Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

There are still many phenomena, especially in continuum physics, that are described by means of parabolic partial differential equations whose solution are not compatible with the causality principle. Compatibility with this principle is required also by the theory of relativity. A general form of hyperbolic operators for the most frequently occurring linear governing equations in mathematical physics is written down. It is then easy to convert any given parabolic equation to the hyperbolic form without necessarily entering into the cause of the inadequacy of the governing equation. The method is verified on the well-known example of Timoshenko’s correction of the Bernoulli–Euler–Rayleigh beam equation for flexural motion. The “Love–Rayleigh” fourth-order differential equations for the longitudinal and torsional wave propagation in the rod is generalized with this method. The hyperbolic version (not to mention others) of the linear Korteweg–de Vries equation and of the “telegraph” equation governing electromagnetic wave propagation through relaxing material are given. Lagrangians of all the equations studied are listed. For all the reasons given we believe the hyperbolic governing equations to be physically and mathematically more realistic and adequate.

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Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity

Applied Sciences ◽

10.3390/app112210948 ◽

2021 ◽

Vol 11 (22) ◽

pp. 10948

Author(s):

Elisabetta Manconi ◽

Sergey V. Sorokin ◽

Garziera Rinaldo ◽

Matheus Mikael Quartaroli

Keyword(s):

Wave Propagation ◽

Forced Response ◽

Polar Coordinates ◽

Element Analysis ◽

Forced Wave ◽

Practical Engineering ◽

Approaches To Study ◽

Flow Through ◽

Waves Propagation ◽

Numerical Approaches

In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship’s deck, an aeroplane fuselage, a satellite antenna, a wall panel. To avoid transmission of the vibration and structure-borne sound, radial or polar periodicity may be used. In these cases, numerical approaches to study free and forced wave propagation close to the excitation source in polar coordinates are desirable. This is the paper’s aim, where a numerical method based on Floquet-theory and the FE discretision of a finite slice of the radial periodic structure is presented and verified. Only a small slice of the structure is analysed, which is approximated using piecewise Cartesian segments. Wave characteristics in each segment are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis, while wave amplitude change due to the changes in the geometry of the slice is accommodated in the model assuming that the energy flow through the segments is the same. Forced response of the structure is then evaluated in the wave domain. Results are verified for an infinite isotropic thin plate excited by a point harmonic force. A plate with a periodic radial change of thickness is then studied. Free waves propagation are shown, and the forced response in the nearfield is evaluated, showing the validity of the method and the computational advantage compared to FE harmonic analysis for infinite structures.

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Electron Microscopy of Shock Loaded Polycrystalline Beryllium

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100052547 ◽

1974 ◽

Vol 32 ◽

pp. 506-507 ◽

Cited By ~ 1

Author(s):

J. M. Galbraith ◽

L. E. Murr ◽

A. L. Stevens

Keyword(s):

Electron Microscopy ◽

Wave Propagation ◽

Structural Change ◽

Single Crystal ◽

Diffraction Pattern ◽

Shock Loading ◽

Slip Systems ◽

Compression Tests ◽

X Ray Diffraction ◽

X Ray

Uniaxial compression tests and hydrostatic tests at pressures up to 27 kbars have been performed to determine operating slip systems in single crystal and polycrystal1ine beryllium. A recent study has been made of wave propagation in single crystal beryllium by shock loading to selectively activate various slip systems, and this has been followed by a study of wave propagation and spallation in textured, polycrystal1ine beryllium. An alteration in the X-ray diffraction pattern has been noted after shock loading, but this alteration has not yet been correlated with any structural change occurring during shock loading of polycrystal1ine beryllium.This study is being conducted in an effort to characterize the effects of shock loading on textured, polycrystal1ine beryllium. Samples were fabricated from a billet of Kawecki-Berylco hot pressed HP-10 beryllium.

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