Showing papers on "Normal mode published in 1988"

Resonance of a fluid-driven crack: Radiation properties and implications for the source of long-period events and harmonic tremor

Abstract: A dynamic source model is presented, in which a three-dimensional crack containing a viscous compressible fluid is excited into resonance by an impulsive pressure transient applied over a small area ΔS of the crack surface. The crack excitation depends critically on two dimensionless parameters called the crack stiffness, C = (b/μ)(L/d), and viscous damping loss, F = (12ηL)/(ρƒd2α), where b is the bulk modulus, η is the viscosity, ρƒ is the density of the fluid, μ is the rigidity, α is the compressional velocity of the solid, L is the crack length, and d is the crack thickness. The first parameter characterizes the ability of the crack to vibrate and shapes the spectral signature of the source, and the second quantifies the effect of fluid viscosity on the duration of resonance. Resonance is sustained by a very slow wave trapped in the fluid-filled crack. This guided wave, called the crack wave, is similar to the tube wave propagating in a fluid-filled borehole; it is inversely dispersive, showing a phase velocity that decreases with increasing wavelength, and its wave speed is always lower than the acoustic velocity of the fluid, decreasing rapidly as the crack stiffness increases. The source spectrum shows many sharp peaks characterizing the individual modes of vibration of the crack; the variation of spectral shape, both in the number and width of peaks, is surprisingly complex, reflecting the interference between the lateral and longitudinal modes of resonance, as well as nodes for these modes. The far-field spectrum is marked by narrow-band dominant and subdominant peaks that reflect the interaction of the various source modes. The frequency of the dominant spectral peak radiated by the source is independent of the radiation direction. The frequency, bandwidth, and spacing of the resonant peaks are strongly dependent on the crack stiffness, larger values of the stiffness factor shifting these peaks to lower frequencies and decreasing their bandwidth. The excitation of a particular mode depends on the position of the trigger and on the extent of the crack surface affected by the pressure transient. Fluid viscosity decreases the amplitudes of the main spectral peaks, smears out the finer structure of the spectrum, and greatly reduces the duration of the radiated signal. The energy loss by radiation is stronger for high frequencies, producing a seismic signature that is marked by a high-frequency content near the onset of the signal and dominated by a longer-period component of much longer duration in the signal coda. Such signature is in harmony with those displayed by long-period events observed on active volcanoes and in hydrofracture experiments. The very low velocity which is possible in a crack with high stiffness (C ≥ 100) also provides an attractive explanation for very long period tremor, such as type 2 tremor at Aso volcano, Japan, without the requirement of an unrealistically large magma container. The standing wave pattern set up on the crack surface by the sustained resonance in the fluid is observable in the near field of the crack, suggesting that the location and extent of the source may be estimated from the mapping of the pattern of nodes and antinodes seen in its vicinity. According to the model, the long-period event and harmonic tremor share the same source but differ in the boundary conditions for fluid flow and in the triggering mechanism setting up the resonance of the source, the former being viewed as the impulse response of the tremor generating system and the latter representing the excitation due to more complex forcing functions.

Normal vibrational modes of buckminsterfullerene

Abstract: The MNDO approximation is employed to compute normal modes of vibration for the proposed C/sub 60/ isomer known as buckminsterullerene. Group theoretical invariance theorems are derived to aid in the interpretation of the normal modes. One particularly interesting mode (the sole A/sub u/ vibration) consists of a rotary oscillation of the pentagonal rings of C/sub 60/, with all rings rotating in the same direction.

Kinetics of channelized membrane ions in magnetic fields

Abstract: The cyclotron resonance model for channel ion transport in weak magnetic fields is extended to include damping losses. The conductivity tensor is obtained for different electric field configurations, including the circuital field E phi normal to the channel axis. The conductivity behavior close to the cyclotron resonance frequency omega c is compared to existing Ca2+-efflux data in the literature. A collision time of .023 s results from this comparison under the assumption that K+ ions are transiting in a 0.35 G field. We estimate a mean kinetic energy of 3.5 eV for this ion at resonance. This model leads to discrete modes of vibration (eigenfrequencies) in the ion-lattice interaction, such that omega n = n omega c. The presence of such harmonics is compatible with recent results by Blackman et al. [1985b] and McLeod et al. [1986] with the interesting exception that even modes do not appear in their observations, whereas the present model has no restriction on n. This harmonic formalism is also consistent with another reported phenomenon, that of quantized multiple conductances in single patch-clamped channels.

