Local vibration of an elastic plate and zero-group velocity Lamb modes

Elastic plates or cylinders can support guided modes with zero group velocity (ZGV) at a nonzero value of the wave number. Using laser-based ultrasonic techniques, we experimentally investigate some fascinating properties of these ZGV modes: resonance and ringing effects, backward wave propagation, interference between backward and forward modes. Then, the conditions required for the existence of ZGV Lamb modes in isotropic plates are discussed. It is shown that these modes appear in a range of Poisson's ratio about the value for which the cutoff frequency curves of modes belonging to the same family intercept, i.e., for a bulk wave velocity ratio equal to a rational number. An interpretation of this phenomenon in terms of a strong repulsion between a pair of modes having a different parity in the vicinity of the cutoff frequencies is given. Experiments performed with materials of various Poisson's ratio demonstrate that the resonance spectrum of an unloaded elastic plate, locally excited by a laser pulse, is dominated by the ZGV Lamb modes.     

Estimation of wall thinning in mild steel using laser ultrasound Lamb waves and a non-steady-state photo-emf detector

In this work, a non-steady-state photo-emf receiver has been used to detect the lower frequency fundamental a0 Lamb waves in mild steel. Experimentally, the Lamb waves are laser-generated in the thermoelastic regime using a Q-switched, 20 ns, pulsed Nd:YAG laser and a line source. Typical Lamb waves had centre frequencies of 250 kHz but with frequency components that extended beyond 1 MHz. In mild steel, higher order Lamb wave modes were not considered to be significant below a frequency thickness product of 1.6 MHz mm. Below this level, associated velocity dispersion curve offered phase velocity changes that were sensitive to thickness change. Samples up to 5 mm in thickness were examined without significant interference from higher order modes. A non-steady-state photo-emf detector used as the ultrasonic detector had the advantage of a lower frequency cut-off at 100 kHz compared to a confocal Fabry-Pérot interferometer (CFPI) of about 2 MHz. Both schemes offered greater stand-off distance (>20 cm) than is possible with EMATs, which have a stand-off distance less than 1.0 mm. Progress made in detecting wall thinning in steel plate with thickness up to 5 mm is reported.     

On the “overlapping resonances” concept of acoustic transmission through an elastic plate,II: Numerical examples and physical implications

In a companion paper [1] the Fiorito, Madigosky and Überall approximate theory of transmittivity of an infinite, isotropic, elastic plate in terms of Lamb mode resonances [2] is examined, and extended by validating coherent addition of these overlapping resonances. The present paper provides numerical and heuristic tests which yield convincing proof of the applicability of the overlapping resonances concept. At various thicknesses of a water-immersed, steel plate, including the half-space and zero thickness limits, the behaviour is shown to be compatible with known exact behaviour. Overlapping resonances explain the reduction towards zero of the transmittivity near the coincidence angles of the zero order Lamb modes as these converge to the Rayleigh angle. Contrary to earlier interpretations, both zero order modes survive at the half-space limit. The A₀ mode is shown to contribute below (fd)_critical, becoming then highly damped and, effectively, non-propagating. The interpretation provided of the transition of the A₀ mode in this region bridges the gap between conventional Lamb mode theory and thin plate theory.     

Elastic constants of a plate from impact-echo resonance and Rayleigh wave velocity

A new method is proposed for calculating the dynamic elastic constants of an isotropic plate from measurements of the impact-echo resonance and Rayleigh wave velocity. Poisson's ratio is shown to be a single-valued function of the ratio between thickness frequency and Rayleigh wave velocity. This dependence is derived theoretically from the condition of resonance at the minimum frequency of the first-order symmetric Lamb mode. A finite element model is developed to determine how this frequency varies with Poisson's ratio. The results obtained by modal analysis and the power-spectral density technique are in good agreement with those calculated as the solution of the S1 Lamb mode equation. The method is verified by impact-echo tests on concrete and methacrylate plates. A laser interferometer is used to detect the vibration. Thickness frequencies are accurately identified by applying the multicross-spectral density to the signals detected at several points close to the impact point. In a separate experiment, Rayleigh waves are generated by the mediator technique. The wave velocities are determined from the arrival times of the surface wave at several points. Finally, the main sources of uncertainty are evaluated.     

