Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

弯曲波彩虹捕获效应及其在能量俘获中的应用

弹性波在色散关系经过设计的梯度结构中传播时会产生空间分频现象和波场能量增强现象, 即不同频率的弹性波会在结构的不同位置停止向前传播并发生能量聚集, 这就是弹性波彩虹捕获效应. 其相关研究成果可以促进结构健康监测、振动控制以及能量俘获等领域的发展. 本文通过所设计的梯度结构梁, 系统地研究了弯曲波彩虹捕获效应及其在压电能量俘获中的应用. 首先, 利用传递矩阵法获得了梯度结构梁元胞能带结构的解析解, 进而分析了弯曲波彩虹捕获效应的产生机理: 不同频率的弯曲波会在不同元胞附近群速度减小到零, 从而停止向前传播并发生反射; 入射波和反射波的叠加, 以及群速度减小带来的能量聚集, 会显著增强反射处的波场能量. 其次, 通过有限元仿真和实验验证了弯曲波彩虹捕获效应的空间分频现象和波场能量增强现象. 最后, 通过有限元多物理场耦合仿真和实验, 研究了粘贴PVDF压电薄膜的梯度结构梁相对于均匀梁的弯曲波能量俘获效果及其随入射波频率的变化规律. 结果表明, 在弯曲波彩虹捕获效应发生频带内, 粘贴在梯度结构梁上的PVDF压电薄膜的输出电压约为粘贴在均匀梁相应位置处的PVDF压电薄膜的输出电压的2倍.      

Free torsional vibrations of an elastic cylinder with laminated periodic structure

The theory of torsional vibrations of a circular cylinder, with a periodic variation of elastic constants and density normal to the axis of the cylinder, is developed in terms of Floquet waves. Floquet waves are quasi-periodic waves, whose amplitude profile has the same periodicity as that of the material and repeats with the periodicity of the cell. Using Floquet's theory, the dispersion spectrum is obtained for time-harmonic waves propagating in a laminated cylinder with periodic structure. It is shown that the dispersion spectrum has a band structure, consisting of passing bands and stopping bands. Motion in the case of grazing incidence, and motion at the end of the zones is discussed. It is also shown that as the radius of the cylinder tends to infinity, the torsional waves in a circular cylinder degenerate to SH-waves in laminated plates.     

一维磁电弹准周期结构带隙特性的研究

采用传递矩阵方法,研究了横波(SV波)垂直入射时压电/（弹性/压磁）和（压电/弹性）/压磁两种Fibonacci准周期结构的频带特性,通过计算局部化因子和位移透射系数,数值揭示了此两种Fibonacci准周期结构频带特性的差异以及与相应理想周期结构频带特性的不同,而且表明（压电/弹性）/压磁Fibonacci准周期结构的频带特性与纯弹性材料Fibonacci准周期结构的频带特性是相似的。     

Floquet–Bloch Waves in Periodic Networks of Rayleigh Beams: Cellular System,Dispersion Degenerations,and Structured Connection Regions

The paper is dedicated to Professor N. F. Morozov on the occasion of his 85th birthday. In the paper, we consider new dispersive properties of elastic flexural waves in periodic structures with rotational inertia. The structure is represented as a lattice with elementary bonds of Rayleightype beams. Although such beams in the semiclassical regime react as the classical Euler–Bernoulli beams, they exhibit new interesting characteristics as the dispersion frequency of flexural waves increases. Special attention is paid to degenerate cases related to the so-called Dirac cones on dispersion surfaces and to the directed anisotropy for the doubly periodic lattice. A comparative analysis accompanied by numerical simulation is carried out for the Floquet–Bloch waves propagating in periodic flexible lattices of different geometry.     

Wave propagation and resonance phenomena in inhomogeneous media

The waveguide and resonance properties of inhomogeneous penetrable one–dimensional–periodical structures that consist of two different media are studied within the framework of a one–dimensional approximation. The pass and stop bands are determined. A dispersion relation for all the waveguide modes is obtained. Explicit expressions for low waveguide frequencies and corresponding phase velocities of waveguide modes for mono– and polydisperse media are found. The influence of the polydispersity of the sizes of heterogeneities on the low frequencies of a pass band is considered. A pass band in the range of low frequencies is detected. It is shown that the polydispersity does not affect the waveguide properties of a medium at low frequencies of the first pass band. The resonance phenomena in periodical media and structures are investigated. The resonance phenomena are shown to occur for an unlimited discrete set of frequencies if the group velocity of the waveguide mode for them is zero; in this case, the growth of the oscillation amplitude is localized in the neighborhood of a source (localization of the resonance). A synchrophasotron resonance at which the infinite chain of oscillation sources has the oscillations phase of a corresponding traveling wave from the pass band is detected.     

