Showing papers on "Normal mode published in 1991"

Collective motions in proteins: a covariance analysis of atomic fluctuations in molecular dynamics and normal mode simulations.

Abstract: A method is described for identifying collective motions in proteins from molecular dynamics trajectories or normal mode simulations. The method makes use of the covariances of atomic positional fluctuations. It is illustrated by an analysis of the bovine pancreatic trypsin inhibitor. Comparison of the covariance and cross-correlation matrices shows that the relative motions have many similar features in the different simulations. Many regions of the protein, especially regions of secondary structure, move in a correlated manner. Anharmonic effects, which are included in the molecular dynamics simulations but not in the normal analysis, are of some importance in determining the larger scale collective motions, but not the more local fluctuations. Comparisons of molecular dynamics simulations in the present and absence of solvent indicate that the environment is of significance for the long-range motions.

Closed-form dynamic model of planar multilink lightweight robots

Abstract: Closed-form equations of motion are presented for planar lightweight robot arms with multiple flexible links. The kinematic model is based on standard frame transformation matrices describing both rigid rotation and flexible displacement, under small deflection assumption. The Lagrangian approach is used to derive the dynamic model of the structure. Links are modeled as Euler-Bernoulli beams with proper clamped-mass boundary conditions. The assumed modes method is adopted in order to obtain a finite-dimensional model. Explicit equations of motion are detailed for two-link case assuming two modes of vibration for each link. The associated eigenvalue problem is discussed in relation with the problem of time-varying mass boundary conditions for the first link. The model is cast in a compact form that is linear with respect to a suitable set of constant parameters. Extensive simulation results that validate the theoretical derivation are included. >

Feedback instability of the ionospheric resonant cavity

Abstract: A model is developed that provides a theoretical basis for previous numerical results showing a feedback instability with frequencies characteristic of Alfven travel times within the region of the large increase of Alfven speed above the ionosphere. These results have been extended to arbitrary ionospheric conductivity by developing a numerical solution of the cavity dispersion relation that involves Bessel functions of complex order and argument. It is concluded that the large contrast between the magnetospheric and ionospheric Alfven speed leads to the formation of resonant cavity modes with frequencies ranging from 0.1 to 1 Hz. The presence of the cavity leads to a modification of the reflection characteristics of Alfven waves with frequencies that compare to the cavity's normal modes.

Potential energy surfaces, quasiadiabatic channels, rovibrational spectra, and intramolecular dynamics of (HF)2 and its isotopomers from quantum Monte Carlo calculations

Abstract: We report analytical representations of the six‐dimensional potential energy hypersurface for (HF)2, the parameters of which are closely adjusted to low energy experimental properties such as hydrogen bond dissociation energy (D0=1062 cm−1 ) and vibrational–rotational spectra in the far and mid infrared. We present a detailed analysis of properties of the hypersurface in terms of its stationary points, harmonic normal mode amplitudes, and frequencies for the Cs minimum and C2h saddle point and effective Morse parameters and anharmonic overtone vibrational structure for the hydrogen bond and the HF stretching vibrations. The comparison between experimental data and the potential energy surface is carried out by means of accurate solutions of the rotational–vibrational Schrodinger equation with quantum Monte Carlo techniques, which include anharmonic interactions between all modes for the highly flexible dimer. Two extensions of the quantum Monte Carlo technique are presented, which are based on the clamped...

Optical Waveguide Concepts

Abstract: 1. Basic Concepts. Maxwell equations. Normal mode theory. Scattering matrices. Coupled mode theory. Variational techniques. 2. Modes of Optical Waveguides - I. The weak guidance approximation. Waveguides embedded in a low index medium. Step-index slab waveguides. Step-index circular fibers. Weakly guiding anisotropic waveguides. 3. Modes of Optical Waveguides - II. Graded slabs or fibers. 3-D planar waveguides. Numerical methods for arbitrary cross-sections. Leaky modes. 4. Perturbation Techniques. Modal expansion method. Variational techniques. Reciprocity techniques. Global perturbation methods. Applications. 5. Non-Uniform Systems. Local mode expansions. Bent waveguides. Bent and twisted waveguides. Non-uniform curvatures. 6. Discontinuities. Excitation of a guided mode through a junction. Abrupt discontinuities. Bibliography. Index.

