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.     

Wave propagation analysis in 2D nonlinear hexagonal periodic networks based on second order gradient nonlinear constitutive models

We analyze the propagation of nonlinear waves in homogenized periodic nonlinear hexagonal networks, considering successively 1D and 2D situations. Wave analysis is performed on the basis of the construction of the effective strain energy density of periodic hexagonal lattices in the nonlinear regime. The obtained second order gradient nonlinear continuum has two propagation modes: an evanescent subsonic mode that disappears after a certain wavenumber and a supersonic mode characterized by an increase of the frequency with the wavenumber. For a weak nonlinearity, a supersonic mode occurs and the dispersion curves lie above the linear dispersion curve (v_p =v_p0). For a higher nonlinearity, the wave changes from a supersonic to an evanescent subsonic mode at s=0.7 and the dispersion curves drops below the linear case and vanish for certain values of the wavenumber. An important decrease in the frequency occurs for both subsonic and supersonic modes when the lattice becomes auxetic, and the longitudinal and shear modes become very close to each other. The influence of the lattice geometrical parameters of the lattice on the dispersion relations is analyzed.     

Transient thermoelastic waves in an anisotropic hollow cylinder due to localized heating

This paper presents a theoretical study of transient ultrasonic guided waves generated by concentrated heating of the outer surface of an infinite anisotropic hollow circular cylinder. Generalized thermoelastic theory proposed by Lord and Shulman is adopted to model the dynamic thermoelastic behavior of the cylinder. The concentrated heat source model used is to represent heating due to a pulsed laser beam, which is focused on the outer surface of the cylinder. A semi-analytical finite element (SAFE) method is employed to evaluate guided wave modes in the cylinder. Using integral transform techniques, the modal wave forms are obtained in frequency and wave number domains. Time histories of the propagating modes are then calculated by applying inverse Fourier transformation in the time domain. Numerical results showing the dispersion curves for the group velocities of the propagating modes and transient radial displacements are presented. For this purpose it is assumed that the cylinder is made of transversely isotropic silicon nitride (Si₃N₄). Attention is focused on the propagation characteristics of longitudinal and flexural modes separately.     

Experimental investigation of transient flexural waves in beams with discontinuities of cross section

There has been little experimental work on flexural wave propagation in general and on flexural wave propagation in beams with discontinuities of cross section in particular. Experimental data are obtained for various test beams subjected to eccentric longitudinal impact. The bending strain versus time results are presented for several positions along a uniform beam and finite beams (of circular cross section) with discontinuities of cross section. Bending strain histories are recorded at several positions before and after the discontinuity. The effect of reflections on the propagated flexural wave is illustrated. The dispersion of the traveling flexural wave caused several alternating peaks within the duration of the original positive input pulse. The importance of investigating discontinuities of cross section in structures subjected to impact loading is clearly manifested.     

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。     

Wave propagation in quasiperiodic structures: stop/pass band distribution and prestress effects

The band structure of dispersion diagrams for axial and flexural waves of quasiperiodic infinite beams is investigated. Every structure is composed of a repeated elementary cell generated adopting the Fibonacci sequence (different terms of this sequence are investigated); the problem is then solved evaluating the transmission matrix of the cell and applying the Floquet–Bloch technique. In the case of axial vibrations, a homogeneous rod where the quasiperiodicity is given by the particular distribution of external springs is considered, while for the flexural problem the quasiperiodicity is defined in terms of relative distance between the simple supports that sustain the beam. In both cases it is shown that, for different Fibonacci sequences, the number of stop/pass bands within a defined range of frequencies changes and follows the Fibonacci recursion rule showing a self-similar pattern. In addition, the overall dispersion characteristic can be interpreted in terms of an invariant function of the circular frequency, independent of the sequence generating the elementary cell. For flexural waves, the effects of the axial prestress on dispersion diagrams is highlighted and the frequency-shift of the stop/pass band positions is quantified. It is noted that a tensile axial prestress promotes length reduction of pass bands while leaves almost unchanged the length of stop-band intervals.     

