Normal Mode Expansion Research Articles

New calculation strategy for acoustic radiation from a thick annular disk

This article proposes a new semianalytical procedure for the calculation of sound radiation from a thick annular disk when it is excited by arbitrary harmonic forces. As the first step, structural eigensolutions for both in-plane (radial) and out-of-plane (flexural) modes are calculated using analytical methods. These are examined by the finite element analyses as well as the experimental investigations. The far-field sound pressure distributions (including directivity) due to selected modes of the disk are obtained from numerically obtained surface velocities using Rayleigh integral solutions based on cylindrical or circular plate radiator formulations. Such formulations define the modal radiation solutions corresponding to the structural eigensolutions of a thick disk. The boundary element analyses and vibro-acoustic experiments validate analytical predictions. Surface velocity and far-field sound pressure due to an arbitrary harmonic force excitation are then obtained from the structural and acoustic normal mode expansions. Based on the far-field sound pressure, acoustic power and radiation efficiency spectra are obtained. The method is also confirmed by comparing analytical results with those from finite and boundary element analyses. Finally, the effect of coupling between modes is also investigated by the proposed procedure.

Efficient computation of the eigenspectrum of viscoelastic flows using submatrix-based transformation of the linearized equations

Linear stability analysis of incompressible viscoelastic flows based on normal mode expansions of the eigenfunctions requires the numerical solution of a generalized eigenvalue problem (GEVP). The complex boundary layer structure of the leading eigenfunctions and the singular character of the continuous set of eigenvalues, necessitate the use of fine mesh sizes, leading to large algebraic GEVPs. In this paper, we present a submatrix-based transformation of the linearized equations (SubTLE) that converts the GEVP into a simple eigenvalue problem (EVP) of half the original dimension for the purely elastic isothermal and non-isothermal flows of an Oldroyd-B liquid. This leads to significant (up to an order of magnitude) reduction in the CPU time and memory required for the solution of the EVP. This is illustrated in the context of isothermal and non-isothermal shear flows.

Time dependent analysis of short-range monopole wake fields driven by accelerated electron beams

Combinations of integral transforms and Green's function are employed in order to obtain the distribution of the short-range monopole wake fields driven by accelerated electron beams in a cylindrical cavity. This constituted an alternative but simpler and more versatile method to the normal mode expansion which we used earlier to solve for such a distribution (Salah and Dolique, Nucl. Instr. and Meth. A 431 (1999) 27). Whereas direct comparisons of the resulting analytical expressions are found difficult to directly compare, numerical illustration of results obtained from both approaches show excellent agreements and hence confirm the fact that both approaches are equivalent.

Optical absorption for parallel cylinder arrays.

We study the long-wavelength electromagnetic resonances of interacting cylinder arrays. By using a normal-modes expansion where the effects of geometry and material are separated, it is shown that two parallel cylinders with different radii have electromagnetic modes distributed symmetrically about depolarization factor 1 / 2. Both sets couple to longitudinal and transverse components of the external field, but amplitudes of symmetric depolarization factors become exchanged when considering longitudinal or transverse polarization. We also find that amplitudes satisfy sum rules that depend on the ratio of the cylinders radii. The main effect of the difference in radii is a spectral shift towards the isolated cylinder resonance as this difference increases.

Numerical analysis on laser-generated guided elastic waves in a hollow cylinder

Numerical analyses are presented for laser-generated guided elastic waves in a hollow cylinder. Time-dependent displacement at the outer surface of a hollow cylinder is expressed by summation of longitudinal and flexural type modes by employing the normal mode expansion (NME) method, then the transient waveforms excited by a single beam of laser pulse and four beams of laser pulse with an axisymmetric spatial distribution are calculated numerically. The influence of the spatial distribution of laser pulses on the waveforms are discussed in detail, and the features of major modes are explained based on dispersion curves. Finally, the total waveform of longitudinal modes obtained by the NME method is compared to that predicted by the finite element method (FEM), and a good agreement is obtained.

Baroclinic Instability in High Latitudes Induced by Polar Vortex: A Connection to the Arctic Oscillation

Abstract In this study, baroclinic instability of the northern winter atmosphere is investigated in the context of the dynamical interpretation of the Arctic oscillation. The unstable solutions, obtained by a method of 3D normal mode expansion, are compared for observed zonal basic states with strong and weak polar vortices in reference to the Arctic oscillation index. As a result of the eigenvalue problem of the linear stability analysis, a characteristic unstable solution is obtained that dominates in high latitudes when the polar vortex is strong. The mode is called a monopole Charney mode M1, which is similar to an ordinary Charney mode MC in midlatitudes. In order to understand the origin of the M1 mode, a hypothetical zonal basic state that has only the polar jet with no subtropical jet is analyzed. It is found that the M1 mode in high latitudes is excited by the baroclinicity associated with the polar vortex. The M1 mode in high latitudes is dynamically the same Charney mode as MC but is excited by...

