# Analysis of Source Influence on Guided Wave Excitation in Cylindrical Structures Using Spatial Fourier Transform Method

Authors: Yunfei Li; Jiang Xu; Qinghua Li; Guang Chen
Journal: Journal of Sensors
 (2020)

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## Abstract

Guided wave transducers, such as electromagnetic acoustic transducers and piezoelectric transducers, generate multimode waves at a given excitation frequency in a cylindrical structure, making it difficult to detect flaws in such structures. To accurately identify the flaws, the transducers must be well designed to suppress the nonaxisymmetric modes. Instead of using the normal mode expansion (NME) method, a spatial Fourier transform (SFT) method is proposed to analyze source influence on the guided wave excitation in a cylindrical structure. A two-dimensional SFT is performed on the spatial distribution function of the surface loading applied to the cylindrical structure. The spatial distribution function is represented in a cylindrical coordinate system. The circumferential-direction SFT is carried out from the angular coordinate to the circumferential orders of the guided wave modes. The axial-direction SFT is carried out from the axial coordinate to the wavenumbers of the guided wave modes. The results of the two-dimensional SFT represent guided wave excitation capabilities for different circumferential orders and wavenumbers. The specific surface loading conditions on the outer surface of a pipe are analyzed to predict source influence on the guided wave excitation. The results are consistent with those obtained using the NME method. Experiments corresponding to the specific surface loading conditions are carried out on a stainless steel pipe. The results confirm the effectiveness of the SFT method.


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## Body

## 1. Introduction

Cylindrical structures, such as rods and pipes, are widely used in petrochemical and transportation industries. Because of the harsh working environment, these structures would deteriorate and thus pose a threat to safety. Therefore, cylindrical structures must be regularly inspected for corrosion to ensure safe operation [1, 2]. The ultrasonic guided wave technology has been used for inspecting cylindrical structures [3, 4]. This nondestructive testing technology is receiving much attention because of the possibility of long-distance inspection from a single probe position, unreachable area inspection, and high efficiency.Guided waves, which travel in the axial direction of cylindrical structures, comprise longitudinal and torsional mode groups [5, 6]. Typical guided wave transducers based on the piezoelectric effect [7, 8], Lorentz effect [9, 10], or magnetostrictive effect [11–15] are used to generate and receive guided waves from cylindrical structures. The surface excitation sources formed by these transducers generate multimode waves comprising axisymmetric and nonaxisymmetric modes at the given excitation frequency because of the limitations of the structure, size, and other parameters of the transducers. There are many nonaxisymmetric modes which are often with serious dispersion. The dispersion makes data analysis difficult for detection. However, it is advantageous for defect localization and focusing. The multimode waves make it difficult to detect flaws. To accurately identify the defects and improve the sensitivity, it is often desirable to obtain a single mode guided wave. Therefore, it is necessary to suppress specific modes when exciting guided waves. In addition to transducers with a single element, transducers with multiple elements have been used to improve the guided wave mode and frequency selection performance [16]. The physics of the generation process whereby the guided wave modes are introduced in the cylindrical structures must be carefully analyzed to determine the design parameters of the transducers influencing the characteristics of the generated field.Two main theoretical methods have been proposed to evaluate the effects of excitation sources on guided wave excitation: the classical integral transform method [17–19] and the normal mode expansion (NME) method [20–22]. The integral transform methods, such as the Fourier transform, have been developed to solve partial differential equations. Pelts and Rose [23] employed this method to analyze source influence parameters affecting the