Showing papers on "Normal mode published in 1983"

Harmonic dynamics of proteins: normal modes and fluctuations in bovine pancreatic trypsin inhibitor.

Abstract: A normal mode analysis making use of an empirical potential function including local and nonlocal (nonbonded) interactions is performed for the bovine pancreatic trypsin inhibitor in the full conformational space of the molecule (1,740 degrees of freedom); that is, all bond lengths and angles, as well as dihedral angles, are included for the 580-atom system consisting of all heavy atoms and polar hydrogens. The heavy-atom frequency spectrum shows a dense distribution between 3 and 1,800 cm-1, with 350 modes below 216 cm-1. Most of the low-frequency modes, of which many have significant anharmonic character, are found to be delocalized over the protein. The root-mean-square amplitudes of the atomic fluctuations are calculated at 300 K from the normal modes and compared with those obtained from a solution molecular dynamics simulation based on the same potential function; very good agreement is obtained for the variation in the main-chain fluctuations as a function of residue number, though larger differences occur for the side chains. The fluctuations are generally, though not always, dominated by frequencies below 30 cm-1, in accord with the results of the dynamics simulation. The vibrational contributions to the thermodynamic properties of the protein are calculated as a function of temperature; the effects of perturbations on the spectrum, suggested for ligand or substrate binding, are examined. The analysis demonstrates that, in spite of the anharmonic contributions to the potential, a normal mode description can provide useful results concerning the internal motions of proteins.

Dynamics of a small globular protein in terms of low-frequency vibrational modes

Abstract: Normal modes of low-frequency vibrations are calculated for a small globular protein, bovine pancreatic trypsin inhibitor. In modes with frequencies below 120 cm-1 the protein molecule behaves like a continuous elastic body. Most modes with frequencies above 50 cm-1 are shown to behave harmonically within the range of thermal fluctuations at room temperature. Those with frequencies below 50 cm-1 show some anharmonicity. Magnitudes of displacements of atoms are mainly determined by the modes with frequencies below 30 cm-1. These very-low-frequency modes contribute significantly to the entropy of the system. The dynamic structure of the globular protein is described as a superposition of harmonic high-frequency motions and coupled anharmonic low-frequency motions of collective variables corresponding to the normal modes of vibration.

Improvement of a Large Analytical Model Using Test Data

Abstract: A method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements. An application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure. The resulting changes in the model are judged to be small compared to expectations of error in the analysis. Thus, the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics. In addition, it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure. Nomenclature - analytical matrix = matrix of changes = identity matrix = full improved stiffness and mass matrices (n x n) = full analytical K, M matrices (n x n) = partitions of KA,MA corresponding to test coordinates = partitions of KA,MA corresponding to coupling elements = partitions of KA,MA corresponding to unmea- sured coordinates = number of measured modes = number of degrees of freedom in the model = measures of changes, Eqs. (15-17) = matrix norm, sum of the squares of all the elements = rectangular modal matrix, normalized (n x m) = /th mode, /th column of $ = measured and unmeasured partitions of ,- = diagonal matrix of measured natural frequencies (m xm) = natural frequency of /th mode = 12 17 = sum of the squares of all elements of matrix ( )

Ballooning mode spectrum in general toroidal systems

Abstract: A WKB formalism for constructing normal modes of short‐wavelength ideal hydromagnetic, pressure‐driven instabilities (ballooning modes) in general toroidal magnetic containment devices with sheared magnetic fields is developed. No incompressibility approximation is made. A dispersion relation is obtained from the eigenvalues of a fourth‐order system of ordinary differential equations to be solved by integrating along a line of force. Higher‐order calculations are performed to find the amplitude equation and the phase change at a caustic. These conform to typical WKB results. In axisymmetric systems, the ray equations are integrable, and semiclassical quantization leads to a growth rate spectrum consisting of an infinity of discrete eigenvalues, bounded above by an accumulation point. However, each eigenvalue is infinitely degenerate. In the nonaxisymmetric case, the rays are unbounded in a four‐dimensional phase space, and semiclassical quantization breaks down, leading to broadening of the discrete eigenvalues and the accumulation point of the axisymmetric unstable spectrum into continuum bands. Analysis of a model problem indicates that the broadening of the discrete eigenvalues is numerically very small, the dominant effect being broadening of the accumulation point.

Dynamics of DNA oligomers.