A theoretical study of the dielectric constant of protein.

Abstract: The dielectric properties of a protein molecule were investigated by calculating a 'local dielectric constant' with the aid of normal mode analysis. This local dielectric constant was calculated from the electronic polarization of atoms and the orientational polarization of local dipoles. The former was obtained from atomic polarizations of the whole atoms in a protein molecule. The latter was determined from the fluctuation-dissipation theorem. The degree of dipole fluctuation was calculated from the positional fluctuation of each atom obtained by the normal mode analysis. Assuming a minimum volume for a continuum model, the resulting local dielectric constants ranged from 1 to 20 inside the protein.

Splitting Functions of Long-Period Normal Modes of the Earth

Abstract: The spectral splitting of the Earth's free oscillation eigenfrequencies departs from that predicted on the basis of ellipticity and rotation because of lateral heterogeneity in the structure of the Earth's mantle and core and because of deviations from hydrostatic ellipticity of the major internal discontinuities. For an isolated multiplet of angular degree l, singlet eigenfrequencies and eigenfunctions are fully determined by the coefficients cst of the splitting function (s = 0,2,4,…, 2l, −s ≤ t ≤ s). These coefficients constitute linear contraints on the Earth's heterogeneous structure of even spherical harmonic degree s and order t; they are analogous, in this and other respects, to the spherical harmonic expansion coefficients of the phase velocity distribution of a surface wave of given frequency. For an earthquake having known source parameters, knowledge of the splitting coefficients is also sufficient to predict, through a nonlinear relationship, the spectra of the multiplet at all stations. This paper addresses the nonlinear inverse problem of estimating the splitting coefficients, cst, using, as data, observed spectra of the multiplet for an array of sources and receivers. The problem is solved by an iterative, least squares procedure in which the individual data are the complex spectral values obtained by Fourier transformation of windowed, long-period accelerograms from the International Deployment of Accelerometers (IDA) network. Using recordings of large events since 1977, the data set consists of approximately 1000 narrow spectral windows, relating to 27 different modes, each spectral window containing one or two multiplets. The splitting function possesses a real part, which is related to heterogeneity in density and elastic properties, and an imaginary part corresponding to asphericity of anelastic parameters. Inversion is limited to the real and imaginary parts of the splitting function for degree s = 0 and to the real part only for degrees s = 2, 4. The complex c00 term provides a very accurate determination of the degenerate eigenfrequency and attenuation of the multiplet, which constitute additional constraints on the spherically averaged Earth. We find that splitting is systematically larger than that predicted due to rotation and ellipticity. This phenomenon is particularly evident for the PKIKP-equivalent modes (e.g., 13S2, 11S4) and for some very long period modes (3S2, 2S3) with significant sensitivity in the inner core; it is also present, however, for modes sensitive primarily to mantle structure. Comparison between splitting functions and the kernels characterizing their sensitivity with depth enables us to identify large-scale patterns of heterogeneity in the Earth's mantle and core. Forward modeling using existing heterogeneous mantle models produces splitting functions in substantial agreement with those obtained from the modal observations if it is assumed that vp and vs heterogeneity are proportional; it is necessary, however, that d In vs/d In vp has a value in the range 2 to 2.5 in the lower mantle, a value much larger than is often supposed and roughly corresponding to the case that lateral variations in shear modulus dominate those in bulk modulus. Because the modal data are principally sensitive to vS and the prior models of the lower mantle are based on short-period P residuals, a possible explanation is that the P heterogeneity at modal periods is greater than that inferred at ∼1s period. However, some mantle modes have nonnegligible sensitivity to vp and provide evidence that the level of vp heterogeneity is the same at modal periods as it is at 1s, thus supporting, using an entirely different kind of data, both the size and the pattern of lower mantle vp deduced from travel times.