采用单个压电传感器的单模式兰姆波激发频率的选择

考虑了压电传感器(PZT)与板之间的耦合作用,从理论上研究了单个压电传感器激发时产生单模式兰姆波的频率调节方法,实验给出了模式选择在兰姆波结构健康监测中的应用. 在板材、板厚以及PZT尺寸一定的情况下,从理论上能够预测到作为频率函数的各兰姆波模式的幅值变化. 根据某特定兰姆波模式的幅值最大而其他模式幅值相对最小时所对应的频率,即可识别出该兰姆波模式优化的激发频率. 数值仿真验证了理论的有效性以及单模式兰姆波选择的可能性. 在不同的激发频率下,分别激发了优化的A0 模式,优化的S0模式以及共存的A0和S0模 关键词： 兰姆波 压电传感器 激发频率 结构健康监测     

Feasibility of bone assessment with leaky Lamb waves in bone phantoms and a bovine tibia

In this study, the effect of cortical thickness variation on the propagation of leaky Lamb waves is investigated by using an axial transmission technique commonly used to characterize long bones. Three Lucite plates with thicknesses of 1, 3, and 5 mm as bone phantoms and one bovine tibia with a cortical thickness of 2 mm were used at various low frequencies. Experimental measurements in bone phantoms show that the peak frequency and amplitude of excited Lamb modes strongly depend on the thickness of the Lucite plate. In the bovine tibia, the S0 and A0 Lamb modes are consistently observed in the frequency-thickness region from 0.2 to 1.0 MHz mm, and can be effectively launched at a frequency of 200 kHz, suggesting 200 kHz to be the optimal signal frequency for in vivo clinical applications. It can be also seen that both modes are affected by the frequency-thickness product, but the effect is greater for the A0 mode. Hence, the A0 Lamb mode seems more sensitive to cortical thickness change due to aging and osteoporosis. This study suggests that the use of leaky Lamb waves is feasible for ultrasonic bone assessment.     

Characterization of mechanical properties of a hollow cylinder with zero group velocity Lamb modes

Hollow cylinders used in the industry must be regularly inspected. Elastic guided waves, similar to Lamb modes in a plate, can propagate in the axial direction or around the circumference. They are sensitive to geometrical and mechanical parameters of the cylindrical shell. The objective of this paper is to show that zero group velocity (ZGV) Lamb modes can be used to bring out anisotropy and to measure elastic constants of the material. This study provides experimental and numerical investigations on a Zirconium alloy tube extensively used by the nuclear industry in reactor core components. A non-contact method, based on laser ultrasound techniques and ZGV Lamb modes, demonstrates that the difference observed between axial and circumferential guided waves cannot be explained by an isotropic model. Then, a transverse isotropic model is used for the Zircaloy tube. Four of the five elastic constants are directly extracted from ZGV resonance frequencies. The last one is deduced from the measured dispersion spectra. With this complete set of constants, a good agreement is obtained between theoretical and experimental dispersion curves for both axially and circumferentially propagating guided waves.     

Numerical description of the edge mode at the beveled extremity of a plate

This paper deals with the reflection of a two-dimensional harmonic Lamb wave at the beveled end of a plate. The existence of a resonant edge mode is described by a numerical model. It is proved that the edge mode is the resonance of different complex modes. The behavior of this mode as a function of the bevel angle is studied. Its amplitude decreases and its resonance frequency shifts as the bevel angle decreases from 90 to 85 deg. An unexpected strong variation of the repartition of the reflected energy is linked to this phenomenon.     

