Normal Mode Expansion Technique Research Articles

Behaviour of circumferential and helical guided waves in two-layered composite cylindrical shell

A model of the acoustic scattering of a plane wave incident obliquely upon an infinite elastic bi-layered cylindrical shell is developed based on normal mode expansion technique. The far field backscattering form function for the composite copper/stainless steel cylindrical shell is calculated for reduced frequency range of k⊥a1=0–70 and for different values of incident angles α (k⊥=k1cos⁡α, k1 is the wave number in the fluid surrounding the shell and a1 is the outer radius of the shell). Numerical calculations show the behaviour of the resonances of circumferential and helical guided waves as the angle of incidence increases. The obtained results are compared with those of copper and stainless steel one-layered cylindrical shells. The use of frequency-angle representation depicts the helical propagation of surface waves along the axis of the shell. This representation, along with the identification plan, enables to describe the evolution of acoustic scattering as a function of incidence angle and vibration mode n. Furthermore, a comparative analysis of the resonance trajectory evolution between mono-layered and composite cylindrical shells is performed. The evolution of the helical guided waves, the bifurcation of the A0 wave into the A0− and A0+ waves, and a region with high backscattering are investigated. The presented results show that the extension of the region with high backscattering depends on the thickness and physical properties of the bi-layered cylindrical shell.

Size Determination of Immersed and Embedded Cylindrical Rods by Inversion of Acoustic Scattering Data

In this paper, a new approach is proposed for nondestructive size determination of immersed and embedded cylindrical rods by inversion of acoustic scattering data. The normal mode expansion technique is used for modelling the scattered field. Also, the experimental backscattered field is measured using short pulse MIIR technique. The Genetic algorithm is the inversion technique used for measuring the diameters of the rods. The inversion technique matches the modelled and experimental scattered fields at resonance points, so the diameters of the rods can be estimated. The numerical results indicate that proper selection of resonance frequencies leads to accurate measurement of diameter. The proposed approach showed very good convergence and the results obtained were found to agree very well with available data.

Pipe Defect Visualization and Quantification Using Longitudinal Ultrasonic Modes

In this study, a new ultrasonic wave based imaging techniques developed using longitudinal ultrasonic waves for detecting defects in pipeline structures. Ultrasonic waves are gaining popularity for pipeline monitoring because of its sensitivity to small defects and a long sensing range. Based on the merits of the ultrasonic waves, several research groups have developed ultrasonic wave based imaging techniques for pipeline monitoring. Conventionally, a pure torsional mode is often generated using shear-mode piezoelectric transducers or electromagnetic acoustic transducers (EMAT) and used for pipe damage detection. In this study, a new ultrasonic wave based imaging technique is developed using a longitudinal wave mode instead of the pure torsional mode. The longitudinal mode is generated using inexpensive macro fiber composite (MFC) transducers attached at one end of the pipe, eliminating the need for the shear-mode transducers or EMATs. Then, the reflections generated by the interaction of the incident longitudinal mode with a defect are measured in a pulse echo manner. Using a normal mode expansion technique, flexural modes are extracted from the reflected signals. When a defect-induced reflected wave mode is propagated back along the longitudinal direction of the pipe, its dispersive nature is minimized and best-compensated at the defection location. Therefore, by virtually propagating each defect-induced flexural mode back in the wave propagation direction, an image which visualizes the focusing of the back-propagated flexural modes can be obtained and the defect location can be identified. Numerical simulations and experimental tests are conducted to demonstrate that a wall-thinning in a steel pipe can be detected and quantified using the proposed imaging technique.

Characterization of a cylindrical rod by inversion of acoustic scattering data

In this paper, a new approach is proposed for nondestructive characterization of immersed and embedded isotropic rod-shaped samples by inversion of acoustic scattering data. The normal mode expansion technique is used for modelling the scattered field and the compression incident and compression scattered waves are considered. Genetic algorithm is the inversion technique used for estimating the elastic wave velocities and density of the rods from their measured backscattered pressure spectrum. The inversion technique is capable of computing the parameter values that best fit a particular set of data. A perturbation study is conducted on the sensitivity of the resonance frequencies to changes in elastic properties and density of the rods. The numerical results indicate that proper selection of resonance frequencies leads to accurate measurement of elastic constants and density. The proposed approach showed very good convergence and the results obtained were found to agree very well with available data.

