Showing papers on "Normal mode published in 1993"

A systematic and efficient method of computing normal modes for multilayered half-space

Abstract: SUMMARY We present a systematic and efficient approach for computing the dispersion curves (i.e. phase velocities for a given frequency) as well as the eigenfunctions of normal modes in a multilayered half-space medium. Our approach is superior to previous approaches in the following aspects. First, it is a simple and self-contained algorithm for simultaneously determining both the phase velocities and the corresponding eigenfunctions. From the basic principle that the normal modes are nun-trivial solutions of the free elastodynamic equation under appropriate boundary conditions, we naturally derive the phase velocities and eigenfunctions. Second, because we use the reflection/transmission coefficients of Luco & Apsel’s version (Luco & Apse1 1983), which intrinsically exclude the growth terms, our algorithm not only exhibits the physical mechanism of the formation of normal modes, i.e. constructive interference, as Kennett’s did (1979,1983), but also is numerically more stable for high-frequency cases. Furthermore, we derive a high-frequency asymptotic solution of the fundamental Rayleigh mode. Therefore, our approach can provide efficient and accurate solutions in any frequency range, and it is expected to be a powerful and useful tool for simulating the LR and RR phases and complete seismograms at regional distances.

Global Alfvén modes: Theory and experiment*

Abstract: It is shown that the theoretical predictions and experimental observations of toroidicity‐induced Alfven eigenmodes (TAE’s) are now in good agreement, with particularly detailed agreement in the mode frequencies. Calculations of the driving and damping rates predict the importance of continuum damping for low toroidal mode numbers and this is confirmed experimentally. However, theoretical calculations in finite‐β, shaped discharges predict the existence of other global Alfven modes, in particular the ellipticity‐induced Alfven eigenmode (EAE) and a new mode, the beta‐induced Alfven eigenmode (BAE). The BAE mode is calculated to be in or below the same frequency range as the TAE mode and may contribute to the experimental observations at high β. Experimental evidence and complementary analyses are presented confirming the presence of the EAE mode at higher frequencies.

Support for anisotropy of the Earth's inner core from free oscillations

Abstract: IN 1983, Poupinet et al.1 observed that compressional seismic waves traversing the inner core along a trajectory parallel to the Earth's rotation axis arrive faster than the same (PKIKP) waves travelling in the equatorial plane. They interpreted this observation as revealing prolate topography of the inner-core boundary. In 1986, Morelli et al.2 and Woodhouse et al.3 suggested that inner-core anisotropy could explain both the travel-time observations and the anomalous splitting of some of the Earth's normal modes. Inner-core anisotropy continues to be the preferred explanation for the travel-time anomalies, although there is disagreement about the magnitude of anisotropy4–6. More recent explanations for the anomalous splitting involve topography of the inner-core and core–mantle boundaries as well as lateral heterogeneity of the core7–11. In particular, Widmer et al.11 dismissed a rather complex recent model of inner-core anisotropy12 because it could not explain the splitting of several previously unidentified modes. Here I show that the anomalous splitting of all currently identified modes can in fact be explained by cylindrical anisotropy of the Earth's inner core that is also compatible with the observed PKIKP travel-time anomalies. The resulting model should be regarded as an upper limit to the amount of anisotropy, as lateral heterogeneity also undoubtedly contributes to the splitting.

Topographic effects in stratified flows resolved by a spectral method

Abstract: The aim of the paper is to validate a particular model formulation in the study of wind-induced response in a wide, rotating channel with varying bottom topography The model consists of a continuous stratified upper layer joined to a homogeneous lower layer For this density profile, standard normal mode solution techniques can be extended to forced models with sloping bottoms in order to separate the vertical dependency The governing equations are then reduced to a time dependent problem by expanding across channel variations in Fourier series and applying periodic boundary conditions along the channel An initial-value problem is solved numerically, using an eigenmode solution technique starting from a state of rest, where a constant wind stress is suddenly imposed and then switched off after a certain period of time The eigenmode solution technique is found to be applicable in the examination of the coupling between vertical modes and the analysis has resulted in a simplified coupling pattern for this specific two layer model Modified baroclinic eigenmodes are introduced in a superposition of each baroclinic and the barotropic mode, according to the eigenvalue problem For these modes the interaction amongst baroclinic modes is found to be insignificant while the barotropic eigenmodes are not found to be affected by the stratification For varying bottom topography these modified baroclinic modes will reflect both stratification and topography Model simulations show that the flow pattern is dominated by long period barotropic oscillations as topographic Rossby waves The waves show a dominance in the second lateral mode, and they are less affected by the stratification The barotropic response to the wind is mirrored in internal displacements and shows long periodic oscillations corresponding to the topographic waves The use of a bottom Ekman layer has shown that bottom friction mainly influences on long-period barotropic modes The baroclinic modes are damped by internal friction DOI: 101034/j1600-08701993t01-1-00003x

