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Center frequency shift in pipe inspection using magnetostrictive guided waves
Sensors and Actuators A 298 (2019) 111583
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Center frequency shift in pipe inspection using magnetostrictive guided waves Chaoyue Hu, Jiang Xu ∗ School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, PR China
a r t i c l e
i n f o
Article history: Received 20 April 2019 Received in revised form 15 August 2019 Accepted 2 September 2019 Available online 3 September 2019 Keywords: Center frequency shift Magnetostrictive effect Guided wave Sensor
a b s t r a c t Magnetostrictive guided wave technology is widely employed to inspect pipes. In general, it is assumed that the center frequency of the exciting signal is the same as that of the receiving signal; however, this assumption does not hold in practice. To elucidate this phenomenon, we investigate the reason behind the center frequency shift (CFS) based on the energy coupling mechanism of magnetostrictive guided waves; we construct a theoretical model to calculate the CFS of the receiving signal. The detection process for guided waves is separated into three stages: excitation, propagation, and reception. At the exciting stage, the width of the coil affects the alternating magnetic field distribution. The vibration generated by the alternating magnetic field with a certain width produces a superimposed strain, which results in the CFS of the generating wave. At the propagating stage, we consider that the center frequency of the guided wave remains unchanged because the propagation is a linear process. At the receiving stage, the coil length determines the size of the induction area and the magnetic induction at a point is determined by the strain around that point. The number of coil turns, the distance of two adjacent wires, and the differential during the process of the electromagnetic induction affect the frequency of the induction signal, which leads to the CFS of the receiving signal. The center frequency of the receiving signal decreases with increasing coil width. Experiments are performed to verify that the theoretical model and the experimental results are in good agreement with the theoretical results. The model developed in this study provides a reference for the design of magnetostrictive guided wave sensors. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Magnetostrictive guided wave technology is widely used to test pipes and cables for many years [1–7]. Compared with other guided wave technologies, this technology has the advantages of large liftoff distance, easy implementation, etc. [8,9]. Longitudinal mode and torsional mode guided waves are the most widely used in practice [9]. Solenoid coils are usually employed as a part of the magnetostrictive sensor and the coil width is usually half the wavelength of the guided wave at the exciting frequency [10]. When the static bias magnetic field is sufficiently strong, the magnetostrictive effect is almost linear [11–13]. In general, researchers consider that the center frequency of the exciting signal is the same as that of the receiving signal. However, in practical applications, we found that the center frequency of the exciting signal was different from that of the receiving signal. The center frequency shift (CFS) of the receiving signal plays little role in macro defect
∗ Corresponding author. E-mail address: [email protected] (J. Xu). https://doi.org/10.1016/j.sna.2019.111583 0924-4247/© 2019 Elsevier B.V. All rights reserved.
detection [2,14]. However, the requirements for frequency accuracy are high in some cases. Nonlinear acoustic parameters are used commonly for micro defect detection such as fatigues, fracture toughness, material degradation, and so on [15–19]. The method for calculating nonlinear acoustic parameters requires that the center frequency of the receiving signal be constant; otherwise, the calculation result is inaccurate. Magnetostrictive position sensor is a type of high-precision position sensor based on the time-of-flight measurement of the guided wave [20–23]. The CFS of the wave affects wave velocity, which results in large measurement errors. In addition, some applications, such as damage evaluation based on the change of the center frequency of the reflection wave [24,25], or thickness measurement by frequency-domain analysis [26], have high requirements of frequency accuracy. Many studies have focused on modelling the magnetostrictive effect. In the 1950s, William [27] developed a model based on magnetostrictive delay lines to calculate the output voltage under the assumption of a linear relationship between the elastic and magnetic variables. Sablik [28,29] proposed a model for a cylindrical structure, which was used to calculate the amplitude and dispersion of longitudinal mode guided waves under an axial bias
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magnetic field. Dapino [30] studied the strain model of a magnetostrictive transducer under an external magnetic field, combined with a quadratic moment rotation model for magnetostriction. In this model, the Jiles–Atherton mean field theory was incorporated for ferromagnetic hysteresis. Zhang [31] built a one-dimensional nonlinear magnetoelastic coupled constitutive model, which was used to calculate the magnetostrictive strain and magnetization curve of a material under bias magnetic field and pre-stress. Cai [32] presented the equivalent mechanical model of a giant magnetostrictive ultrasonic processing system. The vibration amplitude of the system could be calculated using this model if the frequency and amplitude of the exciting current were known. Most of these models were used to calculate the amplitude and dispersion of the waves. However, little research has been performed on calculating the center frequency of the receiving signal. In this paper, we investigate the reason behind the occurrence of CFS based on the energy coupling mechanism of magnetostrictive guided waves. A theoretical model to calculate the center frequency of the receiving signal is presented. Experiments are carried out on a steel pipe to verify the model. This model provides a method to correct the frequency, and it serves as a reference for the design of magnetostrictive guided wave sensors. 2. Theoretical model The size of a ferromagnetic material changes due to the external alternating magnetic field and a wave is generated in the ferromagnetic material, which is called the magnetostrictive effect [33,34]. The magnetic properties of the ferromagnetic material are changed when the wave propagates in the material, which is called the inverse magnetostrictive effect [35]. When the static bias magnetic field is far stronger than the alternating magnetic field, the effects can be described as [11–13] ε = /E H + dH
(1)
B = d∗ + H
(2)
where ε is the strain, is the stress, EH is Young’s modulus at a constant applied magnetic field, d is the magnetostriction coefficient, d* is the inverse magnetostriction coefficient, B is the magnetic induction, is the permeability at a constant stress, and H is the magnetic field. Guided wave testing based on the magnetostrictive effect can be separated into three stages: excitation, propagation, and reception. At the exciting stage, the current is coupled to the magnetic field and then the magnetic field is coupled to the strain. At the propagating stage, the center frequency of the wave remains unchanged because the propagation of the elastic wave is a linear process [36,37]. At the receiving stage, the stress is coupled to magnetic induction, which is coupled to the voltage. We focus on analyzing the CFS of the signal at the exciting and receiving stages.
where ω0 is the center frequency of the current and t is the time. According to the law of electromagnetic induction, the relationship between the current and the magnetic field can be expressed as [12] Ha = NI/WE
(4)
where Ha is the alternating magnetic field in the pipe surface below the exciting coil, which is induced by the exciting current, N is the number of exciting coil turns, and WE is the width of the exciting coil. The right side of Eq. (1) includes two parts: static stress and alternating magnetic field. The generation of a guided wave is only related to the alternating magnetic field. Therefore, combining Eq. (1), Eq. (3), and Eq. (4), the alternating magnetostrictive strain in the pipe surface below each turn of the exciting coil can be expressed as εMS−m (ω0 t) =
d · NI d·N = y(ω0 t) WE WE
where εMS-i is the alternating magnetostrictive strain in the pipe surface below the mth turn wire. We suppose that A is a point on the pipe surface and the distance between A and the 1st turn wire is z0 . Without considering the attenuation during the propagation of the wave, the contribution of the wave generated by the mth turn wire to point A can be expressed as [36] εm = εMS−m (ω0 t − k(ml − l + z0 ))
εm =
d·N y(ω0 t − k(ml − l + z0 )) WE
(7)
The wave at point A is a superimposed strain generated by the entire exciting coil, which can be expressed as εA =
N
d·N y(ω0 t − k(ml − l + z0 )) WE N
εm =
m=1
(8)
m=1
By incorporating Eq. (3), the amplitude spectrum of the wave at point A can be represented as
N−1
−jωkml/ω d · N 0 |εA (ω)| = × I(ω)e−jωkz0 /ω0 × e WE m=0
N−1 d · N −jωkml/ω 0 = × I(ω) × e WE
(9)
m=0
where |I(ω)| is the amplitude spectrum of I. According to the summation formula of the geometric progression, Eq. (9) can be written as
WE
The schematic of the exciting stage is shown in Fig. 1. A static bias magnetic field can prevent the frequency-doubled effect and improve energy coupling efficiency [38]. When an alternating current is applied to the exciting coil, an alternating magnetic field is generated in the pipe surface [39,40], and this leads to the generation of an alternating strain in the pipe surface because of the magnetostrictive effect [41–44]. The exciting current can be expressed as
k = ω0 /c
I = y(ω0 t)
WE = N · l
(3)
(6)
where k is the wavenumber, and l is the center distance of two adjacent wires of the exciting coil. By substituting Eq. (5), Eq. (6) can be expressed as
εA (ω) = d · N × I(ω) ×
2.1. Signal CFS at the exciting stage
(5)
1 − cos(ωNkl/ω0 ) 1 − cos(ωkl/ω0 )
(10)
The relationship between the wavenumber and the exciting frequency can be expressed as (11)
where c is the speed of the guided wave. The relationship between the coil turns and the center distance of two adjacent wires can be expressed as (12)
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
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Fig. 1. Schematic of the exciting stage.
