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Research Papers

# Spectral Distribution of Derailment Coefficient in Non-Linear Model of Railway Vehicle–Track System With Random Track Irregularities

[+] Author and Article Information
Ewa Kardas-Cinal

Faculty of Transport,
Warsaw University of Technology,
Koszykowa 75,
Warsaw 00-662, Poland
e-mail: ekc@it.pw.edu.pl

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received March 30, 2012; final manuscript received December 23, 2012; published online March 21, 2013. Assoc. Editor: Khaled E. Zaazaa.

J. Comput. Nonlinear Dynam 8(3), 031014 (Mar 21, 2013) (9 pages) Paper No: CND-12-1055; doi: 10.1115/1.4023352 History: Received March 30, 2012; Revised December 23, 2012

## Abstract

Improving the running safety and reducing the risk of derailments are the key objectives in the assessment of the running characteristics of railway vehicles. The present study of the safety against derailment is focused on the effect of wheelset hunting on the derailment coefficient $Y/Q$ and, especially, how it is reflected in the power spectral density (PSD) of $Y/Q$. The lateral Y and vertical Q forces at the wheel/rail contact are obtained in numerical simulations for a four-axle railway vehicle moving at a constant velocity along a tangent track with random geometrical irregularities. The PSD of $Y/Q$, calculated as a function of spatial frequency, is found to have a characteristic structure with three peaks for the leading wheelsets and one peak for the trailing wheelsets of the front and rear bogies. The positions of the PSD maxima remain unchanged with increasing ride velocity, while their magnitudes and shapes evolve. One of the PSD peaks occurs for all wheelsets at the same spatial frequency corresponding to the wheelset hunting, while an additional peak at the double hunting frequency is found for the leading wheelsets. Such a peak structure is also found in the PSD of $Y/Q$ determined in simulations with modified parameters of the vehicle primary suspension and for different track sections. The peak at the double hunting frequency is shown, by a detailed analysis of the contact forces, the flange angles and their PSDs, to result from the nonlinear geometry of the wheel/rail contact leading to the second-harmonic term in $Y/Q$. The emergence of this peak is also closely related to the phase difference between the hunting oscillations of the wheelset lateral displacement and the oscillations of its yaw angle, for which the difference is significantly smaller for the leading wheelset than for the trailing one. Finally, the effect of wheelset hunting is also shown to manifest itself in the strong dependence of the running average of $Y/Q$, which is used in the railway technical safety standards for the assessment of the safety against derailment (with the Nadal criterion), on the applied window width.

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## Figures

Fig. 1

Power spectral densities of geometrical track irregularities: (a) lateral (solid line) and vertical (dashed line), and (b) local superelevation (solid line) and half track gauge (dashed line)

Fig. 2

Force components at the wheel/rail contact point: lateral (Y) and vertical (Q), normal force (N), and transverse rolling-friction force (F)

Fig. 3

Contact force ratio Y/Q (derailment coefficient) for the leading wheelset (left wheel) of the front bogie: v=200 km/h. The horizontal lines mark the 99.85-percentile values (Y/Q)Δx|0.9985 of the running average (Y/Q)Δx found for various window widths Δx[m]. The value of (Y/Q)Δx|0.9985 for Δx=0 corresponds to the raw signal Y/Q.

Fig. 4

(a) Contact force ratio Y/Q, and (b) the running average (Y/Q)2m: v=200 km/h

Fig. 5

Power spectral density of Y/Q for the leading wheelset (left wheel) of the front bogie for various ride velocities: v = 80, 120, 160, and 200 km/h

Fig. 6

Power spectral density of Y/Q for the trailing wheelset (left wheel) of the front bogie for various ride velocities: v = 80, 120, 160, and 200 km/h

Fig. 7

Power spectral density of Y/Q for the (a) leading, and (b) trailing wheelsets (left wheel) of the rear bogie for v = 200 km/h

Fig. 8

(a) Wheelset lateral displacement y1*=y1-yw, its yaw angle ψ1*=ψ1-ψw, (c) flange angle γ1L, (e) components Y1L,cos=F1L cosγ1L, Y1L,sin=N1L sinγ1L of the lateral force Y1L at the wheel/rail contact point (see Eq. (2)), and (g) the ratio Y1L/Q1L for the leading wheelset (left wheel) of the front bogie over a 100 m track section: v = 200 km/h. The corresponding power spectral densities are shown in panels (b), (d), (f), and (h).

Fig. 9

(a) Wheelset lateral displacement y2*=y2-yw, its yaw angle ψ2*=ψ2-ψw, (c) flange angle γ2L, (e) components Y2L,cos=F2L cosγ2L, Y2L,sin=N2L sinγ2L of the lateral force Y2L at the wheel/rail contact point (see Eq. (2)), and (g) the ratio Y2L/Q2L for the trailing wheelset (left wheel) of the front bogie over a 100 m track section: v = 200 km/h. The corresponding power spectral densities are shown in panels (b), (d), (f), and (h).

Fig. 10

Power spectral density of Y/Q for the (a),(c) leading, and (b),(d) trailing wheelsets (left wheel) of the front bogie for modified parameters of the primary suspension: (a),(b) lateral stiffness constant (on one side of wheelset) 1000 kN/m (nominal value 1962 kN/m), and (c),(d) zero vertical damping (damper failure)

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