Shaped Input Control of a System With Multiple Modes

Abstract: This paper describes a method for limiting vibration in flexible systems that have more than one characteristic frequency and mode. It is only necessary to have knowledge of the component mode frequencies and damping ratios in order to be able to calculate the timing and magnitudes of the impulse sequence used in the shaping. Only two impulses, in the nonrobust case, or three impulses in a more robust case, are necessary regardless of the number of component frequencies. Simple tests are established to determine when this technique can be used and examples are presented.

Waveforms of long-period body waves in a slightly aspherical earth model

Abstract: Summary Considering the coupling of normal modes of different dispersion branches, we present a formulation for synthesizing waveforms of long-period body waves in a slightly aspherical Earth model. A general asymptotic result, which is valid when angular degrees of modes are much higher than the maximum harmonic degree of lateral heterogeneity, is derived for arbitrary source mechanisms and receiver components. If we restrict the coupling to the modes along a single dispersion branch, this result reduces to the pre-existing theory for surface waves. For body waves, we demonstrate numerically that cross-branch coupling is important to set up the correspondence between normal mode and ray theories. Actually, the coupling of the modes which interfere constructively in the spherically symmetric reference model represents the largest effects of heterogeneity on the waveform of a body wave packet. This would lead to a significant reduction in computation time. Numerical tests show that the formulation is able to bring out the ray character of body waves: only the structural perturbation in the neighbourhood along the geometrical ray affects the seismogram.

Resonant behaviour of MHD waves on magnetic flux tubes. IV: Total resonant absorption and MHD radiating eigenmodes

Abstract: Resonant absorption of MHD waves on a nonuniform flux tube is investigated as a driven problem for a 1D cylindrical equilibrium. The variation of the fractional absorption is studied as a function of the frequency and its relation to the eigenvalue problem of the MHD radiating eigenmodes of the nonuniform flux tube is established. The optimal frequencies producing maximal fractional absorption are determined and the condition for total absorption is obtained. This condition defines an impedance matching and is fulfilled for an equilibrium that is fine tuned with respect to the incoming wave. The variation of the spatial wave solutions with respect to the frequency is explained as due to the variation of the real and imaginary parts of the dispersion relation of the MHD radiating eigenmodes with respect to the real driving frequency.

Shallow-Water Flow past Isolated Topography. Part II: Transition to Vortex Shedding

Abstract: The formation of Karman vortex streets is studied within the framework of single-layer shallow-water dynamics and in absence of surface friction and background rotation. In the first part of this study, steady numerical solutions for flow past circular topography were obtained by imposing a symmetry condition that essentially suppressed vortex shedding. In the second part, this symmetry condition is relaxed in order to study the transition into the vortex-shedding regime. This transition is due to an instability of the symmetric wake pattern. The most unstable global normal mode of this instability is derived by a numerical method. Most of the features of this mode can be understood in terms of the absolute instability theory. The mode is essentially barotropic and relies on a positive feedback between the perturbations located on the two shearlines on either side of the wake. The classical shear modes centered on a single shearline are, on the other hand, shown to be absolutely stable even thoug...

Study of vibrational modes of gold nanostructures by picosecond ultrasonics

Abstract: We have used picosecond ultrasonics techniques to study the vibrations of several nanostructures composed of laterally patterned gold films on fused quartz substrates. The structures include arrays of stripes and dots with lateral repeat distance as small as 2000 A. The frequencies and attenuation rates of the low‐lying normal modes of these structures are measured. We compare these results with those obtained theoretically from a finite element analysis of the dynamics of the structures, and discuss the physical nature of the principal vibrational modes.