Combining Eq. (11) and Eq. (12), Eq. (10) can be written as
εA (ω) = d · N × I(ω) ×
WE
1 − cos(ωWE /c) 1 − cos(ωl/c)
(13)
In Eq. (13), the frequency corresponding to the highest peak of the amplitude spectrum is the center frequency of the generated wave, which can be expressed as
ωc−ε = arg max{ εA (ω) }
(14)
ω
where ωc-ε is the center frequency of the generated wave. Therefore, the wave generated in the pipe is a superimposed strain, which causes the CFS of the wave. 2.2. Signal CFS at the receiving stage The schematic diagram of the receiving stage is shown in Fig. 2. Due to the inverse magnetostrictive effect, an alternating magnetic induction is generated in the pipe surface. Then, an induced voltage is generated in the receiving coil because of the electromagnetic induction effect. According to elastic mechanics, stress is proportional to strain. The right side of Eq. (2) has two parts: alternating stress and static magnetic field. The receiving wave is only related to the alternating stress. When the receiving coil is a single-turn coil, Eq. (2) can be written as BIMS = d∗ R = d∗ E H εR
(15)
where εR is the alternating strain in the pipe surface below the single-turn coil, which originates from the exciting region, and its amplitude spectrum is the same as |εA (ω)|; R is the stress corresponding to εR ; and BIMS is the magnetic induction in the pipe surface below the single-turn coil and it is engendered by εR due to the inverse magnetostrictive effect. There exists a region with a certain width in the pipe surface below the single-turn coil, for which the strain at every point has an influence on the magnetic induction in the pipe surface below the coil. Therefore, the magnetic induction in the pipe surface below the of the magnetic induction in single-turn coil is the superimposition √ this region. The region width is 2/2 times the half wavelength of the guided wave according to the root mean square value of the sinusoidal signal. Similar to Eq. (13), we can write
BIMS−S (ω) = BIMS (ω) ×
1 − cos(ω ·
√ 2 /c) 4
1 − cos(ωd/c)
(16)
where BIMS-S is the superimposed magnetic induction in the pipe surface below the coil, which is engendered by the strain in the
aforementioned region; |BIMS (ω)| is the amplitude spectrum of BIMS ; is the wavelength of the guided wave; and d is the center distance of two adjacent wires of the receiving coil. By substituting Eq. (15), Eq. (16) can be written as
BIMS−S (ω) = d∗ EH × εR (ω) ×
1 − cos(ω ·
√ 2 /c) 4
1 − cos(ωd/c)
(17)
Now, the magnetic induction is coupled to the voltage according to the law of electromagnetic induction. When the receiving coil is a single-turn coil, the induced voltage in the coil is given by [12] 2
V1 = − (r + h) · ∂BIMS−S /∂t
(18)
where V1 is the induced voltage in the single-turn coil. Eq. (18) implies that there exists a time-domain differential during the coupling from magnetic induction to voltage. According to Fourier transform, time-domain differential means that the spectrum is multiplied by the angular frequency and an imaginary number. The amplitude spectrum of the induced voltage in the single-turn coil can be expressed as
V1 (ω) = − (r + h)2 · BIMS−S (ω) · jω = − (r + h)2 · BIMS−S (ω) · ω
(19)
When the receiving coil is a multi-turn coil, the receiving signal is a superimposition of the voltage in each turn wire. Similar to Eq. (13), the amplitude spectrum of the receiving signal can be expressed as
VM (ω) = V1 (ω) ×
1 − cos(ωWR /c) 1 − cos(ωd/c)
(20)
where |VM (ω)| is the amplitude spectrum of the receiving signal, M is the number of the receiving coil turns, and WR is the width of the receiving coil. Combining Eq. (13), Eq. (17), Eq. (19), and Eq. (20), the center frequency of the receiving signal can be expressed as
⎧ 1 − cos ωWE /c) ⎪ ⎪ VM (ω) = p × I(ω) × ω × ⎪ ⎪ 1 − cos(ωl/c) ⎪ ⎪
⎪ √ ⎪ ⎪ ⎪ 1 − cos ω 2 /c) ⎪ ⎪ 1 − cos(ωWR /c) ⎨ 4 ×
1 − cos(ωd/c)
×
⎪ ⎪ ⎪ 2 ⎪ − (r + h) d∗ dE H N ⎪ ⎪ p= ⎪ ⎪ W N ⎪ ⎪ ⎪ ⎪ ⎩ ωc−r = arg max{ VM (ω) } ω
1 − cos(ωd/c)
(21)
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Fig. 2. Schematic diagram of the receiving stage.
Fig. 3. Relationships between the four parameters and the CFS.
where ωc-r is the center frequency of the receiving signal, and p is a constant that has no influence on the center frequency. If the exciting signal and the parameters of the coils are known, the center frequency of the receiving signal can be calculated using Eq. (21).
2.3. Discussions and analyses Eq. (21) indicates that the CFS of the receiving signal occurs because of four reasons: 1) the superposition of the strain during the coupling from the magnetic field to the strain decreases the
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
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Fig. 4. Dispersion curves of the sample pipe.
Fig. 5. Schematic diagram and photograph of the experimental setup.
center frequency of the generated wave; 2) the superposition of the magnetic induction during the coupling from the strain to the magnetic induction decreases the center frequency of the magnetic induction; 3) the superposition of the voltage in each turn wire of the receiving coil decreases the center frequency of the receiving signal; 4) the differential during the coupling from magnetic
induction to voltage increases the center frequency of the induced voltage. From Eq. (21), when the exciting frequency is fixed, the CFS of the receiving signal is determined by four parameters: WE , WR , l and d. WE determines the width of the exciting region, which affects the superposition of the strain, and WR determines the width of
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Table 1 Width and turn of the coils (20 kHz, =256.0mm).
Table 2 Width and turn of the coils (160 kHz, =32.7mm).
No.
Width(mm)
Width()
Turns
No.
Width(mm)
Width()
Turns
1 2 3 4 5 6
32.0 64.0 96.0 128.0 160.0 192.0
0.125 0.250 0.375 0.500 0.625 0.750
29 57 85 113 143 171
1 2 3 4 5 6
4.4 8.8 13.2 16.5 21.2 24.8
0.125 0.250 0.375 0.500 0.625 0.750
4 8 12 16 20 24
the receiving region, which affects the superposition of the magnetic induction and voltage. When the coil width is fixed, l and d determine the coil turns. Considering the 20 kHz and 160 kHz exciting frequencies as the examples, the relationships between the center frequency of the receiving signal and these four parameters are shown in Fig. 3. Figs. 3(a) and (b) indicate that, when l and d are fixed at 1 mm, the center frequency of the receiving signal decreases with the increasing in WE and WR . This is because the increase in WE or WR promotes the superposition of the strain, magnetic induction and voltage, which causes the center frequency to decrease. Additionally, The CFS at the exciting frequency of 160 kHz is larger than the CFS at the exciting frequency of 20 kHz. This is because when the exciting frequency is different, the relative change in center frequency caused by coil width is the same. Fig. 3(c) and (d) indicate that, when WE and WR are fixed at 0.5 , the center frequency of the receiving signal increases with the increasing in l and d as the exciting frequency is 160 kHz. When the exciting frequency is 20 kHz, l and d have no effects on the center frequency of the receiving signal. This is because when obtaining Eq. (21), we divided the exciting region and the receiving region in units of l and d respectively. The increase in l or d leads the number of the divided regions decrease and weakens the superposition of the strain, magnetic induction and voltage, which causes the center frequency to increase. When the exciting frequency is low, the coil width is large, therefore, l and d have a weaker effect on the center frequency. Since FFT itself has resolution, the effects of l and d on the center frequency cannot be observed when the exciting frequency is below a certain value. 3. Experimental setup To verify the aforementioned model, experiments were performed on a steel pipe with 25 mm outer diameter, 3.5 mm wall thickness, and 2500 mm length. Fig. 4 shows the dispersion curves of this pipe. The dispersion renders the signal analysis difficult. In order to mitigate the dispersion effect, we chose 20 kHz (L(0,1)) and 160 kHz (L(0,2)) as the exciting frequency. The experimental setup is shown in Fig. 5. A magnetostrictive guided wave instrument was employed [11]. The exciting sensor consisted of an exciting coil and a bias coil. The receiving sensor was the same as the exciting sensor. Two bias coils with 500 turns, 110 mm width, and 160 mm inner diameter were used. The input currents to the bias coils were about 2.7 A. The exciting signal was a three-cycle sine-wave current. When the exciting frequency was 20 and 160 kHz, the peak amplitude of the current was maintained at 15 and 35 A, respectively. The signal magnification of the instrument was about 2000. The pass frequency of the bandpass filter was 1 kHz – 300 kHz; the sampling rate was 5 MHz. In order to reduce the noise, each signal was repeatedly sampled 200 times to obtain the average. A resistor was used to determine the exciting current. The exciting coils and the receiving coils were made of No. 19 gauge wire and their inner diameters were 32 mm. The widths and the turns of the coils are shown in Tables 1 and 2. From Fig. 4(b), when the exciting frequency is 20 kHz, the wave velocity is about 5238 mm/ms and the wavelength is about 256.0 mm.