Solution to the initial value problem for a high-gain FEL via Van Kampen's method

Using Van Kampen's normal mode expansion, we solve the initial value problem for a high-gain free-electron laser described by the three-dimensional Maxwell–Klimontovich equations. An expression of the radiation spectrum is given for the process of coherent amplification and self-amplified spontaneous emission. It is noted that the input coupling coefficient for either process increases with the initial beam energy spread. The effective start-up noise is identified as the coherent fraction of the spontaneous undulator radiation in one field gain length, and is larger with increasing energy spread and emittance mainly because of the increase in gain length.

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.

Active acoustic control in gradient coils for MRI.

The new principles of active acoustic control in gradient coil design recently introduced by Mansfield and Haywood (MAGMA 1999;8(Suppl 1):55) are further developed theoretically for the far-field acoustic output for a single sector of a coil system comprising four or more flat rectangular coil sectors. Each sector consists of a split plate arrangement in which are embedded two windings, an outer primary winding and a narrow inner re-entrant loop control winding immediately adjacent to and surrounding the split or air gap. The wire spacing of the control winding is made small so as not to affect substantially the magnetic field created by the primary winding. Experimental results are presented for two sectors each made of a different readily available plastic material and tested over a range of frequencies. They both show substantial average reductions in acoustic output over the full output when the control winding is appropriately driven. New theoretical expressions are derived for particular frequencies based on normal mode expansions for the plate. This new approach is better able to explain the acoustic output difference between the full and reduced output modes. Empirical expressions are also developed which include longitudinal as well as transverse plate characteristics and used to fit the experimental acoustic output data as a function of frequency and indicate good agreement with regard to both the form and amplitude of the acoustic output response.

Horizontal dispersion of near-inertial oscillations in a turbulent mesoscale eddy field

We study the dispersion of wind-induced near-inertial oscillations (NIOs) in a fully turbulent baroclinic mesoscale eddy e eld characterized by a continuous wavenumber spectrum. The ine uence of the eddy e eld on the horizontal dispersion of the different NIO modes is analyzed using a vertical normal mode expansion. Previous studies have identie ed two dispersion regimes: trapping and strong dispersion. We examine the modes in physical and spectral space to assess which regime prevails. Numerical and analytical results show the prevalence of a trapping regime. For each NIO mode, there exists a critical horizontal wavenumber, k c, that separates large-scale NIO structures, where trapping dominates, from the much less energetic small-scale NIO structures, where strong dispersion dominates. The maximum efe ciency of dispersion for scales close to k c concentrates NIO kinetic energy at these scales. The wavenumber k c results from a balance between refraction and dispersion. This balance e rst occurs at the highest wavenumber. Thereafter, kc, which has dimensional expression kc 2 5 p/( ftRm 2 ), decreases with time at a rate inversely proportional to the radius of deformation, Rm, of the baroclinic NIO mode considered. As a consequence, at any given time, higher NIO baroclinic mode energy can mostly be found in small-scale negative vorticity structures, such as e laments near sharp vorticity fronts, whereas lower NIO mode energy is concentrated within the core of mesoscale anticyclonic vortices. For large times, a saturation mechanism stops the time-evolution of k c at a value close to the peak of the kinetic energy spectrum of the QG e ow e eld.

Earthquake source model using strong motion displacement as response of finite elastic media

The strong motion displacement records available during an earthquake can be treated as the response of the earth as the a structural system to unknown forces acting at unknown locations. Thus, if the part of the earth participating in ground motion is modelled as a known finite elastic medium, one can attempt to model the source location and forces generated during an earthquake as an inverse problem in structural dynamics. Based on this analogy, a simple model for the basic earthquake source is proposed. The unknown source is assumed to be a sequence of impulses acting at locations yet to be found. These unknown impulses and their locations are found using the normal mode expansion along with a minimization of mean square error. The medium is assumed to be finite, elastic, homogeneous, layered and horizontal with a specific set of boundary conditions. Detailed results are obtained for Uttarkashi earthquake. The impulse locations exhibit a linear structure closely associated with the causative fault. The results obtained are shown to be in good agreement with reported values. The proposed engineering model is then used to simulate the acceleration time histories at a few recording stations. The earthquake source in terms of a sequence of impulses acting at different locations is applied on a 2D finite elastic medium and acceleration time histories are found using finite element methods. The synthetic accelerations obtained are in close match with the recorded accelerations.