Wave propagation in periodic and disordered layered composite elastic materials

A powerful complex transfer matrix approach to wave propagation perpendicular to the layering of a composite of periodic and disordered structure is worked out showing propagating and stopping bands of time-harmonic waves and the singular cases of standing waves. A state ratio of left- and right-going plane waves is defined and interpreted geometrically in the complex plane in terms of fixed points and flow lines. For numerical considerations and extension of the approach to higher dimensional problems a continued fraction expansion of the state ratio mapping is presented. Impurity modes of wave propagation in composites with widely spaced impurity cells of different elastic materials are discussed. Stopping bands in the frequency spectrum of global waves in fully disordered composites are found to exist in the range of frequencies corresponding to common gaps in the spectrum of cnstituent regular periodic composites which are constructed from the cells of the disordered system. For those frequencies, waves propagate only a (short) finite distance and are therefore strongly localized modes in a composite of fairly large extent.     

Lamb wave mode decomposition using concentric ring and circular piezoelectric transducers

Lamb waves using surface-bonded piezoelectric transducers (PZTs) have been widely used for nondestructive testing (NDT). However, the identification of individual Lamb wave modes and the subsequent data interpretation are often difficult due to the dispersive and multimodal natures of Lamb waves. To tackle the problem, a Lamb wave mode decomposition technique using concentric ring and circular PZTs is proposed. Its advantages over the conventional approaches are that (1) PZTs need to be placed only a single surface of a specimen and (2) mode decomposition can be performed at any desired frequency without changing the PZT size and/or spacing configuration. The proposed mode decomposition technique is formulated by solving 3D Lamb wave propagation equations considering the PZT size and shape, and this technique requires a specially designed dual PZT composed of concentric ring and circular PZTs. The effectiveness of the proposed technique for the Lamb wave mode decomposition is investigated through numerical simulation and experimental tests performed on an aluminum plate.     

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.      

Propagation of guided waves in pressure vessel

Pressure vessels usually operate under extremes of high/low temperatures and high pressures. Defect, such as crack and corrosion, can result in leakage or rupture failures, even catastrophic incidents. Guided wave-based structural health monitoring (SHM) technology is one of the most prominent options in non-destructive evaluation and testing (NDE/NDT) techniques. Propagation of guided waves in a typical pressure vessel is systematically investigated in this study for the application of guided wave-based SHM. Shape of the pressure vessel is a cylinder with two end caps. Because of geometric similarity, theory of guided wave propagation in the cylinderical structure is analyzed to study dispersive features of guided waves in the pressure vessel. Dispersion curves of three different types of guided wave modes, viz. the longitudinal, torsional and flexural modes, are calculated using theoretical method. Based on the analyses and experimental wave signals, central frequency and wave parameters of incident wave are optimized. Effect of contained liquid on propagation of guided waves, especially the L(0, 2) mode, in the pressure vessel is further investigated to minimize energy leakage of the wave to the contained liquid. The analytical method, finite element analysis (FEA) and experiments are applied to study propagation characteristics of guided waves in the pressure vessel, so as to demonstrate the feasibility of guided wave-based non-destructive evaluation and testing (NDE/NDT) for such kind of complex structures.     

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.     

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.     

非饱和土-结构动力响应的多耦合周期性有限元法

真实的地基土体-隧道系统中土体及结构性质往往沿线路纵向变化.为考虑土体与结构沿纵向的变化特性,提出了一种非饱和土-结构系统动力响应分析的多耦合周期性有限元法.首先基于非饱和土的实用波动方程,采用Galerkin法推导了单节点5个自由度的非饱和土ub-pl-pg格式有限元表达式,相比于单节点9个自由度的ub-v-w格式有...     

基于波传播方法的埋地管道的振动特性分析

利用波传播方法从理论上分析了有限长埋地管道的振动特性．得到土壤．结构耦合的频散方程．计算了埋地管道振动时的固有频率,通过与相关文献结果的比较,验证了该文理论分析和计算方法的正确性．数值计算结果表明埋地管道的固有频率,并不是在任何轴向和周向模态下都会出现,而是存在截止模态,土壤的剪切模量和管道参数对埋地管道的固有频率影响显著．     

Damage detection based on the propagation of longitudinal guided wave in a bimetal composite pipe