generated field in an orthotropic plate. Koduru and Rose [24] used this method to analyze a transducer array to control an omnidirectional guided wave mode in plate-like structures. In this method, the parameters of the guided wave transducers affecting the generated field cannot be obtained directly. In the NME method, the fields generated in the structures due to loading are expanded in the form of an infinite series of normal modes, analogous to the expansion of a function in terms of any orthogonal (and complete) set of functions. The parameters affecting the amplitude are solved directly and expressed in terms of the physical properties of the guided wave modes and the excitation source. Ditri et al. [25] employed this method to analyze wedge- and comb-type transducers. From their opinion, the wedge-type transducer can be employed to generate the range of the phase velocities by the source which is proportional to the ratio of the wavelength to the size of the loading region. The comb transducer can excite the guided wave modes with small phase velocities over the wedge-type transducer. Zhang et al. [26] used this method to analyze magnetostrictive transducers. In the NME method, the derivation process of the general expression of unknown amplitude is complicated, and the general expression changes depending on the loading conditions.The Fourier transform is a typical time and frequency domain transform method. It decomposes a function of time into the frequencies that make it up. When the method is extended to spatial domains, it is defined as the spatial Fourier transform (SFT). Because the propagating guided waves are sinusoidal in the time and spatial domains [17, 27], the SFT method can be employed to measure the multimode waves. Tamura [28] used a two-dimensional SFT method to measure reflection coefficients. Alleyne and Cawley [29] used a two-dimensional Fourier transform method to convert the received signals from the time and spatial domains to the frequency and wavenumber domains. Zhang et al. [30] used a one-dimensional SFT method to convert the received signals from the spatial domain to the circumferential order domain. The surface loading applied to the cylindrical structure can be considered as a summation over all the guided wave modes [20]. The surface loading applied using the guided wave transducers is distributed in the circumferential and axial directions. The advantage of the SFT method compared to the NME method in mode choosing during guided wave excitation is that the SFT can be carried out on the spatial domain distribution of the applied surface loading without complex analytical calculations. In addition, the spatial domain distribution of the applied surface loading exhibits periodicity in the circumferential direction and nonperiodicity in the axial direction of the cylindrical structure. Therefore, the circumferential-direction SFT can be carried out from the angular coordinate to discrete circumferential orders and the axial-direction SFT from the axial coordinate to continuous wavenumbers.In this paper, an SFT method is proposed to analyze source influence on guided wave excitation in a cylindrical structure. The spatial distribution function of an arbitrary surface loading condition on the cylindrical structure is represented in a cylindrical coordinate system. The spatial distribution function is converted from the spatial domain to circumferential order and wavenumber domains using the two-dimensional SFT. The circumferential-direction SFT is carried out from the angular coordinate to the circumferential orders of the guided wave modes, whereas the axial-direction SFT is carried out from the axial coordinate to the wavenumbers of the guided wave modes. The results of the two-dimensional SFT represent guided wave excitation capabilities for different circumferential orders and wavenumbers. The two separated one-dimensional SFTs are performed on the separable spatial distribution functions to determine the circumferential and axial amplitude factors. Moreover, the specific surface loading conditions on the outer surface of a pipe are analyzed to predict the source influence on guided wave excitation. The results are compared with those obtained using the NME method. Finally, experiments corresponding to the specific theoretical surface loading conditions are carried out on a stainless steel pipe to test and verify the SFT method.