On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects

Abstract: The Hamilton’s principle approach to the calculation of vibrational modes of elastic objects with free boundaries is exploited to compute the resonance frequencies of a variety of anisotropic elastic objects, including spheres, hemispheres, spheroids, ellipsoids, cylinders, eggs, shells, bells, sandwiches, parallelepipeds, cones, pyramids, prisms, tetrahedra, octahedra, and potatoes. The paramount feature of this calculation, which distinguishes it from previous ones, is the choice of products of powers of the Cartesian coordinates as a basis for expansion of the displacement in a truncated complete set, enabling one to analytically evaluate the required matrix elements for these systems. Because these basis functions are products of powers of x, y, and z, this scheme is called the xyz algorithm. The xyz algorithm allows a general anisotropic elastic tensor with any position dependence and any shape with arbitrary density variation. A number of plots of resonance spectra of families of elastic objects are...

The Slow Sea Surface Temperature Mode and the Fast-Wave Limit: Analytic Theory for Tropical Interannual Oscillations and Experiments in a Hybrid Coupled Model

Abstract: A modified shallow water model with simplified mixed layer dynamics and a sea surface temperature (SST) equation is employed to gain a theoretical understanding of the modes and mechanisms of coupled air-sea interaction in the tropics. Approximations suggested by a scaling analysis are used to obtain analytic results for the eigenmodes of the system. A slow time scale, unstable eigenmode associated with the time derivative of the SST equation is suggested to be important in giving rise to interannual oscillations. This slow SST mode is not necessarily linked to conventional equatorial oceanic wave modes. A useful limit of this mode is explored in which the wave speed of uncoupled oceanic wave modes is fast compared to the time scales that arise from the coupling. This is referred to as the fast-wave limit. The dispersion relationship in this limit is used to present a number of coupled feedback mechanisms, which contribute simultaneously to the instability of the SST mode. It is suggested that in...

Theory of ab initio molecular-dynamics calculations

Abstract: The theoretical basis of the first-principles molecular dynamics introduced by Car and Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] is investigated. We elucidate how the classical dynamics generated by the Car-Parrinello Lagrangian approximates efficiently the quantum adiabatic evolution of a system and discuss the role played by the spectrum of the eigenvalues of the Hamiltonian of Kohn and Sham [Phys. Rev. 140, A1133 (1965)]. A detailed characterization of the statistical ensemble sampled in the numerical simulation is given. By combining theoretical arguments and numerical results we demonstrate that the motion of the electronic variables is a superposition of a direct drag due to the ions and of high-frequency normal modes. By making a connection with the averaging methods of classical mechanics, we argue that whenever it is possible to get a large separation between the time scales of these modes and the ionic frequencies, the dynamics of the ions closely approximates that resulting from the adiabatic approximation. We introduce simple n-level models, easily amenable to analytic treatment, to add clarity and study the possible mechanisms of broken adiabaticity encountered in the actual calculations.

Free vibrations of delaminated beams

Abstract: Free vibration of laminated composite beams is studied. The effect of interply delaminations on natural frequencies and mode shapes is evaluated both analytically and experimentally. The equation of motion and associated boundary conditions are derived for the free vibration of a composite beam with a delamination of arbitrary size and location. A generalized variational principle is used to formulate the equation of motion, taking into account the interlaminar stress concentration at the crack-tips. This is accomplished by introducing a 'crack function' into the beam's compatibility relations. This function has its maximum value at the crack tip and decays exponentially in the longitudinal direction. The rate of exponential decay is determined by a least-square fit with the experimental results. The effect of coupling between longitudinal vibration and bending vibration is considered in the present study. This coupling effect is found to significantly affect the natural frequencies and mode shapes of the delaminated beam.

Microwave localization by two-dimensional random scattering

Abstract: WAVEFUNCTIONS of electrons or photons in a strongly scattering random medium may become localized owing to the underlying wave nature of the particles1,2. Particularly surprising and counterintuitive is the prediction that, under appropriate conditions, scatterers placed randomly in space will always produce fully localized states—that is, an energy distribution of the normal modes whose envelope decays exponentially in all directions. In consequence, energy at the resonant frequency of a localized mode, injected into that mode's region of space, cannot diffuse away, but remains trapped until dissipated. Here we report measurements of the electric-field energy density for microwave radiation localized in essentially two-dimensional space by scattering from a random array of dielectric cylinders placed between a pair of parallel conducting plates. We detect regions of high energy density representing the signature of localized modes. The available range of measured variables, scattering materials and cylinder configurations offer the opportunity to provide quantitative answers to important general questions about strong localization. In particular, a better understanding of two-dimensional localization raises the possibility of using localized-mode resonances as a diagnostic tool for situations in which localization phenomenon may occur naturally3—for example, in investigations of the internal distribution of media and defects in geological strata, under-ocean topology or electronic thin films, all of which may exhibit pseudo-two-dimensional characteristics.

