A model reduction procedure for the dynamic analysis of rail vehicles subjected to linear creep forces

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A model reduction procedure for the dynamic analysis of rail vehicles subjected to linear creep forces

McPhee, J.J. / Anderson, R.J.

The set of differential equations governing the motion of an unrestrained coned wheelset travelling on a tangent section of track and acted upon by creep forces arising from the contact between wheel and rail are, in the terminology of numerical analysis, extremely 'stiff'. This stiffness can be attributed to the existence of two real eigenvalues in the solution of the eigenproblem associated with the linearized equations of motion. Compared with the two complex conjugate eigenvalues that complete this solution, the real eigenvalues have large magnitudes and necessitate that relatively small timesteps can be used in order to obtain an accurate numerical integration for the full set of equations of motion. However, by truncating the set of left and right eigenvalues to eliminate these real eigenvalues in modal analysis of the wheelset, it was found that their contribution to the overall dynamic is negligible. The same modal truncation approach was then applied to the substructured equations of motion for a simple rail vehicle system consisting of two wheelsets connected to a main body by linear springs and dampers. Essentially, the physical degrees of freedom for each wheelset substructure were replaced by a simple complex coordinate obtained from the previous normal modes analysis. Using this model reduction procedure, accurate numerical results for the motion of the rail were generated several times faster than the results obtained by numerically integrating the full set of differential equations directly.