Stability of Cracked Rotors in the Coupled Vibration Mode

Abstract: A transverse surface crack is known to add to a shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented, in the general case by way of a 6 × 6 compliance matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by the off-diagonal terms. In addition, when the shaft rotates the crack opens and closes. Then the differential equations of motion have periodically varying stiffness coefficients and the solution can be expressed as a sum of harmonic functions of time. A method for the determination of the intervals of instability of the first and of second kind is developed. The results have been presented in stability charts in the frequency vs. depth of the crack domain. The coupling effect due to the crack leads to very interesting results such as new frequencies and vibration modes.

Isolation of soil‐structure interaction effects by full‐scale forced vibration tests

Abstract: SUMMARY Forced vibration tests designed to isolate the effects of soil-structure interaction are described and the results obtained for the nine-storey reinforced concrete Millikan Library Building are analysed. It is shown that it is possible to determine experimentally the fixed-base natural frequencies and modal damping ratios of the superstructure. These values may be significantly different from the resonant frequencies and damping ratios of the complete structure-foundation-soil system. It is also shown that forced vibration tests can be used to obtain estimates of the foundation impedance functions. In the case of the Millikan Library it is found that during forced vibration tests the rigid-body motion associated with translation and rocking of the base accounts for more than 30 per cent of the total response on the roof and that the deformation of the superstructure at the fundamental frequencies of the system is almost entirely due to the inertial forces generated by translation and rocking of the base. Full-scale forced vibration tests are commonly used to determine the natural frequencies, modal damping values and mode shapes of structures. The frequent practice in interpreting the results of forced vibration tests is to neglect the effects of the interaction between the structure and the soil. Such a simplifying assumption may lead to serious errors in that resonant frequencies, energy dissipation and other dynamic characteristics of the complete structure-foundation-soil system are ascribed to the superstructure. The typical result is that the fixed-base natural frequencies of the structure are underestimated while the energy dissipation in the structure is overestimated. The principal objective of this study is to analyse in detail the effects of soil-structure interaction during forced vibration tests. In particular, an attempt is made at extracting structural characteristics, such as fixed- base natural frequencies and energy dissipation mechanism, as well as foundation-soil characteristics, such as foundation impedance functions, from forced vibration test results which involve the complete structure-foundation-soil system. Although a large number of theoretical studies of the interaction between structures and the supporting soil have been made and a variety of sophisticated analytical models have been proposed, the experimental study of the interaction phenomenon has been very limited. The second objective of this study is associated with the need of illustrating the interaction effects under controlled experimental conditions. For the purpose of this study the nine-storey reinforced concrete Millikan Library Building was selected as the experimental site. The Millikan Library has been the subject of a large number of forced vibration tests (Kuroiwa,12 Jennings and Kuroiwa,9 Trifunac,19 Foutch et al.,' Luco et al.,'3) and ambient vibration tests (Blandford et al.,' Trifunac,19 McLamore," Udwadia and Trifunac20). Accelerograms for the 1968 Borrego Mountain, 1970 Lytle Creek and 1971 San Fernando earthquakes have been recorded in the Library and a

Hamiltonian theory for vibrational dephasing rates of small molecules in liquids

Abstract: A technique is developed for solving the generalized Langevin equation (GLE) describing anharmonic oscillators in the weak coupling limit. The GLE is rewritten as a Hamiltonian with a nonlinear system coupled to an infinite bath of harmonic oscillators. A normal mode transformation followed by a perturbation technique is used to obtain the fluctuating system frequency. When the method is applied to a single oscillator with cubic anharmonicity, both the classical and quantal dephasing rates are shown to be equal to the well‐known result of Oxtoby. The technique is also applied to a system with more than one vibrational degree of freedom (linear triatomic molecules) to obtain the dephasing rates for the symmetric and antisymmetric normal modes. The effects of system anharmonicity on frequency shifts are investigated.

