Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 25 Journal of Zhejiang University-SCIECE A (Applied Physics & Engineering) ISS 673-565X (Print); ISS 862-775 (Online) www.ju.edu.cn/jus; www.springerlink.co E-ail: jus@ju.edu.cn Excitation of axisyetric and non-axisyetric guided waves in elastic hollow cylinders by agnetostrictive transducers * Xiao-wei ZHAG, Zhi-feng TAG 2, Fu-ai LV, Xiao-hong PA ( Institute of Modern Manufacture Engineering, Zhejiang University, Hanghou 327, China) ( 2 Institute of Advanced Digital Technologies and Instruentation, Zhejiang University, Hanghou 327, China) E-ail: tanghifeng@ju.edu.cn Received June 2, 25; Revision accepted Oct. 3, 25; Crosschecked Feb. 5, 26 Abstract: Ultrasonic guided waves have been successfully applied in nondestructive evaluation (DE) and structural health onitoring (SHM) of pressure vessels and pipelines due to their advantages, such as long detection range and high inspection efficiency. Copared with other ultrasonic guided wave actuators, agnetostrictive transducers are ore cost-effective, involve sipler fabrication process, and have higher possible transduction efficiency. The noral ode expansion (ME) ethod is adopted to analye the forced response and perturbation analysis of elastic hollow cylinders with respect to agnetostrictive loadings, including partial loading, axial array loading, and circular array loading. The phase velocity and frequency spectra of axisyetric/non-axisyetric guided waves excited by agnetostrictive transducers are analyed. The theoretically predicted trends are verified by finite eleent nuerical siulations and experients. Key words: Guided waves, Hollow cylinder, oral ode expansion (ME), Magnetostriction http://dx.doi.org/.63/jus.a584 CLC nuber: TB559; O347 Introduction Ultrasonic guided waves have already been deonstrated by various research works to have great potential and a good prospect in online nondestructive evaluation (DE) and long-ter structural health onitoring (SHM) of pipelines of various industries (Rose, 24) due to its various corresponding advantages, such as single-point excitation, long detection range, high inspection efficiency, and % cross-sectional detectability. The three ain types of transducers that are currently in service for guided wave inspection of pipelines are pieoelectric Corresponding author * Project supported by the ational atural Science Foundation of China (os. 62784 and 5275454), and the Fundaental Research Funds for the Central Universities, China ORCID: Xiao-wei ZHAG, http://orcid.org/-2-3995-595; Zhi-feng TAG, http://orcid.org/--559-875 Zhejiang University and Springer-Verlag Berlin Heidelberg 26 transducers (Alleyne and Cawley, 996; Marty, 22), agnetostrictive transducers (Kwun and Bartels, 998; Ki Y.Y. et al., 25; Cho et al., 26; Turcu, 28; Lee et al., 29; Ki Y.G. et al., 2), and electroagnetic acoustic transducers (Ribichini, 2). Magnetostrictive transducers are ore cost-effective, involve relatively sipler fabrication process, and have higher transduction efficiency than other types of transducers. However, it is highly iportant to develop a theoretical approach to consider the forced response and perturbation analysis of elastic pipes under agnetostrictive loadings, and the excitation of axisyetric and non-axisyetric guided waves in elastic hollow cylinders by finite sie agnetostrictive transducers, with the unique feature of agnetostriction taken into consideration. These topics are rarely reported in the literature, but further investigation can provide guidance for the further developent of agnetostrictive guided wave transducers. Probles involving the forced response of a structure are typically solved using one of the
26 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 following two approaches (Rose, 24): integral transfor technique or noral ode expansion (ME) ethod. The integral transfor technique solves the guided wave excitations through transfors, such as Laplace, Fourier, Hankel, and Mellin. The ME ethod, on the other hand, analyes the wave fields in the cylinder in the for of a su of an infinite nuber of noral odes. Gais (959) showed that there exist an infinite nuber of noral odes in an elastic hollow cylinder, each with its own characteristics, such as phase velocity, group velocity, and wave structure profile. He obtained the general solution of haronic waves propagating in an infinite long hollow cylinder, which has been very beneficial for long-range guided wave inspection on widely distributed pipelines. The forced response proble in a hollow cylinder proble was first studied by Ditri and Rose (992) using the ME ethod to obtain the aplitude factors of different guided wave odes. Li and Rose (2a) investigated the field distribution of non-axisyetric longitudinal waves. The angular profile was calculated by taking into account the aplitude factors of every excited ode. The ME ethod was also used by Luo (25), Zhang (25), Mu (28), and Mu