Nondestructive characterization of materials by inversion of acoustic scattering data

In this paper, a new approach is proposed for nondestructive characterization of isotropic materials by inversion of resonance scattering data. The normal mode expansion technique is used for modelling the scattered field and the resonances of the cylinder are identified using resonance scattering theory. Genetic algorithm (GA) and simplex method are the inversion techniques used for estimating the elastic properties of the cylinder from its measured backscattered pressure spectrum. Both of these inversion techniques are capable of computing the parameter values that best fit a particular set of data. It was found that the GA is more suitable for preliminary iterations to find initial estimates of the parameters, while the simplex method is more suitable for obtaining accurate results in later stages of computations. They both showed very good convergence and the results obtained from the proposed approach were found to agree very well with those available in the literature.

Exact analytical modeling of power and energy for multimode lamb waves excited by piezoelectric wafer active sensors

This article presents an analytical model for power and energy transfer between excited piezoelectric wafer active sensors and host structure. This model is based on exact multimodal Lamb waves, normal mode expansion technique, and orthogonality of Lamb waves. Modal participation factors are presented to show the contribution of every mode to the total energy transfer. The model assumptions include the following: (1) straight-crested multimodal ultrasonic guided wave propagation, (2) propagating waves only, (3) ideal bonding (pin-force) connection between piezoelectric wafer active sensors and structure, and (4) ideal excitation source at the transmitter piezoelectric wafer active sensors. Constrained piezoelectric wafer active sensor admittance is reviewed. Electrical active power, mechanical converted power, and Lamb wave kinetic and potential energies are derived in closed-form formulae. Numerical simulations are performed for the case of symmetric and antisymmetric excitation of thin aluminum structure. The simulation results are compared with axial and flexural approximation for the case of low-frequency Lamb waves. In addition, a thick steel structure example is considered to illustrate the case of multimodal guided waves. A parametric study for different excitation frequencies and different transducer sizes is performed to show the best match of frequency and piezoelectric wafer active sensor size to achieve maximum energy transfer into the excited structure.

Vibration Response and Noise Radiation of Engine Block Coupled with the Rotating Crankshaft and Gear Train

This paper presents the theoretical procedure to predict the vibratory response and radiated noise of the engine block coupled with the rotating crankshaft and gear train shafts which drives the fuel injection pump and valve system. The exciting forces acting on the engine block and shaft system are combustion pressure, inertia forces of the moving parts, piston slap forces, fuel injection pressure and valve driving force and torque. Theoretical procedures consist of the following four steps; (1) Dynamic characteristics of the engine block and shafts are determined separately by Finite Element Method or experimental modal analysis, (2) Normal mode expansion technique is employed to derive the equation of motion of the total system in which rotating shafts with gear train are combined to the engine block by the oil film and contact stiffness, (3) The time histories of the vibratory response of the engine block and rotating shafts are calculated by the numerical integration technique, (4) Engine noise radiated from the engine block surface is evaluated using the Boundary Element Method. This method is applied to estimate the effect of the backlash of the gear train on the engine block vibration and radiated noise.

クランク軸・歯車軸系とエンジンブロックの連成振動応答及び騒音放射予測法

This paper presents the theoretical procedure to predict the vibratory response and radiated noise of the engine block coupled with the rotating crankshaft and gear train shafts which drives the fuel injection pump and valve system. The exciting forces acting on the engine block and shaft system are combustion pressure, inertia forces of the moving parts, piston slap forces, fuel injection pressure and valve driving force and torque. Theoretical procedures consist of the following four steps; (1) Dynamic characteristics of the engine block and shafts are determined separately by Finite Element Method or experimental modal analysis, (2) Normal mode expansion technique is employed to derive the equation of motion of the total system in which rotating shafts with gear train are combined to the engine block by the oil film and contact stiffness, (3) The time histories of the vibratory response of the engine block and rotating shafts are calculated by the numerical integration technique, (4) Engine noise radiated from the engine block surface is evaluated using the Boundary Element Method. This method is applied to estimate the effect of the backlash of the gear train on the engine block vibration and radiated noise.

Acoustic resonance scattering from a submerged anisotropic sphere

The resonance scattering theory (RST) is applied to the problem of sound scattering from an elastic transversely isotropic solid sphere suspended in an ideal acoustic fluid medium. The normal mode expansion technique in conjunction with the Frobenius power series solution method is utilized to deal with the material anisotropy. The presented model, which degenerates to the simple isotropic solution in the case of very weak anisotropy, is initially employed to study sensitivity of various resonant modes of vibration to perturbations in elements of the stiffness matrix. Employing a rigid background subtraction, the target’s spectrum of resonances is extracted from the relevant modal backscattering form functions and subsequently traced and discussed through Regge pole trajectory plots. Also, the backscattering form function and resonance spectra, along with the dispersion curves for selected transversely isotropic solid spheres with distinct degrees of material anisotropy, are calculated and discussed. The various modes of propagation associated with the Rayleigh, Whispering Gallery, and fluid-borne Scholte-Stoneley surface waves are identified and examined.

