Modal Analysis of Guided Waves in a Bar with an Arbitrary Cross-section

Abstract

Ultrasonic wave packets propagate in the longitudinal direction by inputting low frequency ultrasonic into such bar-like structures as rails. This ultrasonic mode, called guided wave has become popular as a promising technique for rapid long-range nondestructive inspection for pipes and rails. In guided wave inspection, guided wave velocities (dispersion curves) and wave structures are firstly needed. Dispersion curves and wave structures can be analytically derived for such simple structures as plates and pipes, but not for bar-like structures with complex cross-section such as rails.Authors have developed calculation technique to obtain the dispersion curves and wave structures for such structures using a special finite element method called a semi-analytical finite element method in which dispersion curves and wave structures can be obtained as eigenvalues and eignvectors of an eigensystem. This study developed more accurate calculation technique for dispersion curves and wave structures using mirror relation of guided wave modes and an iteration method for solution of the eigen problem.And experimental studies for JIS 6 kg rail verify that dispersion curves and wave structures were obtained with sufficient accuracy for typical out-of-plain vibration modes. Wave structures were obtained by measuring waveforms at many points on the curved surface of a rail with a laser interferometer controlled by robot arms.

Modal Analysis of Guided Waves and Its Application to Rail Inspection

“…Solving the governing equation for a certain frequency ω gives solution of the displacement vector. On the other hands, in a semi-analytical FEM (4)- (6) , wave field for a wave number ξ is expressed by dividing only a cross-section of a bar-like structure as shown in Fig.1 and associating ) exp( z iξ to longitudinal (z) direction.…”

“…is the eigenvector for the m th eigenvalue m ξ . The 2M eigenvalues and eigenvectors correspond to M forward and M backward waves (4)- (6) . The fact that forward and backward guided wave modes can be obtained in the same number represents the mirror relation between them.…”

“…Authors have studied on derivation of dispersion curves and numerical simulation for guided wave propagation in plates, pipes and rails using a semi-analytical finite element (SAFE) method (4)- (6) , in which we developed derivation technique for guided wave dispersion curves and wave structures, not only for plates and pipes which have already been obtained with analytical solutions, but also for a bar with complex cross-section such as railway rails. And recently we have developed inspection equipments for rails.…”

A Recursive Legendre Polynomial Analytical Integral Method for the Fast and Efficient Modelling Guided Wave Propagation in Rectangular Section Bars of Orthotropic Materials

“…18 Hayashi et al derived the theoretical dispersion curves and wave structures for a bar with an arbitrary cross section. 19 Manzanares et al presented a plane wave expansion method to predict the vibrational elastic modes in bars with square and circular cross sections. 20 Using an analytical superposition method, Krushynska and Meleshko investigated guided waves in a rectangular elastic bar.…”