and Rose (28) in their studies. Several results have been reported on the source influence analysis of hollow cylinders with respect to pieoelectric loadings. However, a coprehensive perturbation analysis on pipes with respect to agnetostrictive loadings, considering unique features of agnetostriction, is essential for further developent of guided wave agnetostrictive transducers and transducer arrays. In this studies, the classical ME ethod is adopted to study the excitation of axisyetric and non-axisyetric guided wave odes in linear elastic hollow cylinders by agnetostrictive actuation. The phase velocity and frequency spectra of axisyetric and non-axisyetric guided wave odes excited by agnetostrictive transducers are odeled with a large static bias agnetic field assuption. The influence of load paraeters is analyed. Several novel results have been found about the source influence of different agnetostrictive loadings. The prediction of guided wave excitation by the ME ethod is verified by finite eleent nuerical siulations and experients. 2 ME ethod 2. oral odes of a hollow cylinder The ME ethod solving the forced loading proble is analogous to the eigenfunction expansion ethod in the field of atheatics, and the noral odes of cylinders serve as the eigenfunctions. The ain idea of the ME ethod is to assue that the sought function can be written in the for of a series of known functions, i.e., the noral odes, each with an unknown aplitude. The goal is then to find a general expression for the unknown aplitudes or a nuerical estiate of the. ote that the efficiency of ME ethod depends on two ain considerations: copleteness and orthogonality of the noral odes (Rose, 24). Consider a hollow cylinder in a cylindrical coordinate syste as shown in Fig.. There exists an infinite nuber of propagating odes in the hollow cylinder. The velocity field due to a noral ode with circuferential order in the th faily can be written as (Ditri and Rose, 992; Li and Rose, 2a): v i( tk ) (, r )e Rr() r r ( ) er R () r ( ) e i( tk ) R () r ( ) e e, () where ω and k are the angular frequency and wave nuber, respectively. Functions R(r) and ϕ(θ) denote the radical and angular field distributions of the velocity coponent of the noral ode with circuferential order in the th faily, respectively. The phase velocity dispersion curves of an elastic pipe with the paraeters given in Table solved using the sei-analytical finite eleent ethod (Marani, 28; Zhang et al., 24) for the axisyetric and non-axisyetric odes are shown in Fig. 2. ote that the curves in different colors denote odes with different circuferential orders. According to the ME ethod, the generated particle velocity can then be expanded as v v (2) it it e A e,,
Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 27 where A is the ME aplitude of the ode with circuferential order in the th faily containing ik wave coponent e. where v and T represent the particle velocity and stress field of a solution to the linear elastic wave propagation equation, respectively. Let v and T, v 2 and T 2 denote different noral odes of the cylinder: ik ik v ve, T T e, M M M ikn M ikn v2 vn e, T2 Tn e. (4) After substituting Eq. (4) into Eq. (3), we obtain Fig. Stress-free hollow cylinder in a cylindrical coordinate syste R i and R o are the inner and outer radii, and D and 2 D are the inner and outer boundaries, respectively v T v T M* M M* M* 4i( k kn ) Pn r ( n n )d, D where D is the cross section and P M P n is defined as M M * M * n D n n (5) d. 4 v T v T e (6) For two different waveguide odes, the orthogonality relation can then be written as M M P, unless M and k k. (7) n n 2.3 ME aplitude Fig. 2 Phase velocity dispersion curves for L(, ), T(, ), and F(, ) odes in a steel pipe. ote: for interpretation of the references to color in this figure, the reader is referred to the web version of this article 2.2 Orthogonality of noral odes Before calculating the aplitude factor A by using the ME ethod, the orthogonality relation between the noral odes should be established. We start with the coplex reciprocity relation (Auld, 99), which is described as ( v T v T ), (3) * * 2 2 In the coplex reciprocity relation equation (3), let v represents the actual particle velocity field in the cylinder and v 2 denotes the particle velocity field of a noral ode, which is given as v v A ( ) v( r, ), M M ikn v2 vn (, r )e. (8) Using Eqs. (3), (7), and (8), after soe interediary transforations, one obtains d 4P i k A ( ) d v ( T n )d s v ( T n )d s, * * D 2D 2 (9) Table Diensions and aterial properties of a steel pipe Diension Material property Outer diaeter, D o () Thickness, h () Density, ρ (kg/ 3 ) Young s odulus, E (GPa) Poisson s ratio, υ 4 4.5 78 2.28