Forced nutations of the Earth: Influence of inner core dynamics: 2. Numerical results and comparisons

Abstract: We apply the theory developed in Paper 1 (Mathews et al., this issue), which includes the solid inner core explicitly in the dynamical equations, to obtain the eigenfrequencies and other characteristics of the Earth's nutational normal modes as well as the amplitudes of forced nutations at various tidal frequencies, for two commonly used Earth models, 1066A and the Preliminary Reference Earth Model (PREM). We also make an evaluation of various procedures for taking account of known deviations of the Earth from models, notably in the dynamical ellipticity e for which the two models yield values which are over 1% smaller than the value e* deduced from the precession constant. On adopting the procedure of simply replacing e by e* in the equations of our theory, the values obtained for some of the nutation amplitudes for model 1066A differ significantly from the corresponding results of Wahr (1981b). The largest of the differences, which occur in the prograde semiannual, retrograde 18.6 year, and retrograde annual nutation terms, amount to −0.59, 0.35, and −0.25 milliarcseconds (mas), respectively, while the standard errors in the very long baseline interferometry (VLBI) determinations are now only about 0.04 mas except in the long period terms. The difference in the procedures used to take account of the discrepancy between e and e* contributes −0.56, 0.81, and −0.17 mas, respectively, to the above-noted differences. For the purpose of comparison with VLBI-observed data, we use the results for a “modified PREM,” defined by a set of Earth model parameters which differ from those of PREM only in having e* for the dynamical ellipticity of the Earth as a whole and a modified value for the dynamical ellipticity eƒ of the fluid outer core. The amplitudes computed for this model, with corrections applied for the effects of ocean tides and mantle anelasticity, are in generally satisfactory agreement with observed values, when the modified eƒ is determined by matching the theoretical and observed values for the retrograde annual term. (The modified eƒ is 0.002665, about 4.6% higher than in PREM, equivalent to an increase, relative to PREM, of about 430 meters in the difference between the equatorial and the polar radii of the core-mantle boundary. We find that contributions from inner-core dynamics to the prograde semiannual and annual, and the retrograde 18.6 year and annual terms, recomputed for modified PREM, amount to −0.09, 0.03, −0.36, and −0.09 mas, respectively.) The largest residual remaining, other than in the long-period terms which still have an uncertainty of about 1 mas, is −0.25 mas in the prograde fortnightly amplitude. Consideration of possible sources of the discrepancies is facilitated by a resonance expansion of the amplitude of forced nutations, as a function of frequency, normalized relative to that for a rigid Earth model. We also provide tables which exhibit the sensitivities of various relevant quantities (the eigenfrequencies and the coefficients which appear in the resonance expansion, as well as the nutation amplitudes at important tidal frequencies) to possible errors in the Earth parameters which enter our theory. Reconciliation of theoretical and experimental values for the prograde fortnightly term, for instance, could be accomplished, without affecting significantly the comparison for other nutation terms, by a decrease of about 10% in the value of the compliance parameter k that represents, in effect, the deformability of the Earth as a whole in response to perturbations of its rotation; but this change in k would have to be produced by some mechanism which does not affect the values of the other compliances.

Resonant interaction of sound wave with internal solitons in the coastal zone

Abstract: Naturally occurring internal solitary wave trains (solitons) have often been observed in the coastal zone, but no reported measurements of such solitary waves include low‐frequency long‐range sound propagation data. In this paper, the possibility that internal waves are responsible for the anomalous frequency response of shallow‐water sound propagation observed in the summer is investigated. The observed transmission loss is strongly time dependent, anisotropic and sometimes exhibits an abnormally large attenuation over some frequency range. The parabolic equation (PE) model is used to numerically simulate the effect of internal wave packets on low‐frequency sound propagation in shallow water when there is a strong thermocline. It is found that acoustic transmission loss is sensitive to the signal frequency and is a ‘‘resonancelike’’ function of the soliton wavelength and packet length. The strong interaction between acoustic waves and internal waves, together with the known characteristics of internal waves in the coastal zone, provides a plausible explanation for the observed anomalous sound propagation in the summer. By decomposing the acoustic field obtained from the PE code into normal modes, it is shown that the abnormally large transmission attenuation is caused by ‘‘acoustic mode‐coupling’’ loss due to the interaction with the internal waves. It is also shown that the ‘‘resonancelike’’ behavior of transmission loss predicted by the PE analysis is consistent with mode coupling theory. As an inverse problem, low‐frequency acoustic measurements could be a potential tool for remote‐sensing of internal wave activity in the coastal zone.

