On the crossing points of the Lamb modes and the maxima and minima of displacements observed at the surface

This article elaborates on the crossing points of the frequency–wavenumber branches for the symmetric and anti-symmetric Lamb modes in a homogeneous plate. It is shown both theoretically as well as experimentally that at these crossing points either the normal or the longitudinal components of modal displacement attain an extreme value, i.e. a maximum or it vanishes. This behavior is assessed herein using a method due to Mindlin, who showed that the dispersion curves for a plate with mixed boundary conditions – which are associated with uncoupled shear and dilatational modes – provide bounds to the spectral lines of the free plate. Therefore, a subset of the crossing points of the symmetric and antisymmetric Lamb modes for a free plate coincide with the crossing points for a plate with mixed boundary conditions.     

Time-frequency analysis of the dispersion of Lamb modes.

Accurate knowledge of the velocity dispersion of Lamb modes is important for ultrasonic nondestructive evaluation methods used in detecting and locating flaws in thin plates and in determining their elastic stiffness coefficients. Lamb mode dispersion is also important in the acoustic emission technique for accurately triangulating the location of emissions in thin plates. In this research, the ability to characterize Lamb mode dispersion through a time-frequency analysis (the pseudo-Wigner-Ville distribution) was demonstrated. A major advantage of time-frequency methods is the ability to analyze acoustic signals containing multiple propagation modes, which overlap and superimpose in the time domain signal. By combining time-frequency analysis with a broadband acoustic excitation source, the dispersion of multiple Lamb modes over a wide frequency range can be determined from as little as a single measurement. In addition, the technique provides a direct measurement of the group velocity dispersion. The technique was first demonstrated in the analysis of a simulated waveform in an aluminum plate in which the Lamb mode dispersion was well known. Portions of the dispersion curves of the A0, A1, S0, and S2 Lamb modes were obtained from this one waveform. The technique was also applied for the analysis of experimental waveforms from a unidirectional graphite/epoxy composite plate. Measurements were made both along and perpendicular to the fiber direction. In this case, the signals contained only the lowest order symmetric and antisymmetric modes. A least squares fit of the results from several source to detector distances was used. Theoretical dispersion curves were calculated and are shown to be in good agreement with experimental results.     

Conventional electromagnetic acoustic transducer development for optimum Lamb wave modes

Lamb waves are normally utilized for inspecting thin metal sheets. Wheel type probes using piezoelectric oscillators have generally been used as the sensors for Lamb waves. Recently, the electromagnetic acoustic transducer (EMAT) has been developed and is beginning to be used as a Lamb wave detector. We have developed a useful type of transducer for Lamb waves. The new EMAT consists of a meander coil with a narrow distance of 2.5 mm and has a symmetrical structure in the vertical direction for both surface sides. The new EMAT can generate Lamb waves with variable wavelengths corresponding to the frequency range from approximately 300 kHz to 2.5 MHz and multiple modes, and can also generate selected symmetrical and anti-symmetrical mode Lamb waves. It is demonstrated that the optimum Lamb wave mode could be produced by the appropriate positioning of the EMATs and controlling the phase (same or inversed) of the electrical signal driving the device. The described EMAT can be used to examine steel (or other material) sheets of different thickness. It is also shown that the S0 (0.3 MHz) mode Lamb wave is the most effective for the deepest (up to 6 mm) penetration.     

Propagation of Lamb waves in pure longitudinal modes

I.IntroductionSinceLambwaveshavewideaPplicationinnondestructiveinspectionandsensors1interestsintheinvestigationofit8propagationaredeveloped.VictorovI1lhaspointedoutthatastohomogeneous,whformsolidplateinunersedillliquld,theverticaldisplacementoftheplate'ssurfaCemaybezerowhenthephasevelocityisequaltotheplatelongitudinalwavespeed.ThisconclusionhasimportantaPplication.RecelltlyPilaskietc.I2]aPproveditbyusing.thefree-boundaryLambwavtheoryandgavetheconditionstoreachit.ItisknownthatthedisaPpeara…     

纯纵体波模式Lamb波的研究

本文导出更为普遍的对称液层负载下板中Lamb波传播公式。并提出在有限层液体负载下,板中有可能激发出一种Lamb波的纯纵波模式。     

Flaw localization using the reassigned spectrogram on laser-generated and detected Lamb modes

Rapid, accurate inspection of metallic plates using broadband guided waves has traditionally been limited by the multi-mode, highly dispersive nature of these waves; current practice typically restricts either the type of mode generated or detected, and/or its frequency range. The current study presents an effective alternative procedure by combining a novel digital signal processing technique, the reassigned spectrogram, with laser generated and detected Lamb waves. The reassigned spectrogram is used to characterize the modal and frequency content of a single ultrasonic signal as a function of time, enabling a procedure to locate flaws in an aluminum plate specimen.     

