Axisymmetric Excitation Research Articles

A finite element analysis of the time-delay periodic ring arrays for guided wave generation and reception in hollow cylinders

A new guided wave transducer model, time-delay periodic ring arrays (TDPRAs), is proposed and investigated in this paper for guided cylindrical wave generation and reception in hollow cylinders with application interests focusing on non-destructive testing (NDT) of piping/tubing. A finite element simulation has been performed for axisymmetric guided-mode excitation and reception with TDPRAs. By arranging a proper configuration of the time-delay profile and the electric-connection pattern of a ring array, unidirectional excitation and reception of guided waves can be achieved. The numerical results are obtained for the first three axisymmetrical modes and are compared with respect to generation efficiency and mode selectivity. Parametric influences on the performance of TDPRAs are discussed, combining a 2-D phase velocity-frequency spectrum approach with the mode dispersion and displacement structure analyses. The identification of converted modes in guided cylindrical wave reflections with a flexible TDPRA receiver has also been studied through sample notch reflection.

Nonlocal density wave theory for gravitational instability of protoplanetary disks without sharp boundaries

The stability of a self-gravitating infinitesimally thin gaseous disk rotating around a central mass is studied. Our global linear analysis concerns marginal stability, i.e. it yields the critical temperature for the onset of instability for any given ratio of the disk mass to the central mass. Both axisymmetric and low-m nonaxisymmetric excitations are analysed. When the fractional disk mass increases, the symmetry character of the instability changes from rings (m = 0) to one-armed trailing spirals (m = 1). The distribution of the surface density along the spiral arms is not uniform, but describes a sequence of maxima that might be identified with forming planets. The number of the mass concentrations decreases with increasing fractional disk mass. We also obtain solutions in the form of global nonaxisymmetric vortices, which are, however, never excited.

Vertical vibrations of a rigid disk embedded in a poroelastic medium

This paper considers the steady-state vertical vibrations of a rigid circular disk embedded at a finite depth below the free surface of a poroelastic medium. Biot's elastodynamic theory for porous media is used in the analysis. General solutions for axisymmetric poroelastic fields are obtained by using Hankel integral transforms. Analytical solutions for influence functions corresponding to four types of buried axisymmetric excitations are derived. The embedded disk problem is fomulated in terms of a set of coupled integral equations for unknown traction and pore pressure jumps across the disk. The kernel functions of the integral equations are the influence functions corresponding to buried vertical, radial and pore pressure ring loads. The system of integral equations is solved numerically by discretizing the disk into several concentric annular rings. Selected numerical solutions for displacements, vertical stress and pore pressure due to a buried fully flexible disk (uniform pressure) are also presented. The vertical compliances of a rigid disk are examined for different depths of embedment, poroelastic materials and hydraulic boundary conditions. Solutions for traction and pore pressure jumps are also examined. The present results are useful in the study of dynamic response of embedded foundations and anchors in poroelastic soils. Copyright © 1999 John Wiley & Sons, Ltd.

Guided waves by axisymmetric and non-axisymmetric surface loading on hollow cylinders

Guided waves generated by axisymmetric and non-axisymmetric surface loading on a hollow cylinder are studied. For the theoretical analysis of the superposed guided waves, a normal mode concept is employed. The amplitude factors of individual guided wave modes are studied with respect to varying surface pressure loading profiles. Both theoretical and experimental focus is given to the guided waves generated by both axisymmetric and non-axisymmetric excitation. For the experiments, a comb transducer and high power tone burst function generator system are used on a sample Inconel tube. Surface loading conditions, such as circumferential loading angles and axial loading lengths, are used with the frequency and phase velocity to control the axisymmetric and non-axisymmetric mode excitations. The experimental study demonstrates the use of a practical non-axisymmetric partial loading technique in generating axisymmetric modes, particularly useful in the inspection of tubing and piping with limited circumferential access. From both theoretical and experimental studies, it also could be said that the amount of flexural modes reflected from a defect contains information on the reflector's circumferential angle, as well as potentially other classification and sizing feature information. The axisymmetric and non-axisymmetric guided wave modes should both be carefully considered for improvement of the overall analysis of guided waves generated in hollow cylinders.

Transient axisymmetric response of a finite cylindrical shell with internal fluid loading. Part 2. Comparison of theory and experiment

An experimental investigation addressing the transient axisymmetric response of a finite length cylindrical shell with a base plate and internal fluid loading is presented. The configuration consists of a steel cylindrical shell with a circular base plate welded to one end. The shell and base plate contain an internal volume of water with a free surface. Data are presented for the special case of an axisymmetric transient excitation due to a force impulse hammer acting on the base plate. The in-vacuo and fluid-loaded response of the shell and base plate measured with an accelerometer will be presented and compared with numerical results. Transient field point pressure measurements taken with a hydrophone will also be presented and compared with numerical results. The agreement between experiment and numerical results is noted to be excellent.

