Showing papers on "Normal mode published in 1969"

Computer Simulation of the Lattice Dynamics of Solids

Abstract: Neutron-diffraction data on dispersion curves for the rare-gas solids has emphasized the need for an anharmonic treatment of lattice dynamics. The molecular-dynamic technique is a powerful way of examining the temperature and volume dependence of phonon properties, since anharmonicity is treated without approximation. Using a Lennard-Jones potential, classical calculations have been made, at different temperatures and volumes, of average phonon properties such as pressure and energy, correlations in velocity and position, frequency distribution of normal modes, and phonon-dispersion curves. The volume dependence of the frequency was used to test the Gruneisen approximation. The magnitude of fluctuations in temperature and in the various correlations were compared with theory. These calculations predict relatively large shifts in frequency at constant volume as the temperature is raised in the regime where the classical approximation is valid.

Free Vibrations of Circular Cylindrical Shells

Abstract: : Essentially a compilation of tables giving frequencies and mode shapes of vibrating cylindrical shells, the book is written for research and development engineers as well as for active scientists working in wave propagation and dynamics of thin shells. The text includes a self-contained treatment of the problem of propagation of plane harmonic waves along a hollow circular cylinder, within the framework of the three-dimensional theory of elasticity. The tables present frequencies of free vibrations of such cylinders for a wide range of axial and circumferential wave-lengths and of shell dimensions. In addition, graphs of frequency spectra and associated mode shapes are included. (Author)

Analysis of Anisotropic Plates II

Abstract: The energy formulation presented earlier is extended to include the analysis of plane anisotropic rectangular plates with non-uniform plate properties and non-uniform in-plane and distributed normal loadings. The formulation includes linear theory stability analysis, the calculation of natural frequencies and mode shapes, and analysis of displacement due to lateral loads. The extension presented in the present paper is formulated in a manner such that the non-uniform properties and loadings can be handled with no appreciable increase in difficulty over the formulations previously presented except for the evaluation of certain more general integrals. An easily automated method of evaluating these more general integrals is presented.

The Normal Modes of a Rotating, Elliptical Earth—II Near-Resonance Multiplet Coupling

Abstract: Summary Certain of the Earth's poloidal and toroidal elastic-gravitational normal mode multiplets may be strongly coupled to another multiplet by the Earth's rotation and ellipticity of figure. This paper uses Rayleigh's variational principle to specify the selection rules for strong mode coupling and to examine the nature and effects of this mode coupling. It is found that some of the normal modes of a slowly rotating, slightly elliptical Earth model may consist of about half poloidal type motion and half toroidal type motion. This strong coupling can severely hinder the problem of proper mode identification and measurement of the angular frequencies of oscillation of the Earth's normal modes.

Large‐Amplitude Damped Free Vibration of a Stretched String

Abstract: The equations of motion of a vibrating string are derive, and it is shown that a coupling exists between the longitudinal and transverse modes of vibration. Free transverse vibration of small order, under sinusoidal initial conditions, is analyzed. Under these conditions, the equations are separable. The time‐dependent parts of the equations are solved by the method of variable amplitude and phase. It is seen that, when the vibration is nonplanar, part of the energy oscillates between the mutually perpendicular transverse components with a frequency proportional to the nonlinearity parameter α. The path of any point on the string is shown to be an ellipse with slowly rotating and shrinking axes.

Dilatational‐Mode Sound Transmission in Sandwich Panels

Abstract: Dilational vibration modes have been shown to produce an undesirable “coincidence type” response in the sound‐transmission spectrum of lightweight‐core sandwich panels in the medium to high‐frequency range. In particular, it was found that the dilatational coincidence region often occurs at considerably lower frequencies than those coincidence effects produced by flexural vibration. The governing characteristic frequency equations were developed by employing the Rayleigh‐Ritz Energy Principle for a sandwich panel with variable dimensions and properties. The present solution also employs a general strain energy relation without restricting the skin materials to be identical. Eight characteristic natural frequencies representing dilatational, flexural, and longitudinal vibration were determined for the fundamental as well as higher order modes as a function of panel dimensions and material properties. Throughout this study particular attention was given to the influence of core properties on the dilatational natural frequencies. Experimental panels with various face and core materials were tested under random incidence sound‐transmission conditions and found to compare favorably with the theoretical predictions.

