Measurement of modal amplitudes of guided waves in rails

One important application of guided wave ultrasound is that of rail condition monitoring where long lengths of rail can be monitored from permanently attached transducer locations. During the development of transducers for such a system it is advantageous to be able to measure the amplitude of the individual modes of propagation on a short length of rail in the laboratory. This paper describes a method of extracting modal amplitudes from measured time domain signals performed at a limited set of points on the waveguide. The method uses the wave propagation characteristics of the waveguide, predicted by a semi-analytical finite element model, to extract the modal amplitudes from experimental measurements. The frequency response at a set of measurement locations is described by a superposition (with unknown amplitude coefficients) of the frequency response of the modes that propagate in the frequency range of interest. Experimental time domain responses are measured and transformed to frequency responses. The amplitude of each mode is estimated using the pseudo-inverse to provide a minimum norm least-squares estimate. The technique is demonstrated on a rail excited by a piezoelectric patch transducer. A laser vibrometer was used to measure displacements at five points around the rail circumference at three distances giving a total of 15 measurements. Eight propagating modes were extracted from these measurements. The extracted modes were then used to predict the response at points further along the waveguide and these predictions were verified by further measurements indicating that the modes of propagation were accurately estimated. The technique requires that the distance between the measurement points be known but does not require that the distance from the transducer be known. This feature and the fact that only a few measurements are required make the method suitable for measuring the propagation of individual modes over long distances in the field.