Propagation of guided waves in fluid-coupled plates of fiber-reinforced composite

Abstract: Guided wave propagation in fluid‐coupled plates of fiber‐reinforced composites has been investigated by studying ultrasonic reflection in these structures. From measurements of ultrasonic reflection on unidirectional graphite‐epoxy plates over a range of incident angles, experimental dispersion curves using a total‐transmission criterion for leaky plate waves have been constructed. Theoretical calculations with no adjustable parameters demonstrate excellent agreement with the experimental data. Unusual behavior observed in the fundamental total‐transmission curve led to a reexamination of the mode identification criteria. Both the total‐transmission curves and the normal modes of the fluid‐coupled plate differ significantly from the result expected on the basis of the widely used Cremer coincidence condition to identify propagating plate waves. It is found that these differences are particularly pronounced in cases where the ratio of fluid to solid densities is high, as for composite materials. The existence of such behavior is also demonstrated numerically in a fluid‐coupled aluminum plate by arbitrarily increasing the density of the fluid.

Unconstrained and Constrained Mode Expansions for a Flexible Slewing Link

Abstract: The linear equations of motion of a uniform flexible slewing link which were derived via Hamilton's Extended Principle are considered. These equations account for the coupling between bending and rigid modes. Unconstrained and constrained mode expansions are investigated and a quantitative comparison is made between the frequency equations and associated mode shapes. A finite dimensional model is derived using the assumed modes method and the theoretical frequencies are verified with an experimental counterbalanced aluminum beam.

Ballooning instabilities in tokamaks with sheared toroidal flows

Abstract: The ballooning-mode eikonal representation is applied to the linearized incompressible magnetohydrodynamic (MHD) equations in axisymmetric systems with toroidal mass flows to obtain a set of initial value partial differential equations in which the time t and the poloidal angle theta are the independent variables. To derive these equations, the eikonal function S is assumed to satisfy the usual condition B.gradS=0 to guarantee that the modes vary slowly along the magnetic field. In addition, to resolve the V.grad operator acting on perturbed quantities, the eikonal must also satisfy the condition dS/dt=0. this induces a Doppler shift in S. This description of the instability, however, is incompatible with normal mode solutions of the MHD equations because the wave vector gradS becomes time dependent when the velocity shear is finite. Nevertheless, the author is able to investigate the effects of the sheared toroidal flows on localized ballooning instabilities because the initial value formulation of the problem developed does not constrain the solutions to evolve as exp (i omega t). Fixed boundary MHD equilibria with isothermal toroidal flows that model the JET device are generated numerically with a variational inverse moments code. As the initial value evaluations are evolved in time, periodic burst of ballooning activity are observed which are correlated with the formation of a ballooning structure at the outside edge of the torus that becomes displaced by 2 pi in the extended poloidal angle domain from one burst to the next. The velocity shear has a stabilizing influence on plasma ballooning.

Time-resolved study of vibrations of a-Ge:H/a-Si:H multilayers

Abstract: We describe experiments in which the vibrations of a-Ge:H/a-Si:H multilayer structures on silica substrates are studied in real time. The vibrations are generated when a picosecond light pulse is absorbed in the structure, thereby setting up an elastic stress. The resulting motion is studied through a measurement of the change in the optical reflectivity as a function of time. We analyze the results in terms of the spectrum of normal modes of the multilayers and show that there exist surface modes at the free surface of the multilayer.

Molecular dynamics with internal coordinate constraints

Abstract: The method of Ryckaert, Ciccotti, and Berendsen [J. Comp. Phys. 23, 327 (1977)] for integrating the Cartesian equations of motion of a system with holonomic constraints, has been extended to allow the independent constraint of arbitrary internal coordinates. To illustrate this new methodology, and to investigate the effects of dihedral angle constraints on the equilibrium and dynamical properties of macromolecules, we have carried out parallel sets of molecular dynamics simulations and normal mode analyses of a small dipeptide: one without constraints, and one with a single backbone dihedral angle constrained. We find that the averages and the fluctuations of the energies, and of the internal degrees of freedom are not significantly modified by the constraint. However, in the region between 100 and 1400 cm−1 of the normal mode spectrum, the constraint shifts the frequencies of the modes, and modifies their contributions to the spectra of the internal coordinates. Except for the lowest frequency torsional ...