28 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 where T n and T n 2 are the loading conditions on the inner and outer boundaries, D and 2 D, respectively, of the cylinder. Considering the traction-free condition on the inner and outer boundaries of the hollow cylinder (Fig. ), Eq. (9) can be integrated with the result, that is, the general for of the forward noral ode aplitude: A ik c e ( ) e 4P ik * ( )ds v T n D * ( 2 )ds v T n d. 2D () Only source condition on the outer boundary will be considered in this study. 3 Modeling of agnetostrictive transducers Typical agnetostrictive transducers are odeled in this study (Hirao and Ogi, 23; Ribichini et al., 2; 2; 22). Ferroagnetic aterials have the property that when placed in a agnetic field, they are echanically defored. This property is called the agnetostrictive effect. The reverse phenoenon in which the agnetic induction of the aterial changes when it is echanically defored is called inverse agnetostrictive effect. The agnetostrictive phenoenon and its inverse effect can be utilied for the generation and detection of ultrasonic guided waves. Magnetostrictive guided wave transducers, including longitudinal wave transducer based on Joule effect and its inverse effect and torsional wave transducer based on Wiedeann effect and its inverse effect, can produce axial and circuferential loading, respectively. The agnetostrictive effect relating the agnetic and echanical aterial states can be assued to take a siilar for as that for the pieoelectricity (Hirao and Ogi, 23; Ribichini et al., 2): S st dh, T B d T μh, () where S and T are strain and stress, while H and B are agnetic field and agnetic induction, respectively, s is the elastic copliance atrix, μ is the agnetic pereability atrix, and d= S/ H is the pieoagnetic coupling atrix. Eq. () assues a linear S H relation, but it is experientally known that ferroagnetic aterials exhibit a highly nonlinear agnetostriction curve. In the case of agnetostrictive guided wave transducers, a sall dynaic agnetic field H is superiposed on a uch larger static bias agnetic field H ; therefore, the linear assuption is locally valid. When the sall dynaic agnetic field H is superiposed on a larger static bias agnetic field H, the resulting strain can be decoposed into a static coponent S and a dynaic coponent S (Ribichini et al., 2): H=H+H, S=S+S. (2) Here only the dynaic coponents of the constitutive equations are considered: S=sT+dH, T B=d T+μH, (3) where the dynaic agnetostriction atrix d depends on the direction of the agnetiation. ote that the linearied equations are valid in the large static bias agnetic field assuption, even when static coponent of linear constitutive equations is invalid. The first equation of Eq. (3) accounts for the direct agnetostrictive effect used in the generation of elastic waves, while the second equation describes the inverse agnetostriction used in the detection process. Both the static bias agnetic field H and sall dynaic agnetic field H of longitudinal wave transducers are applied along the direction, as is shown in Fig. 3. When H is uch saller than H, the coupling atrix can be approxiated as
Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 29 3S t H 3S t d, (4) H 2 2 T Since only the S 4 shear strain coponent is responsible for generation of torsional waves, the constitutive equations for torsional wave transducers can also be siplified as S 4 s44t 4 d43h, B d43t 3 3H. (7) where γ= S 3 / H is the slope of agnetostriction curve and S t is the total agnetostrictive strain. ote that the S 3 strain coponent is solely responsible for the generation of longitudinal waves, so the constitutive equations for longitudinal wave transducers can be siplified and given as S 3 s33t 3 d33h, B d 333 3H. (5) ote that thin agnetostrictive patches such as nickel and ferrocobalt strips are often used to generate and easure elastic waves. The agnetostrictive longitudinal and torsional wave transducers described above can be odeled using coercial finite eleent software packages, such as COMSOL Multiphysics. r H H H θ Longitudinal waves Fig. 4 Scheatic of torsional wave transducers on the outer surface of a pipe Fig. 3 Scheatic of longitudinal wave transducers on the outer surface of a pipe For torsional wave transducers, the static bias agnetic field is applied along the θ direction, while the sall dynaic agnetic field is applied along the direction, as is shown in Fig. 4. Under the large static bias agnetic field assuption, the coupling atrix for torsional wave transducers can also be approxiated as 3S t H d. (6) 2 2 3S t H T 4 Perturbation analysis of cylinders under agnetostrictive loadings As described in Section 3, agnetostrictive transducers are capable of providing circuferential and axial loadings (T=s (S dh)) on the surface of a hollow cylinder. Therefore, only circuferential and axial agnetostrictive loadings on the outer boundary of the cylinder are considered in this study, which can be written as T n, (8) p( ) p2 ( ) e /, L,, r Ro, Tn 2 (9), L,, r Ro, where p (θ) and p 2 () are circuferential and axial loading functions, and L and α are axial and circuferential extent of the loading area, respectively.
