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.     

Mechanisms for saturation of nonlinear pulsed and periodic signals in focused acoustic beams

Acoustic fields of powerful ultrasound sources with Gaussian spatial apodization and initial excitation in the form of a periodic wave or single pulse are examined based on the numerical solution of the Khokhlov-Zabolotskaya-Kuznetsov equation. The influence of nonlinear effects on the spatial structure of focused beams, as well as on the limiting values of the acoustic field parameters is compared. It is demonstrated that pressure saturation in periodic fields is mainly due to the effect of nonlinear absorption at a shock front, while in pulsed fields is due to the effect of nonlinear refraction. The limiting attainable values for the peak positive pressure in periodic fields turned out to be higher than the analogous values in pulsed acoustic fields. The total energy in a beam of periodic waves decreases with the distance from the source faster than in the case of a pulsed field, but it becomes concentrated within much smaller spatial region in the vicinity of the focus. These special features of nonlinear effect manifestation provide an opportunity to use pulsed beams for more efficient delivery of wave energy to the focus and to use periodic beams for attaining higher values of pressure in the focal region.     

Simulation of three-dimensional nonlinear fields of ultrasound therapeutic arrays

A novel numerical model was developed to simulate three-dimensional nonlinear fields generated by high intensity focused ultrasound (HIFU) arrays. The model is based on the solution to the Westervelt equation; the developed algorithm makes it possible to model nonlinear pressure fields of periodic waves in the presence of shock fronts localized near the focus. The role of nonlinear effects in a focused beam of a two-dimensional array was investigated in a numerical experiment in water. The array consisting of 256 elements and intensity range on the array elements of up to 10 W/cm² was considered. The results of simulations have shown that for characteristic intensity outputs of modern HIFU arrays, nonlinear effects play an important role and shock fronts develop in the pressure waveforms at the focus.     

An approximate method of predicting the response of periodically supported beams subjected to random convected loading

The space-averaged response of an infinite, elastically supported, periodic beam subjected to convected random loading has been studied by using an approximate “assumed mode” method. The complex wave motion in the beam is represented by any number of suitably chosen complex modes. With a good, yet simple, choice of mode which satisfies certain boundary conditions on one periodic beam element, a “single mode approximation” can yield very accurate values of the average response. This has been verified for a wide range of the support stiffnesses and loading convection velocities. Consideration has also been given to the ratio of the maximum response in the beam to the space-averaged response. The method has been applied only to uniform beams in this paper, but it should be readily applicable to periodic systems consisting of non-uniform beam elements.     

利用偏置磁极周期会切磁铁产生边聚焦场

 利用理论分析和数值计算的方法,研究了偏置磁极周期会切磁铁产生的、可用于带状电子束宽边聚焦的边聚焦场。结果表明:增加偏置长度,减小电子束通道宽度,增大磁极轴向长度可以在基本不改变边聚焦场在x方向上分布特性的前提下提高幅值;增大电子束通道的高度,增大磁极厚度均可以减小边聚焦场在x方向上分布曲线的曲率,但同时其幅值也会降低,该降低可利用前述方法予以补偿。在对带状电子束宽边聚焦进行束匹配时,可以先进行曲率匹配,再进行幅值匹配。在进行参数选择时,应合理选择束通道高度和磁极厚度的取值,以避免束通道内的边聚焦场在x方向上的分布出现曲率反向。     

周期场聚焦电子束

引言 自从发现“强聚焦效应”可以应用在粒子加速器内以后,人们知道利用周期磁场能把离子束控制得更细,于是可以减少磁铁的需要量。因此近年来为了改建或新设计各种类型粒子加速器,对周期场聚焦离子束或电子束方面曾进行了很多研究工作。另一方面人们也开始研究用周期电场和磁场聚焦各种电子管内的电子束,例如行波管内电子束的聚焦等。聚焦电子束的主要特点在於必要考虑空间电荷。     

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.     

Numerical and experimental investigations on the band-gap characteristics of metamaterial multi-span beams

A novel metamaterial multi-span beam with periodic simple supports and local resonators is designed and investigated. The frequency responses of the proposed metamaterial multi-span beam are computed by the spectral element method (SEM). The accuracy and feasibility of the SEM are verified by the finite element method (FEM) and the vibration experiments. The results show that the metamaterial multi-span beam could generate both the local resonance band-gaps in the low-frequency ranges and the Bragg band-gaps in the medium and high frequency regions. By adjusting the natural frequencies of the local resonators, the thickness of the base beam and the length of the unit-cell, the local resonance and the Bragg band-gaps can be controlled, respectively. The coupling effects of these two kinds of band-gaps are investigated by the parametrical design, which broadens the band-gaps and consequently improves the vibration reduction performance.     