Vibratory linear acceleration and angular rate sensing system

Abstract: An inertial sensing system includes a geometric structure having several independent mechanically resonant modes of vibration such that when the structure is subjected to outside inertial motions that are to be sensed, a driven mode of vibration will couple energy in a specific manner into a pickup mode. The driven mode serves as a reference motion. The system measures angular rotation rate by sensing the vibrations in the pickup mode caused by the action of the Coriolis effect on the balanced driven mode which couples energy into the pickup mode. The system includes a portion of the geometric structure that is subject to strain caused by linear acceleration in one direction. The strain introduces an imbalance in the driven mode which causes energy to be coupled to the pickup mode of vibration in a nominally linear manner. A sensing circuit is provided to sense the amount of energy coupled into the pickup system as a measure of applied angular rotation and applied acceleration. The coupling of energy into the pickup caused by linear acceleration is orthogonal to the coupling of energy caused by angular rotation rate, and the effects of the two sources of coupled energy can be separated by the system circuitry such that independent determination of the two types of input motion is realized. The sensing system is preferably made of a material such as crystalline quartz that is mechanically stable and is also piezoelectric so that the driving and sensing can be accomplished by piezoelectric techniques.

Implicit Normal Mode Initialization

Abstract: It is shown that nonlinear normal mode initialization (NMI) can be implemented without knowing the normal modes of a model. The implicit form of nonlinear NMI is particularly useful for models whose normal modes cannot readily be found; for example, if the underlying linear equations are nonseparable. An implicit nonlinear NMI scheme is formulated for the shallow-water equations on a polar stereographic projection. The linear equations which define the implicit normal modes include most of the beta terms as well as variable Coriolis parameter and map scale factor. Even in this nonseparable case, the equivalence between implicit and conventional nonlinear NMI is shown to be exact. The scheme is implemented in a regional model on a quasi-hemispheric domain, which uses a finite-element discretization on a nonuniform grid. The well-posed lateral boundary conditions of this model lead to consistent boundary conditions for the initialization. Results are presented not only for the implicit form of Mach...

Phase space structure in classically chaotic regions and the nature of quantum eigenstates

Abstract: The authors review recent work concerning phase space structure relevant to energy relaxation and chemical reactions. Another example is shown, and that is for the local to normal mode transition in classically chaotic regions. The main purpose of the paper is to investigate the nature of the quantum eigenstates under such a condition, and they investigate what happens to the local and normal mode character of eigenstates when there is widespread chaos and there is extensive exchange of energy between two local modes and local and normal modes classically. They show that the quantum local-normal transition does not change significantly as classical chaos spreads. They relate this to the size of the flux across various phase space structures and present evidence that the quantum mechanics is essentially regular even when chaos is so widespread that, near the dissociation energy, chaotic regions can support approximately 10 quantum states. To further demonstrate this regularity, they quantitize some of the noninvariant phase space structure.

Matched mode localization

Abstract: In this article, a method of passively localizing a narrow‐band source in range and depth in a waveguide is presented based on ‘‘matching’’ predicted normal mode amplitudes to measured mode amplitudes. The modes are measured by using a vertical array of hydrophones and performing mode filtering. Previous studies of mode filtering have considered only the overdetermined case, i.e., where there are more hydrophones than discrete modes present in the waveguide. In this study, mode filtering is considered for the underdetermined case, i.e., where there are fewer hydrophones than the total number of discrete modes in the waveguide, but only a subset of the total number of modes is to be estimated. Previous studies of matched field localization have been based on matching the entire pressure field. In this study, the pressure field is expressed in terms of normal modes, and only a subset of the total number of modes is used for localization. Using a subset of modes allows trade‐offs to be made between localization accuracy, computational complexity, and sensitivity to environmental mismatch. In this article, the matched mode localization method is presented, and the dependence of its localization accuracy on the number of modes used and environmental conditions is demonstrated.The effects of array length and hydrophone spacing on mode estimation error, and hence on localization accuracy, are also demonstrated for the particular method of mode estimation used here. Other methods of mode estimation may produce different results. Finally, the effects of mismatch between the assumed and actual environment due to water depth variation are explored. It is shown that localization accuracy in range is proportional to the mode interference distance between the lowest and highest modes used to localize, and that as few as six modes can be used for ranging. It is also shown that the array length need not be any longer than the depth extent of the highest mode to be estimated, and that the hydrophone spacing must be no greater than half the vertical wavelength of the highest mode that contributes significantly to the sound field (not just the highest mode to be estimated). Localization is most sensitive to environmental mismatch effects that contribute to changes in the phase of the horizontal component of the mode amplitudes. Because a subset of modes is used for localization instead of the entire pressure field, this method of localization can be fairly insensitive to certain kinds of environmental mismatch.