An efficient numerical modeling of the DC resistivity logging: Theory and applications

The numerical solution of Green’s function for the potential in 2-D arbitrary inhomogeneous media with axial symmetry has been given by use of efficient half-analytical, halfnumerical hybrid method. Then the logging responses of various kinds of the DC resistivity log with axisymmetric excitation have been obtained by using surface integral equation method to match the boundary conditions on the electrodes of the logging sonde. Comparing the results with that obtained by other methods, one can see good precision and efficiency of the given method. Some applications of the numerical modeling have been also discussed.

Response of thin semi-infinite cylindrical elastic shell to axisymmetric impulsive radial excitation

The transient response of a semi-infinite, thin cylindrical elastic shell to an impulsive, axisymmetric radial excitation applied at the origin is investigated in the time and frequency domains using Fourier and Laplace transform methods. Solutions for the radial and axial displacements as functions of time and axial position show the dispersive nature of the wave propagation. High-frequency components propagating at the plate wave speed are followed by slower components occurring near the shell ring frequency. The temporal variation of the spectral components of the transient displacement is examined using a joint time–frequency distribution function. Analytical results are confirmed by experimental measurements on a brass tube.

Excitations of thermoelastic waves in plates by a pulsed laser

The method of the eigenfunction expansion, also known as the expansion in normal modes, is employed to study numerically the axisymmetric excitation of the thermoelastic waves in plates by a pulsed laser. This method gives a systematic treatment and allows one to investigate not only the quasistatic and dynamic thermoelastic responses of pulsed photothermal deformation on the time scale of 1 μs, but also the thermoelastic generation of longitudinal, transverse, and surface acoustic waves in thick materials, as well as the excitations of the Rayleigh-Lamb wave modes in thin plates. The formalism is particularly suitable for waveform analyses of the excitations of transient Lamb waves in thin plates because one needs only to calculate the contributions of several lower eigenmodes. The numerical technique provides a quantitative tool for the experimental determination of material properties, especially the mechanical and elastic properties of free-standing films and thicker sheet materials by thermoelastic detection.

Acoustic harmonic radiation from fluid-loaded spherical shells using elasticity theory

The finite Legendre transform technique is used to address the problem of acoustic harmonic radiation from fluid-loaded elastic spherical shells subjected to axisymmetric force excitations. The partial wave series for the response parameters such as the normal velocity of the shell surface, the pressure on the outer surface of the shell, and the far-field pressure are obtained using the mathematical theory of elasticity. Expressions for the input admittance and the transfer admittances for a ring force are also derived. Numerical results are presented for the velocity of the shell, the pressure on the shell, and the far-field pressure directionality patterns as a function of frequency, the relative thickness of the shell, and the location of the force. For low frequencies, the results are compared to those obtained by using a thin shell theory. The convergence properties of the partial wave series as a function of the frequency are also discussed.

Radial Impulsive Excitation Of Infinite Fluid-Filled Elastic Cylindrical Shells

The impulsive radial excitation of fluid-filled elastic cylinders is analytically studied using a double Fourier transform in the wavenumber and the frequency domains. In particular, the contribution of various waves to the structural response and their propagation paths are investigated by evaluating individual pole contributions to the total response using the method of residues. For axisymmetric excitation, the results surprisingly demonstrate that although the excitation is structural, the major path of vibrational energy to the structural observation point is through the fluid medium. For the beam mode type of excitation, two major paths are identified; the disturbance propagates either through low frequency structural components or through the fluid medium as higher frequency components. The work adds new insight into the behaviour of coupled pipe-fluid systems

Mathematical boundary integral equation analysis of an embedded shell under dynamic excitations

AbstractA boundary integral equation method is presented for the analysis of a thin cylindrical shell embedded in an elastic half‐space under axisymmetric excitations. By virtue of a set of ring‐load Green's functions for the shell and a group of dynamic fundamental solutions for the semi‐infinite medium, the structure–medium interaction problem of wave propagation is shown to be reducible to a set of coupled boundary integral equations. Through the analysis of an auxiliary pair of Cauchy integral equations, the singularities of the contact stress distributions arc rendered explicit. With a direct incorporation of such analytical features into the formulation, an effective computational procedure is developed which involves an interpolation of regular functions only. Typical results for the dynamic contact load distributions, displacements, and complex compliance functions are included as illustrations. In addition to furnishing quantities of direct engineering interest, this treatment is apt to be useful as a foundation for further rigorous as well as approximate developments for various related physical problems and boundary integral methods.

