·Table of Contents ·Methods and Instrumentation | Suppressing of Antisymmetric Mode using Superposition of Lamb Waves Generated by two Laser Beams in a Thin PlateSeung Seok Lee, Sang Whoe Dho, Young Gil Kim, Bong Young Ahn, Se-kyoung LeeNondestructive Testing Group, Korea Research Institute of Standards and Science, P.O.BOX 102, Yusong, Taejon 305-600, Korea Contact |
Lamb waves[1,2,4-8] are the elastic wave motions that can propagate in a solid such as thin plates, and have two basic mode types, symmetrical and antisymmetrical, according to whether the displacements on the two surfaces are in phase or anti-phase. The lowest symmetric( s_{o} ) Lamb wave(LSL wave) propagates with only a little dispersion in low frequency region. When the thickness[1,2,5] of a plate increases and becomes comparable to the ongitudinal and shear wavelengths, the higher order modes begin to propagate.
We commonly use the LSL wave to measure the elastic properties of a plate or to detect defects in a thin plate. Specially, when the LSL wave is used to measure the reflection from sides or defects, the superposition[3] of the reflection waves from edges or defects on antisymmetric Lamb waves makes a serious difficulty in data analysis. So, a technique to suppress undesired antisymmetric Lamb waves is needed.
Degertekin and Khuri-Yakub[5] presented a technique to excite selectively the lowest order symmetric and antisymmetric Lamb wave modes using Hertzian contacts in thin solid plates.
We presented a technique[6] to generate single-mode Lamb wave by the symmetric laser beam technique without special contacts[5] in a thin plate. In the method two symmetric laser beams quasi-simultaneously hit at the same points of both sides of the plate. But, in real applications, we often face up to an accessible situation on only one side of a thin plate.
In this study, we present a suppressing technique of antisymmetric modes by superposition of Lamb waves generated by two laser beams on one side of a thin plate. Two Lamb waves of the same frequencies propagating from the opposite direction simultaneously arrive at the point of measurement and are superposed to compose one Lamb wave. The amplitude of the superposed Lamb wave depends on the distance between two laser beams. The suppressing of the antisymmetric Lamb wave mode is accomplished by selecting the distance between two beams simultaneously satisfies the condition of the anti-node(maximum)[3] for the symmetric mode and the node(minimum) for antisymmetric mode.
Fig 1: Coordinates system used in the experiment |
Consider the situation of Fig. 1 as the propagation of Lamb waves generated by laser-ultrasonic on the plane of the plate. The Z-axis is the direction of thickness of the thin plate. Solutions are required for the wave equation[1,2]
(1) subject to the boundary condition |
(2) where f is the displacement |
The wave solutions may be found by the method of separation of variables for x, y, z in the 3-dimensional solution. From the wave propagation theory[1] in a thin plate, we can easily predict the form of the wave solution as the linear combination of symmetric and antisymmetric modes[1,2].
(3) |
This solution is induced by the laser beam I. The laser beam II also make the same Lamb wave.
The total wave solution by the laser beam I and II at the measurement position on X-axis is the superposition[3] of f_{I}(z,y,x) and f_{II}(z,x,y) . To treat superposition effect of Lamb waves of the same frequency propagating in the opposite direction, we have a concern for Y-axis.
(4) |
The solution of Eq. 4 represents two travelling waves in the ± direction and will be written in terms of exp(ik_{y}y).
Lamb wave f_{I}(y) generated by the laser beam I propagates from the position of y to the direction of -Y axis. Lamb wave f_{II}(y) generated by the laser beam II propagates from the position of -y to the direction of +Y axis. The superposition of f_{I}(y) and f_{II}(y) on the X axis is treated as
where the time term is disregarded for simplicity and A_{i}^{j} is the amplitude of each mode and d_{i} is the initial phase of each mode. Hence Eq. 6 becomes
where initial phases are ignored in the calculation. The cosine term is the interference by the superposition[3] of two Lamb waves. The exponential term implies a traveling component and disappears if A_{II}^{j}=A_{I}^{j}.
Figure 2 shows a schematic diagram of the experimental setup for the suppression of antisymmetric Lamb wave mode using the superposition of Lamb waves generated by two laser beams.
Fig 2: Experimental setup for the suppression of the antisymmetric Lamb wave mode using the superposition of Lamb waves generated by two laser beams |
The waveform of the Lamb wave is detected by the Electro-Magnetic Acoustic Transducer(EMAT) fixed at a position on X-axis as seen in Fig. 1. The waveform averaged 30 pulses is captured on a Lecroy 9310 digital oscilloscope and is stored for subsequent analysis on a notebook computer by GPIB interface.
Figure 3 shows the interference effect of the symmetric Lamb wave and the antisymmetric Lamb wave by the superposition of Lamb waves generated by two laser beams. The interval between each adjacent node of the antisymmetric mode is about eight times bigger than that of the symmetric mode. We use Eq. 6 for fitting of the experimental data. Figure 4 shows the amplitude ratio of the symmetric Lamb wave and the antisymmetric Lamb wave (the first up-peak, P in Fig. 5) versus the distance between two laser beams. When the distance between two laser beams is about 20 mm(about 4 wavelengths for the lowest symmetric Lamb wave mode and about half wavelength for the lowest antisymmetric Lamb wave mode), the ratio of amplitudes of symmetric and antisymmetric mode is the minimum value.
Fig 3: Interference effect of the symmetric Lamb wave(a) and the antisymmetric Lamb wave(b) by the superposition of Lamb waves generated by two laser beams |
Fig 4: Amplitude ratio of the symmetric Lamb wave and the antisymmetric Lamb wave |
The distance is the anti-nodal(maximum)[3] position for the symmetric Lamb wave mode and the minimum position for the antisymmetric Lamb wave mode. In this distance, the antisymmetric Lamb wave mode is suppressed to the degree of 1.4% of the amplitude measured at zero distance between two beams.
Figure 5 shows several Lamb waveforms received by the EMAT when the distance between two beams is changed from zero to 20 mm in Y-direction.
The symmetric mode( s_{o}) and the antisymmetric mode(a_{o}) are seen in the wavefroms of top and middle position of the figure. The LSL wave( s_{o}) and reflection waves from edges are only observed at 20 mm by the suppression of the antisymmetric Lamb wave mode.
Fig 5: Several Lamb waveforms received by the EMAT when the distance between two beams is moved from zero to 20 mm in Y-direction |
In summary, We present a suppressing technique of the antisymmetric mode by the superposition of Lamb waves generated by two laser beams in a thin plate. Two Lamb waves of the same frequency propagating in the opposite direction simultaneously arrive at the point of measurement and are superposed to compose one Lamb wave. The amplitude of the superposed Lamb wave depends on the distance between two laser beams. The suppressing of antisymmetric Lamb wave mode is accomplished by selecting the distance between two beams simultaneously satisfies the condition of the anti-node(maximum) for the symmetric mode and the minimum for the antisymmetric mode. By the method, the antisymmetric Lamb wave mode is suppressed to the degree of 1.4% of the amplitude measured at the zero distance between two beams.
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