Abstract: The techniques of molecular and harmonic dynamics are used to study the internal mobility of threė double-stranded DNA hexamers. A 60 ps molecular dynamics simulation and a normal mode description of d(CpGpCpGpCpG)2 in the B conformation characterize the atomic fluctuations of this structure. A comparison between the two approaches validates the harmonic results at room temperature. Detailed examination of the normal modes indicates that only the low-frequency modes are needed to determine atomic fluctuations. A harmonic analysis is made of d(CpGpCpGpCpG)2 in the Z conformation and of d(TpApTpApTpA)2 in the B conformation using only the low-frequency modes. The atomic fluctuations of the three alternating pyrimidine-purine helices are compared and the dependence on conformation and sequence are discussed. The insights which theoretical calculations can provide for the interpretation of experimental results are explored.

Attenuation in the earth at low frequencies

Abstract: The introduction of global, digitally recording, seismic networks has provided the seismological community with a large quantity of high quality data. At low frequencies the IDA (International Deployment of Accelerometers) network provides the best available data and, in this report, over 500 IDA records have been carefully analysed giving nearly 4000 reliable measurements of centre frequency and apparent attenuation of fundamental spheroidal modes. The attenuation rate of a normal mode of free oscillation of the Earth is measured in terms of its or quality factor and mean Q values for the modes 0 S 8 - 0 S 46 are presented with standard deviations of 2-9% . Mean centre frequencies have relative standard deviation of 5 x 10- 5 to 5 x 10- 4 . The distribution of the centre frequencies reveals a large-scale aspherical structure in velocity and density but the distribution of the apparent attenuation measurements does not reveal a corresponding structure. A total of 26 new measurements of the mean Q of overtone modes with standard deviations of 5-30 % have also been obtained by using single-record and multiple-record techniques. Combining the new data with reliable Q measurements from the literature gives a total of 71 data with which we can infer the radial structure of attenuation inside the Earth. This structure is not well constrained in detail and very simple models are capable of fitting the data. Experiments with synthetic data show that an improvement of an order of magnitude in both the number and quality of the measurements is required to make detailed inferences about the structure of attenuation. The data do constrain the average shear Q- 1 in the inner core to be 1/3500 ( ± 60 %) and the average shear Q- 1 the mantle to be 1/250 ( ± 4 %). These values are appropriate for frequencies less than 5 mHz. Comparison with published values at higher frequencies indicates there is a measurable frequency dependence of attenuation between 3 and 30 mHz. Very little can be inferred about bulk dissipation in the Earth beyond that it must exist to satisfy the attenuation of the radial modes. Experiments show that the data can be satisfied if bulk attenuation is an average 1.3%, or more, of the shear attenuation. Constraining bulk attenuation to be no greater than 2 % of the shear attenuation, and constraining the outer core to have no attenuation, forces bulk attenuation to be concentrated in the upper mantle.

On the three‐dimensional vibrations of the cantilevered rectangular parallelepiped

Abstract: A solution is presented for the three‐dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces. This problem apparently is previously unsolved in the published literature. The Ritz method is used, with displacements assumed in the form of algebraic polynomials. Convergence is studied. Numerical results are given for the first five frequencies of each of the four symmetry classes of vibration, for five thick parallelepiped configurations, including the cube. Contour plots are exhibited for the modal displacements of the cube. The effects of varying Poisson’s ratio are also observed.

Local mode theory for C3v molecules: CH3D, CHD3, SiH3D, and SiHD3

Abstract: A three parameter potential model for the stretching vibrations of AH3D and AHD3 species (A=C or Si) is shown to give good agreement with experimental data thereby justifying predictions for the term values of unobserved levels. The structures of the overtone manifolds for different molecules are rationalized by means of an AH3 local mode to normal mode correlation diagram.

Computation of Normal Modes from Identified Complex Modes

Abstract: A technique is presented to compute a set of normal modes from a set of measured (damped) complex modes. The number of elements in the modal vectors, which is equal to the number of measurements, can be larger than the number of modes under consideration. It is also shown in this paper that the practice of normal mode approximation to complex modes can lead to considerably large errors when the modes are too complex. A numerical example and a simulated experiment are presented to illustrate the concepts discussed and to support the theory presented.

A Nonlinear Vertical Mode Initialization Scheme for a Limited Area Prediction Model

Abstract: A vertical mode initialization scheme for a limited area baroclinic primitive equations prediction model is proposed. The scheme is based on the approach employed in nonlinear normal mode initialization but incorporates a substantial simplification: the linearized equations with respect to which the model modes are defined admit only gravity modes. A variational constraint on induced increments is possible within the scheme. In the framework of the Australian Region Primitive Equations model the approach has been found to effectively control spurious model oscillations. A number of filtering conditions are considered and tested within the overall approach of this vertical mode scheme.