This paper investigates the damage detection based on the propagation of guided wave in bimetal composite pipes, which can identify damage locations in both axial and circumferential directions. The feasibility of the method is showed by numerical simulations using FEM code ANSYS. Mode analysis is used to evaluate the guided wave mode and its structure, which can provide the basis of the mode selection in measurements scheme. The guided wave propagation in a damaged pipe is computed by transient analysis. 16 nodes around the pipe wall, as probes, are used to record the guided wave signal. When Pseudo Margenau—Hill distribution (PMHD) for each signal is carried out, three types of modes could be found, which are led mode, excited mode and lag mode in sequences. Based on the results, the arrival time of the excited mode could be used to locate damage in axial direction, and the energy distribution around the pipe of lag mode is consistent with the damage in circumferential direction. The simulation illustrated the possibility of detecting damage location in both axial and circumferential directions based on longitudinal ultrasonic guided waves only.     

DERIVATION OF NONLINEAR WAVE EQUATION FOR FLEXURAL MOTIONS OF AN ELASTIC BEAM TRAVELLING IN AN AIR-FILLED TUBE

Asymptotic analysis is carried out to derive a nonlinear wave equation for flexural motions of an elastic beam of circular cross-section travelling along the centre-axis of an air-filled, circular tube placed coaxially. Both the beam and tube are assumed to be long enough for end-effects to be ignored and the aerodynamic loading on the lateral surface of the beam is considered. Assuming a compressible inviscid fluid, the velocity potential of the air is sought systematically in the form of power series in terms of the ratios of the tube radius to a wavelength and of a typical deflection to the radius. Evaluating the pressure force acting on the lateral surface of the beam, the aerodynamic loading including the effects of finite deflection as well as of air's compressibility and axial curvature of the beam are obtained. Although the nonlinearity arises from the kinematical condition on the beam surface, it may be attributed to the presence of the tube wall. With the aerodynamic loading thus obtained, a nonlinear wave equation is derived, whereas linear theory is assumed for the flexural motions of the beam. Some discussions are given on the results.     

Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane （with respect to the lattice plane） and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave prop- agation are highlighted in high frequency domains. One im- portant result presented herein is the comparison between the first Bloch wave modes to the membrane and bend- ing/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homog- enized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retro- propagating Bloch wave modes with a negative group veloc- ity, and of the corresponding ＂retro-propagating＂ frequency bands.     

WAVE PROPAGATION IN A TRANSVERSELY ISOTROPIC THERMOELASTIC SOLID CYLINDER OF ARBITRARY CROSS-SECTION

The wave propagation in an infinite, homogeneous, transversely isotropic solid cylinder of arbitrary cross-section is studied using Fourier expansion collocation method, within the frame work of linearized, three-dimensional theory of thermoelasticity. Three displacement potential functions are introduced, to uncouple the equations of motion and the heat conduction. The frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibration and are studied numerically for elliptic and parabolic cross-sectional zinc cylinders. The computed non-dimensional wave numbers are presented in the form of dispersion curves.     

ANALYSIS ON IMPACT RESPONSES OF UNRESTRAINED PLANAR FRAME STRUCTURE(Ⅱ)—NUMERICAL EXAMPLE ANALYSIS

By using the formula derived in Part （ Ⅰ ）, the instant response of an unrestrained planar frame structure subjected to the impact of a moving rigid-body are evaluated and analysed. The impact force-time history between the structure and the moving rigid-body, shear force and bending moment distribution along the beams, axial force distribution along the bars were calculated. The wave propagation phenomena of the longitudinal wave in the bars, the flexural and shear waves in the beams were also analysed. The numerical results show that the time duration of impact force is controlled by the flexural wave and the longitudinal wave ; the shear effect in beams should not be neglected in the impact response analysis of structures.     

ANALYSIS ON IMPACT RESPONSES OF UNRESTRAINED PLANAR FRAME STRUCTURE(Ⅱ)-NUMERICAL EXAMPLE ANALYSIS

By using the formula derived in Part (Ⅰ), the instant response of an unrestrained planar frame structure subjected to the impact of a moving rigid-body are evaluated and analysed.The impact force-time history between the structure and the moving rigid-body, shear force and bending moment distribution along the beams, axial force distribution along the bars were calculated. The wave propagation phenomena of the longitudinal wave in the bars, the flexural and shear waves in the beams were also analysed. The numerical results show that the time duration of impact force is controlled by the flexural wave and the longitudinal wave; the shear effect in beams should not be neglected in the impact response analysis of structures.