When the exciting frequency is 160 kHz, the corresponding values are about 5119 mm/ms and 32.7 mm, respectively. In order to study the effect of the exciting coil width on the frequency, we fixed the width of the receiving coil at 0.5 and increased the width of the exciting coil from 0.125 to 0.75 in steps of 0.125 . Similarly, to study the effect of the receiving coil width on the frequency, we fixed the width of the exciting coil at 0.5 and increased the width of the receiving coil from 0.125 to 0.75 in steps of 0.125 . We repeated each experiment three times in order to reduce the random errors.
4. Results and discussion The exciting current is shown in Fig. 6, which was obtained by measuring the voltage across the resistor. Noise exists in the current owing to the presence of inductive loads, which would affect the center frequency. The actual center frequency of the exciting current is inconsistent with the preset center frequency. The amplitude spectrums of the exciting currents corresponding to two preset center frequencies are shown in Fig. 7. When the preset center frequency is 20.000 kHz and 160.000 kHz, the actual center frequency of the exciting current is 19.941 kHz and 160.336 kHz, respectively. Unless otherwise stated, the signal frequencies described below are preset frequencies. When the width of the exciting coil was 0.125 and that of the receiving coil was 0.5 , the receiving signals of 20 kHz and 160 kHz are shown in Fig. 8. According to the time of the first passing wave, the wave velocities of the signals at 20 kHz and 160 kHz are about 5180 mm/ms and 5250 mm/ms, respectively. From the dispersion curves in Fig. 3, the modes of the wave were confirmed as L(0,1) and L(0,2), respectively. The first passing wave was considered as the receiving signal to calculate its center frequency. Theoretical results were calculated using Eq. (21). The coil widths were obtained from Tables 1 and 2. The center distance of two adjacent wires of the exciting coils is the same as that of the receiving coils, which is about 1.1 mm. The relationship between the coil width and the center frequency of the receiving signal is shown in Fig. 9. The center frequency of the receiving signal is less than the exciting frequency, and it decreases with the increasing coil width. When the exciting frequency is 20 kHz, the experimental results are in good agreement with the theoretical results and the maximum error is 1.0%. On the other hand, when the exciting frequency is 160 kHz, the difference between the experimental results and the theoretical results is relatively larger, and the maximum error is 2.3%. In the theoretical model, we consider the exciting coil width as the width of the alternating magnetic field region at the exciting stage. However, the alternating magnetic field generated by the exciting coil has a fringe effect. The actual width of the alternating magnetic field region is larger than that of the exciting coil width. When the frequency is 160 kHz, the coil width and the alternating magnetic field width are smaller than those when the frequency is 20 kHz. The neglected magnetic field at the coil edge has a considerable influence on the center frequency. The effective width of the alternating
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
Fig. 6. Exciting current in the time-domain.
Fig. 7. Amplitude spectrums of the exciting current.
Fig. 8. Receiving signals in the time domain.
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C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
Fig. 9. Relationship between the coil width and the center frequency of the receiving signal.
magnetic field in the pipe surface should be derived by adding a correction factor, which is expressed as WE-C = WE + C
(22)
where WE-C is the corrected width of the alternating magnetic field, and C is the correction factor. The corrected theoretical results shown in Fig. 8(c) and (d) are obtained when C is considered as 2 mm. After the correction, the maximum error is reduced to 1.3%. 5. Conclusions and future work In this paper, we investigated the CFS phenomenon of the receiving signal for magnetostrictive guided wave. A theoretical model to calculate the CFS of the receiving signal was developed based on the energy coupling mechanism of the magnetostrictive guided waves. Increase of the coil width resulted in the superposition of the strain, magnetic induction, and voltage, which caused the center frequency of the receiving signal to decrease. The differential relationship between the magnetic induction and the voltage caused the center frequency of the receiving signal to increase. The proposed model was verified by conducting experiments on a steel pipe. The model has good adaptability if the coil width is large, and it needs to be corrected if the coil width is small. The model
developed in this study provides a method to modify the frequency for the high requirement of frequency accuracy and to serve as a reference for the design of magnetostrictive guided wave sensors. For the steel wire and cable, the energy coupling mechanism of the magnetostrictive guided waves is the same as the pipe. Therefore, the model is suitable for the steel wire and cable too. However, due to the difference in structure, the model may need to be modified. Additionally, further study of the model is required so that the model can have better adaptability, especially for coils with a small width. In the future, we will improve the model from these two aspects.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No.51575213) and the National Key Research and Development Program of China (Grant No.2016YFC0801904).
References [1] Z. Liu, Y. Hu, J. Fan, W. Yin, X. Liu, C. He, B. Wu, Longitudinal mode magnetostrictive patch transducer array employing a multi-splitting meander coil for pipe inspection, NDT E Int. 79 (2016) 30–37, http://dx.doi.org/10. 1016/j.ndteint.2015.11.009.
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583 [2] S.B. Oh, Y.-M. Cheong, D.-H. Lee, K.-M. Kim, Magnetostrictive guided wave technique verification for detection and monitoring defects in the pipe weld, Materials 12 (2019) 867, http://dx.doi.org/10.3390/ma12060867. [3] J. Zhou, J. Xu, Feasibility study of fatigue damage detection of strands using magnetostrictive guided waves, Int. J. Appl. Electromagn. Mech. 59 (2019) 1313–1320, http://dx.doi.org/10.3233/JAE-171153. [4] F. Lv, P. Zhang, Z. Tang, Y. Yue, K. Yang, A guided wave transducer with sprayed magnetostrictive powder coating for monitoring of aluminum conductor steel-reinforced cables, Sensors 19 (2019) 1550, http://dx.doi.org/ 10.3390/s19071550. [5] J. Wu, J. Zhu, H. Xia, C. Liu, G.Y. Tian, DC-biased magnetization based eddy current thermography for subsurface defect detection, IEEE Trans. Ind. Inform. (2019), http://dx.doi.org/10.1109/TII.2019.2891107. [6] P.W. Tse, X.C. Liu, Z.H. Liu, B. Wu, C.F. He, X.J. Wang, An innovative design for using flexible printed coils for magnetostrictive-based longitudinal guided wave sensors in steel strand inspection, Smart Mater. Struct. 20 (2011), 055001, http://dx.doi.org/10.1088/0964-1726/20/5/055001. [7] J. Xu, C.Y. Hu, J.F. Ding, An improved longitudinal mode guided wave received sensor based on inverse magnetostrictive effect for open end pipes, J. Nondestr. Eval. 38 (2019) 83, http://dx.doi.org/10.1007/s10921-019-0621-6. [8] Y.Y. Kim, Y.E. Kwon, Review of magnetostrictive patch transducers and applications in ultrasonic nondestructive testing of waveguides, Ultrasonics 62 (2015) 3–19, http://dx.doi.org/10.1016/j.ultras.2015.05.015. [9] J. Xu, Y. Sun, J. Zhou, Research on the lift-off effect of receiving longitudinal mode guided waves in pipes based on the villari effect, Sensors 16 (2016) 1529, http://dx.doi.org/10.3390/s16091529. [10] I.K. Kim, Y.Y. Kim, Wireless frequency-tuned generation and measurement of torsional waves using magnetostrictive nickel gratings in cylinders, Sens. Actuators A Phys. 126 (2006) 73–77, http://dx.doi.org/10.1016/j.sna.2005.09. 