Optimum frequency of propagation in shallow water environments

Abstract: The optimum frequency of propagation in shallow‐water environments is the result of competing propagation and attenuation mechanisms at high and low frequencies. It is shown that the optimum frequency is strongly dependent on water depth, that it has some dependence on the sound‐speed profile, while it is only weakly dependent on the bottom type. A comparison between experimental data and normal‐mode theory indicates the importance of shear waves in the bottom, both in determining the optimum frequency of propagation and in determining the actual propagation‐loss levels at lower frequencies.

Testing solar models with global solar oscillations in the 5-minute band

Abstract: Frequencies of solar oscillation for normal modes described by spherical harmonics with l-values between 0 and 4 are computed. The method of computation is discussed and some of the theoretical uncertainties are examined. It is shown that the standard solar model has eigenfrequencies which do not agree with the frequencies observed for the low l-modes to within the estimated accuracy of either the observed or theoretical frequencies. Four non-standard models are considered: (1) the interior Z abundance is lower than the surface abundance; (2) the interior Z abundance is higher than the surface abundance; (3) the interior Z abundance is altered by mixing; and (4) a large primordial magnetic field remains in the solar core. The effect of all these models on the solar neutrino flux is considered, with the result that the high-Z model is rejected. The conclusions of Bahcall and Ulrich (1971) that a primordial magnetic field increases the neutrino flux are disputed.

Discrete Alfvén eigenmode spectrum in magnetohydrodynamics

Abstract: The conditions for the existence of global Alfven eigenmodes below the continuum are investigated, and an analytical dispersion relation describing the modes is obtained. In cylindrical geometry, the curvature together with gradients of the equilibrium current (magnetic shear), and finite ω/ωci effects are responsible for these modes which could play an important part in Alfven wave heating of tokamak plasmas.

Dynamic Modeling of Structures from Measured Complex Modes

Abstract: A technique is presented to use a set of identified complex modes together with an analytical mathematical model of a structure under test to compute improved mass, stiffness and damping matrices. A set of identified normal modes, computed from the measured complex modes, is used in the mass orthogonality equation to compute an improved mass matrix. This eliminates possible errors that may result from using approximated complex modes as normal modes. The improved mass matrix, the measured complex modes and the higher analytical modes are then used to compute the improved stiffness and damping matrices. The number of degrees-of-freedom of the improved model is limited to equal the number of elements in the measured modal vectors. A simulated experiment shows considerable improvements, in the system's analytical dynamic model, over the frequency range of the given measured modal information.

Dynamic Analysis of Light Equipment in Structures: Modal Properties of the Combined System

Abstract: Perturbation methods are employed to determine the dynamic properties of a combined system composed of a multi‐degree‐of‐freedom structure to which is attached a light, less single‐degree‐of‐freedom equipment item. Closed‐form expressions are derived for the natural frequencies, mode shapes, modal dampings, and other modal properties of the combined system in terms of the dynamic properties of the structure alone, and the equipment alone. The effect of tuning and equipment‐structure interaction are included in this analysis. The derived properties can be used in conjunction with a mode‐super‐position procedure to determine the response of the equipment to any excitation of the combined system. For an example structure, the natural frequencies and mode shapes based on the method developed agree well with exact results.

“Non-linear normal modes” and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems

Abstract: In the study of natural vibrations of non-linear elastic systems it is shown that the mode shape of the vibration can vary with the amplitude as well as the frequency, and that the amplitude frequency relation is strongly affected by constraints imposed on the mode shape in an approximate solution. A method is developed which assumes the approximate solution in the form of a truncated series in which, instead of the set of coefficients, the set of functions of spatial variables is unknown and then determined by a procedure that can be regarded as a generalization of the Ritz method. The problem of variations of the normal mode shapes and of the associated natural frequencies with the amplitude is illustrated by two examples of beams with non-linear boundary conditions, and the amplitude-frequency relation is compared to that corresponding to the a priori assumed linear normal mode solution. Further possible consequences of the mode shape amplitude variations in forced, resonant motion of nonconservative systems are also indicated.

Free and Forced Vibrations of Elastically Connected Double-beam Systems

Abstract: The free vibrations and forced vibrations of a system comprising two elastically-connected parallel upper and lower beams having unequal masses and unequal flexible rigidities are analysed using a generalised method of finite integral transformation and the Laplace transformation. The natural frequencies and the amplitude ratios for the normal modes of vibration are presented; it is also shown that a system comprising a lower beam and coupling springs can be used as a dynamic vibration absorber to reduce the vibrations of an upper beam that is subjected to a periodic force.