Lattice dynamics of terbium

Abstract: The frequency-wave-vector dispersion relation for the normal modes of vibration of terbium at room temperature has been measured by means of slow-neutron inelastic scattering techniques. The triple-axis spectrometer at the Oak Ridge high flux isotope reactor was used, mostly in the constant-$Q$ mode of operation. Phonon frequencies for wave vectors along the principal symmetry directions have been determined and, in addition, measurements of phonon frequencies along the boundaries of the Brillouin zone and along a more general direction are reported. The data have been fitted with a Born-von K\'arm\'an force model which includes interactions out to the eighth nearest neighbor. The interactions have been assumed to be general (tensor) out to the fourth neighbor and axially symmetric beyond. The model has been used to calculate a frequency distribution function $g(\ensuremath{ u})$ and related quantities such as the lattice specific heat and Debye temperature.

Surface Thermodynamic Functions for Noble‐Gas Crystals

Abstract: Surface thermodynamic quantities have been calculated as functions of temperature between 0°K and the melting temperature for the (111), (100), and (110) surfaces of the noble‐gas solids Ne, Ar, Kr, and Xe. The method of calculation differs from the methods used in previous treatments of surface thermodynamic functions in that the atomic vibrations are taken into account, and the vibrational frequencies are properly calculated rather than obtained from a Debye model or an Einstein model. The quantities calculated are the static surface energy, vibrational surface energy, surface entropy, vibrational surface free energy, and surface specific heat. In addition, a surface frequency‐distribution function f8(ω) has been calculated; f8(ω) is positive at low frequencies, because of the presence of surface modes of vibration, and negative at higher frequencies. This behavior of f8(ω) produces a narrow peak in the graph of the surface specific heat as a function of temperature.

Ferromagnetic Resonance in Epitaxial Garnet Thin Films

Abstract: The chemical vapor deposition process for growing epitaxial yttrium iron garnet (YIG) has matured to the point where films of varying geometry and thickness from ½ to as great as 40 μ can be readily grown on several orientations of gadolinium gallium garnet, yttrium aluminum garnet, and YIG, itself. As a measure of the quality of the films, line widths (at 9.4 GHz and room temperature) as small as 0.6 and 1.4 Oe were obtained in samples with substrates removed and intact, respectively, without a major effort toward improvement. Magnetostatic modes, exchange modes, and ferromagnetic normal modes having comparable exchange and demagnetization energies were observed in these films and in thin bulk disks and were explained in detail by a new theory which applies not only to films, but also to samples of arbitrary shape and size, and includes the effects of exchange and demagnetization energies. The frequencies of the general normal modes were calculated by casting the linearized equation of motion of the magn...

The lattice dynamics of tin telluride

Abstract: The frequencies of normal modes of vibration propagating in high symmetry directions in tin telluride at 100 °K have been measured by means of inelastic neutron scattering techniques. The shape of the longitudinal optic branches is dominated by the screening effects of the free carriers in the crystal. We have developed a modification of the shell model in which this screening is included in a consistent way, and we are able to obtain a satisfactory description of the dispersion curves using this model with an ionic charge of 134e, and a slightly worse fit using a non-ionic model. Both these models were based on the Hartree dielectric function, and were found to be superior to a model based on the Thomas-Fermi dielectric function.

A method for computation of vibration modes and frequencies of orthotropic thin shells of revolution having general meridional curvature

Abstract: Finite element method for computing natural frequencies and mode shapes of thin shells of revolution

Modal Curves of Pre-Twisted Beams of Rectangular Cross-Section

Abstract: An important aspect of the theoretical study of the vibrational characteristics of turbine and compressor blading is the prediction of the modal curves from which the stresses along the length of the blading can be determined.The accurate prediction of the modal curves allowing for such factors as pre-twist, camber, size of cross-section, centrifugal tensile effects, aerodynamic effects, etc., is still not possible. However, a better understanding of the effects of some of these parameters can be obtained by a study of the modal curves of relatively simple idealized models. In this work the theoretical mode shapes of vibration of pre-twisted rectangular cross-section beams for various width to depth ratios and pre-twist angle in the range 0-90° are examined.The theoretical results are obtained by the transformation method given by Carnegie, Dawson and Thomas (1)† and the accuracy of these results is verified by comparison with results obtained by Dawson (2) using the Ritz method. The theoretical results a...

