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·Methods and Instrumentation
Lamb Wave Analysis of Acousto-Ultrasonic Signals in PlateLIU Zhenqing
Institute of Acoustics, Tongji University Shanghai 200092, P.R.China
In the acousto-ultrasonic technique, a broadband ultrasonic wave is injected onto the surface at one location of the tested plate with the help of a longitudinal transducer and a receiving transducer is coupled to the same surface at another location. Thus, understanding the propagation characteristics of the ultrasonic wave is essential for successful application of the technique. In this paper, it has been found that the dominant acousto-ultrasonic waves produced experimentally in thin plate were multi-mode Lamb waves. The broadband generation and detection of Lamb waves has been performed in aluminum plate and laminatd composite plate using the acousto-ultrasonic method.
Acousto-Ultrasonics is a new method for non-destructive testing and evaluation of the plates, which was proposed by American scientists at the end of the 1970's[1-2]. The term of Acousto-Ultrasonics may be taken as a contraction of "acoustic emission simulation with ultrasonic sources." In acousto-ultrasonic approach, the sender and receiver probes are usually coupled to the same surface of the tested object. This satisfies the need of testing on one side in many cases. In order to excite the stress waves, which propagate parallel to the surface of the tested plate, the sender is coupled directly to a surface with fluid (grease). These stress waves are taken as the simulation of acoustic emission. But in this technique loading is not needed which is necessary in the acoustic emission detect. The transducer receives these stress waves at certain distance from the transmitting transducer. So the acousto-ultrasonic method is also an ultrasonic testing in nature.
In acoustio-ultrasonics, a stress wave factor (SWF) is used to characterize acousto-ultrasonic waves. The SWF is based directly on acoustic emission practice such as energy, peak voltage or ringdown. It indicates the energy of the received signal in the given time domain or frequency domain. The modes of the ultrasonic waves in the plate are complicated because of the arrangement of the transducers in the acousto-ultrasonic approach. It may be thought that, in the tested plates, there are lamb waves and the waves, which are reflected many times between the two surfaces of the plate. But the SWF often used disregards the nature of the ultrasonic propagation. This greatly limits the utilization of the acoustio-ultrasonic technique. We have studied the ultrasonic wave propagation properties in the thick (relative to the wavelength of the ultrasonic wave) plates[3-4]. It was found that the resonance mode that resulted from multiple reflections of the longitudinal waves between the top and bottom surfaces was appropriate for this analysis. This paper analyzed experimentally the ultrasonic wave propagation under the acousto-ultrasonic mode when the tested plates are relatively thin. The work helps to understand the wave propagation when the acousto-ultrasonic method is used to test the thin plates. It is also helpful to use correctly this new ultrasonic non-destructive testing technique
In the acousto-ultrasonic method, both the sender and the receiver are the longitudinal transducers. When the receiver is close enough to the sender, the acousto-ultrasonic approach turns to the traditional pulse-echo ultrasonic testing method. When the tested plates are thick (compared with the wavelength of ultrasonic wave), and the spacing between sending and receiving transducers are greater than the plate thickness, we can explain the main components of the received waves using resonance of ultrasonic waves[3-4]. When the tested plates are thin, the guided waves, namely Lamb waves, may be generated in the plates. At this time the dominant components of the received waves may be Lamb waves.
Because of the dispersion, the phase velocities and group velocities of the Lamb waves vary with the change of frequency, and their phase velocities are different from the group velocities. In an infinite solid plate of thickness d, the dispersion equations can be expressed as:
where,, is the wave-number of Lamb wave. Cl and Ct are the velocities of longitudinal wave and transverse wave in the plate respectively. Cp is the phase velocity of Lamb wave in the plate. w is the angular frequency, and w = 2pf ; f is the frequency. The equation (1a) is the Lamb characteristic equation for the propagation of symmetric waves in a plate, and the equation (1b) is for antisymmetric Lamb waves. Equation (1) elucidates the relationship between the Lamb wave phase velocities Cp and the frequency-thickness product fd. Numerical computations based on the Lamb wave dispersion equation (1) will give the Lamb wave dispersion curses, which graph the relationship between the phase velocity Cp and frequency-thickness product fd. Fig.1 shows the Lamb wave phase dispersion curves in aluminum plate. a0 is the zero-th order of the antisymmetric Lamb wave, s0 is the zero-th order of the symmetric Lamb wave, a1 is the first-order of the antisymmetric Lamb wave,...o The parameters of the aluminum plate are:
Cl =6400m/s , Ct =3170m/s o
The relationship between the group Cg velocity and the phase velocity Cp is: Cg = Cp - lp(¶Cp /¶l p),
Then we have the expression of Cg as follows:
Where, x = f · d. The meanings of f and d are the same as the above. We will obtain the group velocity dispersion curves of Lamb waves by calculating the equation (2). Fig.2 shows group velocity dispersion curves of Lamb wave in the aluminum plate.