Experimental study of thermal noise caused by an inhomogeneously distributed loss

The thermal fluctuation of a mechanical system with an inhomogeneously distributed loss was measured to prove the invalidity of the normal-mode expansion method which is commonly used to calculate the thermal noise of mechanical oscillators. The measured spectrum is inconsistent with the modal expansion model, while they agree well with an evaluation obtained by applying the fluctuation-dissipation theorem to the measured mechanical response. These are the first experimental results which show that mode expansion is invalid.

Excitation and propagation of non-axisymmetric guided waves in a hollow cylinder.

Excitation and propagation of non-axisymmetric guided waves in a hollow cylinder is studied by using the normal mode expansion method (NME). Different sources such as angle beam, tube end excitation with normal beam, and comb transducer possibilities are discussed based on the derivations of the NME method. Numerical calculations are focused on the case of angle beam partial loading. Based on the NME method, the amplitude coefficients for all of the harmonic modes are obtained. Due to the difference of phase velocities for different modes, the superimposed total field varies with propagating distances and hence makes particle displacement distribution patterns (angular profile) change with distance. This varying non-axisymmetric angular profile of guided waves represents a nonuniform energy distribution around the hollow cylinder and thus has an impact on the inspection ability of guided waves. The angular profiles of an angle beam source are predicted by theory and then verified by experiments. The predicted angular profiles also provide information for determining the transducer location to find defects in a certain position on the hollow cylinder.

Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading

Abstract In this paper, a formulation for the dynamic stability analysis of functionally graded shells under harmonic axial loading is presented. A profile for the volume fraction is assumed and a normal-mode expansion of the equations of motion yields a system of Mathieu–Hill equations the stability of which is analyzed by the Bolotin’s method. The present study examines the effects of the volume fraction of the material constituents and their distribution on the parametric response, in particular the positions and sizes of the instability regions.

Dispersion relation for spherical electromagnetic resonances in the atmosphere

An analytic model is developed to describe atmospheric lightning flash fields by use of a normal mode expansion with frequency dependent ionospheric heights. The calculated magnetic field spectra are compared to modeling with the residue series formula, the normal mode expansion with constant ionospheric heights, and measured magnetic field spectra. The proposed new model exhibits the best agreement with the measured magnetic field spectra. The results are summarized in a complex dispersion relation k= ω c S 2 for electrodynamic extremely low frequency wave propagation in the atmosphere and make possible global ionospheric monitoring.

Normal-mode theory for cylinder arrays

We study a set of interacting cylinders under the influence of an electromagnetic field in the long-wavelength limit. Cylindrical harmonics are used as basis functions in order to write the electric potential in terms of multipolar moments of the charge distribution in the cylinders. We get a normal-mode expansion where the effects of geometry and material are separated. It is shown that for a row of identical parallel cylinders the electromagnetic modes are distributed symmetrically about the depolarization factor 1/2, each set coupling to different components of the external field. The amplitudes of these symmetric depolarization factors are the same and satisfy proper sum rules.

An object transport system using flexural ultrasonic progressive waves generated by two-mode excitation

An object transport system using low amplitude and high frequency progressive waves generated by two-mode excitation is presented. A theoretical model for the system was developed using normal mode expansion and the modal participation factor. To identify the factors that affect the transport speed, the changes with the mass of objects on the beam, the input power, the phase difference, and the excitation frequency were experimentally investigated. With a power input of 40 W, a transport speed of 10 cm/s was obtained for an object weighing 30 g. The tests indicate that, not only the phase difference but also the excitation frequency, were the dominant factors in determining the transport speed and direction. Specifically, when the excitation frequency was chosen to be at the exact midpoint of the two modes, the object stopped moving. A slight change of frequency in either direction resulted in change of object transport direction. For actual factory application, a simple stop-go and tracking control using the General Purpose Interface Bus (GPIB) were implemented.

Surface Wave Generation by Buried Forces in a Half Space

ABSTRACTThe method of explicit expansion in normal modes is applied to derive expressions for the Rayleigh waves generated by distributions of buried transient and harmonic forces. The derived expressions are conceptually clear, and relatively simple.

A NORMAL MODE EXPANSION METHOD FOR THE UNDAMPED FORCED VIBRATION OF LINEAR PIEZOELECTRIC SOLID

A normal mode expansion method for the vibrational responses of non-homogeneous linear piezoelectric materials without damping is presented. It can be applied directly to arbitrary piezoelectric composites, which are widely used in vibrational and acoustic sensor/actuator/transmitter applications. In the present article it is shown that if the normal modes are given, the displacement field can be expanded as the linear superposition of normal modes, while the modal coefficients can be represented in terms of surface and volume integrals directly over the six types of distributed excitations without solving the quasi-static solution explicitly. The present treatment is a modification of an earlier work by Liu [11] using a different definition of the so-called quasi-static solution, and the damping effect has been neglected for simplicity. A simple example is given to exemplify the application of the present formulation.

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.