Axisymmetric and beam-type vibrations of thin cylindrical shells.

Abstract: : Considerable attention has been devoted in recent years to the use of shell equations for the prediction of the dynamic behavior of thin cylindrical shells as an element in a missile or spacecraft. The complexity involved in the use of the shell equations must be tolerated for problems that require modes having several circumferential waves (e.g., prediction of panel flutter or response to acoustic loading). Moreover the minimum natural frequency usually corresponds to a mode having two or more circumferential waves. For prediction of overall vehicle behavior, however, one is primarily interested in axisymmetric (n = o) and beam-type (n = 1) modes; in these instances the problem can be considerably simplified by considering the cylinder as a bar for n = 0 modes or a compact beam for n = 1 modes. The present paper examines the accuracy of these engineering approximations as compared with exact solutions from Flugge's shell equations and discusses the error in terms of frequency, mode shape, modal forces, and generalized mass. Consideration is given to the effect of shell bending stiffness and the influence of boundary conditions on these parameters. (Author)

The lattice dynamics of niobium I. Measurements of the phonon frequencies

Abstract: Measurements of the frequencies of the normal modes of vibration of niobium at 296 °K have been made by inelastic neutron scattering techniques. Results are presented for several hundred phonons with wave vectors at general positions in the [11, macron0] reciprocal lattice plane. The phonon dispersion relations in symmetry directions agree well with the previous work of Nakagawa and Woods. The measured frequencies cannot be fitted well by tensor force-constant models even when the range of the forces extends to tenth-nearest neighbours, but the critical points in the phonon density of states derived from the eighth-neighbour model agree with those observed in superconducting tunnelling experiments, and the calculated Debye temperature also agrees quite well with measured values.

Surface Modes of Vibration in Monatomic Crystals

Abstract: In computer-experimental studies of surface vibrations in monatomic crystals, surface modes of a new type have been found near the Brillouin zone boundary in gaps between bands of different kinds of bulk modes (e.g., longitudinal and transverse). Other surface modes, including Rayleigh waves and extra high- or low-frequency modes when adsorbed films are present, have also been found and investigated.

The Symmetry of Lattice Vibrations in the Zincblende and Diamond Structures

Abstract: A simple and practical method is presented for analysing the vibrational symmetries in any perfect crystal. It takes the space spanned by the complete set of normal modes of wave vector $\mathbf{q}$, and factorizes it into a 'cell space' S$\_C$, and a 'complex Euclidean space' S$\_E$. Two simple equations describe the effect of the crystal symmetry operations in each of these factor spaces, so that the symmetry of all the normal modes can readily be determined. This formalism makes it easy to understand several qualitative features of vibrational spectra: for example, if the unit cell contains only two atoms and they are dissimilar, the irreducible representations inside the Brillouin zone always appear doubled. Similarly, it is shown that the number of different types of atom vibrating in any mode cannot exceed the number of times that modes of the same symmetry appear in the vibrational spectrum: so that if a mode has a unique symmetry, only one type of atom is vibrating in that mode, and all other atoms are stationary. Vibrations in the perovskite structure are discussed briefly by way of an example, and this is followed by a fuller treatment of the zineblende and diamond structures.

Thickness‐Twist Vibrations in Beveled AT‐Cut Quartz Plates

Abstract: Frequencies and mode shapes of thickness‐twist vibrations in beveled, rectangular AT‐cut quartz plates have been determined from an infinite‐strip model. In addition, the possible use of beveling for mode suppression has been studied. Experimental results from beveled plates verify the theoretical predictions. Descriptions of the calculations and comparisons with the experimental results are included. Thickness‐shear frequencies and mode shapes were also calculated using a similar model and are compared with thickness‐twist results.

Low-frequency waves and instabilities on the positive column in a magnetic field. i. analysis and study of axisymmetric modes.