Mode Analysis and Stabilization of a Spatial Power Combining Array with Strongly

Abstract: In order to attain stabilized power-combining opera- tions of a strongly coupled active antenna array, mode analysis for the multi-moding problem of the array is presented, and an effective method for exciting only the in-phase power-combining mode is proposed. In a one-dimensional array of active antennas coupled mutually through microstrip lines, the frequencies and the voltage distributions of the normal modes are obtained. Stable modes of the array are identified using the averaged potential theory. Time evolutions of the mode amplitudes are also calculated. In order for only the desired power-combining mode to oscillate, appropriate resistors are introduced at the midpoints of the coupling lines between the active antennas. An experiment for the arrays with up to eight active antennas has been carried out using Gunn diodes. It has been confirmed that the introduction of the resistors is effective for stable excitation of the in-phase power-combining mode.

Relation between main and normal mode relaxations for polyisoprene studied by dielectric spectroscopy

Abstract: Broadband dielectric spectroscopy (10 -2 -10 6 Hz) is employed to study the relation between α relaxation and normal mode process for oligomeric polyisoprene samples of different molecular weights. It is shown that α relaxation and normal mode process merge above the Vogel temperature of the α relaxation. This means that the Rouse model is not applicable to the normal mode relaxation near the glass transition. Further, this result is discussed from a microscopic point of view in terms of the different temperature dependences of the two relevant length scales, the correlation length for the α relaxation ξ and the end-to-end vector of the polymer chain

Electrostatic modes of ion-trap plasmas

Abstract: The electrostatic modes of a non-neutral plasma confined in a Penning or Paul (rf) trap are discussed in the limit that the Debye length is small compared to the plasma dimensions and the plasma dimensions are small compared to the trap dimensions. In this limit the plasma shape is spheroidal and analytic solutions exist for all of the modes. The solutions for the modes of a Paul-trap plasma are a special case of the modes of a Penning-trap plasma. A simple derivation of some of the low-order quadrupole modes is given. Experimental measurements of these mode frequencies on plasmas of laser-cooled ${\mathrm{Be}}^{+}$ ions in a Penning trap agree well with the calculations. A general discussion of the higher-order modes is given. The modes provide a nondestructive method for obtaining information on the plasma density and shape. In addition, they may provide a practical limit to the density and number of charged particles that can be stored in a Penning trap.

Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum

Abstract: We present a comprehensive normal-mode decomposition analysis for the recently introduced [ J. Opt. Soc. Am. A10, 95 ( 1993)] class of twisted Gaussian Schell-model fields in partially coherent beam optics. The formal analogies to quantum mechanics in two dimensions are exploited. We also make effective use of a dynamical SU(2) symmetry of these fields to achieve the mode decomposition and to determine the spectrum. The twist phase is nonseparable in nature, rendering it nontrivially two dimensional. The consequences of this, resulting in the need to use Laguerre–Gaussian functions rather than products of Hermite–Gaussians, are carefully analyzed. An important identity involving these sets of special functions is established and is used in deriving the spectrum.

The thermodynamic stability of clathrate hydrate. II. Simultaneous occupation of larger and smaller cages

Abstract: The thermodynamic stability of a clathrate hydrate encaging methane or xenon has been investigated by examining the free energy of formation. The total free energy is divided into several contributions: the interaction between water and guest molecules, the entropic contribution arising from the combinations of cage occupancy, and also the free energy arising from the guest molecular motions inside cages. Our method is based on the generalized van der Waals and Platteeuw theory used for the study of the hydrate encaging propane and is free from some of the fundamental assumptions in the original theory. This enables us to evaluate separately the factors which contribute significantly to the thermodynamic stability of the hydrate, and to address a question as to what extent the original theory is applicable to the prediction of the phase diagrams. While the potential energy curve of the guest molecule with surrounding water molecules in a smaller cage has a single minimum and the molecular motion can be accurately approximated to a collection of harmonic oscillators strongly coupled with the host molecules, the guest molecular motion in a larger cage does not couple with the host. To show evidence that the fixed lattice approximation is sufficient to describe molecular motions inside the larger cage, two kinds of molecular dynamics simulations have been carried out. In one simulation, both host water and guest molecules move according to the classical equations of motion. In another simulation only guest molecules are allowed to move, interacting with fixed host molecules. We perform two kinds of analyses on those motions. In the first analysis, the velocity autocorrelation functions are calculated from molecular dynamics simulations at several temperatures and the power spectra are obtained by the Fourier transform of the correlation functions. In the second, a so‐called normal mode analysis is performed and the densities of state for intermolecular vibrations are obtained. The densities of state (corresponding to 0 K) are compared with the power spectra. It is revealed that the fixed lattice approximation can be applied when describing the molecular motions of methane and xenon in larger cages. The free energy for the accommodation of an extremely large CF4 or small argon guest is also examined.