22 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 Substituting Eqs. (8) and (9) into Eq. () gives ik ()e b ik 2 4P R A (), p () p (),e, (2) L (, ), where, p ( ) and 2 ( ),e i p k are the circuferential and axial aplitude factors, respectively, which are defined as 2π p H b p, ( ) ( ) ( )d, (2) ik ik 2 2 p ( ),e H p ( )e d. (22) 4. Magnetostrictive partial loading A agnetostrictive partial loading is nonaxisyetric and only covers a portion of the pipe surface over soe circuferential angle, as is shown in Fig. 5. The loading distribution functions p (θ) and p 2 () are given as (Shin and Rose, 999) P,, p ( ),, P2, L, p2 ( ), L. (23) (24) Fig. 5 A hollow cylinder with partial loading of a transducer with 2L width and 2α circuferential coverage angle Substituting Eqs. (23) and (24) into Eqs. (2) and (22), the source functions can be obtained: H bp (2 ),, sin( ) 2 bp,, H 2P2 sin( k L). k (25) (26) According to Eq. (25), the circuferential aplitude factor H θ is a function of circuferential order and loading angle α, but is independent of phase velocity and frequency. There is a linear relation between the axisyetric circuferential aplitude factor and loading angle α, while the nonaxisyetric circuferential aplitude factor follows a sinusoidal trend as a function of the loading angle α for a certain circuferential order. Also, H θ = ( ) when 2α=2π, which indicates that only axisyetric odes will be excited if the agnetostrictive load is axisyetric. If α=pπ, p=,, 2,, then H θ = ( ), which iplies that noral odes with circuferential order will not be excited. The relative circuferential aplitude factors for different loading angles of 45, 9, 8, and 36 are shown in Fig. 6. It is evident that the aplitude of non-axisyetric flexural odes decreases, while the loading angles of agnetostrictive partial loading increase. The flexural odes are copletely suppressed when the loading angle approaches 36. According to Eq. (26), the axial aplitude factor H is a function of loading length 2L and wavenuber k of ode with circuferential order in the th faily. Furtherore, H will reach the axiu when k L =(2p+)π/2, p=,, 2, ; thus, 2 L(2p ) /2, p=,, 2,, are efficient loading lengths for a ode at a particular frequency. The axial aplitude factor can be rewritten as a function of frequency f, phase velocity Cp, and loading length 2L such that H 2P2 sin 2π f ( Cp ) L. (27) 2π f ( Cp ) The phase velocity spectra (f=32 kh) of H with different loading lengths 2L are shown in Fig. 7. It can be observed that H has a very broad phase
Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 22.8 2α=45 (a).6.4.2 2 3 4 5 6 7 8 9 Circuferential order.8.6 2α=9 (b).4.2 2 3 4 5 6 7 8 9 Circuferential order.8.6 2α=8 (c).4.2 2 3 4 5 6 7 8 9 Circuferential order.8.6 2α=36 (d).4.2 2 3 4 5 6 7 8 9 Circuferential order Fig. 6 Relative circuferential aplitude factors for axisyetric and non-axisyetric odes at loading angles of 2α=45 (a), 9 (b), 8 (c), and 36 (d) Fig. 7 Phase velocity spectra of the axial aplitude factor at different paraeters: (a) 2L=2.7 ; (b) 2L= 25.4 ; (c) 2L=5.8 ; (d) 2L=.6 velocity bandwidth, which eans it has poor phase velocity selectivity. The frequency spectra of H for axisyetric odes T(, ) and L(, 2) are shown in Fig. 8, which indicates that the axial aplitude