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.     

Nondestructive evaluation of helicopter rotor blades using guided Lamb modes

This paper presents an application for turning and direct modes in a complex composite laminate structure. The propagation and interaction of turning modes and fundamental Lamb modes are investigated in the skin, spar and web sections of a helicopter rotor blade. Finite element models were used to understand the various mode conversions at geometric discontinuities such as web-spar joints. Experimental investigation was carried out with the help of air coupled ultrasonic transducers. The turning and direct modes were confirmed with the help of particle displacements and velocities. Experimental B-Scans were performed on damaged and undamaged samples for qualitative and quantitative assessment of the structure. A strong correlation between the numerical and experimental results was observed and reported.     

Numerical predictions and experiments for optimizing hidden corrosion detection in aircraft structures using Lamb modes

To increase the sensitivity of Lamb waves to hidden corrosion in aircraft structures, a preliminary step is to understand the phenomena governing this interaction. A hybrid model combining a finite element approach and a modal decomposition method is used to investigate the interaction of Lamb modes with corrosion pits. The finite element mesh is used to describe the region surrounding the corrosion pits while the modal decomposition method permits to determine the waves reflected and transmitted by the damaged area. Simulations make easier the interpretation of some parts of the measured waveform corresponding to superposition of waves diffracted by the corroded area. Numerical results permit to extract significant information from the transmitted waveform and thus to optimize the signal processing for the detection of corrosion at an early stage. Now, we are able to detect corrosion pits down to 80-μm depth distributed randomly on a square centimeter of an aluminum plate. Moreover, thickness variations present on aircraft structures can be discriminated from a slightly corroded area. Finally, using this experimental setup, aircraft structures have been tested.     

铌锌酸铅-钛酸铅薄层中Lamb波模式的交叉特性

以[001]_c和[011]_c极化的铌锌酸铅-钛酸铅晶体为研究对象, 利用子波理论对其无限大自由薄层中传播的Lamb波的色散及模式交叉特性进行了研究. 发现只有[001]_c极化的晶体中的对称与反对称模式Lamb波之间出现了多次交叉, 并且变化规律与铌镁酸铅-钛酸铅的情形相同. Lamb波的A₀和S₀模式的交叉是由准纵向剪切波慢度曲线的多值关系引起的, 此时其x₃方向的波数在一定范围内存在一对非纯虚数的复共轭根. 利用此结论推导出A₀和S₀模式交叉时弹性常数需要满足的条件, 为判断正交、四方对称性晶体中Lamb波的A₀和S₀模式是否交叉提供了一种直观、简便的方法.     

Piezoelectric transducer parameter selection for exciting a single mode from multiple modes of Lamb waves

Excitation and propagation of Lamb waves by using rectangular and circular piezoelectric transducers surface-bonded to an isotropic plate are investigated in this work. Analytical stain wave solutions are derived for the two transducer shapes, giving the responses of these transducers in Lamb wave fields. The analytical study is supported by a numerical simulation using the finite element method. Symmetric and antisymmetric components in the wave propagation responses are inspected in detail with respect to test parameters such as the transducer geometry, the length and the excitation frequency. By placing only one piezoelectric transducer on the top or the bottom surface of the plate and weakening the strength of one mode while enhancing the strength of the other modes to find the centre frequency, with which the peak wave amplitude ratio between the S0 and A0 modes is maximum, a single mode excitation from the multiple modes of the Lamb waves can be achieved approximately. Experimental data are presented to show the validity of the analyses. The results are used to optimize the Lamb wave detection system.     

Edge resonance and zero group velocity Lamb modes in a free elastic plate

The local resonances of a free isotropic elastic plate are investigated using laser ultrasonic techniques. Experimental results are interpreted in terms of zero group velocity Lamb modes and edge mode. At a distance from the edge larger than the plate thickness a sharp resonance is observed at the frequency where the group velocity of the first symmetrical Lamb mode vanishes. Close to the edge of the plate, the resonance due to the edge mode dominates. Both zero group velocity and edge resonances appear at the theoretically predicted frequencies. These frequencies do not vary with the distance from the edge of the plate and the transition between the two modes of vibration, at about the plate thickness, is abrupt. Using a laser excitation on the edge, the amplitude profile of the normal displacement at the edge resonance frequency was determined.     