Low-frequency vibrations and relaxations in glasses

Abstract: Using molecular dynamics a glass of soft spheres, quenched to zero temperature, is produced and its vibrational behaviour investigated by an analysis of the normal modes. In addition to localized modes with high frequencies, one finds (quasi)localized modes with low frequencies which are centred around structural anomalies of the glass. The glass is then heated in stages to 12.5% of the glass temperature and is allowed to relax. In the relaxed glasses, again (quasi)localized low-frequency modes are found, with effective masses ranging upward from about 10 atomic masses as in the original glasses. The observed relaxations have similar effective masses and are closely related to the (quasi)localized vibrations.

Ab initio vibrational frequencies of the triflate ion, (CF3SO3)-

Abstract: The optimized geometry, harmonic vibrational frequencies and infrared intensities of the trifluoromethanesulfonate (triflate) ion, CF3SO3-, have been determined with ab initio self-consistent Hartree-Fock theory by using 3-21G* and 6-31G* Gaussian basis sets. Second-order Moller-Plesset perturbation calculations were also carried out with 6-31G* basis. A normal mode analysis shows that the vibrations assigned as symmetric and antisymmetric CF3 stretching involve other internal coordinates as well, viz., CF3 bending and CS stretching. The corresponding SO3 stretching modes, on the other hand, are almost entirely described with SO stretching coordinates. The assignments of the symmetric and antisymmetric SO3 and CF3 stretching vibrations from Moller-Plesset theory are seen to be different from those reported in the literature. Recent infrared spectroscopic experiments of the triflate ion coordinated to the zinc or lead ion in poly(ethylene oxide) complexes support the conclusions from second-order perturbation theory. The vibrational frequencies and infrared intensities show a strong dependence on basis set and electron correlation.

Low-frequency waves in magnetized dusty plasmas

Abstract: A systematic analysis of low-frequency waves such as hydromagnetic and acoustic waves in a magnetized dusty plasma containing electrons, ions and dust particles is presented. Starting with the three-fluid equations and the Maxwell equations, we derive, retaining finite ion mass, an effective two-fluid model incorporating deviations from the frozen-in-fluid approximation for the ion and electron fluid motions. We show that normal modes exist in two widely separated frequency regimes. In addition to obtaining hydromagnetic waves that are generalizations of the usual Alfven and magneto-acoustic modes in a two-component electron–ion plasma, we demonstrate the existence of a new class of magneto-acoustic waves (both fast and slow type) called ‘dustmagneto-acoustic waves’. These modes have qualitatively different dependences on the equilibrium parameters such as density, magnetic field and temperature when compared with the usual magneto-acoustic waves. The new modes arise owing to finite ion mass (compared with the dust-particle mass), leading to an effective inertial resistivity that inhibits the ion (as well as the electron) fluid from being frozen to the magnetic field lines. The fast branch of the dust-magneto-acoustic waves is shown to be the electromagnetic generalization of the electrostatic dust-acoustic wave recently obtained in unmagnetized dusty plasmas. In the two-component limit the new modes degenerate into the usual type of magneto-acoustic waves. Dispersion relations for various other modes are also presented.

Diagonalization in a Mixed Basis: A Method to Compute low-Frequency Normal Modes for large Macromolecules

