Topic

# Normal mode

About: Normal mode is a research topic. Over the lifetime, 15642 publications have been published within this topic receiving 353226 citations. The topic is also known as: Characteristic vibration, normal mode.

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01 Jan 19792,002 citations

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Duke University

^{1}TL;DR: In this article, a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure, and it is shown that the absolute changes in the curvature shape are localized in the region of damage and hence can be used to detect damage.

1,924 citations

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TL;DR: It is shown that a single-parameter potential is sufficient to reproduce the slow dynamics of proteins obtained with vastly more complex empirical potentials, which inevitably leads to unstable modes which must be eliminated through elaborate methods, and which cast doubts on the validity of the analysis.

Abstract: Normal mode analysis (NMA) is a leading method for studying long-time dynamics and elasticity of biomolecules. The method proceeds from complex semiempirical potentials characterizing the covalent and noncovalent interactions between atoms. It is widely accepted that such detailed potentials are essential to the success of NMA’s. We show that a single-parameter potential is sufficient to reproduce the slow dynamics in good detail. Costly and inaccurate energy minimizations are eliminated, permitting direct analysis of crystal coordinates. The technique can be used for new applications, such as mapping of one crystal form to another by means of slow modes, and studying anomalous dynamics of large proteins and complexes. [S0031-9007(96)01063-0] PACS numbers: 87.15.By, 87.15.He Thermal equilibrium fluctuations of the x-ray crystal coordinates of proteins provide a basis for understanding the complex dynamics and elasticity of biological macromolecules [1]. Analysis of the normal modes of globular proteins shows an interesting anomaly. The density of the slow vibrational modes is proportional to their frequency, gsv d, v, rather than gsv d, v 2 as predicted by Debye’s theory [2]. Yet, the atoms in globular proteins are packed as tightly as in solids. We show that a single-parameter potential reproduces the slow elastic modes of proteins obtained with vastly more complex empirical potentials. The simplicity of the potential permits greater insight and understanding of the mechanisms that underlie the slow, anomalous motions in biological macromolecules such as proteins. To date, normal modes of globular proteins have been used to reproduce crystallographic temperature factors [3] and diffuse scatter [4]. Normal mode analyses (NMA’s) shed light on shear and hinge motions necessary for catalytic reactions, and have been used with some success to map one crystal form of a protein into another [5]. Finally, NMA’s yield macroscopic elastic moduli of large protein assemblies, based on their microscopic structure [6]. NMA studies of macromolecules are handicapped, however, by the complex phenomenological potentials used to model the covalent and nonbonded interactions between atom pairs. The necessary initial energy minimization requires a great deal of computer time and memory, and is virtually impossible for even moderately large proteins (with typically thousands of degrees of freedom) with a reasonable degree of accuracy. This inevitably leads to unstable modes which must be eliminated through elaborate methods, and which cast doubts on the validity of the analysis. Moreover, partly because the minimization is carried out in vacuo, the final configuration disagrees with the known crystallographic structure, complicating the interpretation of the results of NMA. A typical example of a semiempirical potential used in molecular dynamics studies and NMA’s has the form [7] Ep › 1 X bonds Kbsb 2 b0d 2 1 1 X angles Kus u2u 0 d 2

1,620 citations

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TL;DR: In this paper, a theoretical investigation has been undertaken to study diffraction of electromagnetic waves in Fabry-Perot interferometers when they are used as resonators in optical masers.

Abstract: A theoretical investigation has been undertaken to study diffraction of electromagnetic waves in Fabry-Perot interferometers when they are used as resonators in optical masers. An electronic digital computer was programmed to compute the electromagnetic field across the mirrors of the interferometer where an initially launched wave is reflected back and forth between the mirrors. It was found that after many reflections a state is reached in which the relative field distribution does not vary from transit to transit and the amplitude of the field decays at an exponential rate. This steady-state field distribution is regarded as a normal mode of the interferometer. Many such normal modes are possible depending upon the initial wave distribution. The lowest-order mode, which has the lowest diffraction loss, has a high intensity at the middle of the mirror and rather low intensities at the edges. Therefore, the diffraction loss is much lower than would be predicted for a uniform plane wave. Curves for field distribution and diffraction loss are given for different mirror geometries and different modes. Since each mode has a characteristic loss and phase shift per transit, a uniform plane wave which can be resolved into many modes cannot, properly speaking, be resonated in an interferometer. In the usual optical interferometers, the resolution is too poor to resolve the individual mode resonances and the uniform plane wave distribution may be maintained approximately. However, in an oscillating maser, the lowest-order mode should dominate if the mirror spacing is correct for resonance. A confocal spherical system has also been investigated and the losses are shown to be orders of magnitude less than for plane mirrors.

1,420 citations

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TL;DR: In this article, the authors show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail, and the shortest length scale which the given data can resolve at any particular depth.

Abstract: A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of interia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail; and how to determine the shortest length scale which the given data can resolve at any particular depth. We apply the general theory to the linear problem of finding the depth-variation of a frequency-independent local Q from the observed quality factors Q of a finite number of normal modes. We also apply the theory to the non-linear problem of finding density vs depth from the total mass, moment, and normal-mode frequencies, in case the compressional and shear velocities are known.

1,371 citations