Wave transmission characteristics in periodic media of finite length: multilayers and fiber arrays

Wave transmission characteristics in elastic media that have periodic microstructure over a finite spatial length are examined theoretically as well as numerically. Two classes of such media are demonstrated, namely, one-dimensional multilayered media with finite-length periodicity and two-dimensional composite media with square arrays of aligned fibers within a finite length. From these one-dimensional and two-dimensional analyses, the influence of the finite-length periodicity on the wave transmission characteristics is discussed. In these media, there are frequency bands (stop bands) where the energy transmission coefficient appears to vanish or takes very low values, while in pass bands it oscillates with the frequency due to the finite-length periodicity. It is theoretically demonstrated in the one-dimensional case of multilayers how the frequency intervals of the oscillation in the transmission spectrum depend on the repeating number of the periodic cells as well as other acoustic and geometrical parameters. The results of the two-dimensional fiber arrays, which are obtained numerically by solving the equations of the SH wave multiple scattering, are shown to fit well in the one-dimensional framework of multilayered structures up to a certain frequency encompassing the first stop band. This similarity between two classes of problems is demonstrated by appropriately identifying the one-dimensional reduced transfer matrix for a single cell that is representative of the periodic fiber array.     

A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion

The goal of this study is to understand the physical meaning and evaluate the intrinsic length scale parameters, featured in the theories of gradient elasticity, by deploying the analytical treatment and experimental measurements of the dispersion of elastic waves. The developments are focused on examining the propagation of longitudinal waves in an aluminum rod with periodically varying cross-section. First, the analytical solution for the dispersion relationship, based on the periodic cell analysis of a bi-layered laminate and Bloch theorem, is compared to two competing models of gradient elasticity. It is shown that the customary gradient elastic model with two length-scale parameters is able to capture the dispersion accurately up to the beginning of the first band gap. On the other hand, the gradient elastic model with an additional length scale (affiliated with the fourth-order time derivative in the field equation) is shown to capture not only the first dispersion branch before the band gap, but also the band gap itself and the preponderance of the second branch. Closed form relations between the microstructure parameters and the intrinsic length scales are obtained for both gradient elasticity models. By way of the asymptotic treatment in the limit of a weak contrast between the laminae, a clear physical meaning and scaling of the length-scale parameters was established in terms of: (i) the microstructure (given by the size of the unit cell and the contrast between the laminae), and (ii) thus induced dispersion relationship (characterized by the location and the width of the band gap). The analysis is verified through an experimental observation of wave dispersion, and wave attenuation within the band gap. A comparison between the analytical treatment, the gradient elastic model with three intrinsic length scales, and experimental measurements demonstrates a good agreement over the range of frequencies considered.     

SYMPLECTIC ANALYSIS FOR WAVE PROPAGATION OF HIERARCHICAL HONEYCOMB STRUCTURES

Wave propagation in two-dimensional hierarchical honeycomb structures with twoorder hierarchy is investigated by using the symplectic algorithm. By applying the variational principle to the dual variables, the wave propagation problem is transformed into a two-dimensional symplectic eigenvalue problem. The band gaps and spatial filtering phenomena are examined to find the stop bands and directional stop bands. Special attention is directed to the effects of the relative density and the length ratio on the band gaps and phase constant surfaces. This work provides new opportunities for designing hierarchical honeycomb structures in sound insulation applications.     

Low-frequency band gaps in chains with attached non-linear oscillators

The aim of this article is to investigate the wave propagation in one-dimensional chains with attached non-linear local oscillators by using analytical and numerical models. The focus is on the influence of non-linearities on the filtering properties of the chain in the low frequency range. Periodic systems with alternating properties exhibit interesting dynamic characteristics that enable them to act as filters. Waves can propagate along them within specific bands of frequencies called pass bands, and attenuate within bands of frequencies called stop bands or band gaps. Stop bands in structures with periodic or random inclusions are located mainly in the high frequency range, as the wavelength has to be comparable with the distance between the alternating parts. Band gaps may also exist in structures with locally attached oscillators. In the linear case the gap is located around the resonant frequency of the oscillators, and thus a stop band can be created in the lower frequency range. In the case with non-linear oscillators the results show that the position of the band gap can be shifted, and the shift depends on the amplitude and the degree of non-linear behaviour.     