Imaging of group velocity surfaces in a cuspidal region of zinc by laser-generated ultrasonic waves

An image of group velocity surfaces within and outside a cuspidal region of a (0001) oriented zinc crystal is obtained with a pointlike scanning laser source and a fixed longitudinal (L) mode piezoelectric detector. Because of axisymmetric excitation and sensing, the group velocity surface of the pure transverse (PT) mode, being shear horizontally (SH) polarized, is absent in the obtained image. In the cuspidal region the quasi transverse (QT) mode has three group velocity branches in a given direction: the fast QT (FQT) branch, the intermediate speed QT (IQT) branch, and the slow QT (SQT) branch. The laser imaging clearly shows the wave fronts of the quasi longitudinal (QL), IQT, and SQT rays in the cuspidal region. Near the epicentral [0001] direction, the arrivals of the head wave (HW) and SQT rays are almost simultaneous and their wave fronts are not distinct. However, farther from the epicentral direction, they separate and outside the cuspidal region the imaging shows the group velocity surfaces of the QL, HW, and SQT rays.

Extension of the surface variational principle to nonaxisymmetric response of submerged thin shells of revolution

Previously, the surface variational principle (SVP) was implemented for rigid and elastic submerged structures under axisymmetric excitations. More recent developments extended SVP to address nonaxisymmetric problems of acoustic radiation and scattering situations for rigid bodies of revolution. In this presentation, SVP is applied jointly with Hamilton’s principle governing nonaxisymmetric responses of elastic thin shells of revolution. Complex Fourier expansions of the azimuthal dependence are combined with Ritz expansions of the spatial dependence to represent the displacement field and surface pressure. This representation is equivalent to a wave-number decomposition of the surface response into a series of helical-type waves. By enforcing continuity of normal velocity on the fluid–structure interface, a set of coupled algebraic equations is formed for each azimuthal harmonic. Problems addressed here are radiation due to arbitrary loading, and scattering associated with arbitrary incidence of a plane wave on a stationary elastic thin shell. The rigid body modes are included in the scattering case. Numerical results are assessed for the convergence qualities of an SVP simulation, particularly regarding series length requirement as the frequency is increased. [Work support by ONR.]

Mid-frequency endcap reflection and radiation coefficient extraction from FEM data

The response of a simple cylindrical shell terminated by rigid endcaps to various axisymmetric mechanical excitations is determined using the SARA 2-D finite element code. By assuming that the response of the shell is primarily composed of structural waves, which satisfy the fluid-loaded shell equations, the structural wave reflection matrix of the endcap is determined through an over-constrained system of equations. Results for the frequency range, 2<k0a<6, show that for this type of endcap, the reflection process may be parametrized with a smooth function in frequency that is dependent on a small number of variables. Using a similar approach, the far-field radiation patterns of the endcap due to each incident structural wave type are also determined and parametrized. This work suggests that at mid frequencies and away from trace-matched angles, many of the essential elements of the axisymmetric elastic structural response of simple shells may be modeled using a one-dimensional ray picture where all the structural wave coupling and radiation is accounted for by a single reflection and radiation matrix at each endcap. [Work supported by ONR.]

Focusing of fast transverse modes in (001) silicon at ultrasonic frequencies

This paper presents observations of the focusing of fast transverse (FT) ultrasonic waves in a (001) oriented, disk-shaped silicon single crystal. These modes are almost perfectly shear horizontally (SH) polarized and were absent from earlier reported observations of focusing of ultrasonic waves based on axisymmetric excitation and sensing. In an experiment the FT modes are generated and detected at room temperature by two small [100] polarized PZT piezoelectric shear transducers. The source transducer is fixed on the bottom surface of the specimen and the detector scans the top surface in the [100] direction along lines that intersect the [010] axis at various distances from epicenter. The observed focusing pattern indicates a strong concentration of the FT mode flux in a narrow band about the (100) plane containing the source. Because of the specific way in which the monopolar source acts, the radiated acoustic flux pattern breaks the fourfold symmetry associated with cubic media. While it shows a strong concentration of FT flux toward the (100) plane, it suppresses FT modes propagating very near the (010) plane passing through the source. The spatial variation of the Fourier components of the detected signal at approximately 2, 6, and 8 MHz has been examined, and there is good accountability for this variation on the basis of the computed frequency domain elastodynamic Green’s function for silicon.

