A modelling framework for simulation of ultrasonic guided wave-based inspection of welded rail tracks.

A modelling framework for ultrasonic inspection of waveguides with arbitrary discontinuities, excited using piezoelectric transducers, is developed. The framework accounts for multi-modal, dispersive and damped one dimensional propagation over long distances. The proposed model is applied to simulate a realistic guided wave-based inspection of a welded rail. The framework models the excitation, propagation and scattering of guided waves from welds by respectively employing a hybrid model that couples a 3D FEM model of a piezoelectric transducer with a 2D SAFE model of the rail; a 2D SAFE model of the rail; and another hybrid method which couples a 3D FEM model of the arbitrary discontinuity (weld) with two SAFE models of the rail to represent the semi-infinite incoming and outgoing waveguides. Optimal damping parameters for hysteretic and viscous damping, respectively, are determined using a model updating procedure to approximate attenuation in the rail. Good agreement between the experimental measurement and simulation is demonstrated, even for weld reflections originating over 640 m from the transducer location. The proposed physics-based framework can be used to efficiently perform multiple analyses considering different numbers and locations of welds, different excitation signals or to investigate the effects of changes in parameters such as transducer geometry, or material property variations caused by temperature fluctuations. The framework could therefore be used in future to set up a digital twin of a section of rail track, or in the development of a rail monitoring system by predicting reflections from defects which cannot readily be measured, but which can be simulated.

Estimation of rail properties using semi-analytical finite element models and guided wave ultrasound measurements.

Guided wave based monitoring systems require accurate knowledge of mode propagation characteristics such as wavenumber and group velocity dispersion curves. These characteristics may be computed numerically for a rail provided that the material and geometric properties of the rail are known. Generally, the rail properties are not known with sufficient accuracy and these properties also change due to temperature, rail wear and rail grinding. An automated procedure is proposed to estimate material and geometric properties of a rail by finding the properties which, when input into a Semi-Analytical Finite Element (SAFE) model, accurately reproduce measured dispersion characteristics. Pulse-echo measurements were performed and spectrograms show the reflections from aluminothermic welds of three modes of propagation. The SAFE method was used to solve the forward problem of predicting the dispersion characteristics for specified rail properties. Dispersion curves are computed for different combinations of Poisson's ratio and three geometric parameters. These dispersion curves are scaled to cover a range of longitudinal speeds of sound of the rail material. A technique is developed to determine which SAFE model provided the best fit to the experimental measurements. The technique does not require knowledge of the distances to the reflectors; rather these distances are estimated as part of the proposed procedure. A SAFE model with the estimated rail parameters produced dispersion curves and distances in very good agreement with the measured spectrograms. In addition, the estimated mean geometric parameters agreed with the measured profile of the rail head.

Mode repulsion of ultrasonic guided waves in rails.

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.

Laser vibrometer measurement of guided wave modes in rail track.

The ability to measure the individual modes of propagation is very beneficial during the development of guided wave ultrasound based rail monitoring systems. Scanning laser vibrometers can measure the displacement at a number of measurement points on the surface of the rail track. A technique for estimating the amplitude of the individual modes of propagation from these measurements is presented and applied to laboratory and field measurements. The method uses modal data from a semi-analytical finite element model of the rail and has been applied at frequencies where more than twenty propagating modes exist. It was possible to measure individual modes of propagation at a distance of 400 m from an ultrasonic transducer excited at 30 kHz on operational rail track and to identify the modes that are capable of propagating large distances.

Semi-analytical finite element analysis of elastic waveguides subjected to axial loads.

Predicting the influence of axial loads on the wave propagation in structures such as rails requires numerical analysis. Conventional three-dimensional finite element analysis has previously been applied to this problem. The process is tedious as it requires that a number of different length models be solved and that the user identify the computed modes of propagation. In this paper, the more specialised semi-analytical finite element method is extended to account for the effect of axial load. The semi-analytical finite element method includes the wave propagation as a complex exponential in the element formulation and therefore only a two-dimensional mesh of the cross-section of the waveguide is required. It was found that the stiffness matrix required to describe the effect of axial load is proportional to the mass matrix, which makes the extension to existing software trivial. The method was verified by application to an aluminium rod, where after phase and group velocities of propagating waves in a rail were computed to demonstrate the method.

Simulation of piezoelectric excitation of guided waves using waveguide finite elements.

A numerical method for computing the time response of infinite constant cross-section elastic waveguides excited by piezoelectric transducers was developed. The method combined waveguide finite elements (semi-analytical finite elements) for modeling the waveguide with conventional 3-D piezoelectric finite elements for modeling the transducer. The frequency response of the coupled system was computed and then used to simulate the time response to tone-burst electrical excitation. A technique for identifying and separating the propagating modes was devised, which enabled the computation of the response of a selected reduced number of modes. The method was applied to a rail excited by a piezoelectric patch transducer, and excellent agreement with measured responses was obtained. It was found that it is necessary to include damping in the waveguide model if the response near a "cut-on" frequency is to be simulated in the near-field.

Analysis of piezoelectric ultrasonic transducers attached to waveguides using waveguide finite elements.

A finite-element modeling procedure for computing the frequency response of piezoelectric transducers attached to infinite constant cross-section waveguides, as encountered in guided wave ultrasonic inspection, is presented. Two-dimensional waveguide finite elements are used to model the waveguide. Conventional three-dimensional finite elements are used to model the piezoelectric transducer. The harmonic forced response of the waveguide is used to obtain a dynamic stiffness matrix (complex and frequency dependent), which represents the waveguide in the transducer model. The electrical and mechanical frequency response of the transducer, attached to the waveguide, can then be computed. The forces applied to the waveguide are calculated and are used to determine the amplitude of each mode excited in the waveguide. The method is highly efficient compared to time integration of a conventional finite-element model of a length of waveguide. In addition, the method provides information about each mode that is excited in the waveguide. The method is demonstrated by modeling a sandwich piezoelectric transducer exciting a waveguide of rectangular cross section, although it could be applied to more complex situations. It is expected that the modeling method will be useful during the optimization of piezoelectric transducers for exciting specific wave propagation modes in waveguides.