Formula omitted] solutions of radiative heat transfer in a slab with reflective boundaries

The spherical harmonics method is used to obtain solution for the radiative heat transfer equation for a slab with reflective boundaries. An absorbing, emitting, non-isothermal, gray medium is considered with linearly anisotropic scattering. Under the condition of the thermal equilibrium, the slab boundaries are subjected to specular and diffuse reflection. The analytical form of solutions is obtained for both conservative and non-conservative cases. The accuracy of the method was verified by benchmark comparisons against the solutions of an earlier work performed by the normal-mode expansion technique. The present predictions of heat flux were found to be in good agreement with the benchmark data.

Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium

AbstractScattering of an elastic wave by a cylindrical shell embedded in poroelastic medium is investigated theoretically with the assumption that the shell material is also a porous elastic medium. The porous medium is modellized via Biot's theory and the scattering by cylindrical shell is expressed by the definition of scattering matrix. The normal mode expansion technique is employed for analysing the scattering field, and the asymptotic solutions of displacements, stresses and pore pressure are derived. Two limiting cases‐scattering by a poroelastic cylinder in Biot medium and a elastic cylindrical shell in elastic medium are obtained from the general solutions. The dispersion curves of displacement amplitude at the interface of shell and medium is compared with the case of elastic shell. The scattering amplitude associated with the fast, slow and transverse waves are identified by numerical simulation. Furthermore, the influence of the poroelastic property of shell material on scattering amplitude is analysed. Copyright © 2005 John Wiley & Sons, Ltd.

Mode extraction technique for guided waves in a pipe

Guided waves propagating in a pipe consist of many modes with different velocities and dispersions. To analyze these complex guided waves through a normal mode expansion technique, that is the fundamental theory on guided waves, we must first extract guided wave modes from received signals. In this study, we develop a mode extraction technique in which many received signals at different circumferential positions can be processed based on the fact that guided wave modes have different displacement distributions in the circumferential direction. After discussing the relevant theory, we verify our mode extraction technique experimentally using eight signals at eight different circumferential positions. Moreover, we show that the circumferential position of an excitation transducer, as well as the distance between an excitation transducer and a receiver in a pitch-catch configuration can be identified using the mode extraction technique.

Three-Dimensional Defect in a Plate Boundary Element Modeling for Guided Wave Scattering

A hybrid three-dimensional boundary element normal mode expansion technique is developed to study the guided wave scattering from an arbitrary shape defect in a plate. Lamb wave incident into a circular through plate hole was studied as an example. The scattered wave displacement field and its normal mode component are given. Mode conversion from Lamb wave to SH wave at the defect boundary were observed.

Scattering of plane acoustic waves by a transversely isotropic cylindrical shell—application to material characterization

Scattering of obliquely incident plane acoustic wave by an air-filled, transversely isotropic cylindrical shell immersed in water is investigated theoretically and experimentally. The normal mode expansion technique is employed for analyzing the scattering field, and then resonance modes of the shell appearing in modal scattering form functions are identified performing the resonance scattering analysis. Dispersion curves for Sholte–Stoneley, SH and Lamb waves are obtained and their characteristics are interpreted. Calculated backscattering and resonance spectra as well as dispersion curves are compared with those from ultrasonic experiments for two composite samples having the same nominal composition but fabricated under different conditions. Sensitive change of the dispersion curves is observed for both normal and oblique incidences, which demonstrates the feasibility of systematic inverse evaluation of damage or elastic constants of the composite shell using data from the acoustic scattering measurements.

Boundary element modeling for defect characterization potential in a wave guide

Wave scattering analysis implemented by boundary element methods (BEM) and the normal mode expansion technique is used to study the sizing potential of two-dimensional shaped defects in a wave guide. Surface breaking half-elliptical shaped defects of three opening lengths (0.3, 6.35 and 12.7 mm) and through-wall depths of 10–90% on a 10 mm thick steel plate were considered. The reflection and transmission coefficients of both Lamb and shear horizontal (SH) waves over a frequency range 0.05–2 MHz were studied. A powerfully practical result was obtained whereby the numerical results for the S 0 mode Lamb wave and n 0 mode SH wave at low frequencies showed a monotonic increase in signal amplitude with an increase in the defect through-wall depth. At high frequency (usually above the cut-off frequency of the A 1 mode for Lamb waves and the n 1 mode for SH waves, respectively), the monotonic trend does not hold in general due to the energy redistribution to the higher order wave modes. Guided waves impinging onto an internal stringer-like an inclusion were also studied. Both the Lamb and SH waves were shown to be insensitive to the stringer internal inclusions at low frequency. Experiments with piezoelectric Lamb wave transducers and non-contact SH wave electro-magnetic acoustic transducers (EMAT) verified some of the theoretical results.