027. [11] J. Xu, Z. Xu, X. Wu, Research on the lift-off effect of generating longitudinal guided waves in pipes based on magnetostrictive effect, Sens. Actuators A Phys. 184 (2012) 28–33, http://dx.doi.org/10.1016/j.sna.2012.06.014. [12] D. Zhang, Z. Zhou, J. Sun, E. Zhang, Y. Yang, M. Zhao, A magnetostrictive guided-wave nondestructive testing method with multifrequency excitation pulse signal, IEEE Trans. Instrum. Meas. 63 (2014) 3058–3066, http://dx.doi. org/10.1109/TIM.2014.2324812. [13] R. Ribichini, Modelling of Electromagnetic Acoustic Transducers, Imperial College London, 2011. [14] J. Xu, G. Chen, J. Zhou, Y. Li, Effect of fatigue damage on the notch frequency of the prestressing strand based on magnetostrictive guided waves, Int. J. Appl. Electromagn. Mech. 59 (2019) 1341–1348, http://dx.doi.org/10.3233/JAE171109. [15] X. Liu, L. Bo, Y. Liu, Y. Zhao, J. Zhang, N. Hu, et al., Detection of micro-cracks using nonlinear lamb waves based on the Duffing-Holmes system, J. Sound Vib. 405 (2017) 175–186, http://dx.doi.org/10.1016/j.jsv.2017.05.044. [16] I.K. Park, H.M. Kim, K.Y. Jhang, Nondestructive Evaluation of Degraded 2.25Cr-1Mo Steel and Estimation of Nonlinear Acoustic Effect Using Bispectral Analysis, 2003, pp. 65–71, http://dx.doi.org/10.1115/PVP2003-1854. [17] V. Marcantonio, D. Monarca, A. Colantoni, M. Cecchini, Ultrasonic waves for materials evaluation in fatigue, thermal and corrosion damage: a review, MSSP 120 (2019) 32–42, http://dx.doi.org/10.1016/j.ymssp.2018.10.012. [18] H. Jeong, S.-H. Nahm, K.-Y. Jhang, Y.-H. Nam, A nondestructive method for estimation of the fracture toughness of CrMoV rotor steels based on ultrasonic nonlinearity, Ultrasonics 41 (2003) 543–549, http://dx.doi.org/10. 1016/S0041-624X(03)00, 154-9. [19] K.Y. Jhang, K.C. Kim, Evaluation of material degradation using nonlinear acoustic effect, Ultrasonics 37 (1999) 39–44, http://dx.doi.org/10.1016/ S0041-624X(98)00045-6. [20] E. Hristoforou, P.D. Dimitropoulos, J. Petrou, A new position sensor based on the MDL technique, Sens. Actuators A Phys. 132 (2006) 112–121, http://dx. doi.org/10.1016/j.sna.2006.04.015. [21] X. Li, Z. Wu, X. Shu, X. Gao, M. Li, Parameter Optimization of Magnetostrictive Linear Displacement Transducer, 2015 IIICEC, Atlantis Press, 2015, http://dx. doi.org/10.2991/iiicec-15.2015.405. [22] T. König, T. Greiner, A. Zern, Z. Kántor, A. Szabó, A. Hetznecker, Using eddy currents within magnetostrictive position sensors for velocity estimation, IEEE Sens. J. (2019), http://dx.doi.org/10.1109/JSEN.2019.2910635. [23] T. König, T. Greiner, A. Zern, Z. Kántor, A. Szabó, Introducing a New Method for Temperature Measurement by Magnetostrictive Position Sensors, 2017, pp. 1392–1397, http://dx.doi.org/10.1109/ISIE.2017.8001449, IEEE 26th ISIE. [24] W. Jian Jun, S. Jian Hua, Y. Shang Yu, F. Yao Rong, L. Kai, Tubing string thread sealing surfaces damage evaluation based on acoustic elasticity theory, Mater. Sci. Forum. 944 (2019) 828–834, http://dx.doi.org/10.4028/www.scientific. net/MSF.944.828. [25] L. Wei, J.L. Rose, Guided wave thickness measurement with EMATs, Insight-Non-Destruct. Test. Condition Monit. 45 (2003) 735–739, http://dx. doi.org/10.1784/insi.45.11.735.52961.
9
[26] Z. Fang, P.W. Tse, Axial magnetized patch for efficient transduction of longitudinal guided wave and defect identification in concrete-covered pipe risers, Struct. Control. Health Monit. 25 (2018) e2231, http://dx.doi.org/10. 1002/stc.2231. [27] R.C. William, Theory of magnetostrictive delay lines for pulse and continuous wave transmission, IRE Trans. Ultrason. Eng. (1959) 16–32, http://dx.doi.org/ 10.1109/T-PGUE.1959.29250, PGUE-7. [28] M.J. Sablik, Y. Lu, G.L. Burkhardt, Modeling magnetostrictive generation of elastic waves in steel pipes, II. Comparison to experiment, Int. J. Appl. Electromagn. Mech. 10 (1999) 167–176, http://dx.doi.org/10.3233/JAE-1999138. [29] M.J. Sablik, S.W. Rubin, Modeling magnetostrictive generation of elastic waves in steel pipes, I. Theory, Int. J. Appl. Electromagn. Mech. 10 (1999) 143–166, http://dx.doi.org/10.3233/JAE-1999-137. [30] M.J. Dapino, R.C. Smith, A.B. Flatau, Structural magnetic strain model for magnetostrictive transducers, IEEE Trans. Magn. 36 (2000) 545–556, http:// dx.doi.org/10.1109/20.846217. [31] D.-G. Zhang, M.-H. Li, H.-M. Zhou, A general one-dimension nonlinear magneto-elastic coupled constitutive model for magnetostrictive materials, AIP Adv. 5 (2015), 107201, http://dx.doi.org/10.1063/1.4933024. [32] W. Cai, J. Zhang, D. Yu, P. Feng, J. Wang, A vibration amplitude model for the giant magnetostrictive ultrasonic processing system, J. Intell. Mater. Syst. Struct. 29 (2018) 574–584, http://dx.doi.org/10.1177/1045389X17711818. [33] A. Grunwald, A.G. Olabi, Design of a magnetostrictive (MS) actuator, Sens. Actuators A Phys. 144 (2008) 161–175, http://dx.doi.org/10.1016/j.sna.2007. 12.034. [34] J. Zheng, S. Cao, H. Wang, Modeling of magnetomechanical effect behaviors in a giant magnetostrictive device under compressive stress, Sens. Actuators A Phys. 143 (2008) 204–214, http://dx.doi.org/10.1016/j.sna.2007.10.063. ´ R. Szewczyk, The possibility of utilizing the high permeability [35] A. Bienkowski, magnetic materials in construction of magnetoelastic stress and force sensors, Sens. Actuators A Phys. 113 (2004) 270–276, http://dx.doi.org/10.1016/j.sna. 2004.01.010. [36] K.H. Matlack, J.Y. Kim, L.J. Jacobs, J. Qu, Review of second harmonic generation measurement techniques for material state determination in metals, J. Nondestruct. Eval. 34 (2014) 273, http://dx.doi.org/10.1007/s10921-0140273-5. [37] J. Rostami, P.W.T. Tse, Z. Fang, Sparse and dispersion-based matching pursuit for minimizing the dispersion effect occurring when using guided wave for pipe inspection, Materials 10 (2017) 622, http://dx.doi.org/10.3390/ ma10060622. [38] L. Laguerre, J.C. Aime, M. Brissaud, Magnetostrictive pulse-echo device for non-destructive evaluation of cylindrical steel materials using longitudinal guided waves, Ultrasonics 39 (2002) 503–514, http://dx.doi.org/10.1016/ S0041-624X(01)00088-9. [39] F. Seco, J.M. Martin, A.R. Jimenez, Improving the accuracy of magnetostrictive linear position sensors, IEEE Trans. Instru. Meas. 58 (2009) 722–729, http:// dx.doi.org/10.1109/TIM.2008.2005260. [40] F. Baronti, G. Fantechi, R. Roncella, R. Saletti, A new and accurate system for measuring cruising yacht freeboards with magnetostrictive sensors, IEEE Trans. Instru. Meas. 60 (2011) 1811–1819, http://dx.doi.org/10.1109/TIM. 2011.2113050. [41] J. Xu, Y. Li, G. Chen, Effect of tensile force on magnetostrictive sensors for generating and receiving longitudinal mode guided waves in steel wires, J. Sensors (2019), 9512190, http://dx.doi.org/10.1155/2019/9512190. [42] S. Djili, F. Benmeddour, E. Moulin, J. Assaad, F. boubenider, Notch detection in copper tubes immersed in water by leaky compressional guided waves, NDT E Int. 54 (2013) 183–188, http://dx.doi.org/10.1016/j.ndteint.2012.10.009. [43] Z. Fan, M.J.S. Lowe, Interaction of weld-guided waves with defects, NDT E Int. 47 (2012) 124–133, http://dx.doi.org/10.1016/j.ndteint.2012.01.001. [44] R. Carandente, P. Cawley, The effect of complex defect profiles on the reflection of the fundamental torsional mode in pipes, NDT E Int. 46 (2012) 41–47, http://dx.doi.org/10.1016/j.ndteint.2011.11.003.
Biographies Chaoyue Hu received his B.S. in measurement technology and instruments from Wuhan University of Technology of China in 2018. After that, he has been a postgraduate student in Huazhong University of Science and Technology of China. His research interests include magnetostrictive transducer and signal processing. Jiang Xu received his B.S. in electrical engineering, M.S. in mechatronic engineering and Ph.D. in measurement technology and instruments from Huazhong University of Science and Technology of China in 2000, 2005 and 2009. Now he is an associate professor in School of Mechanical Science and Engineering at Huazhong University of Science and Technology of China. His research interests include guided waves, magnetoacoustic transducer and signal processing.
