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Using periodic permanent magnet (PPM) electromagnetic acoustic transducers (EMATs), different shear horizontal (SH) guided wave modes can form simultaneously in some situations, which can interfere with the inspection. The main cause of this phenomenon (typically named multiple modes) is related to the frequency bandwidth of excitation signals and the transducer spatial bandwidth. Simply narrowing the frequency bandwidth cannot effectively limit the number of different SH modes. Previous researches showed that unnecessary SH wave modes can be eliminated by using dual EMATs. However, in practical applications, it is more convenient to change the excitation frequency than to use dual EMATs. In this paper, the stress boundary conditions of the PPM-EMAT are analyzed, the analytical expression of SH guided wave is established, and the magnitude of SH guided wave mode under continuous tone and tone-burst input is obtained. A method to generate a single SH mode by re-selecting an operating point is proposed. Furthermore, the influence of the frequency bandwidth of the tone-burst signal is analyzed. Finally, a single SH mode excitation is achieved with tone-burst input.
Ultrasonic guided waves have been widely studied and applied to nondestructive testing because of their low attenuation and large volume coverage. Shear horizontal (SH) guided waves have simple dispersion characteristics. Being nondispersive and having the particle motion in the plane of the plate, the SH0 mode is sensitive to defects that are either perpendicular or parallel to the wave vector.[1–3] These good characteristics enable SH guided waves to be applied to increasing applications such as crack detection,[2–4] corrosion detection,[5,6] weld detection,[7–9] and the evaluation of interfacial adhesion.[10]
SH guided waves can be generated by piezoelectric transducers, but piezoelectric transducers need to be specially designed and attached to plates by a very viscous or solid couplant,[11] making it difficult to apply to online detection. Therefore, SH guided waves are commonly generated by electromagnetic acoustic transducers (EMATs) which are simple in structure and need no couplant.[12]
According to the dispersion characteristics of SH guided waves, the frequency-thickness product is negatively correlated with the wavelength-thickness ratio. If the frequency-thickness product is very low or the wavelength-thickness ratio is very large, only the SH0 mode can be generated. However, in some applications, there are the cases of high frequency-thickness product or low wavelength-thickness ratio, such as (i) high frequency in order to improve accuracy;[13] (ii) the thickness of samples is large but the wavelength of SH guided waves generated by EMATs cannot be too large.[14] Once the excitation frequency exceeds the cut-off frequency of a certain mode, multiple modes occur.[15] The appearance of the multi-mode phenomenon will lead to some adverse phenomena: i) the superposition of multiple modes will weaken the vibration at some positions, where defects may not be able to be detected;[15] ii) multiple SH modes will be superimposed when the detection range is small. The waveform is difficult to analyze in this circumstance.
The multi-modes phenomenon is caused by the frequency bandwidth of exciting signals and the transducer spatial bandwidth.[16] Simply narrowing the frequency bandwidth of exciting signals cannot effectively reduce the production of multiple modes, meaning that the influence of transducer spatial bandwidth cannot be neglected, because the continuous sine tone (namely, a single frequency) can still excite multiple modes of SH guided waves.[15] However, in previous studies, the effect of spatial bandwidth on multi-modes was not evident.
A single and pure SH guided wave mode can be effectively generated by dual EMATs.[17,18] However, the dual EMATs cannot work well in practice and there are few other efficient ways to generate a single SH mode.
According to the expression of SH modes’ amplitude coefficients, which relates to the excitation frequency and the EMAT’s geometrical parameters, in this paper we study the method of suppressing unwanted modes by selecting an appropriate operating point and the frequency bandwidth of the excitation signal. Theoretical analyses and experimental results prove that a single SH mode can be obtained.
SH guided waves are usually generated and detected by EMATs chiefly via two kinds of physical mechanisms: the Lorentz force mechanism when the material is conductive and the magnetostriction mechanism when the material is ferromagnetic.[12] Only the Lorentz force mechanism is considered in this paper.
Periodic permanent magnet (PPM) EMATs are generally used to generate SH guided waves. The structure of a PPM EMAT is shown in Fig.
Figure
For the SH modes, the particle displacement vectors have only a y component. That is, the particle displacement in the x direction is zero, and so is it in the z direction. The particle displacement in the y direction is uy(x,z).[1] The traction component τzy is given by[1]
Since the skin depth of eddy current is far less than the acoustic wavelength at the same frequency and the thickness of sample, the Lorentz force distributed in the skin depth can approximately be regarded as the surface stress at z = h (h = d/2).[19–21] Due to the surface stress, the upper surface of the plate is not a free boundary, while the lower surface is still a free one. Assuming that the distribution function of the upper surface stress is g(x), the boundary conditions at the upper and lower surfaces of the plate are
In the classical Lorentz force model of EMATs, within the transducer area the magnetic field and the eddy current are thought to be constant and uniformly distributed.[11,19–21] And outside the transducer area, the magnetic field and the eddy current are thought to be zero. Therefore, the distribution function g(x) can approximately be regarded as a function shown in Fig.
In Ref. [1], the stress boundary conditions of piezoelectric transducers are
The displacement equation of SH guided wave is
![](cpb_29_5_054303/cpb_29_5_054303_ieqn1.gif)
![](cpb_29_5_054303/cpb_29_5_054303_ieqn2.gif)
![](cpb_29_5_054303/cpb_29_5_054303_ieqn3.gif)
Comparing Eqs. (
The dispersion equation of SH guided wave is
Equation (
It can be concluded from Eq. (
Equation (
![](cpb_29_5_054303/cpb_29_5_054303_ieqn4.gif)
The analytical expression of SH guided wave described above is based on a single frequency signal expressed as a single frequency continuous sine tone in practice. However, in practical applications, transient excitation signals are generally used and the commonly used transient excitation signal is the tone-burst signal.
