applied are to be found in the section on the numerical results, where special kinematic constraints are described. The paper ends with a section that summarizes the main ideas of the paper and establishes the main conclusions as well as the future work.

The main purpose for developing the formulation presented in this paper is to implement a method based on moving mode shapes [2], i.e. modes of vibration that accompanies the motion of the TCS, which avoids the issue concerning the spatial convergence of the mode shapes describing the track flexibility arising from the motion of the loads (wheels) along the track [3]. It is therefore the aim of this paper to present a dynamic formulation that will allow for capturing the dynamic interaction between the railroad vehicle and the flexible track with no increase in the number of elastic coordinates for the flexible track length.

1

A.A. Shabana, K. Zaazaa, H. Sugiyama, "Railroad vehicle dynamics. A computational approach", CRC Press, Boca Raton, 2008.

2

R. Chamorro, J.L. Escalona, M. González, "An approach for modeling long flexible bodies with application to railroad dynamics", Multibody System Dynamics, 26, 135-152, 2011. doi:10.1007/s11044-011-9255-x

3

C. Rathod, R. Chamorro, J.L. Escalona, M. El-Sibaie, A.A Shabana, "Validation of Three-dimensional Multi-body System Approach for Modelling Track Flexibility", Proc. IMechE Part K: J. Multi-Body Dynamics, 223, 269-282, 2009. doi:10.1243/14644193JMBD212

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