Numerical predictions and experiments on the free-plate edge mode

In this paper, te edge mode variation is studied with three different methods: the reciprocal work method, already used by Torvik [J. Acoust. Soc. Am. 41 (1967) 346] to model this phenomenon, the S-parameter method and a finite element model that are applied for the first time to the study of the edge resonance. Moreover, laser probe measurements of the edge mode have also been performed and compared to the numerical predictions. The good agreement between the numerical predictions and the experimental data allows full understanding of the resonant phenomenon. The edge resonance is linked to the strong increase in amplitude of two complex Lamb waves, and the edge mode is proved to radiate into the plate as the first symmetrical Lamb mode S(0). Displacements at the edge and away from the edge have been computed and measured to evaluate the spatial and temporal behaviour of the edge mode. The dependence of the edge resonance frequency and amplitude on the Poisson coefficient has also been studied.     

Reflection of Lamb waves obliquely incident on the free edge of a plate

The reflection of obliquely incident symmetric and anti-symmetric Lamb wave modes at the edge of a plate is studied. Both in-plane and Shear-Horizontal (SH) reflected wave modes are spawned by an obliquely incident in-plane Lamb wave mode. Energy reflection coefficients are calculated for the reflected wave modes as a function of frequency and angle of incidence. This is done by using the method of orthogonal mode decomposition and by enforcing traction free conditions at the plate edge using the method of collocation. A PZT sensor network, affixed to an Aluminum plate, is used to experimentally verify the predictions of the analysis. Experimental results provide support for the analytically determined results.     

用于储罐底板缺陷检测的超声兰姆波模式研究

从超声兰姆波的声场方程出发,得到板材表面离面位移(法向位移)为零的条件,即当超声兰姆波的相速度等于板材介质的纵波声速时,在板材表面的离面位移为零。在给定适当频厚积的条件下,分别数值模拟了仅有切向位移而无离面位移的A1、S1、A2和S2兰姆波模式在有液体负载的单层钢板中的传播情形。结果表明:离面位移为零的S2模式频散较小且对板中缺陷更为敏感。     

On the “overlapping resonances” concept of acoustic transmission through an elastic plate,I: An examination of properties

The Fiorito, Madigosky and Überall approximate representation [1] of the acoustic transmission coefficient of an infinite, isotropic, elastic plate by resonances associated with Lamb modes is examined and clarification of its properties is obtained. Some theoretical extension is provided which validates coherent addition of these overlapping resonances. Minor limitations shown are that the theory is inapplicable near grazing incidence and that it gives only a partial solution at incidence beyond the second critical angle when the frequency thickness product is the variable. Existing overlapping resonances theory is shown not to yield a unique relationship between the incident wave and a set of physical motions within the plate, and an approach to overcome this non-uniqueness is suggested. An equivalence is demonstrated between the 3 dB resonance half-width of each Lamb mode and the imaginary part of the modal wavenumber per unit wavenumber of the fluid, and this facilitates the numerical checking of the theory which is carried out in a companion paper [2] and yields convincing proof of the applicability of the approach.     

Magnetoelastic waves in an in-plane magnetized ferromagnetic plate

The spectrum of magnetoelastic waves propagating along the magnetic field in an in-plane magnetized ferromagnetic plate is numerically investigated in the exchangeless approximation. No restrictions are imposed either on the field pattern of backward volume magnetostatic waves (BVMSWs) or elastic waves supported by a plate of a given geometry across the plate or on the relationship between the sound velocity v _S and the phase velocity of the magnetoelastic waves v=ω/q (ω is the frequency, q is the wave number). The resonance interaction of the BVMSWs and elastic waves is accompanied, as a rule, by the formation of “stop” bands δω that are proportional to the magnetoelastic coupling constant b. When the BVMSWs are in resonance with Lamb and shear elastic modes the values of the magnetoelastic gaps δω at v≈v _S turn out to be of the same order. For v≫v _S , the efficiency of the interaction between the BVMSWs and transverse Lamb modes is almost one order of magnitude higher. If the frequency spacing Δω between the elastic modes is smaller than the mag-netoelastic gap in the spectrum (Δω≤δω), which takes place, particularly, in the region of crowding the elastic mode spectrum (v≈v _S), the resonant interaction results in mixing the dispersion laws for the elastic modes. Namely, a surface mode may transform into a volume one and a shear mode, into the Lamb mode or into a shear mode with another number. The resonance interaction of the shear and Lamb elastic modes not only forms the magnetoelastic gaps δω∼b ² but also changes the efficiency of elastic wave coupling with the magnetic subsystem. This may show up as the coexistence of the effects of “repulsing” both the dispersion laws and the damping decrements of the elastic waves at the resonance frequency. It is shown that magnetostriction splits the cutoff frequencies of both transverse Lamb modes and shear modes, as well as the long-wave (q → 0) frequency limits f ₀ of the BVMSW modes. This may cause the resonance interaction between BVMSW modes of equal evenness in a narrow frequency band Δ∼b near f ₀.     