基于微形态模型的颗粒材料中波的频散现象研究

基于颗粒材料冲击与波动响应特性的调控波传播行为的超材料设计受到广泛关注,设计这类材料需要对颗粒材料的波传播机制及调控机理有深入认识. 波在颗粒材料中传播的频散现象及频率带隙等行为与材料的非均匀性密切相关,通常讨论频散现象是基于弹性理论框架建立微结构连续体或高阶梯度连续体等广义连续体模型来进行. 本研究基于细观力学给出了一个颗粒材料的微形态连续体模型. 在该模型中,考虑了颗粒的平动和转动,且颗粒间的相对运动分解为两部分:即宏观平均运动和细观真实运动. 基于此分解,提出了一个完备的变形模式,得到了对应于不同应变及颗粒间运动的宏细观本构关系. 结合宏观变形能的细观变形能求和表达式,获得了基于细观量表示的宏观本构模量. 应用所建议模型考察了波在弹性颗粒介质的传播行为,给出了不同形式的波的频散曲线,结果显示此模型具有预测频率带隙的能力.      

Harmonic vibrations and waves in a cylindrical helically anisotropic shell

A Kirchhoff-Love type applied theory is used to study the specific characteristics of harmonic waves and vibrations of a helically anisotropic shell. Special attention is paid to axisymmetric and bending vibrations. In both cases, the dispersion equations are constructed and a qualitative and numerical analysis of their roots and the corresponding elementary solutions is performed. It is shown that the skew anisotropy in the axisymmetric case generates a relation between the longitudinal and torsional vibrations which is mathematically described by the amplitude coefficients of homogeneous waves. In the case of a shell with rigidly fixed end surfaces, the dependence of the first two natural frequencies on the shell length and the helical line slope α, i.e., the geometric parameter of helical anisotropy, is studied. A boundary value problem in which longitudinal vibrations are generated on one of the end surfaces and the other end is free of forces and moments is considered to analyze the degree of transformation of longitudinal vibrations into longitudinally torsional vibrations. In the case of bending vibrations, two problems for a half-infinite shell are studied as well. In the first problem, the waves are excited kinematically by generating harmonic vibrations of the shell end surface in the plane of the axial cross-section, and it is shown that the axis generally moves in some closed trajectories far from the end surface. In the second problem, the reflection of a homogeneous wave incident on the shell end is examined. It is shown that the “boundary resonance” phenomenon can arise in some cases.     

Nonlinear normal modes and band zones in granular chains with no pre-compression

We study standing waves (nonlinear normal modes—NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are homogeneous dynamical systems in the notation of Rosenberg (Adv. Appl. Mech. 9:155–242, 1966), we show that the discontinuous nature of the dynamics leads to interesting effects such as separation between beads, NNMs that appear as traveling waves (these are characterized as pseudo-waves), and localization phenomena. In the limit of infinite extent, we study band zones, i.e., pass and stop bands in the frequency–energy plane of these dynamical systems, and classify the essentially nonlinear responses that occur in these bands. Moreover, we show how the topologies of these bands significantly affect the forced dynamics of these granular media subject to narrowband excitations. This work provides a classification of the coherent (regular) intrinsic dynamics of one-dimensional homogeneous granular chains with no pre-compression, and provides a rigorous theoretical foundation for further systematic study of the dynamics of granular systems, e.g., the effects of disorders or clearances, discrete breathers, nonlinear localized modes, and high-frequency scattering by local disorders. Moreover, it contributes toward the design of granular media as shock protectors, and in the passive mitigation of transmission of unwanted disturbances.     

Asymptotic and numerical models of waves in tunable multi-scale structures

The paper presents asymptotic models and numerical illustrations of periodic systems which possess band gaps and support standing waves at low frequencies. The structures considered here include periodic systems of defects (cracks or resonators of different types). Tuning mechanisms are described to control the position of band gaps in dispersion diagrams.     

Symmetric and anti-symmetric Rayleigh–Lamb modes in sinusoidally corrugated waveguides: An analytical approach

The wave propagation analysis in corrugated waveguides is considered in this paper. Elastic wave propagation in a two-dimensional periodically corrugated plate is studied here analytically. The dispersion equation is obtained by applying the traction free boundary conditions. Solution of the dispersion equation gives both symmetric and anti-symmetric modes. In a periodically corrugated waveguide all possible spectral order of wave numbers are considered for the analytical solution. It has been observed that the truncation of the spectral order influences the results. Truncation number depends on the degree of corrugation and the frequency of the wave. Usually increasing frequency requires increasing number of terms in the series solution, or in other words, a higher truncation number. For different degrees of corrugation the Rayleigh–Lamb symmetric and anti-symmetric modes are investigated for their non-propagating ‘stop bands’ and propagating ‘pass bands’. To generate the dispersion equation for corrugated plates with a wide range of the degree of corrugation, appropriate truncation of the spectral orders has to be considered. Analytical results are given for three different degrees of corrugation in three plates. Resonance of symmetric and anti-symmetric modes in these plates, their ‘cut-off’, ‘cut-on’, ‘branch-point’, ‘change-place’, ‘mode conversion’ and ‘pinch points’ at various frequencies are also studied.     

