Using the singularity frequencies of guided waves to obtain a pipe's properties and detect and size notches

Abstract

A survey of relevant literature on the topic of wave propagation and scattering in pipes is given first. This review is followed by a theoretical framework which is pertinent to wave propagation in homogeneous, isotropic, pipes. Emphasis is placed on approximate solutions stemming from a computer based, Semi-Analytical Finite Element (SAFE) formulation. A modal analysis of the dynamic response of homogeneous, isotropic pipes, when subjected to a transient ultrasonic excitation, demonstrates that dominant features, i.e., singularities in an unblemished pipe’s displacement Frequency Response Function (FRF) coincide with its cutoff frequencies. This behaviour is confirmed experimentally. A novel technique is developed to deduce such a pipe’s wall thickness and elastic properties from three cutoff frequencies. The resulting procedure is simulated numerically and verified experimentally. Agreement between the new ultrasonic procedure and traditional destructive tests is within experimental uncertainty. Then a hybrid-SAFE technique is used to simulate waves scattered by various open rectangular notches. The simulations show, for the first time, that singularities distinct from the unblemished pipe’s cutoff frequencies arise in a displacement FRF when an axisymmetric notch is introduced. They also suggest that the new singularities depend on the properties of the parent pipe and the finite element region but effects are local to a notch. It is demonstrated further that the difference between the frequency at which a singularity introduced by a notch occurs and the nearest corresponding unblemished pipe’s cutoff frequency is a function of the notch’s dimensions. By plotting contours of constant frequency differences, it is shown that it is usually possible to characterize the notch’s dimensions by using two modes. However, the frequency difference for a third mode may be also needed occasionally. The more general case of nonaxisymmetric notches is shown to be a straightforward extension of the axisymmetric case.