## 2. Theoretical Background

As shown in Figure1, for the convenience of analysis, the central axis and radial direction of a pipe or rod are made consistent with the longitudinal and polar axes of the cylindrical coordinate system, respectively.Figure 1
Cylindrical structure axis configuration.Generally, because the surface loading applied to the cylindrical structure can be considered as a summation over all the guided wave modes, an arbitrary surface loading condition formed by guided wave transducers on the cylindrical structure to generate guided waves can be expressed in the cylindrical coordinate system as follows:(1)g=∑n=0∞∑m=1∞ϕAmnpr,θ,zqte→ξ,where g denotes the total distribution function of an arbitrary surface loading condition; pr,θ,z and qt denote the spatial and time-domain distribution functions, respectively; ∣Φ∣ is a constant, representing the amplitude coefficient of the pulse excitation; and ∣Amn∣ denotes the amplitude coefficient corresponding to the different guided wave modes and is a function of the position on the dispersion curve of each guided wave mode. The integers n and m denote the circumferential and group orders of the guided wave modes, respectively; eξ→ (ξ=r,θ,z) denotes the dominant vibration direction, corresponding to the unit coordinate vector in the cylindrical coordinate system; and r, θ, and z represent the radial, angular, and axial coordinates of the cylindrical coordinate system, respectively.The elastic waves propagate in the axial direction of the cylindrical structure, with time-domain harmonic solutions. Gazis has given the expressions of the displacement componentsUξξ=r,θ,z of the different guided wave modes in the cylindrical structure in the cylindrical coordinate system as follows [27]:
(2)Ur=Arrcosnθeikze−iωt,Uθ=Aθrsinnθeikze−iωt,Uz=Azrcosnθeikze−iωt,where Aξrξ=r,θ,z denotes the displacement amplitude composed of Bessel functions; k=ω/cp denotes the wavenumbers of the guided wave modes; cp represents the phase velocity; ω=2πf denotes the angular frequency; and i denotes the imaginary unit.As shown in Equation (3), the Fourier transform decomposes a function of time ft into frequencies that compose it. If the original function is a function of the spatial coordinate fx, the SFT can be defined as in Equation (4):
(3)Fω=∫−∞+∞fte−iωtdt,(4)Hτ=∫−∞+∞fxe−iτxdx,where Fω and Hτ denote the Fourier transform of ft and fx; x denotes the spatial coordinate; τ denotes the corresponding domain after the SFT.The propagating guided waves in the cylindrical structure are sinusoidal in both the time and spatial domains, as shown in Equation (2). Generally, a temporal Fourier transform of the time-domain distribution function can be carried out from the time domain to the frequency domain. Therefore, a two-dimensional SFT of the spatial distribution function can be carried out from spatial to circumferential order and wavenumber domains. In the cylindrical coordinate system, the circumferential-direction SFT can be carried out from the angular coordinate to the circumferential orders, and the axial-direction SFT can be carried out from the axial coordinate to the wavenumbers. The results of the two-dimensional SFT represent guided wave excitation capabilities for different circumferential orders and wavenumbers.The following equation is obtained by performing the SFT on the spatial distribution function of Equation (1):
(5)Hn,k=∫−∞+∞∫−∞+∞pθ,ze−inθ+kzdθdz,where Hn,k denotes guided wave excitation capabilities for different circumferential orders and wavenumbers of the guided wave modes.Because the unrolled, infinitely long cylindrical structure exhibits periodicity in the circumferential direction, the spatial distribution function is a periodic function with a period of2π under the angular coordinate, as shown in the following equation:
(6)pθ,z=pθ+2π,z.Therefore, the Fourier series expansion of the spatial distribution function is carried out along the angular coordinate. As such, Equation (5) can be written as follows:
(7)Hn,k=12π∫−∞+∞∫φφ+2πpθ,ze−inθ+kzdθdz,n=0,1,2,3,⋯,where φ denotes an arbitrary initial angle. The axisymmetric modes have the circumferential order n=0; the nonaxisymmetric modes have the circumferential order n≥1.Furthermore, Ditri and Rose give the decomposition formula forpθ,z [20]; the spatial distribution function pθ,z can be represented as a product of two functions, as shown in Equation (8), one describing the angular distribution function, i.e., p1θ, and the other describing the axial distribution function, i.e., p2z.
(8)pθ,z=p1θp2z.The separated one-dimensional SFT can be carried out as shown in Equation (9) by substituting Equation (8) into Equation (7).
(9)Hn,k=H1nH2k,H1n=12π∫φφ+2πp1θe−inθdθ,n=0,1,2,3,⋯,H2k=∫−∞+∞p2ze−ikzdz,where H1n denotes the guided wave excitation capabilities for different circumferential orders and is called the circumferential amplitude factor and H2k denotes the guided wave excitation capabilities for different wavenumbers and is called the axial amplitude factor. The circumferential and axial amplitude factors are related to the design parameters of the guided wave transducers.


## 3. Specific Loading Conditions

The results given in the previous section are applied to a case where the loading condition is due to an equally sized and spaced transducer array, each element of which radiates a uniform pressure distribution across the outer surface of the cylindrical structure, as shown in Figure2.Figure 2
Schematic of an equally sized and spaced transducer array.I1 and I2 denote the number of elements in the circumferential and axial directions of the cylindrical structure, respectively; α denotes the angular extent of each element; β=2π/I1−α denotes the angular extent of each gap; zφ denotes an arbitrary initial axial position; Lα denotes the axial extent of each element; Lβ denotes the axial extent of each gap.The angular distribution function in a cycle fromφ to φ+2π can be written as follows:
(10)p1θ=1,φ+η1−12πI1≤θ≤φ+η1−12πI1+α,0,otherwise,where η1=1,2,3,⋯,I1.The axial distribution function can be written as follows:(11)p2z=1,zφ+η2−1Lα+Lβ≤z≤zφ+η2−1Lα+Lβ+Lα,0,otherwise,where η2=1,2,3,⋯,I2.