Assuming that the function of the tone-burst signal is f(t) and its Fourier transform is F(ω). The magnitude of the amplitude coefficients is
In practical applications, the parameters of detecting transducer are always the same as those of transmitting transducer. By taking this into account, the magnitude of the amplitude coefficient of each SH guided wave mode received by the detecting transducer is
In this paper, to make the distinction, Φn is referred to as the single-frequency magnitude, Φnw as the transient magnitude, and Φnwr as the received magnitude.
It can be concluded from Eqs. (
![]() | Table 1.
Parameters of PPM-EMAT and plate used in this paper. . |
In general, the operating point is chosen according to Eq. (
![]() | Fig. 5. Normalized maximum displacement at each point in plate at input frequency of (a) 0.52 MHz and (b) 0.465 MHz. |
It can be seen from Fig.
Therefore, the excitation frequency is changed into 0.465 MHz. It is calculated that the single-frequency magnitude of the SH1 mode is only 2.8% of that of the SH0 mode. Similarly, the maximum displacement of each position in the plate is calculated, and the normalized maximum displacement at each point is shown in Fig.
In practice, the excitation signal is always the tone-burst signal. When the transient excitation signal is an eight-cycle sinusoidal signal with a center frequency of 0.465 MHz modulated by the Hanning window, the received magnitude curves of SH modes are shown in Fig.
It can be seen from Fig.
The relationship between the excitation function and the frequency-thickness product obtained by solving Eq. (
![]() | Fig. 7. Displacements versus frequency-thickness for (a) excitation function and (b) transducer spatial function. |
The relationship between the transducer spatial function and frequency-thickness product is obtained by solving Eq. (
It can be seen from Fig.
The reason that two SH1 modes are generated at the same time is that the excitation signal has a wide frequency bandwidth. Therefore, reducing the frequency bandwidth of the excitation signal is an effective way to implement the mode suppression. As can be seen from Fig.
The new excitation signal is a 25-cycle sinusoidal tone with a center frequency of 0.465 MHz modulated by the Hanning window. The frequency bandwidth of the new excitation signal is about 0.465 MHz ± 0.04 MHz. At this time, the relationship between the received magnitude and frequency is shown in Fig.
Figure
According to the theoretical analysis, the received signal should include an SH0 mode with a central frequency of 0.476 MHz, an SH1 mode of 0.418 MHz, and an SH1 mode of 0.525 MHz. According to the dispersion curves of SH guided wave, the group velocities of the three wave modes were 3114 m/s, 2078 m/s, and 2507 m/s, respectively. The theoretical arrival times of the three wave modes were about 128 μs, 192 μs, and 159 μs, respectively.
The actually received signal and its short-time Fourier transform (STFT) spectrogram are shown in Fig.
![]() | Fig. 11. (a) The receiving signal of the detector and (b) its STFT spectrogram when excitation signal has 8 cycles. |
The cycle number of the excitation signal was modified to be 25. The frequencies of excitation signals were 0.52 MHz, 0.44 MHz, and 0.465 MHz. The resulting waveforms are shown in Fig.
![]() | Fig. 12. Received signal of detector when excitation signal has 25 cycles at a frequency of (a) 0.52 MHz, (b) 0.44 MHz, (c) 0.465 MHz. |
The experimental results show that (I) both the main lobe and side lobes of the transducer spatial function can generate guided waves; (II) when the frequency bandwidth of the excitation signal is wide, the magnitude of SH guided wave generated by left-side lobes is large, which will aggravate the multi-mode phenomenon; (III) the unwanted SH modes cannot be suppressed at too high or too low a frequency and thus can be suppressed only by selecting an appropriate operating point and the frequency bandwidth of the excitation signal.
We have analyzed the stress boundary conditions of PPM-EMATs, established an analytical expression of SH guided wave generated by EMAT, and obtained the magnitude of SH guided wave modes under continuous tone and tone-burst input. It is proved that the magnitude of each SH mode is related to not only the excitation frequency, but also the parameters of the PPM EMATs and the thickness of plates. The analytical expression provides a theoretical basis for mode suppression.
Taking a specific transducer for example, in order to obtain a single SH0 mode and suppress other SH modes, the relationship between the single-frequency magnitude and frequency of each SH mode is investigated. If the excitation frequency of the SH0 mode is selected according to the traditional method, the SH1 mode is generated at the same time. However, by changing the operating point to a point where the magnitude of the SH1 mode is almost zero, the excitation of a single SH0 mode can be achieved.
The influence of the frequency bandwidth is analyzed here in this work. When the excitation signal is a tone-burst signal with a wide frequency bandwidth, in addition to the SH0 mode, there are also two SH1 modes with different center frequencies, because the SH guided waves generated by the left-side lobes and the main lobe coexist due to similar magnitude. This is a special phenomenon of SH guided wave.
Finally, the frequency bandwidth of the excitation signal is controlled. The two SH1 modes are effectively suppressed and the excitation of a single SH0 mode is achieved, which not only proves that the theoretical analysis is correct, but also confirms that a single SH guided wave mode can be generated by selecting an appropriate operating point and a frequency bandwidth of the excitation signal.
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[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] |