On the Development of Controllable Sources of Single-Photon States with an Orbital Angular Momentum on the Basis of Spontaneous Parametric Down-Conversion of Light

Methods for beam shaping with nonzero orbital angular momentum are studied using diffraction optical elements with the purpose of developing a source of single-photon states based on spontaneous parametric down-conversion of light in the LiNbO₃ crystal in the cavity resonator. The probability of the coincidence of the number of photocounts in detecting signal and idle fields under pumping by a beam with the orbital angular momentum is simulated.     

A parametrically driven harmonic analysis of a non-linear stretched string with time-varying length

This paper is concerned with the first, second and third harmonic lateral vibrations of a string having a forced, periodic, small length variation. Account is taken of the non-linear term arising from the correction to the tension of the string due to local elongation. It is found that increasing the amplitude of the length variation of the string results in a narrowing of the frequency response range of each of the harmonic oscillation components. Both numerical and analytical calculations of the harmonic frequency responses have been carried out The relationships for initial values between the original differential equation and the approximate differential equations for the harmonic responses are discussed in detail.     

Passive reduction of gear mesh vibration using a periodic drive shaft

In this paper, a passive approach to reduce transmitted vibration generated by gear mesh contact dynamics is presented. The approach utilizes the property of periodic structural components that creates stop band and pass band regions in the frequency spectra. The stop band regions can be tailored to correspond to regions of the frequency spectra that contain harmonics and sub-harmonics of the gear mesh frequency, attenuating the response in those regions. A periodic structural component is comprised of a repeating array of cells, which are themselves an assembly of elements. The elements may have differing material properties as well as geometric variations. For the purpose of this research, only geometric variations are considered and each cell is assumed to be identical. A periodic shaft is designed and machined in order to reduce transmitted vibration of a pair of spur gears. Analytical and experimental results indicate that transmitted vibrations from gear mesh contact to the bearing supports are reduced at a variety of operational speeds under static torque preload.     

The effect of localization on interference. I. Calculated intensities for a feasible optical experiment

A simple geometry utilizing a laser-excited atomic beam as light source, and a nearby oscillating mirror, would permit the observation of a two-channel optical interference effect involving photons which can be localized predominantly in one channel by coincidence observations of the recoiling source atom. A sacrifice of the optimum conditions for photon interference is necessary even when photon localization in one channel is accomplished by an observation of the recoil atom. This necessity arises because the width of the slit defining the atomic beam, and with it the important optical source dimension, must be comparable to the optical wavelength to obtain the small uncertainty in initial total momentum needed for significant localization. Quantum mechanical calculations of the coincidence intensity and singles intensity as a function of mirror position are made for a source width of five quarter wavelengths and are compared to a classical optics calculation of the singles intensity. The coincidence intensity calculation, as expected, predicts a smaller interference effect than classical optics due to the photon localization. The singles intensity calculation also predicts the same reduction in the classical interference effect, as a consequence of the orthoganality of the final atom states.     

Molecular dynamics study of temperature-dependent ripples in monolayer and bilayer graphene on 6H—SiC surfaces

Using classical molecular dynamics and a simulated annealing technique,we show that microscopic corrugations occur in monolayer and bilayer graphene on 6H-SiC substrates.From an analysis of the atomic configurations,two types of microscopic corrugations are identified,namely periodic ripples at room temperature and random ripples at high temperature.Two different kinds of ripple morphologies,each with a periodic structure,occur in the monolayer graphene due to the existence of a coincidence lattice between graphene and the SiC terminated surface(Si-or C-terminated surface).The effect of temperature on microscopic ripple morphology is shown through analysing the roughness of the graphene.A temperature-dependent multiple bonding conjugation is also shown by the broad distribution of the carbon-carbon bond length and the bond angle in the rippled graphene on the SiC surface.These results provide atomic-level information about the rippled graphene layers on the two polar faces of the 6H-SiC substrate,which is useful not only for a better understanding of the stability and structural properties of graphene,but also for the study of the electronic properties of graphene-based devices.     

Transverse or off-axis localization of electromagnetic waves in random one- and two-dimensional dielectric systems which are periodic on average

We study the transverse or off-axis localization of electromagnetic waves for several different random dielectric systems which are periodic on average. Unlike previous scalar wave treatments of transverse localization, in the present work we present results based on a full vector treatment of the electromagnetic fields based on Maxwell's equations. In a first system, we consider a random semi-infinite array of slabs with plane waves or finite beams of electromagnetic waves obliquely incident on the slab surfaces. The localization of the fields in a region near the surface of illumination is studied as a function of the oblique angle of incidence. In a second system, an array of semi-infinite slabs with random thickness is considered with an incident finite beam of electromagnetic waves initially directed parallel to the slab surfaces. The spreading of the beam width is computed as it propagates through the array of semi-infinite slabs. In a final system, we consider a semi-infinite array of random dielectric rods (2D system) with obliquely incident plane waves. The localization length of the plane-wave fields is computed as a function of the oblique angle of incidence and as a function of the strength of the disorder of the dielectric medium. All the random media we consider, when averaged over their randomness, are periodic on average. The above systems are studied for both p- and s-polarizations of incident electromagnetic waves, and the difference in the transverse localization of the electromagnetic field for these two polarizations is determined.     