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Center frequency shift in pipe inspection using magnetostrictive guided waves Chaoyue Hu, Jiang Xu ∗ School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, PR China
a r t i c l e
i n f o
Article history: Received 20 April 2019 Received in revised form 15 August 2019 Accepted 2 September 2019 Available online 3 September 2019 Keywords: Center frequency shift Magnetostrictive effect Guided wave Sensor
a b s t r a c t Magnetostrictive guided wave technology is widely employed to inspect pipes. In general, it is assumed that the center frequency of the exciting signal is the same as that of the receiving signal; however, this assumption does not hold in practice. To elucidate this phenomenon, we investigate the reason behind the center frequency shift (CFS) based on the energy coupling mechanism of magnetostrictive guided waves; we construct a theoretical model to calculate the CFS of the receiving signal. The detection process for guided waves is separated into three stages: excitation, propagation, and reception. At the exciting stage, the width of the coil affects the alternating magnetic field distribution. The vibration generated by the alternating magnetic field with a certain width produces a superimposed strain, which results in the CFS of the generating wave. At the propagating stage, we consider that the center frequency of the guided wave remains unchanged because the propagation is a linear process. At the receiving stage, the coil length determines the size of the induction area and the magnetic induction at a point is determined by the strain around that point. The number of coil turns, the distance of two adjacent wires, and the differential during the process of the electromagnetic induction affect the frequency of the induction signal, which leads to the CFS of the receiving signal. The center frequency of the receiving signal decreases with increasing coil width. Experiments are performed to verify that the theoretical model and the experimental results are in good agreement with the theoretical results. The model developed in this study provides a reference for the design of magnetostrictive guided wave sensors. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Magnetostrictive guided wave technology is widely used to test pipes and cables for many years [1–7]. Compared with other guided wave technologies, this technology has the advantages of large liftoff distance, easy implementation, etc. [8,9]. Longitudinal mode and torsional mode guided waves are the most widely used in practice [9]. Solenoid coils are usually employed as a part of the magnetostrictive sensor and the coil width is usually half the wavelength of the guided wave at the exciting frequency [10]. When the static bias magnetic field is sufficiently strong, the magnetostrictive effect is almost linear [11–13]. In general, researchers consider that the center frequency of the exciting signal is the same as that of the receiving signal. However, in practical applications, we found that the center frequency of the exciting signal was different from that of the receiving signal. The center frequency shift (CFS) of the receiving signal plays little role in macro defect
∗ Corresponding author. E-mail address: [email protected] (J. Xu). https://doi.org/10.1016/j.sna.2019.111583 0924-4247/© 2019 Elsevier B.V. All rights reserved.
detection [2,14]. However, the requirements for frequency accuracy are high in some cases. Nonlinear acoustic parameters are used commonly for micro defect detection such as fatigues, fracture toughness, material degradation, and so on [15–19]. The method for calculating nonlinear acoustic parameters requires that the center frequency of the receiving signal be constant; otherwise, the calculation result is inaccurate. Magnetostrictive position sensor is a type of high-precision position sensor based on the time-of-flight measurement of the guided wave [20–23]. The CFS of the wave affects wave velocity, which results in large measurement errors. In addition, some applications, such as damage evaluation based on the change of the center frequency of the reflection wave [24,25], or thickness measurement by frequency-domain analysis [26], have high requirements of frequency accuracy. Many studies have focused on modelling the magnetostrictive effect. In the 1950s, William [27] developed a model based on magnetostrictive delay lines to calculate the output voltage under the assumption of a linear relationship between the elastic and magnetic variables. Sablik [28,29] proposed a model for a cylindrical structure, which was used to calculate the amplitude and dispersion of longitudinal mode guided waves under an axial bias
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magnetic field. Dapino [30] studied the strain model of a magnetostrictive transducer under an external magnetic field, combined with a quadratic moment rotation model for magnetostriction. In this model, the Jiles–Atherton mean field theory was incorporated for ferromagnetic hysteresis. Zhang [31] built a one-dimensional nonlinear magnetoelastic coupled constitutive model, which was used to calculate the magnetostrictive strain and magnetization curve of a material under bias magnetic field and pre-stress. Cai [32] presented the equivalent mechanical model of a giant magnetostrictive ultrasonic processing system. The vibration amplitude of the system could be calculated using this model if the frequency and amplitude of the exciting current were known. Most of these models were used to calculate the amplitude and dispersion of the waves. However, little research has been performed on calculating the center frequency of the receiving signal. In this paper, we investigate the reason behind the occurrence of CFS based on the energy coupling mechanism of magnetostrictive guided waves. A theoretical model to calculate the center frequency of the receiving signal is presented. Experiments are carried out on a steel pipe to verify the model. This model provides a method to correct the frequency, and it serves as a reference for the design of magnetostrictive guided wave sensors. 2. Theoretical model The size of a ferromagnetic material changes due to the external alternating magnetic field and a wave is generated in the ferromagnetic material, which is called the magnetostrictive effect [33,34]. The magnetic properties of the ferromagnetic material are changed when the wave propagates in the material, which is called the inverse magnetostrictive effect [35]. When the static bias magnetic field is far stronger than the alternating magnetic field, the effects can be described as [11–13] ε = /E H + dH
(1)
B = d∗ + H
(2)
where ε is the strain, is the stress, EH is Young’s modulus at a constant applied magnetic field, d is the magnetostriction coefficient, d* is the inverse magnetostriction coefficient, B is the magnetic induction, is the permeability at a constant stress, and H is the magnetic field. Guided wave testing based on the magnetostrictive effect can be separated into three stages: excitation, propagation, and reception. At the exciting stage, the current is coupled to the magnetic field and then the magnetic field is coupled to the strain. At the propagating stage, the center frequency of the wave remains unchanged because the propagation of the elastic wave is a linear process [36,37]. At the receiving stage, the stress is coupled to magnetic induction, which is coupled to the voltage. We focus on analyzing the CFS of the signal at the exciting and receiving stages.
where ω0 is the center frequency of the current and t is the time. According to the law of electromagnetic induction, the relationship between the current and the magnetic field can be expressed as [12] Ha = NI/WE
(4)
where Ha is the alternating magnetic field in the pipe surface below the exciting coil, which is induced by the exciting current, N is the number of exciting coil turns, and WE is the width of the exciting coil. The right side of Eq. (1) includes two parts: static stress and alternating magnetic field. The generation of a guided wave is only related to the alternating magnetic field. Therefore, combining Eq. (1), Eq. (3), and Eq. (4), the alternating magnetostrictive strain in the pipe surface below each turn of the exciting coil can be expressed as εMS−m (ω0 t) =
d · NI d·N = y(ω0 t) WE WE
where εMS-i is the alternating magnetostrictive strain in the pipe surface below the mth turn wire. We suppose that A is a point on the pipe surface and the distance between A and the 1st turn wire is z0 . Without considering the attenuation during the propagation of the wave, the contribution of the wave generated by the mth turn wire to point A can be expressed as [36] εm = εMS−m (ω0 t − k(ml − l + z0 ))
εm =
d·N y(ω0 t − k(ml − l + z0 )) WE
(7)
The wave at point A is a superimposed strain generated by the entire exciting coil, which can be expressed as εA =
N
d·N y(ω0 t − k(ml − l + z0 )) WE N
εm =
m=1
(8)
m=1
By incorporating Eq. (3), the amplitude spectrum of the wave at point A can be represented as
N−1
−jωkml/ω d · N 0 |εA (ω)| = × I(ω)e−jωkz0 /ω0 × e WE m=0
N−1 d · N −jωkml/ω 0 = × I(ω) × e WE
(9)
m=0
where |I(ω)| is the amplitude spectrum of I. According to the summation formula of the geometric progression, Eq. (9) can be written as
WE
The schematic of the exciting stage is shown in Fig. 1. A static bias magnetic field can prevent the frequency-doubled effect and improve energy coupling efficiency [38]. When an alternating current is applied to the exciting coil, an alternating magnetic field is generated in the pipe surface [39,40], and this leads to the generation of an alternating strain in the pipe surface because of the magnetostrictive effect [41–44]. The exciting current can be expressed as
k = ω0 /c
I = y(ω0 t)
WE = N · l
(3)
(6)
where k is the wavenumber, and l is the center distance of two adjacent wires of the exciting coil. By substituting Eq. (5), Eq. (6) can be expressed as
εA (ω) = d · N × I(ω) ×
2.1. Signal CFS at the exciting stage
(5)
1 − cos(ωNkl/ω0 ) 1 − cos(ωkl/ω0 )
(10)
The relationship between the wavenumber and the exciting frequency can be expressed as (11)
where c is the speed of the guided wave. The relationship between the coil turns and the center distance of two adjacent wires can be expressed as (12)
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3
Fig. 1. Schematic of the exciting stage.
