Elsevier

Wear

Volumes 450–451, 15 June 2020, 203150
Wear

Complex eigenvalue analysis and parameters analysis to investigate the formation of railhead corrugation in sharp curves

https://doi.org/10.1016/j.wear.2019.203150Get rights and content

Abstract

Rail corrugation, a quasi-sinusoidal irregularity of the rail head, is a common issue experienced throughout the railway networks worldwide. It generally leads to high wheel-rail dynamic loads, increased noise emission, and poor ride comfort. Most commonly, rail corrugation is likely to develop on sharp curves. This paper aims to study the numerical feasibility of the prediction of self-excited vibrations for the study of rutting corrugation formation without excitation from initial rail roughness. A finite element model for the prediction of the self-excited vibrations of the leading wheelset-rails system in a sharp curve has been developed in ABAQUS. The friction coupling between the wheel and rail is taken into account. It is assumed that the lateral creep forces between wheel and rail are quasi-saturated. The proposed model is applied to investigate the effect of several structural factors on self-excited vibrations occurrence. The obtained numerical results match closely with the typical wavelength of rutting corrugation observed in the field sites and the experimental evidence on rutting corrugation. It has been found out that the interaction effect of the wheelset cross-section and the track gauge has a significant influence on self-excited vibrations. For a typical European wheelset cross-section, self-excited vibrations only occurred under the widened track gauge. For the studied Chinese wheelset, self-excited vibrations occurred, however, only under the standard track gauge. Therefore, following the assumptions underlying the analysis, wheelset cross-section might be an inhibitor factor at a particular track gauge. A parameter sensitivity analysis shows that the friction coefficient is linearly correlated with the system's instability and the frequency of the unstable modes of vibration.

Introduction

Corrugations are a type of quasi-sinusoidal irregularities developing in both rails and wheels. The term is used with specific reference to short-wavelength irregularities. The classification used by Alias [1] considers corrugations as irregularities having wavelengths between 30 and 80 mm, in contrast to short waves (with wavelengths between 150 and 300 mm), and long waves (with wavelengths up to 2000 mm). Rail corrugation has been constituting a serious issue ever since rail transportation is utilized as a transportation means. In the 1930s, up to 46% of the German Railway network was seriously corrugated [2]. Rail corrugation gives rise to poor comfort, dynamic loading of both vehicle and track components, and high levels of noise. Measurements by Vadillo et al. [3] indicated that even in its initial phase and almost invisible, corrugation raised noise emission levels by 6 dB(A). Despite the extensive research on the topic, rail corrugation continues to be a problem with rail grinding as its principal means of control. However, grinding is rather a palliative than a solution since corrugation eventually redevelops. In the 1980s, the global annual cost of rail grinding was of the order of at least US$108 [4]. It was reported that corrugation typically occurs primarily on the low rail in sharp curves (usually curves with radius smaller than 450 m) and to a lesser extent in tangent tracks at traction or braking sites [4].

Rail corrugation mechanisms are presented by S.L. Grassie and J. Kalousek in their review [5]. This review is considered to be the most complete work on the subject, which was later revisited by Grassie [6]. Their corrugation formation mechanism is composed of a wavelength fixing mechanism and a damage mechanism. The wavelength fixing mechanism represents the vibration behavior of the system in terms of vibration modes. The mechanism claims that the vibration frequency is directly related to the corrugation formation wavelength. The damage mechanism is the rail material removal mechanism. The most common damage mechanism is wear, followed by plastic flow and plastic bending. The frequencies associated with the wavelength fixing mechanisms are directly related to the corrugation wavelength on account of the above-mentioned wear mechanism. This relation can be expressed as [6]:λ=vfwhere λ is the corrugation wavelength, v is the speed of those trains that give rise to the corrugation and f the frequency of the associated wavelength fixing mechanism.

Among the many known rail corrugation morphologies, the most common ones are the pinned-pinned resonance corrugation, P2 resonance corrugation, and rutting corrugation [5]. They are described by means of their wavelength fixing mechanism. The pinned-pinned resonance corrugation is characterized by the vertical vibration of the rails as if they were pinned at the sleepers and is related to frequencies around 1200 Hz. P2 resonance corrugation is associated with the vibration of the unsprung mass over the track stiffness and is related to frequencies between 50 and 100 Hz. These two corrugation types are commonly found in straight lines or high rail in curves.

