Keywords: eigenvalue analysis, stability, railroad vehicle, track flexibility.

The lateral stability of railroad vehicles is a classical problem of vehicle dynamics that is very important for the design of such vehicles. Above a certain speed the centred position of the vehicle in the track becomes unstable and undamped lateral oscillations appear. The so-called hunting motion must be avoided by keeping the forward velocity of the railroad vehicle below that speed. Methods for stability analysis can be grouped into three types: the brute force approach; nonlinear stability analysis of limit cycles; and eigenvalue analysis.

This paper deals with the eigenvalue analysis of multibody models of railroad vehicles moving steadily on periodic tracks. The following three methods are explained in the paper:

• Linear models of the lateral dynamics of railroad vehicles.
• Nonlinear multibody models of the three-dimensional dynamics of railroad vehicles based on global reference coordinates.
• Nonlinear multibody models of the three-dimensional dynamics of railroad vehicles based on trajectory coordinates.

For the linear models the steady motion calculation is simply the solution of a system of linear equations and the eigenvalue analysis does not require linearization for obvious reasons.

Nonlinear multibody models based on global coordinates show periodic orbits associated with the vehicle steady motions. In order to simplify the stability analysis, coordinate transformations can be used to get the steady motions in terms of a constant set of coordinates [1]. In this case the calculation of the steady motion requires the solution of non-linear algebraic equations. If the differential-algebraic equations (DAE) of motion are used in these calculations, a non-standard direct eigenvalue analysis can be used to obtain the system dynamics. If the ordinary differential equation (ODE) form of the equations of motion written in terms of independent coordinates is used, the eigenvalue analysis is the standard for ODE systems, however this method requires further computations.

Nonlinear multibody models based on trajectory coordinates are more convenient for the eigenvalue analysis than those based on global coordinates. The reason in that the vehicle steady motions are directly obtained as a set of constant coordinates. After this point the calculation of the steady motions and the eigenvalue analysis is completely equivalent. The use of the trajectory coordinate system allows the description of the track deformation with the moving shape functions method [2]. This method can be used to study the coupled vehicle-track dynamics including the eigenvalue analysis for stability calculations.

1

J. Escalona, R. Chamorro, "Stability analysis of vehicles on circular motions using multibody dynamics", Nonlinear Dynamics, 53, 237-250, 2008. doi:10.1007/s11071-007-9311-5

2

R. Chamorro, J. 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

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