222 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 factor has different frequency sensitivities for different loading lengths with the extree point corre- sponding to the condition 2 L(2p ) /2, p=,, 2,. Substituting Eqs. (25) and (26) into Eq. (2), the aplitude of noral ode with circuferential order in the th faily of an elastic hollow cylinder under agnetostrictive partial loading is obtained. A saple calculation of aplitude factor is carried out and shown in Fig. 9 for T(, ) and F(, 2) odes generated by a 45 circuferential loading at 28 kh in an elastic pipe with the paraeters given in Table. oralied aplitude.8.6.4.2 2 3 4 5 6 7 8 9 Circuferential order Fig. 9 Aplitude factors for T(, ) and F(, 2) odes generated by a 45 agnetostrictive circuferential loading at 28 kh in an elastic pipe 4.2 Magnetostrictive axial array loading Consider the case of axial array loading in a cob transducer (Ditri et al., 993; Rose et al., 998; Hay and Rose, 22; Philtron and Rose, 24), as is shown in Fig., where η agnetostrictive eleents are equally spaced along the axial direction of a cylinder. Here, 2β is the eleent width and 2δ is the spacing. The loading distribution functions p (θ) and p 2 () (without tie delay) can be written as Fig. 8 Frequency spectra of the axial aplitude factor at different paraeters: (a) T(, ), 2L=25.4 ; (b) T(, ), 2L=5.8 ; (c) L(, 2), 2L=25.4 ; (d) L(, 2), 2L=5.8 where p ( ) P, [,2π), (28) P2, G, p2 ( ), G, G /2, (29) g :(2 ) 2( ) (2 ) 2. g Substituting Eqs. (2) and (22) into Eqs. (4) and (5), the source functions are obtained as:
Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 223 H g 2π bp,,,, /2 ik sin( ) k H P2e d P2 k D k D cos [( ) ] cos [( ) ], cos(2 k D) where the eleent distance 2D=2β+2δ. (3) (3) According to Eq. (3), the axial aplitude factor depends on the eleent width 2β (H β ), the eleent distance 2D, and the nuber of eleents η (H D ). The extreu conditions of H β and H D are tan( k )= k and tan( k D)= tan( k D), 2β 2D respectively, where k pπ/ D, p=,, 2,, and 2δ Fig. Scheatic of elastic hollow cylinders under agnetostrictive axial array loading 2 D/ p, p=,, 2,. The phase velocity spectra (f=64 kh) of axial aplitude factor at different values of 2β, 2D, and η are shown in Fig.. It is obvious that agnetostrictive axial array loading has good phase velocity selectivity. Furtherore, it can be seen fro Figs. a and b that the center phase velocity is ainly dependent on eleent distance 2D; a larger eleent distance corresponds to a higher phase velocity. Fro Figs. a and c, a conclusion can be drawn that the phase velocity bandwidth is ainly dependent on the nuber of eleents η. Moreover, it can be seen fro Figs. b and d that the eleent width 2β akes a contribution to the aplitude of H at the center phase velocity. oralied aplitude oralied aplitude oralied aplitude oralied aplitude.8.6.4.2 2 4 6 8 Phase velocity (/s).8.6.4.2 2 4 6 8 Phase velocity (/s).8.6.4.2 2 4 6 8 Phase velocity (/s).8.6.4.2 η=4 2D=4 2β=25.4 η=4 2D=6 2β=25.4 η=8 2D=4 2β=25.4 η=4 2D=6 2β=2.7 2 4 6 8 Phase velocity (/s) Fig. Phase velocity spectra of axial aplitude factor for axial array loading at different paraeters: (a) η=4, 2D=4, 2β=25.4 ; (b) η=4, 2D=6, 2β= 25.4 ; (c) η=8, 2D=4, 2β=25.4 ; (d) η=4, 2D= 6, 2β=2.7 The frequency spectra of H for axisyetric odes T(, ) and L(, 2) are shown in Fig. 2. The (a) (b) (c) (d)
224 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 influence of array paraeters 2β, 2D, and η on the frequency spectru is siilar to that on the phase velocity spectru. Furtherore, if tie delay e iω(ξ td) is applied to each eleent, the cob-type agnetostrictive transducer will provide even better phase velocity, frequency selectivity, and guided wave ode controllability. 4.3 Magnetostrictive circular array loading Consider the circular array loading case (Li and Rose, 2b; 22), as is shown in Fig. 3, where η equally sied (2L 2α) agnetostrictive eleents are equally spaced along the circuferential direction of a cylinder with their circuferential position at θ= (2ξ )π/η, ξ=, 2,, η. The loading distribution functions p (θ) and p 2 () can be written as P,, G p ( ) (32), G, where g G g (2 )π (2 )π :, p ( ) 2 P, L,, 2 (33), otherwise. Fig. 3 Scheatic of elastic hollow cylinders under agnetostrictive circular array loading Substituting Eqs. (32) and (33) into Eqs. (2) and (22), the source functions are obtained as follows: Fig. 2 Frequency spectra of axial aplitude factor for axial array loading at different paraeters: (a) T(, ), 2D=25.4 ; (b) T(, ), 2D=5.8 ; (c) L(, 2), 2D= 25.4 ; (d) L(, 2), 2D=5.8 bp (2 ),, H sin( ) (2 )π 2bP cos,, sin( p) p 2 bp ( ), p, p, p, where p,, 2,, (34)
Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 225 H 2P2 sin( k L). k (35) Eq. (35) is the sae as Eq. (26), and thus, the conclusions are not repeated here. According to Eq. (34), only odes with circuferential order =pη, p=,, 2, can be excited in a hollow cylinder with a agnetostrictive circular array loading. The relative circuferential aplitude factor with respect to that of axisyetric odes can be defined as H sin( p) sin( pπ) =, (36) H p pπ where Δ=η(2α)/2π, Δ[, ] is the loading area ratio. As can be seen fro Fig. 4, the relative aplitude of higher haronics decreases as the loading area ratio Δ increases and the relative aplitude decreases at a faster rate for higher circuferential orders, which indicates that agnetostrictive circular array loading can be designed to generate a specific circuferential order of the guided wave odes. In addition, bea steering (Wilcox, 23) and bea focusing (Hayashi et al., 25; Sun et al., 25) can be achieved when tie delays are applied to the array eleents. In this section, suaried finite eleent nuerical evaluations (Zhu, 2; Drod, 28; Moreau et al., 22) and experients deonstrate the generation of axisyetric and non-axisyetric guided wave odes in elastic hollow cylinders under agnetostrictive loadings. A steel pipe, with paraeters given in Table, is considered in this study with the corresponding phase velocity dispersion curves shown in Fig. 2. 5. Magnetostrictive partial loading 5.. Axial partial loading siulation The first nuerical evaluation involves the axial partial loading at f=28 kh, with a loading area of axial length 5.8 and a circuferential angle 45. According to the ME ethod, the angular profile for L(, 2) and F(, 3) odes at.5 fro the source was calculated and is shown in Fig. 5. Meanwhile, a odel with the sae paraeters is established and calculated by using the finite eleent software Abaqus/Explicit. The axial displaceents (U ) of nodes at =.5 are recorded, and their axial displaceent angular distribution is plotted, which is shown in Fig. 5. The results deonstrated that these two odels are in good agreeent...8.6.4.2 p=4 p=3 p=2 p=.2.4.6.8. Loading area ratio, Fig. 4 Relative aplitudes of higher haronics for circular array loading 5 uerical siulation and experient Fig. 5 Angular profiles of L(, 2) and F(, 3) odes generated by 45 agnetostrictive axial loading at 28 kh in an elastic pipe: (a) theoretical prediction; (b) nuerical siulation result 5..2 Circuferential partial loading experient The experient setup is shown in Fig. 6, where a sall agnetostrictive transducer with a width of 5.8 is bonded on the surface of the pipe as a transitter that covers an angle of 45 along the circuference of the pipe. The central transitting frequency is f=25 kh. Thirty-two receivers were placed around the circuference of the pipe at a distance of.6 fro the transitter. The theoretical and experiental results of angular