Prolonged acousto-optic interaction with Lamb waves in crystalline plates

The propagation and acousto-optic interaction of Lamb modes in an anisotropic plate of tellurium dioxide (TeO2) are studied numerically and analytically. In the case of a Y-cut X-propagating TeO2 plate, the very high elastic anisotropy of the crystal greatly modifies the dispersion curves, giving rise to their multiple oscillations. The existence ranges of backward Lamb modes increase with the mode order contrary to the case of isotropic plates. The quasi-collinear light scattering by Lamb waves is considered. Owing to the structure of Lamb wave field, a simultaneous light diffraction at two different optical frequencies can take place while Lamb waves are excited only at the single frequency. It is demonstrated with the Z-cut (110)-propagating plate that a small change in the acoustic frequency can result in a significant shift in the frequency of the scattered light.     

时频域独立元分析方法实现超声兰姆波的多模式分离

为方便兰姆波信号分析与模式定征,提出一种将短时傅里叶变换(Short-Time Fourier Transform,STFT)与独立元分析(Independent Component Analysis,ICA)相结合的多模式超声兰姆波识别方法。首先通过STFT将兰姆波时域信号投影至时频域,基于各模式信号在时频域相对独立的特点,利用ICA实现混叠模式分离。根据分离模式时频能量脊提取各模式群速度曲线,进而估计板厚。将方法运用于时域有限差分(Finite-Difference Time-Domain,FDTD)法仿真与钢板实验,分离得到A0、A1和S0三种模式。仿真与实验中平均群速度估计误差约为1.5%和2.0%,板厚估计误差约为0.3%和2.0%。仿真结果表明,在信噪比(Signal-to-Noise Ratio,SNR)不小于0 dB的情况下,时频域独立元分析方法均可实现兰姆波多模式分离、群速度曲线提取及板厚估计。      

Radiation efficiency of submerged rectangular plates

The average radiation efficiency of point-excited submerged rectangular plates is investigated in two methods, deterministic analysis and statistical approach, respectively. In the deterministic analysis, the effect of mutual impedance by water loading on the velocity of the plate is illustrated analytically by using a modal summation method. The cross-modal contributions to the average radiation efficiency are averaged to zero by averaging over all possible excitation positions on the plate. In the statistical approach, by analyzing the engineering formulae of the average radiation efficiency in air, this paper modifies the formulae to be applicable in water. The numerical comparisons show that the modified formulae reflect the average level in the frequency region controlled by corner modes and are accurate enough in the region controlled by monopole and edge modes. On this basis, approximate expressions for predicting the average radiation efficiency of the submerged beam-stiffened rectangular plates are proposed. The main differences between air loading and water loading are considered. Firstly, as dry modes taken in analysis instead of real vibration modes in water, the vibration of the stiffened plate is not only determined by the first mode but by several modes simultaneously at low frequencies. Secondly, the “corner mode region” becomes inappropriate as the plate is stiffened. The proposed formulae are validated numerically in different size, thickness, and stiffener amount of the stiffened plate.     