Transient modeling of Lamb waves emitted by integrated transducers using a hybrid finite element–normal mode expansion technique

For a few years, the concept of integrated health monitoring of aeronautic structures has become an important issue. A promising solution is the emission of ultrasonic Lamb waves using thin piezoelectric transducers integrated to the structure. Indeed Lamb waves are known to be able to propagate over long distance and to interact with inner structural defects. One important aspect when applying Lamb waves to health monitoring is the control and optimization of their generation. Therefore, a hybrid finite element–normal mode expansion modeling technique adapted to piezoelectric transducers has been recently developed. This technique allows the determination of the amplitude of each Lamb mode as a function of the transducer parameters. So far, it had been successfully applied to the case of harmonic excitation only. In this paper, adaptation of this hybrid technique to transient excitation is presented, which enables to simulate actual experimental conditions. Contrary to other techniques presented in the literature, the present technique allows to take into account the actual electromechanical behavior of the integrated transducer. Predicted results have been successfully compared to experimental measurements performed on different types of materials.

Numerical simulation of Lamb wave scattering in semi‐infinite plates

AbstractA finite element formulation is applied to study Lamb wave scattering in homogeneous and sandwich isotropic plates. Dispersion curves are calculated in a simple and automatic way by solving a quadratic eigenproblem. A meshing criterion to obtain accurate results with linear and quadratic elements is provided. An absorbing boundary condition for semi‐infinite plates is derived from this formulation by means of a truncated normal mode expansion technique, where the finite element eigenvectors are used instead of the analytical expressions for the normal modes. This non‐reflecting boundary condition is directly applicable to study Lamb wave reflection by simple obstacles such as a flat edge. In order to tackle Lamb wave diffraction problems by defects with more complex geometries, a hybrid boundary element‐finite element formulation is developed. The validity and accuracy of both formulations are checked thoroughly with a series of test problems studied by other researchers. Copyright © 2001 John Wiley & Sons, Ltd.

An elastodynami`c hybrid boundary element study for elastic guided wave interactions with a surface breaking defect

Elastic guided wave interactions with various defects are explored for investigating defect characterization possibilities by using a hybrid boundary element method (BEM) in combination with an elastodynamic boundary integral equation and the Lamb wave normal mode expansion technique. The BEM code accuracy is verified based on energy conservation and available bench marking data for guided wave scattering problems. Through two-dimensional (2-D) parametric studies for an arbitrarily shaped defect, from a surface breaking crack model to a round surface defect, a waveguide cross-section including a defect is locally selected as a model for a given incident mode, frequency and a specific set of material properties. Mode reflection and transmission factors are numerically calculated to evaluate mode sensitivities and to obtain the potentially good classification features. It turns out that the guided wave scattering profiles show quite different behaviors as functions of incident mode, frequency, defect shape and size in providing us with enough rich feature extraction information for defect classification and sizing analysis. The theoretical analysis can be used to establish efficient guidelines for both data acquisition and feature selection in a pattern recognition analysis program of study. Sample results are presented.

Computation of higher spherical harmonics moments of the angular flux for neutron transport problems in spherical geometry

The integral form of one-speed, spherically symmetric neutron transport equation with isotropic scattering is considered. Two standard problems are solved using normal mode expansion technique. The expansion coefficients are obtained by solving their singular integral equations. It is shown that these expansion coefficients provide a representation of all spherical harmonics moments of the angular flux as a superposition of Bessel functions. It is seen that large errors occur in the computation of higher moments unless we take certain precautions. The reasons for this phenomenon are explained. They throw some light on the failure of spherical harmonics method in treating spherical geometry problems as observed by Aronsson.

Ultrasonic Guided Wave NDT for Hidden Corrosion Detection

An experimental study of hidden corrosion detection by using ultrasonic guided waves is presented combined with a BEM numerical simulation. Both corrosion simulation specimens by machine cutting and real corrosion specimens by electrochemical processing were used in the investigation, with a range of corrosion depths from 0.02 to 0.4 mm (1.5 to 20% of the original plate thickness). Various wave modes were subsequently generated on these specimens to examine the implications of thinning on mode cutoff, group velocity, and transmission and reflection amplitudes. The transmission and reflection of guided waves upon entering the corrosion zone were simulated by a hybrid BEM calculation that combines a normal mode expansion technique of Lamb waves for far fields with the boundary element representation for the scattered near fields. A quantitative technique for hidden corrosion depth with guided waves is developed based on a frequency compensation concept. The estimated depth of the real corrosion by this method shows good agreement with that by an optical microscope.