Combining Eq. (11) and Eq. (12), Eq. (10) can be written as
εA (ω) = d · N × I(ω) ×
WE
1 − cos(ωWE /c) 1 − cos(ωl/c)
(13)
In Eq. (13), the frequency corresponding to the highest peak of the amplitude spectrum is the center frequency of the generated wave, which can be expressed as
ωc−ε = arg max{ εA (ω) }
(14)
ω
where ωc-ε is the center frequency of the generated wave. Therefore, the wave generated in the pipe is a superimposed strain, which causes the CFS of the wave. 2.2. Signal CFS at the receiving stage The schematic diagram of the receiving stage is shown in Fig. 2. Due to the inverse magnetostrictive effect, an alternating magnetic induction is generated in the pipe surface. Then, an induced voltage is generated in the receiving coil because of the electromagnetic induction effect. According to elastic mechanics, stress is proportional to strain. The right side of Eq. (2) has two parts: alternating stress and static magnetic field. The receiving wave is only related to the alternating stress. When the receiving coil is a single-turn coil, Eq. (2) can be written as BIMS = d∗ R = d∗ E H εR
(15)
where εR is the alternating strain in the pipe surface below the single-turn coil, which originates from the exciting region, and its amplitude spectrum is the same as |εA (ω)|; R is the stress corresponding to εR ; and BIMS is the magnetic induction in the pipe surface below the single-turn coil and it is engendered by εR due to the inverse magnetostrictive effect. There exists a region with a certain width in the pipe surface below the single-turn coil, for which the strain at every point has an influence on the magnetic induction in the pipe surface below the coil. Therefore, the magnetic induction in the pipe surface below the of the magnetic induction in single-turn coil is the superimposition √ this region. The region width is 2/2 times the half wavelength of the guided wave according to the root mean square value of the sinusoidal signal. Similar to Eq. (13), we can write
BIMS−S (ω) = BIMS (ω) ×
1 − cos(ω ·
√ 2 /c) 4
1 − cos(ωd/c)
(16)
where BIMS-S is the superimposed magnetic induction in the pipe surface below the coil, which is engendered by the strain in the
aforementioned region; |BIMS (ω)| is the amplitude spectrum of BIMS ; is the wavelength of the guided wave; and d is the center distance of two adjacent wires of the receiving coil. By substituting Eq. (15), Eq. (16) can be written as
BIMS−S (ω) = d∗ EH × εR (ω) ×
1 − cos(ω ·
√ 2 /c) 4
1 − cos(ωd/c)
(17)
Now, the magnetic induction is coupled to the voltage according to the law of electromagnetic induction. When the receiving coil is a single-turn coil, the induced voltage in the coil is given by [12] 2
V1 = − (r + h) · ∂BIMS−S /∂t
(18)
where V1 is the induced voltage in the single-turn coil. Eq. (18) implies that there exists a time-domain differential during the coupling from magnetic induction to voltage. According to Fourier transform, time-domain differential means that the spectrum is multiplied by the angular frequency and an imaginary number. The amplitude spectrum of the induced voltage in the single-turn coil can be expressed as
V1 (ω) = − (r + h)2 · BIMS−S (ω) · jω = − (r + h)2 · BIMS−S (ω) · ω
(19)
When the receiving coil is a multi-turn coil, the receiving signal is a superimposition of the voltage in each turn wire. Similar to Eq. (13), the amplitude spectrum of the receiving signal can be expressed as
VM (ω) = V1 (ω) ×
1 − cos(ωWR /c) 1 − cos(ωd/c)
(20)
where |VM (ω)| is the amplitude spectrum of the receiving signal, M is the number of the receiving coil turns, and WR is the width of the receiving coil. Combining Eq. (13), Eq. (17), Eq. (19), and Eq. (20), the center frequency of the receiving signal can be expressed as
⎧ 1 − cos ωWE /c) ⎪ ⎪ VM (ω) = p × I(ω) × ω × ⎪ ⎪ 1 − cos(ωl/c) ⎪ ⎪
⎪ √ ⎪ ⎪ ⎪ 1 − cos ω 2 /c) ⎪ ⎪ 1 − cos(ωWR /c) ⎨ 4 ×
1 − cos(ωd/c)
×
⎪ ⎪ ⎪ 2 ⎪ − (r + h) d∗ dE H N ⎪ ⎪ p= ⎪ ⎪ W N ⎪ ⎪ ⎪ ⎪ ⎩ ωc−r = arg max{ VM (ω) } ω
1 − cos(ωd/c)
(21)
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C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
Fig. 2. Schematic diagram of the receiving stage.
Fig. 3. Relationships between the four parameters and the CFS.
where ωc-r is the center frequency of the receiving signal, and p is a constant that has no influence on the center frequency. If the exciting signal and the parameters of the coils are known, the center frequency of the receiving signal can be calculated using Eq. (21).
2.3. Discussions and analyses Eq. (21) indicates that the CFS of the receiving signal occurs because of four reasons: 1) the superposition of the strain during the coupling from the magnetic field to the strain decreases the
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
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Fig. 4. Dispersion curves of the sample pipe.
Fig. 5. Schematic diagram and photograph of the experimental setup.
center frequency of the generated wave; 2) the superposition of the magnetic induction during the coupling from the strain to the magnetic induction decreases the center frequency of the magnetic induction; 3) the superposition of the voltage in each turn wire of the receiving coil decreases the center frequency of the receiving signal; 4) the differential during the coupling from magnetic
induction to voltage increases the center frequency of the induced voltage. From Eq. (21), when the exciting frequency is fixed, the CFS of the receiving signal is determined by four parameters: WE , WR , l and d. WE determines the width of the exciting region, which affects the superposition of the strain, and WR determines the width of
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C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
Table 1 Width and turn of the coils (20 kHz, =256.0mm).
Table 2 Width and turn of the coils (160 kHz, =32.7mm).
No.
Width(mm)
Width()
Turns
No.
Width(mm)
Width()
Turns
1 2 3 4 5 6
32.0 64.0 96.0 128.0 160.0 192.0
0.125 0.250 0.375 0.500 0.625 0.750
29 57 85 113 143 171
1 2 3 4 5 6
4.4 8.8 13.2 16.5 21.2 24.8
0.125 0.250 0.375 0.500 0.625 0.750
4 8 12 16 20 24
the receiving region, which affects the superposition of the magnetic induction and voltage. When the coil width is fixed, l and d determine the coil turns. Considering the 20 kHz and 160 kHz exciting frequencies as the examples, the relationships between the center frequency of the receiving signal and these four parameters are shown in Fig. 3. Figs. 3(a) and (b) indicate that, when l and d are fixed at 1 mm, the center frequency of the receiving signal decreases with the increasing in WE and WR . This is because the increase in WE or WR promotes the superposition of the strain, magnetic induction and voltage, which causes the center frequency to decrease. Additionally, The CFS at the exciting frequency of 160 kHz is larger than the CFS at the exciting frequency of 20 kHz. This is because when the exciting frequency is different, the relative change in center frequency caused by coil width is the same. Fig. 3(c) and (d) indicate that, when WE and WR are fixed at 0.5 , the center frequency of the receiving signal increases with the increasing in l and d as the exciting frequency is 160 kHz. When the exciting frequency is 20 kHz, l and d have no effects on the center frequency of the receiving signal. This is because when obtaining Eq. (21), we divided the exciting region and the receiving region in units of l and d respectively. The increase in l or d leads the number of the divided regions decrease and weakens the superposition of the strain, magnetic induction and voltage, which causes the center frequency to increase. When the exciting frequency is low, the coil width is large, therefore, l and d have a weaker effect on the center frequency. Since FFT itself has resolution, the effects of l and d on the center frequency cannot be observed when the exciting frequency is below a certain value. 3. Experimental setup To verify the aforementioned model, experiments were performed on a steel pipe with 25 mm outer diameter, 3.5 mm wall thickness, and 2500 mm length. Fig. 4 shows the dispersion curves of this pipe. The dispersion renders the signal analysis difficult. In order to mitigate the dispersion effect, we chose 20 kHz (L(0,1)) and 160 kHz (L(0,2)) as the exciting frequency. The experimental setup is shown in Fig. 5. A magnetostrictive guided wave instrument was employed [11]. The exciting sensor consisted of an exciting coil and a bias coil. The receiving sensor was the same as the exciting sensor. Two bias coils with 500 turns, 110 mm width, and 160 mm inner diameter were used. The input currents to the bias coils were about 2.7 A. The exciting signal was a three-cycle sine-wave current. When the exciting frequency was 20 and 160 kHz, the peak amplitude of the current was maintained at 15 and 35 A, respectively. The signal magnification of the instrument was about 2000. The pass frequency of the bandpass filter was 1 kHz – 300 kHz; the sampling rate was 5 MHz. In order to reduce the noise, each signal was repeatedly sampled 200 times to obtain the average. A resistor was used to determine the exciting current. The exciting coils and the receiving coils were made of No. 19 gauge wire and their inner diameters were 32 mm. The widths and the turns of the coils are shown in Tables 1 and 2. From Fig. 4(b), when the exciting frequency is 20 kHz, the wave velocity is about 5238 mm/ms and the wavelength is about 256.0 mm.