### 3.1. Circumferential Amplitude Factor

The circumferential distribution function can be written as follows:(12)H1n=1−2nπie−inφe−inα−11−e−in2π/I1I11−e−in2π/I1≡0,n≠μI1,I1−2nπie−inφe−inα−1,n=μI1,where μ=0,1,2,3,⋯.Equation (12) shows that only modes that have a circumferential order n equal to some integer μ times the number of elements I1 in the circumferential direction of the cylindrical structure can be generated under such a loading condition. In this case, the circumferential amplitude factors of the axisymmetric guided wave modes can be determined as follows:
(13)H10=I1α2π.This is1/2π times the total angular extent of the elements I1α in the circumferential direction of the cylindrical structure. This result shows that the circumferential amplitude factor of the axisymmetric guided wave modes is the same for the I1-element transducers as long as the total angular extent of the elements (I1α) remains the same. The major difference with respect to the number of elements I1 in the circumferential direction of the cylindrical structure is in deciding which of the higher harmonics can be generated and the relative importance of these harmonics. Generally, for a transducer array with I1 elements in the circumferential direction of the cylindrical structure, the next family of modes that can be generated (after the axisymmetric guided wave modes) is those with a circumferential order n equal to I1, the next are those with n=2I1, etc. The same conclusions were obtained by Ditri and Rose [20]. The decibel scale ratio of the circumferential amplitude factor for the μth harmonics (i.e., n=μI1) to the zero-th harmonics is defined as follows:
(14)χμ=−20lgH1μI1H10.Figure3 shows a plot of Equation (14) in terms of the gap-to-element spacing ratio δ, which is defined in
(15)δ=βα.Figure 3
Results of circumferential amplitude factor ratio compared with those obtained using the NME method.The results are consistent with those obtained by Ditri and Rose [20]. When the gap-to-element spacing ratio δ is equal to 1, the next family of modes with a circumferential order n=2I1 cannot be generated, as shown in Figure 3. In Ditri and Rose’s paper [20], they think the second generated harmonics would have an amplitude factor less than -20 dB of the axially symmetric modes at the 0.66 gap-to-element ratio, which had been neglected. However, the calculated results based on NME and SFT methods as shown in Figure 3 are larger than -20 dB when the gap-to-element ratio is less than 0.66. These results mean the second harmonic modes cannot be neglected when the gap-to-element ratio is small.


### 3.2. Axial Amplitude Factor

The axial amplitude factors can be obtained by substituting Eq. (11) into Eq. (9) as follows.
(16)H2k=1−ike−ikzφe−ikLα−11−e−ikLα+LβI21−e−ikLα+Lβ,k≠2hπLα+Lβ,I2−ike−ikzφe−ikLα−1,k=2hπLα+Lβ,where h=0,1,2,3,⋯.Equation (16) goes through several maxima and minima, and the local maxima (and/or minima) occur when the wavenumbers of a given guided wave mode satisfy the following relationship:
(17)k=2hπLα+Lβ.Considering the wavelength of a given guided wave mode, Equation (17) can be rewritten as follows:
(18)λ=Lα+Lβh.The result of the expressionLα+Lβ represents the spacing between the centers of the adjacent elements in the axial direction of the cylindrical structure. Therefore, Equation (18) shows that the axial transducer array has better capabilities in generating guided wave modes which, at the given excitation frequency, have wavelengths that are equal to the spacing between the centers of the adjacent elements in the axial direction of the cylindrical structure divided by some integer. The same conclusions were drawn by Ditri et al. [25].Figure4 shows a normalized-scale plot of Equation (16) in terms of the wavenumbers. The results are consistent with those obtained by Ditri et al. [25].Figure 4
Results of axial amplitude factor compared with those obtained using the NME method withI2=4, zφ=10 mm (arbitrary value), Lα=6 mm, and Lβ=14 mm.


