Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Mono-coupled periodic systems with a single disorder: Response to convected loadings

This paper presents a general theory of the forced response under convected loading of mono-coupled periodic systems with a single disorder. The general expressions derived have been used to study the response of an infinite periodic beam on simple supports with one of the support spacings different from all the others. Convected harmonic pressure fields and frozen random pressure fields have been considered. Computer studies are presented showing the moment response at supports and the space-time-averaged responses in the disorder and in the nearby periodic beam elements. High response levels can occur due to (i) resonances of the beam length disorder against the stiffness of the attached periodic systems and (ii) hydrodynamic coincidence vibration occurring in the periodic beam. The frequency zones in which these high responses may occur are identified. The high response due to the resonance (ii) is restricted to the vicinity of the disorder, whereas that due to coincidence occurs throughout the system. Computed results show that the highest response levels do not necessarily occur in the beam length disorder, but may occur in one of the nearby periodic beam elements. The dependence of the maximum response levels on the magnitude of the disorder has been investigated. The conditions under which small disorders may be neglected have been pointed out.     

Allowance for shear distortion and rotatory inertia of ship hulls

The response of a ship to waves and to propeller excitation (as well as its distortion in still water) may be analysed in modal form by using a linear theory. For symmetric responses the approach has been discussed in terms of a “hull girder” treated as a simple beam. The same is true of uncoupled bending in antisymmetric motion. Simple beam theory has also been adapted for use in “coupled bending and twisting” responses of hulls with large deck openings. The theory has not, hitherto, embodied an allowance for the effects of shear distortion or rotatory inertia. It is shown in this paper how those effects may be allowed for in the analysis of symmetric response, that they do not alter the form of the more rudimentary analysis and that the response lends itself to a convenient matrix formulation.     

Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements

This paper employs the numerical assembly method (NAM) to determine the “exact” frequency–response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force. First, the coefficient matrices for the intermediate concentrated elements, pinned support, applied force, left-end support and right-end support of a beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix. The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency–response curve for any point of a multiple-span beam carrying a number of various concentrated elements. For the case of zero harmonic force, the above-mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained.     

Dispersive elastodynamics of 1D banded materials and structures: analysis

The elastodynamics of 1D periodic materials and finite structures comprising these materials are studied with particular emphasis on correlating their frequency-dependent characteristics and on elucidating their pass-band and stop-band behaviors. Dispersion relations are derived for periodic materials and are employed in a novel manner for computing both pass-band and stop-band complex mode shapes. Through simulations of harmonically induced wave motion within a finite number of unit cells, conformity of the frequency band structure between infinite and finite periodic systems is shown. In particular, only one or two unit cells of a periodic material could be sufficient for “frequency bandedness” to carry over from the infinite periodic case, and only three to four unit cells are necessary for the decay in normalized transmission within a stop band to practically saturate with an increase in the number of cells. Dominant speeds in the scattered wave field within the same finite set of unit cells are observed to match those of phase and group velocities of the infinite periodic material within the most active pass band. Dynamic response due to impulse excitation also is shown to capture the infinite periodic material dynamical characteristics. Finally, steady-state vibration analyses are conducted on a finite fully periodic structure revealing a conformity in the natural frequency spread to the frequency band layout of the infinite periodic material. The steady-state forced response is observed to exhibit mode localization patterns that resemble those of the infinite periodic medium, and it is shown that the maximum localized response under stop-band conditions could be significantly less than in an equivalent homogenous structure and the converse is true for pass-band conditions.     

Simple interference method of diffraction grating generation for integrated optics by the use of a Fresnel mirror

A very simple and compact arrangement is proposed for generation of surface periodic structures for integrated optics using laser exposition and photolithographic technique. The method applied is based on the division of a laser beam into two “half-wavefronts” by a Fresnel mirror (90°) with one arm formed by an exposed sample, and on interference of divided wavefronts on the sample surface.     

