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PROCEEDINGS OF THE FIRST INTERNATIONAL CONFERENCE ON RAILWAY TECHNOLOGY: RESEARCH, DEVELOPMENT AND MAINTENANCE
Edited by: J. Pombo
Stochastic Railway Dynamics: Modelling and Simulation
KIT-Universität Karlsruhe, Germany
W.V. Wedig, "Stochastic Railway Dynamics: Modelling and Simulation", in J. Pombo, (Editor), "Proceedings of the First International Conference on Railway Technology: Research, Development and Maintenance", Civil-Comp Press, Stirlingshire, UK, Paper 29, 2012. doi:10.4203/ccp.98.29
Keywords: random rail surfaces, stochastic numeric, extended covariance analysis, critical vehicle speeds, overcritical damping.
This paper investigates the dynamics of railway vehicles running on random track surfaces with constant velocity. The vehicle model of interest  possesses two degrees of freedom describing the vertical vibrations of the wheel and car mass. Both masses are connected by a linear dashpot and an elastic upper spring with nonlinear progressive characteristics. This system is linked to the rail surface by a second lower spring with linear characteristics. According to extended measurements the rail surface profiles are described by stationary processes with linear power density distributions in double logarithm scaling . In a first model, they are derived from white noise by means of low-pass filter equations which are transformed from way to the time domain using way and time increments of the constant vehicle velocity and the corresponding noise relation.
In the linear case, stationary vehicle vibrations are studied by means of covariance analysis and classical spectral methods. The calculated root-means-squares related to the rms-values of the base excitations are one for vanishing velocity, they become resonant when the vehicle velocity times the railway frequency reaches the natural frequencies of the two-degree-of-freedom system and vanish when the speed tends to infinity. Singular vibrations are generated when the upper damping is vanishing or increases infinitely, respectively. In between these two extremes there is a critical damping where the vehicle vibrations become minimal.
To avoid frequency-limited simulations like harmonic analysis, higher-order time-discrete schemes are introduced to the drift and diffusion terms of the vehicle and excitation equations applying iterated integrals  and their approximations by time and noise increments. Systematic errors of the applied discrete integration schemes are discussed in dependence on the vehicle speed and compared with strong solutions calculated by an extended covariance analysis. Typical results show that first order Euler schemes are not applicable in case of stiff stochastic differential equations when the upper vehicle frequency is much higher than the lower one. The vehicle models are extended to nonlinear wheel suspensions with cubic-progressive springs. Introducing relative coordinates, rmsvalues of stationary vibrations are numerically computed using higher order simulation schemes and associated nonlinear covariance equations. It is shown that increasing nonlinearities reduce the vehicle vibrations in case of weak damping. For strong overcritical damping, however, both increasing mechanisms, the system damping and the suspension nonlinearity, lead to magnifications and destabilizing effects of the stationary vehicle vibrations.
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