小样本字典学习的兰姆波模态识别方法*

针对兰姆波多模态识别问题,提出了基于小样本字典学习的模态识别方法。将多层复合板的频散特性看作一个线性时不变系统,首先,根据频散知识模拟各个模态传播特定距离后的信号,提取走时和能量特征创建字典;其次,获取待测信号的走时特征,通过查询字典来识别兰姆波模态;最后,根据能量参数估计结果,实现待测信号中各模态信号的分离和重构。通过对三层粘接的AAA板(铝板-亚克力板-铝板,每层厚度为2mm)中传播距离为0.3m、0.5m的直达波和反射波的实验验证,结果显示该方法对A0、S0模态的有效识别和各个波包信号的准确重建。     

Sound absorption and transmission through flexible micro-perforated panels backed by an air layer and a thin plate

This paper describes theoretical and experimental investigations into the sound absorption and transmission properties of micro-perforated panels (MPP) backed by an air cavity and a thin plate. A fully coupled modal approach is proposed to calculate the absorption coefficient and the transmission loss of finite-sized micro-perforated panels-cavity-panel (MPPCP) partitions with conservative boundary conditions. It is validated against infinite partition models and experimental data. A practical methodology is proposed using collocated pressure-velocity sensors to evaluate in an anechoic environment the transmission and absorption properties of conventional MPPCPs. Results show under which conditions edge scattering effects should be accounted for at low frequencies. Coupled mode analysis is also performed and analytical approximations are derived from the resonance frequencies and mode shapes of a flexible MPPCP. It is found that the Helmholtz-type resonance frequency is deduced from the one associated to the rigidly backed MPPCP absorber shifted up by the mass-air mass resonance of the flexible non-perforated double-panel. Moreover, it is shown analytically and experimentally that the absorption mechanisms at the resonances are governed by a large air-frame relative velocity over the MPP surface, with either in-phase or out-of-phase relationships, depending on the MPPCP parameters.     

Mode selection of Lamb waves for the evaluation of solid plates with liquid loading

This paper studies the mode selection of Lamb waves for evaluating solid plates with liquid loading. For this purpose, the Lamb wave selected should have the features such as zero normal displacement components at the plate surface in contact with liquid, small dispersion, and maximum group velocity. It is found that when the phase velocity of Lamb wave is equal to the longitudinal wave velocity of the plate material, its normal displacement at the plate surface is always zero. Through the numerical analyses, the specific S₂ Lamb wave that has zero normal displacement component at the plate surface, small dispersion and maximum group velocity compared with the other Lamb waves has been found. With respect to the specific S₂ Lamb wave, some experimental examinations have been carried out. It is found that the liquid loading on the plate surface has less influence on the specific S₂ Lamb wave signal but it can effectively eliminate the other signals. Moreover, the specific S₂ Lamb wave selected exhibits the capability of detecting multiple defects in the solid plate with the liquid loading. It can be concluded that the specific S₂ Lamb wave selected is suitable for the evaluation of solid plates with liquid loading.     

Negative group velocity Lamb waves on plates and applications to the scattering of sound by shells

Symmetric Lamb waves on plates exhibit anomalies for certain regions of frequency. The phase velocity appears to be double-valued [M. F. Werby and H. Uberall, J. Acoust. Soc. Am. 111, 2686-2691 (2002)] with one of the branches having a negative group velocity relative to the corresponding phase velocity. The classification of the symmetric plate modes for frequencies appearing to have a double-valued phase velocity is reviewed here. The complication of a double-valued velocity is avoided by examining mode orthogonality and the complex wave-number spectra. Various authors have noted an enhancement in the backscattering of sound by elastic shells in water that occurs for frequencies where symmetric leaky Lamb waves (generalized to case of a shell) have contra-directed group and phase velocities. The ray diagram for negative group velocity contributions to the scattering by shells [G. Kaduchak, D. H. Hughes, and P. L. Marston, J. Acoust. Soc. Am. 96, 3704-3714 (1994)] is unusual since for this type of mode the energy on the shell flows in the opposite direction of the wave vector. Circumnavigation of the shell is not required for the leaky ray to be backward directed.