## 3.1. Circumferential Amplitude Factor

The circumferential distribution function can be written as follows:(12)H1n=1−2nπie−inφe−inα−11−e−in2π/I1I11−e−in2π/I1≡0,n≠μI1,I1−2nπie−inφe−inα−1,n=μI1,where μ=0,1,2,3,⋯.Equation (12) shows that only modes that have a circumferential order n equal to some integer μ times the number of elements I1 in the circumferential direction of the cylindrical structure can be generated under such a loading condition. In this case, the circumferential amplitude factors of the axisymmetric guided wave modes can be determined as follows:
(13)H10=I1α2π.This is1/2π times the total angular extent of the elements I1α in the circumferential direction of the cylindrical structure. This result shows that the circumferential amplitude factor of the axisymmetric guided wave modes is the same for the I1-element transducers as long as the total angular extent of the elements (I1α) remains the same. The major difference with respect to the number of elements I1 in the circumferential direction of the cylindrical structure is in deciding which of the higher harmonics can be generated and the relative importance of these harmonics. Generally, for a transducer array with I1 elements in the circumferential direction of the cylindrical structure, the next family of modes that can be generated (after the axisymmetric guided wave modes) is those with a circumferential order n equal to I1, the next are those with n=2I1, etc. The same conclusions were obtained by Ditri and Rose [20]. The decibel scale ratio of the circumferential amplitude factor for the μth harmonics (i.e., n=μI1) to the zero-th harmonics is defined as follows:
(14)χμ=−20lgH1μI1H10.Figure3 shows a plot of Equation (14) in terms of the gap-to-element spacing ratio δ, which is defined in
(15)δ=βα.Figure 3
Results of circumferential amplitude factor ratio compared with those obtained using the NME method.The results are consistent with those obtained by Ditri and Rose [20]. When the gap-to-element spacing ratio δ is equal to 1, the next family of modes with a circumferential order n=2I1 cannot be generated, as shown in Figure 3. In Ditri and Rose’s paper [20], they think the second generated harmonics would have an amplitude factor less than -20 dB of the axially symmetric modes at the 0.66 gap-to-element ratio, which had been neglected. However, the calculated results based on NME and SFT methods as shown in Figure 3 are larger than -20 dB when the gap-to-element ratio is less than 0.66. These results mean the second harmonic modes cannot be neglected when the gap-to-element ratio is small.


## 3.2. Axial Amplitude Factor

The axial amplitude factors can be obtained by substituting Eq. (11) into Eq. (9) as follows.
(16)H2k=1−ike−ikzφe−ikLα−11−e−ikLα+LβI21−e−ikLα+Lβ,k≠2hπLα+Lβ,I2−ike−ikzφe−ikLα−1,k=2hπLα+Lβ,where h=0,1,2,3,⋯.Equation (16) goes through several maxima and minima, and the local maxima (and/or minima) occur when the wavenumbers of a given guided wave mode satisfy the following relationship:
(17)k=2hπLα+Lβ.Considering the wavelength of a given guided wave mode, Equation (17) can be rewritten as follows:
(18)λ=Lα+Lβh.The result of the expressionLα+Lβ represents the spacing between the centers of the adjacent elements in the axial direction of the cylindrical structure. Therefore, Equation (18) shows that the axial transducer array has better capabilities in generating guided wave modes which, at the given excitation frequency, have wavelengths that are equal to the spacing between the centers of the adjacent elements in the axial direction of the cylindrical structure divided by some integer. The same conclusions were drawn by Ditri et al. [25].Figure4 shows a normalized-scale plot of Equation (16) in terms of the wavenumbers. The results are consistent with those obtained by Ditri et al. [25].Figure 4
Results of axial amplitude factor compared with those obtained using the NME method withI2=4, zφ=10 mm (arbitrary value), Lα=6 mm, and Lβ=14 mm.