Abstract: Low-frequency collective motions in proteins are generally very important for their biological functions. To study such motions, harmonic dynamics proved most useful since it is a straightforward method; it consists of the diagonalization of the Hessian matrix of the potential energy, yielding the vibrational spectrum and the directions of internal motions. Unfortunately, the diagonalization of this matrix requires a large computer memory, which is a limiting factor when the protein contains several thousand atoms. To circumvent this limitation we have developed three methods that enable us to diagonalize large matrices using much less computer memory than the usual harmonic dynamics. The first method is approximate; it consists of diagonalizing small blocks of the Hessian matrix, followed by the coupling of the low-frequency modes obtained for each block. It yields the low-frequency vibrational spectrum with a maximum error of 20%. The second method consists, after diagonalizing small blocks, of coupling the high- and low-frequency modes using an iterative procedure. It yields the exact low-frequency normal modes, but requires a long computational time with convergence problems. The third method, DIMB (Diagonalization in a Mixed Basis), which has the best performance, consists of coupling the approximate low-frequency modes with the mass-weighted cartesian coordinates, also using an iterative procedure. It reduces significantly the required computer memory and converges rapidly. The eigenvalues and eigenvectors obtained by this method are without significant error in the chosen frequency range. Moreover, it is a general method applicable to any problem of diagonalization of a large matrix. We report the application of these methods to a deca-alanine helix, trypsin inhibitor, a neurotoxin, and lysozyme. © 1993 John Wiley & Sons, Inc.

Mode analysis and stabilization of a spatial power combining array with strongly coupled oscillators

Abstract: In order to attain stabilized power-combining operation of a strongly coupled active antenna array, a mode analysis for the multimoding array problem is presented, and an effective method for exciting only the in-phase power-combining mode is proposed. In a one-dimensional array of active antennas coupled mutually through microstrip lines, the frequencies and the voltage distributions of the normal modes are obtained. Stable modes of the array are identified using the averaged potential theory. Time evolutions of the mode amplitudes are also calculated. In order for only the desired power-combining mode to oscillate, appropriate resistors are introduced at the midpoints of the coupling lines between the active antennas. An experiment for arrays with up to eight active antennas has been carried using Gunn diodes. It has been confirmed that the introduction of the resistors is effective for stable excitation of the in-phase power-combining mode. >

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

Abstract: Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1 : 1 resonance in the (1, 2) and (3, 1) plate modes.

Mode localization in a class of multidegree-of-freedom nonlinear systems with cyclic symmetry

Abstract: The free oscillations of n-degree-of-freedom (DOF) nonlinear systems with cyclic symmetry and weak coupling between substructures are examined. An asymptotic methodology is used to detect localized nonsimilar normal modes, i.e., free periodic motions spatially confined to only a limited number of substructures of the cyclic system. It is shown that nonlinear mode localization occurs in the perfectly symmetric, weakly coupled structure, in contrast to linear mode localization, which exists only in the presence of substructure “mistuning.” In addition to the localized modes, nonlocalized modes are also found in the weakly coupled system. The stability of the identified modes is investigated by means of an approximate two-timing averaging methodology, and the general theory is applied to the case of a cyclic system with three-DOF. The theoretical results are then verified by direct numerical integrations of the equations of motion.

Lattice vibrations of Y1−xPrxBa2Cu3O7: theory and experiment

Abstract: We report measurements of the far-infrared reflectance of Y 1− x Pr x Ba 2 Cu 3 O 7 thin films. Detailed understanding of the experimental lineshapes is achieved with the proper treatment of the SrTiO 3 substrate, the finite film thicknesses, the background contribution of free carriers and the mid-infrared band, and the phonon modes. Shell-model calculations predict the frequencies of the infrared-active phonons to be very similar for YBa 2 Cu 3 O 7 and PrBa 2 Cu 3 O 7 with the exception of rare-earth-related vibrations. The experimental frequencies of the in-plane as well as z -polarized infrared-active phonons of PrBa 2 Cu 3 O 7 agree satisfactorily with the calculated ones. An in-plane mode at 270 cm -1 , assigned to vibrations confined to the Y/Pr and neighboring oxygen ions, is found to be sensitive to the rare-earth-ion substitution. We report also the frequencies of the Raman-active B 2g and B 3g phonons of PrBa 2 Cu 3 O 7 , thus giving a nearly complete picture of all k =0 eigenmode frequencies in the RBa 2 Cu 3 O 7 crystal structure.

Nonradial oscillations of neutron stars: A new branch of strongly damped normal modes.

Abstract: We study small, nonradial oscillations of neutron stars in a general relativistic perturbation treatment, considering different values for the central density of the star. We adapt two techniques used previously for the determination of quasinormal modes of black holes, allowing us for the first time to determine without approximation solutions which satisfy purely outgoing boundary conditions at spatial infinity. We confirm the existence of strongly damped complex normal modes found by Kokkotas and Schutz (w modes). In addition, we identify a new branch of strongly damped modes (w II modes). Our new modes are much more similar to quasinormal modes of black holes than the w modes known before