MODEL-ORDER REDUCTION AND PASS-BAND BASED CALCULATIONS FOR DISORDERED PERIODIC STRUCTURES

This paper is concerned with the dynamics of disordered periodic structures. The free vibration problem is considered. A method akin to the Rayleigh method is presented. This method is particularly suitable for the study of periodic structures as it exploits the nominal periodicity leading to an approximation that greatly reduces the order of the model. The method is used to calculate the natural frequencies and mode shapes for a pass-band by treating the unknown phases between the nominally identical bays as the generalized co-ordinates of the problem. An illustrative example of a cyclically coupled beam model is presented. In spite of a very large reduction in the computational effort, the results obtained are very accurate both for frequencies and mode shapes even when strong mode localization is observed. To test the performance of the proposed approximation further, a situation where two pass-bands are brought close to each other is considered (a coupled beam model having inherent bending-torsion coupling). The method presented here is general in its formulation and has the potential of being used for more complex geometries.     

Two modes of laser lithography fabrication of three-dimensional submicrometer structures

The modes of laser lithography fabrication of three-dimensional submicrometer structures have been studied. The method is based on the effect of threshold two-photon polymerization of a photosensitive material at the laser beam focus. To determine the lithograph workspace in the coordinates “laser power-speed of the sample displacement with respect to the laser focus,” a series of photonic crystals with the woodpile structure is prepared. Two methods for fabricating three-dimensional structures, i.e., raster scanning and vector graphics (or the vector method) are analyzed in detail. The advantages of the vector method for fabricating periodic structures are demonstrated using crystals of inverted yablonovite as an example. The prepared samples are studied by scanning electron microscopy.     

Flexural wave motion in beam-type disordered periodic systems: Coincidence phenomenon and sound radiation

This paper deals with flexural wave motion in uniform beam-type periodic systems whose repeating units are identical finite beams with multiple beam-length disorders. A general expression derived for the propagation constants has been employed to study its variation with frequency for a beam system having 4-span disordered repeating units. This is helpful in understanding flexural wave motion in disordered periodic beams. Free flexural waves have been studied as wave groups consisting of a large number of harmonic components of different wavelengths, phase velocities and directions. Phase velocities have been computed and plotted for different frequencies in the propagation zones in which the free waves progress without attenuation. This has been found to be useful in understanding and predicting the coincidence phenomenon in disordered periodic beams under convected pressure field loading. The excitation of wave groups in disordered periodic beam-type systems by a slow (subsonic) convecting pressure field can include fast (supersonic) moving flexural wave components which can radiate sound. It has been pointed out that sound radiation from a disordered periodic beam (or plate) can be quite different as compared to that from a periodic beam under similar convected pressure field loading.     

Approximate reconstruction of torsional potential energy surface based on voronoi tessellation

Torsional modes within a complex molecule containing various functional groups are often strongly coupled so that the harmonic approximation and one-dimensional torsional treatment are inaccurate to evaluate their partition functions. A family of multi-structural approximation methods have been proposed and applied in recent years to deal with the torsional anharmonicity. However, these methods approximate the exact “almost periodic” potential energy as a summation of local periodic functions with symmetric barrier positions and heights. In the present theoretical study, we illustrated that the approximation is inaccurate when torsional modes present non-uniformly distributed local minima. Thereby, we proposed an improved method to reconstruct approximate potential to replace the periodic potential by using information of the local minima and their Voronoi tessellation. First, we established asymmetric barrier heights by introducing two periodicity parameters and assuming that the exact barrier positions are at the boundaries of Voronoi cells. Second, we used multiplicatively weighted Voronoi tessellation to refine the barrier heights and positions by defining a structure-related distance metric. The proposed method has been tested for a few higher-dimensional cases, all of which show promising improved accuracy.     

Parametric channeling and collapse of beams of charged particles with internal structure in crystals: 1. Features of parametric channeling of molecular ions in crystal channels

A new type of self-organized orientational motion of charged particles with internal energy structure (molecules, atoms, and in some cases nuclei and electrons) in crystals, i.e., parametric channeling, is proposed and studied in detail. A feature of this mode of motion is associated with the strong parametric coupling of orientational oscillations controlled by the interaction of the moving particle charge with averaged fields of crystal plane axes and vibrations caused by intraparticle processes. The features of parametric processes during channeling of molecular diatomic ions are studied. It is shown that this effect can cause “parametric collapse,” i.e., beam self-cooling, compression, and a decrease in its phase-space volume. For molecules whose axis is perpendicular to the channel axes, periodic parametric collapse occurs. For molecules oriented along the channel, beam self-cooling is an asymptotic, aperiodic, and irreversible process.     