Normal modes and local modes in H2 X: beyond the x, K relations

Abstract: The so-called ‘x, K relations’, which interrelate the normal mode and local mode descriptions of the stretching vibrations of polyatomic molecules, demand simple numerical relationships between the...

Origin and characteristics of the optical properties of general twisted nematic liquid‐crystal displays

Abstract: The origin and characteristics of the field‐off state optical properties of the general twisted nematic liquid‐crystal displays (LCD) are discussed. The optical field propagating normally in a uniformly twisted nematic structure can be represented by two mutually orthogonal elliptically polarized normal modes, one with left elliptical polarization and the other with right elliptical polarization. The interference between the two normal modes explains precisely, for the first time, the working mechanism for the field‐off state optical performance of all twisted nematic liquid‐crystal displays. The sum of the intensities of the two modes is independent of the polarizer orientation and always constitutes one half of the total intensity. The interference between the two modes contributes the second half of the total intensity and gives rise to two separate terms, each of which can be controlled by the entrance polarizer.Each of their contributions to the transmitted intensity can be minimized or maximized by ...

Projection methods for obtaining intramolecular energy transfer rates from classical trajectory results: Application to 1,2‐difluoroethane

Abstract: A general method for analyzing the results of classical trajectory calculations to obtain the details of intramolecular energy transfer is described. The method is based on the determination of the time dependence of the normal mode velocities by projection of the instantaneous Cartesian velocities onto the normal mode vectors. It is shown that the method obviates the need to arbitrarily define a ‘‘bond’’ or ‘‘mode’’ energy as a means of following the energy flow. Average mode energies are computed using the virial theorem. For a given potential surface, the results are exact within the framework of the classical approximation. The method is applied to a study of intramolecular energy transfer in 1,2‐difluoroethane. Decay rates and pathways of energy flow for initial excitation of each of the 18 vibrational modes are reported. The results obtained from the time variation of the normal mode velocities are used to extract a first‐order, mode‐to‐mode energy transfer rate coefficient matrix. The mode‐to‐mode coefficients are shown to provide an excellent means of collating the energy transfer information. Their values yield a quantitative description of the energy transfer rates and a clear picture of the relative importance of the available pathways for energy flow in the system.

Normal mode spectrum of the parallel‐chain β‐sheet

Abstract: Normal mode calculations have been carried out for parallel-chain β-sheet structures These include the parallel-chain pleated sheet of poly(L-alanine) and the parallel-chain rippled sheet of polyglycine Dipole derivative coupling has been included for amide I and II modes, and the effects of parallel-sheet and antiparallel-sheet arrangements of varying separation have been examined for the poly(L-alanine) case Some amide and nonamide modes are distinctly different from their antiparallel-chain counterparts, thus providing a basis for distinguishing between such structures from their ir and Raman spectra As in our previous studies, these results emphasize the need for both kinds of spectral data in order to draw definitive conclusions about conformation

Local mode and normal mode interpretations of the CH and CD stretching vibrational manifolds in C2H4 and C2D4

Abstract: The CH and CD stretching vibrational manifolds in C2H4 and C2D4 have been studied up to six and four quanta of excitation, respectively. They are interpreted in each case in terms of a minimum number of local mode parameters. In C2H4, perturbations arising from Fermi resonances are complex and involve both the C=C stretching and the CH2 scissoring vibrations, with the former itself being in close resonance with the in‐phase CH2 rocking overtone. Nevertheless, once all major resonances are taken into account, a self‐consistent set of vibrational parameters is determined, which reproduces observations well in both infrared and Raman spectra. In C2D4, no significant Fermi resonance complications are apparent, spectroscopic interpretation is straightforward, and approximations inherent in the simple local mode model may be tested. The local mode parameters determined for C2H4 and C2D4 obey closely their expected isotopic interdependences. Values determined for the interbond harmonic coupling parameters in each case are in excellent agreement with those predicted through the harmonic force constants of ethylene.