A combined finite element and modal decomposition method to study the interaction of Lamb modes with micro-defects

This paper presents a combined finite element and modal decomposition method to study the interaction of Lamb waves with damaged area. The finite element mesh is used to describe the region around the defects. On the contrary to other hybrid models already developed, the interaction between Lamb waves and defects is computed in the temporal domain. Then, the modal decomposition method permits to determine the wave reflected and transmitted by the damaged area. Modal analysis allows also identifying the mode conversions induced by the defects. These numerical results agree with previous finite element results concerning the interaction of Lamb modes with a notch. Experiments, carried out with gauged defects on an aluminum plate, are also compared to numerical predictions to validate the simulation. Compared to classical techniques of simulation, this new method allows us to investigate the interaction of Lamb modes generated at high frequency-thickness product with micro-defects as corrosion pitting.     

The propagation in metal tubing of ultrasonic wave modes equivalent to Lamb waves

The generation of ultrasonic wave modes in thin-walled metal tubing has been investigated experimentally using piezoelectric ultrasonic probes. It is shown that the L(0, 1) mode, which is comparable to the A₀ Lamb wave mode in flat plate, can be generated with acceptable efficiency. The L(0, 2) mode (compare the S₀ Lamb wave mode) is generated rather less efficiently, while its greater group velocity tends to enhance the resolution problems so caused.The initial probe designs were inefficient in that a considerable amount of ultrasonic ‘noise’ was also present which could mask ultrasonic pulses of interest. This would be especially so for short range operation. An improved probe design reduced this background noise to a more acceptable level.The propagation of the L(0, 1) mode around a length of bent (U-form) tubing has also been investigated and it is shown that the ultrasonic pulse is both attenuated and lengthened, while structure has also been introduced. The theoretical implications of this are discussed. Finally a small amount of work is presented on the interaction of both modes with artificial defects.     

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.     

Separation of 3-D displacement components in the white light speckle method

Measurement of in-plane and out-of-plane displacement components using the white light speckle method is considered for three different cases. The first is applicable when in-plane components are small and stresses are directly related to the second differential of the out-of-plane component. The other cases involve components of the same order of magnitude but require recording of either one or two specklegrams. Evidence of experimental demonstration of each case is given.     

Existence of axisymmetric modes of thin cylindrical shells with axial displacement restraint

A necessary and sufficient condition is derived for the existence of the axisymmetric mode of cylindrical shells with a radial displacement having one half wave along the axis and axial displacement restrained at both ends. The condition is purely geometric and consists of an upper bound on the mean radius to thickness ratio for a given fixed value of the length to mean radius ratio, above which the mode with one half wave in radial displacement along the axis ceases to exist. The proof is based on enforcing two basic lemmas concerned with the simplicity of the eigenvalues of the shell and the uniform ordering of these eigenvalues with the number of nodes of their corresponding radial displacement eigenfunctions.     

Modeling of integrated Lamb waves generation systems using a coupled finite element-normal modes expansion method

As part of the research work on Smart Materials and Structures, the development of self-monitoring materials is an emerging issue. In the case of plate-shaped structures, Lamb waves can be used for their relevant properties: long-range propagation, sensitivity to internal flaws and whole-thickness interrogation. This concept requires the use of thin piezoelectric transducers integrated to the structure. Since it is of primary importance to be able to control the generated modes, a suitable modeling technique of this kind of system has been tested on different cases of practical interest. The model uses a coupled finite element-normal modes expansion approach, which allows one to consider either the case of bonded or embedded transducers. The results presented deal with examples of multi-element transducers integrated to composite materials. The influence of parameters such as the dimensions, positions and relative excitation delays of the transducers is studied.     

Linear spatial displacement of spectral components in the light reflected from multilayer structures

Based on an analogy with thin-film dispersion compensators used in femtosecond lasers, a linear dependence of the spatial displacement of reflected beams with different wavelengths is obtained at the output surface of a one-dimensional metal-dielectric multilayer at oblique incidence of a wave packet with λ₀ = 1.5 μm with a view to application in compact demultiplexers of optical communication systems.