多模式离子推力器栅极系统三维粒子模拟仿真

栅极系统是离子推力器推力产生的主要部件,推力器的性能和寿命都与栅极系统密切相关.对于具有多种工作模态的离子推力器,基于电流电压入口的仿真可以有效评估推力器的工作状况.采用三维粒子模拟方法对两栅极系统等离子体输运过程进行了仿真,获得了不同模式下的推力器性能参数,对比NSTAR的在轨测试参数,验证了模型的正确性;分析了工作模式变化对栅极区域电场分布和束流状态的影响以及离子推力器多模式设计需求.分析结果表明:远离栅极系统的外凸型屏栅鞘层和内凹型零等势面、低鞍点电势值和平缓的下游电势分布,有利于提高栅极系统离子通过率,抑制电子返流,减小Pits-and-Grooves腐蚀,是离子推力器工作模式的设计方向;提高束流电压会导致发散角损失增大,但可扩展栅极工作电流范围,在束流强度较大的模式下,使束流具有较好的聚焦状态,有利于减小Barrel腐蚀.研究结果为多模式离子推力器工作模式设计提供了参考.     

DLA光源质量对光栅-外腔谱合成系统光束质量的影响

在二极管激光阵列(DLA)光栅-外腔谱合成系统中,由于DLA存在子单元光束发散角、“smile”效应的位置偏差及指向性偏差等因素的综合作用,将导致合成光束的光束质量降低。综合考虑DLA子单元光束发散角、“smile”效应等因素对谱合成系统中光束传输特性的影响,建立了DLA光栅-外腔谱合成系统的光传输模型,进而对谱合成系统中DLA子单元光束发散角、“smile”效应的位置偏差及指向性偏差等因素对合成光束的光束质量影响进行了定量分析。结果表明,DLA光源质量会明显影响合成光束的光束质量：DLA子单元光束发散角和“smile”效应引入的指向性偏差越大,合成光束的光束质量就越差;“smile”效应引入的位置偏差在合束方向上对合成光束的光束质量没有影响,而在非合束方向上引入的位置偏差将会明显降低合成光束的光束质量。在实际工作应用中,需要采取措施提高DLA光源质量,以减小对合成光束的光束质量影响。     

Free vibrations of a mono-coupled periodic system

Vibration problems of periodic systems can be analyzed efficiently by means of the transfer matrix method. The frequency equation for the whole system is shown to be obtained in terms of the eigenvalues, or their natural logarithms, which are often called “propagation constants”, of the transfer matrix for a single periodic subsystem. In case of a mono-coupled system this frequency equation may be solved graphically by using the propagation constant curve, thereby saving a great deal of computational effort. Two types of mono-coupled systems are considered as numerical examples: a spring-mass oscillating system and a continuous Timoshenko beam resting on regularly spaced knife-edge supports. Depending on whether the transfer matrix is derived by an analytical procedure or by the finite element method, the numerical solutions become either exact or approximate.     

Asymmetry of the time dynamics of breathers in the electroconvection twist structure of a nematic

Dynamics of breather defects in the periodic structures of rolls arising at electroconvection in nematic liquid crystals twisted by π/2 has been studied experimentally and theoretically. The axial components of the velocity of a hydrodynamic flow in the twist structures of nematics are opposite in the neighboring rolls. Dynamics of the breather defect is the periodic creation and annihilation of a pair of classical dislocations with the topological indices “+1” and “-1.” The annihilation occurs faster than the creation. It has been shown that the asymmetric time dynamics of the breather defect is described well by the solution of the perturbed sine-Gordon equation in the form of an interacting soliton and antisoliton.     

Generation of phase nuclei by solitons of the new “undulator” type in martensitic phase transitions of crystalline materials

The microdynamics of large-amplitude nonlinear lattice vibrations of plutonium and uranium materials has been investigated at high reactor temperatures in the ranges of martensitic phase transitions. Topologically new large-amplitude solitons of the “undulator” type have been revealed. Transverse and longitudinal “undulator” solitons in crystals with hexagonal and cubic symmetry, depending on the direction of motion, have different kinematic and amplitude characteristics, which differ from the characteristics of the previously known solitons. The transverse “undulator” solitons, like electrons in undulators, are observed with periodic atomic displacements orthogonal to the direction of soliton propagation. The longitudinal “undulator” solitons with displacements of atoms in the direction of soliton propagation are characterized by periodic delays with two-step velocities on the trajectory in a certain analogy with two-period engineering undulator devices. It has been shown that, at high energies, such “undulator” solitons of two types generate nuclei of a new phase in early stages of structural phase transitions.     