When the exciting frequency is 160 kHz, the corresponding values are about 5119 mm/ms and 32.7 mm, respectively. In order to study the effect of the exciting coil width on the frequency, we fixed the width of the receiving coil at 0.5 and increased the width of the exciting coil from 0.125 to 0.75 in steps of 0.125 . Similarly, to study the effect of the receiving coil width on the frequency, we fixed the width of the exciting coil at 0.5 and increased the width of the receiving coil from 0.125 to 0.75 in steps of 0.125 . We repeated each experiment three times in order to reduce the random errors.
4. Results and discussion The exciting current is shown in Fig. 6, which was obtained by measuring the voltage across the resistor. Noise exists in the current owing to the presence of inductive loads, which would affect the center frequency. The actual center frequency of the exciting current is inconsistent with the preset center frequency. The amplitude spectrums of the exciting currents corresponding to two preset center frequencies are shown in Fig. 7. When the preset center frequency is 20.000 kHz and 160.000 kHz, the actual center frequency of the exciting current is 19.941 kHz and 160.336 kHz, respectively. Unless otherwise stated, the signal frequencies described below are preset frequencies. When the width of the exciting coil was 0.125 and that of the receiving coil was 0.5 , the receiving signals of 20 kHz and 160 kHz are shown in Fig. 8. According to the time of the first passing wave, the wave velocities of the signals at 20 kHz and 160 kHz are about 5180 mm/ms and 5250 mm/ms, respectively. From the dispersion curves in Fig. 3, the modes of the wave were confirmed as L(0,1) and L(0,2), respectively. The first passing wave was considered as the receiving signal to calculate its center frequency. Theoretical results were calculated using Eq. (21). The coil widths were obtained from Tables 1 and 2. The center distance of two adjacent wires of the exciting coils is the same as that of the receiving coils, which is about 1.1 mm. The relationship between the coil width and the center frequency of the receiving signal is shown in Fig. 9. The center frequency of the receiving signal is less than the exciting frequency, and it decreases with the increasing coil width. When the exciting frequency is 20 kHz, the experimental results are in good agreement with the theoretical results and the maximum error is 1.0%. On the other hand, when the exciting frequency is 160 kHz, the difference between the experimental results and the theoretical results is relatively larger, and the maximum error is 2.3%. In the theoretical model, we consider the exciting coil width as the width of the alternating magnetic field region at the exciting stage. However, the alternating magnetic field generated by the exciting coil has a fringe effect. The actual width of the alternating magnetic field region is larger than that of the exciting coil width. When the frequency is 160 kHz, the coil width and the alternating magnetic field width are smaller than those when the frequency is 20 kHz. The neglected magnetic field at the coil edge has a considerable influence on the center frequency. The effective width of the alternating
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
Fig. 6. Exciting current in the time-domain.
Fig. 7. Amplitude spectrums of the exciting current.
Fig. 8. Receiving signals in the time domain.
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C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583
Fig. 9. Relationship between the coil width and the center frequency of the receiving signal.
magnetic field in the pipe surface should be derived by adding a correction factor, which is expressed as WE-C = WE + C
(22)
where WE-C is the corrected width of the alternating magnetic field, and C is the correction factor. The corrected theoretical results shown in Fig. 8(c) and (d) are obtained when C is considered as 2 mm. After the correction, the maximum error is reduced to 1.3%. 5. Conclusions and future work In this paper, we investigated the CFS phenomenon of the receiving signal for magnetostrictive guided wave. A theoretical model to calculate the CFS of the receiving signal was developed based on the energy coupling mechanism of the magnetostrictive guided waves. Increase of the coil width resulted in the superposition of the strain, magnetic induction, and voltage, which caused the center frequency of the receiving signal to decrease. The differential relationship between the magnetic induction and the voltage caused the center frequency of the receiving signal to increase. The proposed model was verified by conducting experiments on a steel pipe. The model has good adaptability if the coil width is large, and it needs to be corrected if the coil width is small. The model
developed in this study provides a method to modify the frequency for the high requirement of frequency accuracy and to serve as a reference for the design of magnetostrictive guided wave sensors. For the steel wire and cable, the energy coupling mechanism of the magnetostrictive guided waves is the same as the pipe. Therefore, the model is suitable for the steel wire and cable too. However, due to the difference in structure, the model may need to be modified. Additionally, further study of the model is required so that the model can have better adaptability, especially for coils with a small width. In the future, we will improve the model from these two aspects.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No.51575213) and the National Key Research and Development Program of China (Grant No.2016YFC0801904).
References [1] Z. Liu, Y. Hu, J. Fan, W. Yin, X. Liu, C. He, B. Wu, Longitudinal mode magnetostrictive patch transducer array employing a multi-splitting meander coil for pipe inspection, NDT E Int. 79 (2016) 30–37, http://dx.doi.org/10. 1016/j.ndteint.2015.11.009.
C. Hu and J. Xu / Sensors and Actuators A 298 (2019) 111583 [2] S.B. Oh, Y.-M. Cheong, D.-H. Lee, K.-M. Kim, Magnetostrictive guided wave technique verification for detection and monitoring defects in the pipe weld, Materials 12 (2019) 867, http://dx.doi.org/10.3390/ma12060867. [3] J. Zhou, J. Xu, Feasibility study of fatigue damage detection of strands using magnetostrictive guided waves, Int. J. Appl. Electromagn. Mech. 59 (2019) 1313–1320, http://dx.doi.org/10.3233/JAE-171153. [4] F. Lv, P. Zhang, Z. Tang, Y. Yue, K. Yang, A guided wave transducer with sprayed magnetostrictive powder coating for monitoring of aluminum conductor steel-reinforced cables, Sensors 19 (2019) 1550, http://dx.doi.org/ 10.3390/s19071550. [5] J. Wu, J. Zhu, H. Xia, C. Liu, G.Y. Tian, DC-biased magnetization based eddy current thermography for subsurface defect detection, IEEE Trans. Ind. Inform. (2019), http://dx.doi.org/10.1109/TII.2019.2891107. [6] P.W. Tse, X.C. Liu, Z.H. Liu, B. Wu, C.F. He, X.J. Wang, An innovative design for using flexible printed coils for magnetostrictive-based longitudinal guided wave sensors in steel strand inspection, Smart Mater. Struct. 20 (2011), 055001, http://dx.doi.org/10.1088/0964-1726/20/5/055001. [7] J. Xu, C.Y. Hu, J.F. Ding, An improved longitudinal mode guided wave received sensor based on inverse magnetostrictive effect for open end pipes, J. Nondestr. Eval. 38 (2019) 83, http://dx.doi.org/10.1007/s10921-019-0621-6. [8] Y.Y. Kim, Y.E. Kwon, Review of magnetostrictive patch transducers and applications in ultrasonic nondestructive testing of waveguides, Ultrasonics 62 (2015) 3–19, http://dx.doi.org/10.1016/j.ultras.2015.05.015. [9] J. Xu, Y. Sun, J. Zhou, Research on the lift-off effect of receiving longitudinal mode guided waves in pipes based on the villari effect, Sensors 16 (2016) 1529, http://dx.doi.org/10.3390/s16091529. [10] I.K. Kim, Y.Y. Kim, Wireless frequency-tuned generation and measurement of torsional waves using magnetostrictive nickel gratings in cylinders, Sens. Actuators A Phys. 126 (2006) 73–77, http://dx.doi.org/10.1016/j.sna.2005.09. 