Investigation of flexural wave band gaps in a locally resonant metamaterial with plate-like resonators

This paper presents an investigation of flexural wave band gaps in locally resonant metamaterials (LRMs). An LRM is a periodic structure consisting of repeated unit cells containing a local resonator. Due to the local resonance occurring in the unit cell, the LRM induces a band gap (a frequency band in which no waves propagate). Discrete-like or beam-like resonators have generally been used to realise LRMs in previous research. By extending the beam-like resonator configuration, this paper studies LRMs with a plate-like resonator to exploit its advantages with respect to large design freedom. In order to understand flexural wave band gaps in an LRM with plate-like resonators, parametric studies are conducted with the development of a finite element model. Further, the influences of the plate-like resonator design parameters on flexural wave band gaps are investigated. Based on the parametric studies, the rules governing band gap properties are determined. Finally, tailoring flexural wave band gaps by adjusting the parameters is discussed.     

Wave propagation in two-dimensional anisotropic acoustic metamaterials of K4 topology

An acoustic metamaterial is envisaged as a synthesised phononic material the mechanical behaviour of which is determined by its unit cell. The present study investigates one aspect of mechanical behaviour, namely the band structure, in two-dimensional (2D) anisotropic acoustic metamaterials encompassing locally resonant mass-in-mass units connected by massless springs in a K4 topology. The 2D lattice problem is formulated in the direct space (r-space) and the equations of motion are derived using the principle of least action (Hamilton’s principle). Only proportional anisotropy and attenuation-free shock wave propagation have been considered. Floquet–Bloch’s principle is applied, therefore a generic unit cell is studied. The unit cell can represent the entire lattice regardless of its position. It is transformed from the direct lattice in r-space onto its reciprocal lattice conjugate in Fourier space (k-space) and point symmetry operations are applied to Wigner–Seitz primitive cell to derive the first irreducible Brillouin Zone (BZ). The edges of the first irreducible Brillouin Zone in the k-space have then been traversed to generate the full band structure. It was found that the phenomenon of frequency filtering exists and the pass and stop bands are extracted. A follow-up parametric study appreciated the degree and direction of influence of each parameter on the band structure.     

Wave dispersion in gradient elastic solids and structures: A unified treatment

Analytical wave propagation studies in gradient elastic solids and structures are presented. These solids and structures involve an infinite space, a simple axial bar, a Bernoulli–Euler flexural beam and a Kirchhoff flexural plate. In all cases wave dispersion is observed as a result of introducing microstructural effects into the classical elastic material behavior through a simple gradient elasticity theory involving both micro-elastic and micro-inertia characteristics. It is observed that the micro-elastic characteristics are not enough for resulting in realistic dispersion curves and that the micro-inertia characteristics are needed in addition for that purpose for all the cases of solids and structures considered here. It is further observed that there exist similarities between the shear and rotary inertia corrections in the governing equations of motion for bars, beams and plates and the additions of micro-elastic (gradient elastic) and micro-inertia terms in the classical elastic material behavior in order to have wave dispersion in the above structures.     

分数阶粘弹性地基上薄板波动和振动特性分析

本文研究了粘弹性地基上薄板的波动和振动问题．主要讨论了基于分数导数理论的粘弹性地基模型上 薄板弯曲波的传播特性以及固有频率对地基的依赖特性．推导了三种经典粘弹性地基模型的复模量．并利用分 数导数的性质得到分数阶粘弹性地基上 Kirchhoff板中弯曲波的传播速度、衰减系数以及自由振动的复固有频 率．数值算例表明粘弹性地基对弯曲波传播特性存在显著影响,不同粘弹性模型所对应的色散和衰减特性也存 在较大差别．分数阶导数可以实现相邻整数阶导数之间的光滑过渡．利用分数导数的本构关系可以更加真实地 描述粘弹性地基的历史依赖行为,更准确地表现出粘弹性地基板中弯曲波的色散和衰减特性．     

基于导波技术的螺柱轴力无损检测

根据弹性动力学理论,采用纵向导波与弯曲导波相结合的方法对螺柱所受轴向应力进行无损检测。计算了M22螺柱中纵向导波和弯曲导波的群速度频散曲线。根据频散曲线,确定了采用导波对螺柱轴向应力进行无损检测的最优检测信号频率范围(50~80 kHz),此频率范围的纵向导波与弯曲导波模态单一,并且频散性较低。分别计算了不同轴向应力σ作用下,多种频率的纵向导波和和弯曲导波在螺柱中传播的群速度值cgσrL和cgσrF。结果表明,随着轴向应力的增大,同频率纵向导波和弯曲导波的群速度皆呈线性递减趋势。利用纵向导波和弯曲导波群速度与轴向应力的线性关系及纵向导波和弯曲导波在轴向应力作用螺柱端面的反射时间tLσ和tσF,可以迅速确定螺柱所受轴向应力值。