|Fig 1: Lamb wave phase velocity dispersion cures for aluminum plate||Fig 2: Lamb wave group velocity dispersion cures for aluminum plate|
We can see that in Fig.1 all Lamb waves have a cut-off frequency-thickness product fd with the exceptions of a0 and s0 modes. Then when the ultrasound frequency of excitation is very low or the plate is very thin, there are only two Lamb modes can be excited in the plate, namely a0 and s0 mode. According to the common methods of Lamb wave excitation, the longitudinal transducers should be fixed in a given angle at a given frequency to excite the Lamb wave. If the longitudinal velocity of the media between the sender and the tested plate is Co, the incidence angle a required for the excitation of the desired mode was calculated from :
|Fig 3: Lamb wave dispersion curves plotted as a function of incidence angle|
The Lamb wave dispersion curves plotted as a function of incidence angle can be attained from the equation (3). Fig.3 demonstrates these curves. Here Co used to calculate the angle of incidencea is equal to 5000m/s. From Fig.3, We can see that a great incidence angle a is required to excite the s0 mode, and that the a0 mode can not be excited at all. But this conclusion is drawn from the ideal conditions that the direction of the transducer is very perfect and the ultrasonic wave is only incident in only one angle. As well known, the general pulse longitudinal transducers emit pulse ultrasonic waves which include direct longitudinal waves, edge longitudinal waves, edge transverse waves, head waves and surface waves and so on. Since Lamb waves are generated by the combination of the multiple reflection of the longitudinal waves and transverse waves between the top and bottom boundaries of the plate, it is possible to excite certain kinds of the Lamb wave modes (including a0 mode) in acousto-ultrasonics.
The arrangement used in the experimental investigation is shown schematically in Fig.4. The apparatus used to excite, receive and amplify ultrasonic waves was Panametrics-5052UA Ultrasonic Wave Analyzer. The senders and the receivers produced by Panametrics corporation were the broadband longitudinal transducers. Using single pulse excited the senders to generate the narrow pulse ultrasonic waves. The tested specimens were aluminum plates. The sending and receiving transducers were directly coupled to one surface of the plate with a thin film of grease. Enough pressure had to be applied to eliminate unwanted reverberations within the couplant. The received signals were amplified and then sent to HP-54600 Digital Oscilloscope for A/D conversion and display. The signals from the oscilloscope were sent to the computer via IEEE488 bus to be edited and analyzed.
|Fig 4: Schematic diagram of experimental setup||Fig 5: the acousto-ultrasonic waveform in the aluminum plate of thickness 0.95mm. The central frequency of the sender is 1.0Mhz, and the central frequency of the receiver is 0.5Mhz, and the distance between the two transducers is 20cm.|
Experimental steps and results
The first set of tests was carried out to test an aluminum plate of thickness 0.95mm. The sender used in this experiment was a broadband transducer of 38.1mm diameter, operating at a nominal central frequency of 1.0Mhz. The corresponding parameters of the receiver were 0.5Mhz and 38.1mm respectively. Distance between two probes was 20cm.