226 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 profiles for the T(, ) and F(, 2) odes at the frequency of 25 kh are shown in Fig. 7, which are in good agreeent. Since both theoretical and experiental results are in good agreeent, the earlier described ethod involving agnetostrictive partial loading was verified by nuerical evaluation and experient. The wave structures of L(, ) and L(, 2) at f=28 kh are shown in Fig. 9. It can be observed that radial displaceent (U r ) plays a doinant role for L(, ) ode, whereas the axial displaceent (U ) for L(, 2) ode at f=28 kh. Both r-coponent and -coponent of the displaceent field on the external surface of the pipe at the distance of.7 fro the source are recorded and converted into video signals, which are shown in Fig. 2. The above observation is confired once again, but it also shows that the odes, especially L(, ), are not as pure as expected, as radial displaceent is doinant for L(, ) while the external load is in the axial direction. Therefore, radial load, which the agnetostrictive transducer cannot provide, ay have the best chance to excite pure L(, ) ode. Fig. 6 A setup of circuferential partial loading experient Fig. 8 Snapshot of the wave propagation of L(, ) (a) and L(, 2) (b) odes in the steel pipes (a) Paraeter (): 2β=2, 2δ=4 ; (b) Paraeter (2): 2β=5, 2δ=6 Fig. 7 Angular profiles of T(, ) and F(, 2) odes generated by 45 agnetostrictive circuferential loading at 25 kh in an elastic pipe: (a) theoretical prediction; (b) experiental result 5.2 Magnetostrictive axial array loading Another nuerical evaluation involved an axial array loading at f=28 kh. Two sets of array paraeters were designed for the purpose of L(, ) excitation and L(, 2) excitation, respectively: () 2β=2, 2δ=4 and (2) 2β=5, 2δ=6. Finite eleent odels were established and calculated. Fig. 8 shows the snapshot for wave propagation of L(, ) ode and L(, 2) ode. The L(, ) ode is doinant on the condition of paraeter () and L(, 2) ode is doinant on the condition of paraeter (2), which deonstrates the ode controllability of agnetostrictive axial array load. Fig. 9 Wave structure of L(, ) (a) and L(, 2) (b) odes at 28 kh in the steel pipe
Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 227 A two-eleent agnetostrictive transducer is bonded around the circuference of the pipe with the loading area ratio Δ=.5 and the excitation frequency f=25 kh. Thirty-two receivers are placed around the circuference of the pipe at a distance of.6 fro the transitter. The received signals fro 32 receivers for a 2D data atrix. A 2D Fourier transforation is perfored on the data atrix, and the circuferential order of excited odes is extracted. Fig. 22 verifies the theoretical prediction once again. Fig. 2 Video signals extracted fro the excitation of L(, ) and L(, 2) odes (a) Paraeter (): 2β=2, 2δ=4 ; (b) Paraeter (2): 2β=5, 2δ=6 Fig. 2 Circuferential orders of excited odes for circular array axial load nuerical siulation 5.3 Magnetostrictive circular array loading 5.3. Circular array axial load siulation The last nuerical evaluation involves circular array axial load at f=64 kh, where the nuber of eleents η=2 and the loading area ratio Δ=.5. The finite eleent odel is established and used for calculation. Displaceents of all the nodes at the distance of.5 fro the source are recorded. The received signals for a 2D tie-circuference data atrix. A Fourier transforation is perfored on the data atrix in the circuferential direction, and the circuferential order of the excited odes is extracted. Fig. 2 shows that only odes with circuferential order =2p, p=, 2, are excited, which is in accordance with the theoretical prediction. 5.3.2 Circular array circuferential load experient Fig. 22 Circuferential orders of excited odes for circular array circuferential load experient 6 Conclusions This paper has adopted and developed the classical ME ethod to analye forced response and perturbation analysis of a hollow cylinder considering the finite sie agnetostrictive loadings, including partial loading, axial array loading, and circular array loading, with the unique feature of agnetostriction taken into consideration. The phase velocity and frequency spectra of guided wave aplitude
