Mode repulsion of ultrasonic guided waves in rails

Abstract

Accurate computation of dispersion characteristics of guided waves in rails is important during the development of inspection and monitoring systems. Wavenumber versus frequency curves computed by the semi-analytical finite element method exhibit mode repulsion and mode crossing which can be difficult to distinguish. Eigenvalue derivatives, with respect to the wavenumber, are used to investigate these regions. A term causing repulsion between two modes is identified and a condition for two modes to cross is established. In symmetric rail profiles the mode shapes are either symmetric or antisymmetric. Symmetric and antisymmetric modes can cross each other while the modes within the symmetric and antisymmetric families do not appear to cross. The modes can therefore be numbered in the same way that Lamb waves in plates are numbered, making it easier to communicate results. The derivative of the eigenvectors with respect to wavenumber contains the same repulsion term and shows how the mode shapes swop during a repulsion. The introduction of even a small asymmetry appears to lead to repulsion forces that prevent any mode crossings. Measurements on a continuously welded rail track were performed to illustrate a mode repulsion.

Introduction

Rail track inspection and monitoring systems have been developed based on guided wave ultrasound [1]. These systems operate at frequencies where dozens of guided wave modes can propagate. Some of these modes can propagate large distances (1 – 2 km) making it possible to monitor a long section of rail from a single transducer location. In addition to being multi-modal, signals can exhibit significant dispersion. Modelling of the wave propagation characteristics is required during the development of these systems and is also used in the analysis of signals obtained. As the rail has a complex cross-section, numerical modelling is required and the semi-analytical finite element (SAFE) method is used by various research groups. This method is very powerful for constant cross-section waveguides such as rail. Dispersion characteristics can be computed from the SAFE model at discrete wavenumbers or discrete frequencies. The behaviour of a selected mode of propagation is often required over a range of frequencies and it is then necessary to track this mode from one solution frequency to the next. This need arises in a number of situations including: designing a transducer to excite a particular mode; computing the reflection, transmission and mode coupling of selected modes by a defect or when performing phased array signal processing to preferentially transmit and receive selected mode combinations over the range of frequencies included in the excitation signal.

Dispersion curves, computed by the SAFE method, can show sudden deviations as two modes approach each other. In some cases the wavenumber versus frequency curves for two modes cross each other but in other cases they converge and then suddenly diverge without crossing. Sometimes it is difficult to determine whether two modes have crossed or not and this makes it difficult to track a particular mode. Similar behaviour has been observed for Lamb waves in plates and this mode repulsion was analysed by Überall et al. [2] who showed that repulsion occurs between modes within the symmetric or antisymmetric families but that dispersion curves of symmetric and antisymmetric modes cross each other.

The SAFE method produces a system of equations of motion and an eigenvalue problem is solved to compute the frequencies and mode shapes. The repulsion and crossing of wavenumber versus frequency curves for rail models is investigated in this paper by analysing the second derivative of the eigenvalues with respect to wavenumber and the first derivative of the eigenvectors with respect to wavenumber. Rail is manufactured to be symmetric and is modelled as such. Wear that occurs in use will generally not be symmetric and the influence of the introduction of a small asymmetry is investigated.

In Section 2 the SAFE analysis is briefly described before the eigenvalue problem is presented. A repulsion region and a crossing region in a symmetric rail model are investigated in Section 3 by analysing the second derivative of the eigenvalue with respect to wavenumber. A small asymmetry is then introduced and analysed in the same regions. Section 4 describes an experimental measurement performed to illustrate a mode repulsion.

Section snippets

Safe modelling of guided wave propagation

The propagation of guided waves in one-dimensional waveguides of arbitrary but constant cross-section may be modelled efficiently by using the semi-analytical finite element (SAFE) method. The SAFE method uses finite element discretization over the cross-section of the waveguide and includes the wave propagation along the waveguide analytically in the element formulation. The method is relatively simple to implement and has become popular for modelling guided wave propagation in rails [3], [4],

Analysis of mode repulsion and mode crossing

Consider two curves approaching as we increase wavenumber. We can compute the derivatives of the eigenvalues λ with respect to wavenumber κ and also the derivatives of the eigenvectors ψ with respect to wavenumber.

The first derivatives of the eigenvalues and eigenvectors for a system with symmetric mass and stiffness matrices were derived by Fox and Kapoor [9]. Two methods of calculating the derivatives of the eigenvectors were presented. The second method is used in Appendix A where the first

Measurement of guided wave repulsion

Measurements were performed on an operational rail track to detect a mode repulsion. The repulsion shown in Fig. 3 was selected for experimental measurement because this repulsion involves a mode that has little motion in the foot of the rail and it is therefore expected that this mode will have relatively low attenuation and be detectable over large distances, making dispersion easier to observe. The measurement was performed by bonding a small piezoelectric sandwich transducer to the outside

Conclusions

The mode repulsion and crossing behaviour of approaching wavenumber versus frequency curves was studied by analysing the second derivative of the eigenvalue with respect to wavenumber. A term describing the repulsion force between two eigenvectors was obtained. A condition for this force to be zero and for mode crossing to occur was identified. If this condition is not satisfied the repulsion force is inversely proportional to the distance between the eigenvectors, which prevents mode crossing

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