027. [11] J. Xu, Z. Xu, X. Wu, Research on the lift-off effect of generating longitudinal guided waves in pipes based on magnetostrictive effect, Sens. Actuators A Phys. 184 (2012) 28–33, http://dx.doi.org/10.1016/j.sna.2012.06.014. [12] D. Zhang, Z. Zhou, J. Sun, E. Zhang, Y. Yang, M. Zhao, A magnetostrictive guided-wave nondestructive testing method with multifrequency excitation pulse signal, IEEE Trans. Instrum. Meas. 63 (2014) 3058–3066, http://dx.doi. org/10.1109/TIM.2014.2324812. [13] R. Ribichini, Modelling of Electromagnetic Acoustic Transducers, Imperial College London, 2011. [14] J. Xu, G. Chen, J. Zhou, Y. Li, Effect of fatigue damage on the notch frequency of the prestressing strand based on magnetostrictive guided waves, Int. J. Appl. Electromagn. Mech. 59 (2019) 1341–1348, http://dx.doi.org/10.3233/JAE171109. [15] X. Liu, L. Bo, Y. Liu, Y. Zhao, J. Zhang, N. Hu, et al., Detection of micro-cracks using nonlinear lamb waves based on the Duffing-Holmes system, J. Sound Vib. 405 (2017) 175–186, http://dx.doi.org/10.1016/j.jsv.2017.05.044. [16] I.K. Park, H.M. Kim, K.Y. Jhang, Nondestructive Evaluation of Degraded 2.25Cr-1Mo Steel and Estimation of Nonlinear Acoustic Effect Using Bispectral Analysis, 2003, pp. 65–71, http://dx.doi.org/10.1115/PVP2003-1854. [17] V. Marcantonio, D. Monarca, A. Colantoni, M. Cecchini, Ultrasonic waves for materials evaluation in fatigue, thermal and corrosion damage: a review, MSSP 120 (2019) 32–42, http://dx.doi.org/10.1016/j.ymssp.2018.10.012. [18] H. Jeong, S.-H. Nahm, K.-Y. Jhang, Y.-H. Nam, A nondestructive method for estimation of the fracture toughness of CrMoV rotor steels based on ultrasonic nonlinearity, Ultrasonics 41 (2003) 543–549, http://dx.doi.org/10. 1016/S0041-624X(03)00, 154-9. [19] K.Y. Jhang, K.C. Kim, Evaluation of material degradation using nonlinear acoustic effect, Ultrasonics 37 (1999) 39–44, http://dx.doi.org/10.1016/ S0041-624X(98)00045-6. [20] E. Hristoforou, P.D. Dimitropoulos, J. Petrou, A new position sensor based on the MDL technique, Sens. Actuators A Phys. 132 (2006) 112–121, http://dx. doi.org/10.1016/j.sna.2006.04.015. [21] X. Li, Z. Wu, X. Shu, X. Gao, M. Li, Parameter Optimization of Magnetostrictive Linear Displacement Transducer, 2015 IIICEC, Atlantis Press, 2015, http://dx. doi.org/10.2991/iiicec-15.2015.405. [22] T. König, T. Greiner, A. Zern, Z. Kántor, A. Szabó, A. Hetznecker, Using eddy currents within magnetostrictive position sensors for velocity estimation, IEEE Sens. J. (2019), http://dx.doi.org/10.1109/JSEN.2019.2910635. [23] T. König, T. Greiner, A. Zern, Z. Kántor, A. Szabó, Introducing a New Method for Temperature Measurement by Magnetostrictive Position Sensors, 2017, pp. 1392–1397, http://dx.doi.org/10.1109/ISIE.2017.8001449, IEEE 26th ISIE. [24] W. Jian Jun, S. Jian Hua, Y. Shang Yu, F. Yao Rong, L. Kai, Tubing string thread sealing surfaces damage evaluation based on acoustic elasticity theory, Mater. Sci. Forum. 944 (2019) 828–834, http://dx.doi.org/10.4028/www.scientific. net/MSF.944.828. [25] L. Wei, J.L. Rose, Guided wave thickness measurement with EMATs, Insight-Non-Destruct. Test. Condition Monit. 45 (2003) 735–739, http://dx. doi.org/10.1784/insi.45.11.735.52961.
9
[26] Z. Fang, P.W. Tse, Axial magnetized patch for efficient transduction of longitudinal guided wave and defect identification in concrete-covered pipe risers, Struct. Control. Health Monit. 25 (2018) e2231, http://dx.doi.org/10. 1002/stc.2231. [27] R.C. William, Theory of magnetostrictive delay lines for pulse and continuous wave transmission, IRE Trans. Ultrason. Eng. (1959) 16–32, http://dx.doi.org/ 10.1109/T-PGUE.1959.29250, PGUE-7. [28] M.J. Sablik, Y. Lu, G.L. Burkhardt, Modeling magnetostrictive generation of elastic waves in steel pipes, II. Comparison to experiment, Int. J. Appl. Electromagn. Mech. 10 (1999) 167–176, http://dx.doi.org/10.3233/JAE-1999138. [29] M.J. Sablik, S.W. Rubin, Modeling magnetostrictive generation of elastic waves in steel pipes, I. Theory, Int. J. Appl. Electromagn. Mech. 10 (1999) 143–166, http://dx.doi.org/10.3233/JAE-1999-137. [30] M.J. Dapino, R.C. Smith, A.B. Flatau, Structural magnetic strain model for magnetostrictive transducers, IEEE Trans. Magn. 36 (2000) 545–556, http:// dx.doi.org/10.1109/20.846217. [31] D.-G. Zhang, M.-H. Li, H.-M. Zhou, A general one-dimension nonlinear magneto-elastic coupled constitutive model for magnetostrictive materials, AIP Adv. 5 (2015), 107201, http://dx.doi.org/10.1063/1.4933024. [32] W. Cai, J. Zhang, D. Yu, P. Feng, J. Wang, A vibration amplitude model for the giant magnetostrictive ultrasonic processing system, J. Intell. Mater. Syst. Struct. 29 (2018) 574–584, http://dx.doi.org/10.1177/1045389X17711818. [33] A. Grunwald, A.G. Olabi, Design of a magnetostrictive (MS) actuator, Sens. Actuators A Phys. 144 (2008) 161–175, http://dx.doi.org/10.1016/j.sna.2007. 12.034. [34] J. Zheng, S. Cao, H. Wang, Modeling of magnetomechanical effect behaviors in a giant magnetostrictive device under compressive stress, Sens. Actuators A Phys. 143 (2008) 204–214, http://dx.doi.org/10.1016/j.sna.2007.10.063. ´ R. Szewczyk, The possibility of utilizing the high permeability [35] A. Bienkowski, magnetic materials in construction of magnetoelastic stress and force sensors, Sens. Actuators A Phys. 113 (2004) 270–276, http://dx.doi.org/10.1016/j.sna. 2004.01.010. [36] K.H. Matlack, J.Y. Kim, L.J. Jacobs, J. Qu, Review of second harmonic generation measurement techniques for material state determination in metals, J. Nondestruct. Eval. 34 (2014) 273, http://dx.doi.org/10.1007/s10921-0140273-5. [37] J. Rostami, P.W.T. Tse, Z. Fang, Sparse and dispersion-based matching pursuit for minimizing the dispersion effect occurring when using guided wave for pipe inspection, Materials 10 (2017) 622, http://dx.doi.org/10.3390/ ma10060622. [38] L. Laguerre, J.C. Aime, M. Brissaud, Magnetostrictive pulse-echo device for non-destructive evaluation of cylindrical steel materials using longitudinal guided waves, Ultrasonics 39 (2002) 503–514, http://dx.doi.org/10.1016/ S0041-624X(01)00088-9. [39] F. Seco, J.M. Martin, A.R. Jimenez, Improving the accuracy of magnetostrictive linear position sensors, IEEE Trans. Instru. Meas. 58 (2009) 722–729, http:// dx.doi.org/10.1109/TIM.2008.2005260. [40] F. Baronti, G. Fantechi, R. Roncella, R. Saletti, A new and accurate system for measuring cruising yacht freeboards with magnetostrictive sensors, IEEE Trans. Instru. Meas. 60 (2011) 1811–1819, http://dx.doi.org/10.1109/TIM. 2011.2113050. [41] J. Xu, Y. Li, G. Chen, Effect of tensile force on magnetostrictive sensors for generating and receiving longitudinal mode guided waves in steel wires, J. Sensors (2019), 9512190, http://dx.doi.org/10.1155/2019/9512190. [42] S. Djili, F. Benmeddour, E. Moulin, J. Assaad, F. boubenider, Notch detection in copper tubes immersed in water by leaky compressional guided waves, NDT E Int. 54 (2013) 183–188, http://dx.doi.org/10.1016/j.ndteint.2012.10.009. [43] Z. Fan, M.J.S. Lowe, Interaction of weld-guided waves with defects, NDT E Int. 47 (2012) 124–133, http://dx.doi.org/10.1016/j.ndteint.2012.01.001. [44] R. Carandente, P. Cawley, The effect of complex defect profiles on the reflection of the fundamental torsional mode in pipes, NDT E Int. 46 (2012) 41–47, http://dx.doi.org/10.1016/j.ndteint.2011.11.003.
Biographies Chaoyue Hu received his B.S. in measurement technology and instruments from Wuhan University of Technology of China in 2018. After that, he has been a postgraduate student in Huazhong University of Science and Technology of China. His research interests include magnetostrictive transducer and signal processing. Jiang Xu received his B.S. in electrical engineering, M.S. in mechatronic engineering and Ph.D. in measurement technology and instruments from Huazhong University of Science and Technology of China in 2000, 2005 and 2009. Now he is an associate professor in School of Mechanical Science and Engineering at Huazhong University of Science and Technology of China. His research interests include guided waves, magnetoacoustic transducer and signal processing.