On the other hand, rutting corrugation is associated with torsional/bending vibrations of the leading wheelsets caused by roll-slip vibrations. Here the roll-slip vibrations are related to the saturation of the traction forces, i.e., where the traction ratio (T/N) is close to the friction limit and is related to frequencies between 250 and 400 Hz. T is the tangential contact force while N is the normal contact force at the wheel-rail contact. In fact, the largest contact force values occur in the leading wheelset. Rutting corrugation is typical on sharp curves, though it can also be found in straight lines when the traction or braking forces are sufficiently high. It appears most commonly on the low rail. ‘Rutting’ has been defined as a specific type of corrugation by S. L. Grassie [5]. Discrete irregularities such as welds and joints are common ‘triggers’ for rutting corrugation, which often fix the position of corrugation along the rail. Usually, the wheel on the leading wheelset in a bogie is involved in the formation of rutting corrugation because of the high tangential force during curving. The damage mechanism for rutting is most commonly wear [6]. The effect of lateral creep force on rail corrugation on the low rail at sharp curves has been studied by Ishida et al. [7]. Their experimental investigations showed the prime relevance of lateral creep forces and rail joints on the formation of rutting corrugation, with roll-slip between rail and wheel as a possible physical cause. A non-linear time domain model of a bogie running in a curve with a modal description of the wheelset has been used for the analysis of rail corrugation in curves by Daniel et al. [8]. The analysis showed clear evidence of roll-slip phenomena, especially on the leading axle. The sliding oscillation that causes wear is mainly a roll-slip oscillation of lateral creep.

Friction modifiers (FMs) are known to be useful for reducing corrugation formation, although their effect on the system is still not fully understood [9]. Other effective corrugation mitigation methods are rail hardening, as well as employing soft railpads in the railway track [6,10,11].

The wavelength fixing mechanisms can be excited in different ways. Two main schools of thought have been hitherto developed in scientific literature. The first one considers the excitation of the system in a broad frequency range caused by the longitudinal rail head irregularity. It is assumed always present in the form of a random profile with relatively low amplitude. The fluctuation of the friction forces at the same wavelength fixing mechanism frequency leads to differential wear and, in the long run, to corrugation. As such, this theory has been implemented by using time domain accumulation wear models [10,12,13].

The second school of thought considers the instability of the wheel-rail system in specific situations, i.e., self-excited vibrations, as the cause of excitation of wavelength fixing mechanisms. The instability is considered to be caused by the interaction of the wheel and the rail through the contact patch when the creep forces are near saturation, i.e., when the maximum available tangential contact force μN is approached. A first method used to analyze the instability of the system is developed in time domain. It takes into account the roll-slip vibrations between wheel and rail, i.e., the periodic passage of the contact condition from rolling to sliding. The roll-slip vibrations are considered due to either the periodic variation of the normal contact force between wheel and rail near saturation, or to the negative slope of the friction-creepage relationship after saturation [13,14]. Recently, the instability of the system has been analyzed in the frequency domain by considering the friction-induced dynamic instabilities of the system due to the quasi-saturated friction forces in the neighborhood of a steady-state working condition of the system [9,15,16]. This method may be suitable for the prediction of rutting corrugation in sharp curves or in braking or traction sites. It is also suitable for performing multi-case analyses since frequency domain analyses are generally less time consuming compared to time domain models.