Acoustic transient radiation from fluid-loaded shells of revolution using time-dependent in vacuo eigenvector expansions

A new method is presented to evaluate the acoustic transient radiation from fluid-loaded shells of revolution which are excited by axisymmetric mechanical or acoustical excitations. This method is based on the use of an in-vacuo modal vector (eigenvector) expansion with time-dependent coefficients to describe the velocity field of the fluid-loaded shell. Modal acoustic impulse responses which account for the coupling between the in-vacuo modal vectors for the fluid-loaded shell are introduced in the formulation of the general solution. The time-dependent modal velocity coefficients are expressed as the solution of a set of coupled convolution integral equations which are readily solved by marching forward in time. The associated time-dependent surface pressure is subsequently expressed as a modal vector expansion in which the time-dependent coefficients are readily related to the modal velocity coefficients. Since the surface velocity and pressure are then known, the time-dependent pressures in the field are simply obtained via quadrature methods from the space-time Kirchhoff integral solution. The special case of a fluid-loaded spherical shell is addressed to illustrate the application of the method to determine the general characteristics of the transient velocity response of the shell and the associated pressure field resulting from axisymmetric mechanical force excitations of spherical shells. Numerical results are presented to illustrate the effects on the velocity response and pressure field of time-dependent fluid coupling between the modal vectors corresponding to the upper and lower branches of the in-vacuo frequency spectrum of the spherical shell.

Variational formulation of acoustic radiation from submerged spheroidal shells

The surface variational principle (SVP) governing acoustic interaction between a vibrating surface and a surrounding fluid is combined with the dynamic equations governing response of a shell of revolution subjected to axisymmetric harmonic excitation. Ritz series representations are used to represent the spatial dependence of the surface pressure and shell displacement components. Two formulations are presented, with the difference being whether an intermediate computation of the in-vacuo modes is performed. The direct approach, in which the Ritz coefficients are determined directly from the coupled formulation, is used to validate the SVP approach for the case of a spherical shell, whose response is known analytically. For the case of a slender spheroidal shell, the direct approach is compared to the results obtained from the modal approach, in which a truncated set of in-vacuo modes forms the basis functions representing displacement. The aspect ratio and shell properties for this evaluation are selected to be in a range in which eigenvalue veering phenomena between two modes are encountered in the absence of fluid loading. The excitation is taken to be a uniform internal pressure having harmonic temporal variation. It is shown that truncations should include the modes involved in the eigenvalue veering process, even when the excitation frequency is sufficiently low to expect modal truncation to be reasonably accurate for predicting in-vacuo response.

Disk microstrip antenna with axisymmetric excitation

Disk microstrip antenna with axisymmetric excitation

Axisymmetric impulsive excitation of fluid-filled elastic cylindrical shells−Propagation paths.

The transient response of an infinite fluid-filled cylinder to an axisymmetric radial impact applied on the shell wall was derived, using the Donnell–Mushtari shell equations coupled with interior acoustic field and computing a double Fourier integration in the wave number and the frequency domain. The radial displacement amplitude of the shell as well as the instantaneous energy in both media were evaluated at several distances from the input force and for different structural damping coefficients and shell thicknesses. The contribution of each generated wave, either fluid-like or structural-like waves, was extracted from the total response in order to determine the propagation paths of the impulse into the coupled system. The results reveal that most of the response of the shell is due to low-frequency energy carried by an ‘‘acoustically slow’’ wave, very close to a fluid wave in a rigid-walled tube. This study is a further advance in the understanding of noise propagation along practical piping systems as, for example, the impulsive force can model a constraint and disturbance applied to the pipe wall by a hanger. [Work supported by ONR/DARPA.]

Finite Element Modeling of Remote-Field Eddy Current Problems for the Nondestructive Testing of Metal Tube

Abstract The remote-field eddy current effect refers to a low frequency eddy current nondestructive testing (NDT) phenomenon in tubular conductors (magnetic or nonmagnetic) in which the behavior of both amplitude and phase of induced magnetic field are in apparent contradiction to the conventional eddy current theory. Equal detection sensitivity across the wall thickness, linear relationship between the output signal phase-lag and the wall thickness, and absence of the lift-off problem are some of the attractive and contradicting features of this technique. Despite its early recognition and useful application in down-hole inspection of oil-well casing, no development of adequate scientific base that could explain this phenomenon has been reported. This paper describes modeling of this phenomenon using the finite element numerical analysis technique. Amplitude and phase of the sensor coil output have been predicted using this model for different axisymmetric test geometries, material properties, and excitation frequencies.