The definition of reaction coordinates for reaction‐path dynamics

Abstract: We present equations for generalized‐normal‐mode vibrational frequencies in reaction‐path calculations based on various sets of coordinates for describing the internal motions of the system in the vicinity of a reaction path. We consider two special cases in detail as examples, in particular three‐dimensional atom–diatom collisions with collinear steepest descent paths and reactions of the form CX3+YZ→CX3 Y+Z with reaction paths having C3v symmetry. We then present numerical comparisons of the differences in harmonic reaction‐path frequencies for various coordinate choices for three such systems, namely, H+H2→H2+H, O+H2→OH+H, and CH3+H2→CH4+H. We test the importance of the differences in the harmonic frequencies for dynamics calculations by using them to compute thermal rate constants using variational transition state theory with semiclassical ground‐state tunneling corrections. We present a new coordinate system for the reaction CH3+H2 that should allow for more accurate calculations than the Cartesian ...

Diskoseismology: Probing Accretion Disks. I. Trapped Adiabatic Oscillations

Abstract: The normal modes of acoustic oscillations within thin accretion disks which are terminated by an innermost stable orbit around a slowly rotating black hole or weakly magnetized compact neutron star are analyzed. The dominant relativistic effects which allow modes to be trapped within the inner region of the disk are approximated via a modified Newtonian potential. A general formalism is developed for investigating the adiabatic oscillations of arbitrary unperturbed disk models. The generic behavior is explored by way of an expansion of the Lagrangian displacement about the plane of symmetry and by assuming separable solutions with the same radial wavelength for the horizontal and vertical perturbations. The lowest eigenfrequencies and eigenfunctions of a particular set of radial and quadrupole modes which have minimum motion normal for the plane are obtained. These modes correspond to the standard dispersion relation of disk theory.

Localized low-frequency vibrational modes in glasses.

Abstract: This is the publisher's version, also available electronically from http://journals.aps.org/prb/abstract/10.1103/PhysRevB.44.6746.

Projection of Monte Carlo and molecular dynamics trajectories onto the normal mode axes: human lysozyme.

Abstract: A method is presented to describe the internal motions of proteins obtained from molecular dynamics or Monte Carlo simulations as motions of normal mode variables. This method calculates normal mode variables by projecting trajectories of these simulations onto the axes of normal modes and expresses the trajectories as a linear combination of normal mode variables. This method is applied to the result of the molecular dynamics and the Monte Carlo simulations of human lysozyme. The motion of the lowest frequency mode extracted from the simulations represents the hinge bending motion very faithfully. Analysis of the obtained motions of the normal mode variables provides an explanation of the anharmonic aspects of protein dynamics as due first to the anharmonicity of the actual potential energy surface near a minimum and second to trans-minimum conformational changes.

Instabilities in self-gravitating gaseous discs

Abstract: The non-linear evolution of thin self-gravitating gaseous discs with unstable non-axisymmetric normal modes are calculated. The instabilities occur when the initially axisymmetric disc is stable to axisymmetric modes. The modes can have corotation situated either inside or just outside the disc's outer boundary. The instable modes are found to be important for redistributing the disc's mass and angular momentum on dynamical time-scale appropriate to the outer disc. Unstable non-axisymmetric modes may be crucial in driving mass accretion on to central galactic black holes or protostars embedded by self-gravitating gaseous discs.

Hydrodynamic surface modes on concentrated polymer solutions and gels

Abstract: We present a coupled two‐fluid model for hydrodynamic surface modes on concentrated polymer solutions and polymer gels. This model is used to investigate such surface modes in the limit of strong coupling. The surface mode dispersion relation ω(k) and the structure factor S(k,ω) of thermally induced surface fluctuations are derived in this limit for materials with a general constitutive relation η(ω). We argue for the existence of several types of surface modes, including Rayleigh elastic waves, capillary waves, and overdamped modes resulting from viscous dissipation in the solvent as well as from polymer diffusion in the case of polymer solutions. The properties of surface modes are discussed separately for the cases of polymer gels and concentrated polymer solutions. Detailed predictions for the form of the normal modes ω(k), and the surface mode structure factor S(k,ω) are given and their dependence on relevant material properties are discussed in each case. Possible experimental scenarios are anticipa...