ABAQUS provides an implemented algorithm for the detection of friction-induced self-excited unstable vibration modes, i.e., complex eigenvalue analysis (CEA). In 2006, AbuBakar and his co-authors [17,18] used the CEA in combination with a transient dynamic analysis for predicting brake squeal frequencies, and the effect of temperature-dependent friction of the brake stability. In 2007, Liu et al. [19] performed a parametric analysis of brake squeal by using the same method. In 2009, Nouby et al. [20] applied the CEA in combination with a design of experiment analysis (DOE) in order to perform a parametric analysis of disc brake squeal by using statistical regression techniques. The CEA has been used by Chen et al. to study both rail corrugation and wheel-rail squeal due to self-excited vibrations under saturated creep forces [9,15,16]. In these works, a transient dynamic analysis together with the CEA was used to predict rail corrugation induced by self-excited vibrations. The analysis showed how self-excited vibrations lead to fluctuation of the contact forces. However, the overall physical mechanism of generation of rail corrugation is still not fully understood.

The general aim of this paper is to further expand knowledge on the deployment of the CEA for studying rail corrugation formation in sharp curves. A numerical procedure aiming at the prediction of rutting corrugation on sharp curves is carried out through the detection of the self-excited vibrations of the wheelset-track system. The numerical model includes a full track section and a leading wheelset. It is fully implemented in a Finite Element (FE) environment. Furthermore, a parametric analysis is performed by using the Design of Experiment methodology to identify parameters having significant influence on rail corrugation formation. The overall analysis methodology adopted in this paper combines multibody simulations for correctly defining the boundary conditions of the FE model and the steady-state position of the wheelset. The FE stability analyses are conducted in ABAQUS 6.14.

Section snippets

Wheel-rail contact model

The FE contact interaction model, which considers a surface-based approach, is applied to model the wheel-rail contact. The numerical discretization method considers the shape of both master and slave surfaces, such that the contact condition is distributed over the regions near the contacting nodes. The normal interaction between the contacting surfaces is modeled as a hard relationship between the normal pressure and the surfaces overclosure, i.e., the surfaces penetration. This means that no

Complex eigenvalue analysis (CEA)

The analysis used the official Abaqus User's Manual [21] as technical reference. The complex eigenvalue analysis (CEA) has been widely used mainly as a tool for predicting unstable frequencies in friction-induced vibration problems. The method consists of the computation of system eigenvalues, which in general are complex-valued functions, and their corresponding mode shapes. The real part of the eigenvalues is deployed as parameter correlated to the stability of the particular mode shape. In

Wheelset-track model

The wheelset-track assembly represents a sharp curve section with curve radius 300 m, superelevation or cant 100 mm, rail inclination 1:20, and sleeper distance 600 mm. Two different track gauges are considered in the analysis: the standard track gauge 1435 mm and a widened track gauge 1455 mm. Considering the track curvature, rails are modeled with a rail curvature angle β=l/r, where l is the rail track length and r is the curve radius.

The analysis considers standard rail, sleeper, wheelset

Results and discussion

The results of the analysis are given in terms of complex eigenvalues, complex eigenvectors, i.e., mode shapes, and their associated effective damping ratio in the frequency range of interest. In ABAQUS, the effective damping ratios are collected in the output variable DAMPRATIO. Fig. 5a shows the typical distribution of the complex eigenvalues of the system in terms of effective damping ratio in the frequency range of interest [0–1200 Hz]. The distribution is similar, although not identical,

Conclusions

This paper focuses on the still open research question of rail corrugation. Friction-induced self-excited vibration is one of the mechanisms proposed earlier as the cause of corrugation, but was devoted less attention in research work compared to other proposed explanations of this phenomenon. This work aims at expanding the knowledge of this less-researched theory/mechanism. A 3D finite element model, considering the wheelset and the track, was developed in ABAQUS. The effective damping ratio

Acknowledgments

The authors gratefully thank Prof. G.X. Chen of the Tribology Research Institute, State Key Laboratory of Traction Power, Southwest Jiaotong University for valuable comments and discussions.

References (29)

  • M. Oregui et al.

    An investigation into the modeling of railway fastening

    Int. J. Mech. Sci.

    (2015)
  • J. ~Alias

    Characteristics of wave formation in rails

    Rail Int.

    (1986)
  • J.K. Oostermeijer

    Review on short pitch rail corrugation studies

    Wear

    (2008)
  • S.L. Grassie et al.

    Rail corrugation on north american transit systems

    Veh. Syst. Dyn.

    (1998)
  • Cited by (0)

    View full text