Acoustically induced vibration of,and sound radiation from,beams inside an enclosure

The velocity response of a single cylindrical beam to pure tone sound in a rectangular reverberation chamber is analysed. The variation of the response with position within the chamber and the effect of proximity of other similar, parallel beams is considered. A reciprocity relation is established between the response of a beam to sound in the chamber and the sound radiation by such motion when otherwise generated; this relation is found to be valid even for non-diffuse fields, below the Schroeder “large room” frequency (i.e., above this frequency, the average frequency spacing between modes is less than one-third of the bandwidth of a typical mode [1]). The mean square velocity response, averaged over all possible beam positions within the chamber, is found to be directly proportional to the radiation loss factor of that vibrational mode, and inversely proportional to (frequency)³. The maximum normalized response occurs when the beam is excited in its fundamental vibrational mode. One application is to the estimation of fatigue life of heat exchanger tubes in gas-cooled nuclear reactors; the large number of tubes necessitates an estimate of the variation of the tube response with position within the exchanger casing. Another important application is in a reverberation chamber; in many cases a suitable direct excitation experiment is very difficult, or even impossible to perform, while the corresponding reciprocal experiment is relatively easy.     

Applying the stochastic parallel gradient algorithm in the auto alignment problem for the amplifier channel of the UFL-2M facility

The procedure of automatic alignment of the four-pass amplifier channel of a UFL-2M facility has been simulated numerically in the presence of aberrations in the optical path. The alignment procedure is based on the “marker” or “reference” method. The control elements are directed by the stochastic parallel gradient algorithm. The order of executing the automatic alignment is determined. The numerical simulation shows the possibility of positioning the beam at the exit from the channel with an accuracy of 1% of the pinhole size in the far field and of 0.1% of the beam aperture size in the near field. It was established that the centering accuracy of the alignment beam in inner pinholes in the presence of optical inhomogeneities in the amplifier channel can be worse than at the exit of the channel. The possibility of symmetrizing the picture of far-field markers has been considered in the case where the position of the optical axis of the channel is unknown.     

Band structure of collective modes and effective properties of binary magnonic crystals

In this paper a theoretical study of the band structure of collective modes of binary ferromagnetic systems formed by a submicrometric periodic array of cylindrical cobalt nanodots partially or completely embedded into a permalloy ferromagnetic film is performed. The binary ferromagnetic systems studied are two-dimensional periodic, but they can be regarded as three-dimensional, since the magnetization is non uniform also along the z direction due to the contrast between the saturation magnetizations of the two ferromagnetic materials along the thickness. The dynamical matrix method, a finite-difference micromagnetic approach, formulated for studying the dynamics in one-component periodic ferromagnetic systems is generalized to ferromagnetic systems composed by F ferromagnetic materials. It is then applied to investigate the spin dynamics in four periodic binary ferromagnetic systems differing each other for the volume of cobalt dots and for the relative position of cobalt dots within the primitive cell. The dispersion curves of the most representative frequency modes are calculated for each system for an in-plane applied magnetic field perpendicular to the Bloch wave vector. The dependence of the dispersion curves on the cobalt quantity and position is discussed in terms of distribution of effective “surface magnetic charges” at the interface between the two ferromagnetic materials. The metamaterial properties in the propagative regime are also studied (1) by introducing an effective magnetization and effective “surface magnetic charges” (2) by describing the metamaterial wave dispersion of the most representative mode in each system within an effective medium approximation and in the dipole-exchange regime. It is also shown that the interchange between cobalt and permalloy does not necessarily lead to an interchange of the corresponding mode dispersion. Analogously to the case of electromagnetic waves in two-dimensional photonic crystals, the degree of localization of the localized collective modes is expressed in terms of an energy concentration factor.     

On the two frequency spectra of Timoshenko beams

This paper is concerned with the question of whether there are, indeed, two distinct spectra of frequencies for the transverse vibrations of Timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and, more recently, for beams supported in an arbitrary manner. Elementary analysis leads to the conclusion that there is only a single frequency spectrum; in the particular case of the simply supported beam the “two frequency spectra” viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex, dynamical behavior of Timoshenko beams.