Theory of electrostatic fluid modes in a cold spheroidal non-neutral plasma.

Abstract: The normal modes of a magnetized spheroidally shaped pure ion plasma have recently been measured. Here the theory of these modes is presented. Although one might expect that a numerical solution is required (because the plasma dielectric is anisotropic and the plasma is inhomogeneous), the problem is actually separable in an unusual coordinate system. The result is a simple electrostatic fluid dispersion relation for modes in a cloud of any spheroidal shape.

Large‐scale three‐dimensional even‐degree structure of the Earth from splitting of long‐period normal modes

Abstract: Spectra of the Earth's free oscillations, which depart significantly from those predicted for spherically symmetric Earth models, contain important information on the large-scale aspherical structure of the Earth. In this paper we present theory, techniques, and numerical results for retrieval of Earth models (including mantle heterogeneity, topography of the core-mantle boundary (CMB), and inner core anisotropy), using such data. The inversions for Earth models in this study are based upon spectral splitting of 33 isolated multiplets, observed in long-period accelerograms of 10 large events recorded by the International Deployment of Accelerometers network. Approximately 1000 spectra are involved. Although the data set is insufficient to yield independent results for perturbations in P velocity, S velocity, and density, it is demonstrated that it is possible to obtain large-scale (spherical harmonic degrees s = 0, 2, 4) models of mantle heterogeneity from such a data set under the constraint that aspherical perturbations in seismic velocities and density are proportional to one another. The mantle models developed from modal data are remarkably similar to preexisting models based upon other kinds of seismic data, demonstrating that heterogeneity in seismic velocities is, at most, weakly dependent on frequency. The pattern of the inferred CMB topography is consistent with geodynamic predictions and agrees to a fair extent with results based on travel time anomalies of PcP and PKP, indicating that modal data can add independent constraints on CMB topography. The anomalous splitting of core modes is attributed to inner core anisotropy which is assumed to possess cylindrical symmetry about the Earth's rotation axis. We consider a relatively simple model which varies smoothly with radius (only with radially constant terms and terms varying with r2). Theoretically, 14 parameters are required to describe such an anisotropic tensor field if it is restricted to spherical harmonic degrees 2 and 4 and if analyticity of the field is required. The inversion for such an anisotropic inner core yields a model which can explain the splitting of anomalously split modes, without violating the constraints imposed by PKIKP travel time information. We solve the inverse problem by following two approaches: (1) using splitting function coefficients as data, we derive Earth models by solving linear inverse problems for each spherical harmonic degree and order; and (2) we directly solve the nonlinear inverse problem in which the data are observed modal spectra and the unknowns are the structural parameters. The second procedure has advantages in the case that there are insufficient data to obtain stable results for the splitting functions of some modes. The mantle models generated in these two ways are essentially identical, verifying that splitting functions can serve as a very convenient intermediate stage in modeling Earth structure using the split spectra of free oscillations.

Measurement of the mechanical properties of silicon microresonators

Abstract: A novel technique for the simultaneous determination of the strain and Young's modulus of a silicon microresonator is reported. The lowest two symmetric vibration modes of a single resonator are excited and their frequencies measured. The measured frequencies are compared with a theoretical model. The strain is determined by substituting the ratio of the two frequencies into the model. The calculated strain is then substituted into a characteristic equation to yield Young's modulus. The measured value for Young's modulus is 1.31×1011 Pa, which is close to that of pure silicon, 1.30×1011 Pa.

First-principles quantum molecular-dynamics study of the vibrations of icosahedral C60.

Abstract: Using first-principles real-space quantum-molecular-dynamics, we compute the structure and normal modes of icosahedral ${\mathrm{C}}_{60}$. The many-electron ground-state energy and atomic forces are computed for each atomic configuration, and the results are used in molecular-dynamics calculations of the modes. We focus here on the 14 modes which are either first-order Raman active or infrared active, and we compare our calculated mode frequencies with experimental data. Using no free parameters, we compute the frequencies to between 4% and 16% of their measured values, depending on the mode.