周期会切磁场聚焦强流相对论带状电子束的理论分析和粒子模拟

对低幅值、短周期、有偏置磁极的周期会切磁场中300 keV,3 kA带状电子束的传输进行了理论分析和粒子模拟。给出了将波导宽度考虑在内的轴向均匀带状电子束的空间电荷场和作用在有限厚度的带状电子束的短边和宽边上的聚焦力的表达式,并利用束匹配的方法得到了磁场的幅值和周期,以及电子束通道的宽度和高度等参数。最后根据理论计算的结果进行了3维粒子模拟验证,结果表明:束宽边上聚焦良好,而在短边上带状电子束的轴向有限长效应使得聚焦力与散焦力沿轴向不能完全匹配,在束包络上产生了Betatron振荡,但在300 mm的距离上传输效率仍能达到98%以上,说明有偏置磁极的周期会切磁场聚焦强流带状电子束在理论上是可行的。     

COUPLED WAVES ON A PERIODICALLY SUPPORTED TIMOSHENKO BEAM

A mathematical model is presented for the propagation of structural waves on an infinitely long, periodically supported Timoshenko beam. The wave types that can exist on the beam are bending waves with displacements in the horizontal and vertical directions, compressional waves and torsional waves. These waves are affected by the periodic supports in two ways: their dispersion relation spectra show passing and stopping bands, and coupling of the different wave types tends to occur. The model in this paper could represent a railway track where the beam represents the rail and an appropriately chosen support type represents the pad/sleeper/ballast system of a railway track. Hamilton's principle is used to calculate the Green function matrix of the free Timoshenko beam without supports. The supports are incorporated into the model by combining the Green function matrix with the superposition principle. Bloch's theorem is applied to describe the periodicity of the supports. This leads to polynomials with several solutions for the Bloch wave number. These solutions are obtained numerically for different combinations of wave types. Two support types are examined in detail: mass supports and spring supports. More complex support types, such as mass/spring systems, can be incorporated easily into the model.     

Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement,Part II: Experiment

To compare with the theoretical results of Part I, detailed experimental results have been obtained on the non-linear response of a clamped beam under a uniformly distributed periodic load. A duralumin test specimen with breadth 26 mm, thickness 0·5 mm and length 150 mm was used and tests were conducted by pulsating the base frame with a constant peak accerelation and measuring the relative displacement of the beam to the frame. It is found that the theoretical results are generally is reasonable agreement with the experimental ones, the effects of initial axial displacements being included. Besides the results for comparison with theoretical predictions, various non-linear responses were observed in connection with the internal resonance, combination resonance and dynamic snap-through phenomena, for which the present experimental results seem to provide effective data facilitating further theoretical analyses.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.     

Analysis on band gap properties of periodic structures of bar system using the spectral element method

In this paper, the band gap properties of the periodic structures of bar system, which include the rod-joint, truss, and frame structures, are studied using spectral element method (SEM). The spectral equations of the rod, beam, and joint elements are established and the spectral equations of the whole structures are further assembled. The frequency responses of the whole structures are calculated and the results are compared with those calculated by the finite element method (FEM). It can be observed that the SEM is more accurate in high frequency ranges. The band gap properties of the three types of periodic structures are studied, respectively. Furthermore, the effects of structural length, unit cell number, structural configurations, load conditions, and structural damping on the band gap properties are investigated.     

Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various boundary conditions

The dynamic analysis of a three-layered symmetric sandwich beam with magnetorheological elastomer (MRE) embedded viscoelastic core and conductive skins subjected to a periodic axial load have been carried out under various boundary conditions. As the skins of the sandwich beam are conductive, magnetic loads are applied to the skins during vibration. Due to the field-dependent shear modulus of MRE material, the stiffness of the MRE embedded sandwich beam can be changed by the application of magnetic fields. Using extended Hamilton’s principle along with generalized Galarkin’s method the governing equation of motion has been derived. The free vibration analysis of the system has been carried out and the results are compared with the published experimental and analytical results which are found to be in good agreement. The parametric instability regions of the sandwich beam have been determined for various boundary conditions. Here, recently developed magnetorheological elastomer based on natural rubber containing iron particles and carbon blacks have been used. The effects of magnetic field, length of MRE patch, core thickness, percentage of iron particles and carbon blacks on the regions of parametric instability for first three modes of vibration have been studied. These results have been compared with the parametric instability regions of the sandwich beam with fully viscoelastic core to show the passive and active vibration reduction of these structures using MRE and magnetic field. Also, the results are compared with those obtained using higher order theory.