Abstract: An analysis is presented for the normal modes of low‐frequency wave propagation on the positive column in an axial magnetic field. Dispersion characteristics are obtained from a hydrodynamic treatment by perturbing a steady‐state discharge model which includes generation and ion inertia, and is valid for arbitrary pressure. The resulting complex eigenvalue problem is solved numerically by use of a Fourier‐Bessel expansion. In general, two types of waves are found for a given frequency: ion‐acoustic waves and electron waves. For the regions of parameter space appropriate to laboratory discharges, the ion‐acoustic wave branches are predicted to be heavily damped, expect at low pressures where the computer solutions are in general agreement with available experimental data. The electron wave branches do not show instability for axisymmetric modes (m = 0) , but can be unstable for asymmetric modes (m ≠ 0) . Stable modes only are considered and it is shown that some hitherto unexplained damped waves, observed experimentally, are identifiable as symmetric electron‐wave modes (m = 0) .

Non-divergent oscillations in the solar atmosphere

Abstract: Non-divergent oscillations having the form of deep water waves are shown to form normal modes or free oscillations of the solar atmosphere under two approximations: the chromosphere-coronal interface behaves like a free surface, and the density scale height is sufficiently large in the convective zone. These modes show the temporal and spatial characteristics of the ‘300 second’ chromospheric oscillations.

Vibrational Spectra and Molecular Potential Fields of Mercurous Chloride, Bromide, and Iodide

Abstract: The Raman and infrared spectra of solid mercurous chloride, bromide, and iodide have been recorded and the vibrational fundamentals assigned on the basis of D∞h molecular symmetry. Normal modes were found at 279, 250, 169, 107, and 42 cm−1 for the chloride, 220, 184, 136, 71, and 37 cm−1 for the bromide, and at 194, 138, 113, 49, and 31 cm−1 for the iodide. The potential fields of the molecules have been analyzed assuming the most general valence forces. The potential constants and normal modes are described and discussed in terms of the assignments. The mixing of the Hg–Hg and the Hg–X stretching modes is considered in detail.

Infra-red absorption of beryllium in cadmium telluride: II

Abstract: The dispersion curves and normal modes of vibration of CdTe have been calculated using a shell model. The single-phonon and combined two-phonon density of states have been computed and the latter is compared with measured two-phonon infra-red absorption. The infra-red absorption induced in CdTe by Be impurity is also calculated and compared with experiment. A resonance mode of Be is observed at 61 cm-1 and a local mode at 391 cm-1; calculated values of these modes obtained using a defect mass approximation are in good agreement with experiment. The structure of observed side bands arising from a simultaneous excitation of the local Be mode and lattice band modes is discussed.

Excitation of Electronic and Nuclear Spin Waves in a Flopped Antiferromagnet

Abstract: Critical rf fields for parallel pumping in a flopped antiferromagnet are derived, and experimental results on excitation of electronic and nuclear spin waves in RbMn${\mathrm{F}}_{3}$ are reported. The theory includes expressions for the critical fields necessary for the excitation of two electronic spin waves, two nuclear spin waves, and one nuclear and one electronic spin wave. This involves computation of normal-mode frequencies and eigenvectors for the coupled nuclear and electronic spin systems. These normal modes for the flopped configuration show a high degree of ellipticity associated with the electronic motion, thus making the system ideal for spin-wave studies by parallel pumping. In particular, we predict that in the flopped state two nuclear spin waves can be excited in RbMn${\mathrm{F}}_{3}$ with critical rf fields of only about 0.1 Oe. Simultaneous excitation of one electronic and one nuclear spin wave by parallel pumping in RbMn${\mathrm{F}}_{3}$ has been performed in the temperature range from 2.4 to 13\ifmmode^\circ\else\textdegree\fi{}K. Wave vectors up to ${10}^{5}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ are excited at a pumping frequency of 8.469 GHz. Using the experimental data to infer the product relaxation rates $\ensuremath{\eta}_{k}^{e}\ensuremath{\eta}_{k}^{n}$, we find order-of-magnitude agreement with Richards's theory for $\ensuremath{\eta}_{k}^{n}$ and with Cole and Courtney's experimental value for $\ensuremath{\eta}_{k}^{e}$. In the product relaxation rates, there appears a minimum at a given $k$ whose value increases linearly with temperature. There is no current theory which predicts such behavior. Excitation of two electronic spin waves could not be detected, since simultaneous electronic plus nuclear excitation was always the first process to become unstable.