## 4. Experimental Setup

Experiments corresponding to the specific surface loading conditions were conducted to demonstrate the generation of axisymmetric and nonaxisymmetric guided wave modes in the elastic hollow cylindrical structure. The experiments were conducted on a stainless steel pipe with an outer diameter of 73 mm, a wall thickness of 3 mm, a length of 2505 mm, a density of 7930 kg/m3, a Young’s modulus of 194 GPa, and a Poisson’s ratio of 0.29. Figure 5 shows the dispersion curves of the pipe.Figure 5
Phase velocity dispersion curve of the sample pipe.The excitation signal is a three-cycle sinewave tone-burst modulated by a Hanning window. The center frequency is 64 kHz. Figure6 shows the arrangement of the transmitter and receiver. The central distance between the transmitter and the receiver is 0.6 m, and the center of the transmitter is 0.9 m from the end of the pipe. The axial length of the receiver is 25 mm, and the receiver is divided into 16 channels in the circumferential direction to receive the displacement signals of the different angular positions and extract the nonaxisymmetric guided wave modes. Figure 7 shows the different transmitters corresponding to the specific outer surface loading conditions. The arbitrary initial angle φ and axial position zφ are set to zero.Figure 6
Schematic of the installation position of the transducers on the sample pipe.Figure 7
Schematics of the transmitter arrangements corresponding to the specific loading conditions. (a) No. 1 with ring coil turns: 74. (b) No. 2 with ring coil turns: 140. (c) No. 3 with ring coil turns: 280. (d) No. 4 with ring coil turns: 148. (e) No. 5 with ring coil turns: 66 (f). No. 6 with ring coil turns: 222.
(a)
(b)
(c)
(d)
(e)
(f)The cross-coiled magnetostrictive guided wave transducers [31] comprising solenoid coils, ring coils, and iron-cobalt belts were used in the experiments. The solenoid and ring coils made of enamel-insulated wires were used to provide the axial static bias magnetic field and circumferential dynamic magnetic field in the iron-cobalt belts, respectively. Based on the Wiedemann and Matteucci effects, the transducers can realize the conversion between the magnetic field signal and the circumferential displacement signal in the iron-cobalt belts. Ultrasonic vibration is transmitted between the pipe and the iron-cobalt belt via the couplant. Because of the evenly distributed (largely) static bias magnetic field and dynamic magnetic field, each element of the equally spaced transducer array is assumed to radiate a uniform pressure distribution across the outer surface of the pipe. The iron-cobalt belts of the transmitters have different sizes to meet the different surface loading conditions. Each of the 16 equally spaced channels of the receiver is a 40-turn ring coil, the angular extent of which is 22.5°. The diameters of the enamel insulated wires of the transmitting coils and receiving coils are 0.49 and 0.21 mm, respectively.The RPR-4000 (manufactured by RITEC, Inc.) is employed to generate high-power excitation signals and amplify the received induced voltage signals. The DC power IPD-3012SLU (manufactured by Interlock) is employed to supply DC current to the solenoid coils. The oscilloscope HDO4034 (manufactured by Teledyne LeCroy) is employed to display, average, and store the received signals. Figure8 shows the experimental platform used to collect the data. The peak-to-peak voltage of the excitation signal changes with the change in the transmitter; the frequency range of the band-pass filter is set to 50-800 kHz; the constant current output is set to 5.6 A; the sampling frequency is set to 125 MS/s (megasamples per second); the received signal is averaged 50 times to reduce the white noise; and the signal amplification is set to 36.8 dB. The induced voltage signals are amplified and recorded independently for each channel of the receiver. The total received signals from the 16 channels of the receiver form a 2D data matrix.Figure 8
Photo of the experimental platform.The circumferential and axial amplitude factors are calculated by substituting the parameter settings of the transmitters, shown in Figure7, into Equations (12) and (16). Four types of angular distribution functions and three types of axial distribution functions are observed. Figures 9(a) and 9(b) show the results of the circumferential and axial amplitude factors at a normalized scale, respectively.Figure 9
Normalized circumferential and axial amplitude factors corresponding to the transmitter arrangements shown in Figure7: (a) circumferential amplitude factors; (b) axial amplitude factors.
(a)
(b)Overall, the wave structures of the modes in the same group are similar [17]. The torsional mode group T (n, m) has dominant particle vibration in the angular direction. Therefore, at the given excitation frequency, T (n, 1) (n=0, 1, 2, 3) is mainly considered based on the dispersion curves and wave structures. The wavelengths of T (0, 1), T (1, 1) T (2, 1), and T (3, 1) are 50.000 mm, 51.672 mm, 57.531 mm, and 71.984 mm at 64 kHz, respectively; the corresponding wave numbers are 0.126 mm-1, 0.122 mm-1, 0.109 mm-1, and 0.087 mm-1, respectively.