The received waveform is shown in Fig.5. The amplitude in the Fig.5 has been normalized. According to Fig.2, there were only a0 and s0 modes in 0.95mm thick plate when the incidence frequency was less than 1.8Mhz. We could distinguish the arrivals of two modes, a0 and s0, distinctly in Fig.5. Then we decreased the distance between two transducers by 1cm and 2cm respectively, and received the other two waveforms. Comparing the latter two waveforms with the waveform in Fig.5 respectively, we gained the velocities of the received waves using phase unwrapping algorithm. Therefore we obtained the velocity dispersion curves of a0 and s0 Lamb wave modes experimentally. Fig.6 and Fig.7 show these results. The former represented the a0 mode phase velocity dispersion curves and the latter showed the s0 mode phase velocity dispersion curves. The lines in these two Figures are the results of decreasing the distance between two probes by 1 mm. and the dash lines are the results of decreasing the distance by 2 cm.
|Fig 6: a0 mode phase elocity dispersion curves derived from experiments||Fig 7: s0 mode phase velocity dispersion curves derived from experiments|
In most of the acousto-ultrasonic investigations, the distances between the senders and the receivers were set about 10cm. therefore, we set the distance 7.5cm. And the received waveform is shown in Fig.8. We could also distinguish the comparatively pure a0 and s0 modes. So the distances between two transducers do not influence the exciting and receiving of the ultrasonic wave modes greatly. But in Fig.8, we can see that the two modes are very close to each other in time domain, so they overlap partly. Because of this reasons discussed, we set the distance between the sending and receiving transducers approximate 20cm in all our experiments. The conclusions drawn from our experiments also apply to the acousto-ultrasonic testing of the close distance between two transducers.
|Fig 8: The acousto-ultrasonic waveform in the aluminum plate of thickness 0.95mm. The central frequency of the sender was 1.0Mhz, and the central frequency of the receiver is 0.5Mhz, and the distance between the two transducers is 7.5cm.|
|Fig 9: the acousto-ultrasonic waveform in the aluminum plate of thickness 0.95mm.. The central frequency of the sender is 5.0Mhz, and the central frequency of the receiver is 3.5Mhz, and the distance between the two transducers is 19cm.|
Then we tested the same plate, but the central frequencies of the sender and receiver were greater than before. A 5.0Mhz transducer of 25.4mm diameter was used here as the sender, and the corresponding parameters of the receiver were 3.5Mhz and 25.4mm. In this test the distance between two transducers was 19cm. The waveform received is shown in Fig.9. It is obvious that the signal contains the higher order Lamb waves besides a0 and s0 modes. Since these higher frequency signals overlapped s0 mode, the s0 mode was not so pure as that in Fig.5. As the same steps as above, we decreased the distance between two transducers by 1cm and 2cm respectively, and received other two waveforms. And then we gained the experimental velocities of a0 and s0 Lamb wave modes using phase unwrapping. The results are shown in Fig.10 and Fig.11 respectively.
|Fig 10: a0 mode phase elocity dispersion curves derived from experiments||Fig 11: s0 mode phase velocity dispersion curves derived from experiments|
Moreover, we tested laminatd composite plate of thickness 2.05mm. Two 0.5Mhz transducer of 38.1mm diameter was used here as the sender and the receiver. The waveform received is shown in Fig.12 when the distance between two transducers was 10.5cm. It was the s0 mode with comparatively low frequency that arrived the receiver earliest. The reason is that the group velocity of s0 mode is higher than other modes when the frequency is relatively low. We can also distinguish the arrival of a0 mode in Fig.12.
|Fig 12: the acousto-ultrasonic waveform in the laminatd composite plate of thickness 2.05mm. The central frequency of the sender and the receiver is 0.5Mhz, and the distance between the two transducers is 10.5cm.|
It is theoretically predicted that multiple Lamb wave modes can be excited in thin plates using acousto-ultrasonic approach. This prediction was verified by a series of experiments conducted on aluminum plate and laminatd composite plate. In all signals the s0 mode Lamb wave, whose group velocity is higher than others, can be distinguished easily from acousto-ultrasonic signals. And we have separated a0 and s0 modes under acousto-ultrasonic condition. The experiments are easily conducted since the two transducers can be directly coupled to the same surface of the plates. Because only the single surface testing is satisfied in most industrial process, and the high speed of testing is required, it is appealing to use the method discussed in this paper successfully.
The project is supported by Natural Science Foundation of China (grant No. 19604010) and Trans-Century Training Programme Foundation for the Talents by the State Education Commission
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