On the undamped natural frequencies and mode shapes of a finite‐element model of the cat eardrum

Abstract: This paper presents a three‐dimensional finite‐element model of the cat eardrum which includes inertial effects. The model is implemented using a hierarchical modeling scheme which permits the mesh resolution to be varied. The static behavior of the model is calculated as a function of mesh resolution in order to check the validity of an earlier model. The first six undamped natural frequencies, and the corresponding modal vibration patterns, are then calculated. They are found to lie between about 1.8 and 3.2 kHz for the standard values chosen for the model parameters. The effects on the natural frequencies of varying seven parameters of the model are described.

Fractional calculus in the transient analysis of viscoelastically damped structures

Abstract: Fractional calculus is used to model the viscoelastic behavior of a damping layer in a simply supported beam. The beam is analyzed by using both a continuum formulation and a finite element formulation to predict the transient response to a step loading. The construction of the finite element equations of motion and the resulting nontraditional orthogonality conditions for the damped mode shapes are presented. Also presented are the modified forms of matrix iteration required to calculate eigenvalues and mode shapes for the damped structure. The continuum formulation, also incorporating the fractional calculus model, is used to verify the finite element approach. The location of the poles (damping and frequency) are found to be in satisfactory agreement, as are the modal amplitudes for the first several modes.

Rays and local modes in a wedge-shaped ocean

Abstract: Conventional normal mode theory cannot be applied to the nonseparable problem of wave propagation in an ocean with sloping bottom. For small bottom slopes, the sound field may be expressed approximately in terms of adiabatic modes, but this description fails when a mode propagating upslope passes through cutoff. An alternative solution by ray acoustics, valid at high frequencies, contains many multiply reflected contributions, and also undergoes difficult trapped to leaky ray transitions upslope. To address these transition problems, the conventional ray solution is used here as a convenient starting point for collective treatment of ray fields and their conversion into local modes. First, the ray fields are generalized by associating with each ray trajectory from source to observer a bundle of local plane waves that is multiply reflected between the boundaries and remains valid also in the transition regions. When the generalized ray series is subjected to Poisson summation and subsequent asymptotics on ...

Propagating beam analysis of multimode waveguide tapers

Abstract: We analyze both slowly and rapidly varying tapers with the aid of the propagating beam method. Our results demonstrate that while the optimal longitudinal profile for slowly varying profiles is independent of the taper length, the optimum profile of a taper which varies rapidly with respect to the inverse of the average propagation constant difference of adjacent modes must be determined computationally in each case from the given values of its length and initial and final width parameters. We also analyze the region of validity of local normal mode perturbation theory in representative taper configurations.

Vibrational analysis of pyruvate ion molecules and estimation of equilibrium constants for their hydrogen isotopic exchange reactions

Abstract: The infrared spectra (4000-300 cm/sup -1/) of CH/sub 3/COONa and its five isotopic substitutions including D- and /sup 13/C-labeled modifications suspended in KBr disks were measured at 80 and 290 K. Excellent resolutions were obtained by recording the spectra at low temperature. Complete vibrational assignments were established on the basis of correlations of group modes and the /sup 13/C shifts of the fundamentals. From a normal coordinate analysis a general valence force field involving 26 force constants has been determined, which reproduces 82 experimental frequencies with a root-mean-squares deviation of 3.3 cm/sup -1/. The composition of the normal vibrations from the symmetry coordinates has been given in terms of the potential energy distribution which makes it possible to inspect the relation between the normal modes and the group vibrations. The valence force field was used to predict fundamental frequencies which were not available from experiments, and the reduced partition function ratios of acetate ion molecules were calculated by using the spectroscopic data including the predicted values of the fundamentals. Calculated equilibrium constants of some hydrogen isotopic exchange reactios between the acetate ion molecules are reported.

Calculation of the missing mode effect frequencies from Raman intensities

Abstract: Regularly spaced vibronic structure in the luminescence spectra of large molecules often does not correspond to any ground-state normal-mode vibration of the molecule. This missing mode effect (MIME) is explained in terms of the time-dependent picture of electronic transitions. Simple equations from which the MIME frequency can be calculated are derived. The parameters which are needed to calculate the MIME frequency are independently determined by using preresonance Raman spectroscopy. Good agreement between the experimental spectrum and the spectrum calculated by using the independently determined parameters is observed. The normal modes contributing to the MIME frequency are identified and discussed. 12 references, 2 figures, 1 table.