## 5. Results and Discussions

There are 16 channels in each specific loading conditions; for example, Figure10 is the signal of experimental channel one with α=180 and I1=1. The focus is on the passing signal of each channel.Figure 10
Received channel 1 signal ofα=180 and I1=1.By adding the signals of 16 channels, a nonaxisymmetric guided wave mode with a circumferential ordern=0 can be obtained. To extract the nonaxisymmetric guided wave modes having circumferential orders n≥1, a phase delay of nθk/2π is added to the received signal of each angular position before summation [32]. θk denotes the angular distance from the reference angular position. Figure 11 shows signals of order 0 to 2 with α=180 and I1=1. The maximum amplitude of the absolute value of the intercepted time-domain signal segment is used to indicate the guided wave excitation capabilities.Figure 11
Passing signals of order 0 to 2 withα=180 and I1=1.Figure12 shows the experimental results at a normalized scale. As shown in Equation (2), the final displacement amplitude not only depends on the circumferential and axial amplitude factors, but also on the installation accuracy of the transducers, the sample pipe quality, the noise signals, and the uneven pressure distribution. These factors influence the final received signals and cause errors. The components of circumferential order guided waves in the signal are of primary concern to different arrangements of the transmitter.Figure 12
Experimental results of different transmitter arrangements (corresponding to Figure7) at a normalized scale after signal processing.The experimental results corresponding to the transmitter arrangements, such as in Figures7(a)–7(c), show that the nonaxisymmetric modes are suppressed with the increase in the angular extent of each element. The axisymmetric modes occupy the main component when the angular extent of each element reaches 360°. The experimental results corresponding to Figures 7(a) and 7(d) show that only nonaxisymmetric modes with circumferential order n=2 occupy the main component in addition to the axisymmetric modes because of the two-element transducer array in the circumferential direction of the pipe. These results shown in Figure 12 verify the theoretical prediction of the circumferential amplitude factor as shown in Figure 9(a).The experimental results corresponding to Figures7(a) and 7(e) show that nonaxisymmetric modes with circumferential order n≥2 are enhanced when the axial extent of each element reaches 50 mm, because at the given excitation frequency, the axial amplitude factors of the T (0, 1) and T (1, 1) modes are very low. The experimental results corresponding to Figures 7(a) and 7(f) show that the nonaxisymmetric modes are suppressed because at the given excitation frequency, only the T (0, 1) mode has a wavelength equal to the spacing between the centers of the adjacent elements of the axial transducer array; the other nonaxisymmetric modes do not. These results shown in Figure 10 verify the theoretical prediction of the axial amplitude factor as shown in Figure 9(b).


## 6. Conclusions

The proposed two-dimensional SFT method can be used to analyze source influence on guided wave excitation in cylindrical structures. The advantage of this method is that only the spatial distribution function of an arbitrary surface loading condition on the cylindrical structure needs to be considered. The SFT method is easy to implement in practice owing to the simple algorithm. The results of the two-dimensional SFT represent guided wave excitation capabilities for different circumferential orders and wavenumbers. The circumferential and axial amplitude factors can be obtained by performing the separated one-dimensional SFT on the angular and axial distribution functions, respectively. The theoretical results of the source influence on the guided wave excitation of specific surface loading conditions on the outer surface are consistent with those obtained using the NME method. The experimental results corresponding to the specific surface loading conditions are consistent with the theoretical predictions made using the SFT method. To suppress the nonaxisymmetric modes and enhance the axisymmetric modes, the angular extent of each transducer needs to be as close as possible to2π/I1 rad; the number of transducer arrays in the circumferential direction of the cylindrical structure should be greater than the maximum circumferential order of the guided wave modes at the given excitation frequency; the axial extent of each transducer and the spacing between the centers of the adjacent elements of the axial transducer array should match with the wavelengths of the axisymmetric modes at the given excitation frequency. The method provided in the paper can also be applied to guide the design process of guided wave transducers with different surface loading conditions. A time and spatial Fourier transform method which includes the time-delay function will be conducted in future works.



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