228 Zhang et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 26 7(3):25-229 excited by agnetostrictive loading are analyed in detail, and soe basic conclusions have been reached. Soe novel results of a coprehensive perturbation analysis on elastic hollow cylinders under agnetostrictive loadings are suaried as follows. Magnetostrictive partial loading is able to excite guided wave odes of the sae faily, but has poor phase velocity and frequency selectivity. Axisyetric loading excites axisyetric odes only. Angular profiles in a hollow cylinder are sensitive to the circuferential loading length. Magnetostrictive axial array loading has good phase velocity and frequency selectivity and guided wave ode controllability. According to phase velocity and frequency spectra, center phase velocity and center frequency are ainly deterined by eleent distance. Magnetostrictive circular array load only excites guided wave odes whose circuferential order is integer ultiple of the nuber of eleents. References Alleyne, D.., Cawley, P., 996. The excitation of Lab waves in pipes using dry-coupled pieoelectric transducers. Journal of ondestructive Evaluation, 5(): -2. http://dx.doi.org/.7/bf733822 Auld, B.A., 99. Acoustic Fields and Waves in Solids, Vol. 2. Krieger Publishing Copany, Malabar, Florida, USA. Cho, S.H., Han, S.W., Park, C., et al., 26. oncontact torsional wave transduction in a rotating shaft using oblique agnetostrictive strips. Journal of Applied Physics, ():493. http://dx.doi.org/.63/.2386942 Ditri, J.J., Rose, J.L., 992. Excitation of guided elastic wave odes in hollow cylinders by applied surface tractions. Journal of Applied Physics, 72(7):2589-2597. http://dx.doi.org/.63/.35558 Ditri, J.J., Rose, J.L., Pilarski, A., 993. Generation of guided waves in hollow cylinders by wedge and cob type transducers. In: Thopson, D.O., Chienti, D.E. (Eds.), Review of Progress in Quantitative ondestructive Evaluation. Springer US, ew York, 2:2-28. http://dx.doi.org/.7/978--465-2848-7_26 Drod, M.B., 28. Efficient Finite Eleent Modeling of Ultrasonic Wave in Elastic Media. PhD Thesis, Iperial College London, London, UK. Gais, D.C., 959. Three diensional investigation of the propagation of waves in hollow circular cylinders. I. Analytical foundation. The Journal of the Acoustical Society of Aerica, 3(5):568-573. http://dx.doi.org/.2/.97753 Hay, T.R., Rose, J.L., 22. Flexible PVDF cob transducers for excitation of axisyetric guided waves in pipe. Sensors and Actuators A: Physical, ():8-23. http://dx.doi.org/.6/s924-4247(2)44-4 Hayashi, T., Kawashia, K., Sun, Z.Q., et al., 25. Guided wave focusing echanics in pipe. Journal of Pressure Vessel Technology, 27(3):37-32. http://dx.doi.org/.5/.9929 Hirao, M., Ogi, H., 23. EMATs for Science and Industry: oncontacting Ultrasonic Measureents. Kluwer Acadeic, Boston, USA. http://dx.doi.org/.7/978--4757-3743- Ki, Y.G., Moon, H.S., Park, K.J., et al., 2. Generating and detecting torsional guided waves using agnetostrictive sensors of crossed coils. DT & E International, 44(2):45-5. http://dx.doi.org/.6/j.ndteint.2..6 Ki, Y.Y., Park, C., Cho, S.H., et al., 25. Torsional wave experients with a new agnetostrictive transducer configuration. The Journal of the Acoustical Society of Aerica, 7(6):3459-3468. http://dx.doi.org/.2/.9434 Kwun, H., Bartels, K.A., 998. Magnetostrictive sensor technology and its applications. Ultrasonics, 36(-5): 7-78. http://dx.doi.org/.6/s4-624x(97)43-7 Lee, J.S., Ki, Y.Y., Cho, S.H., 29. Bea-focused shearhoriontal wave generation in a plate by a circular agnetostrictive patch transducer eploying a planar solenoid array. Sart Materials & Structures, 8(): 59. http://dx.doi.org/.88/964-726/8//59 Li, J., Rose, J.L., 2a. Excitation and propagation of nonaxisyetric guided waves in a hollow cylinder. The Journal of the Acoustical Society of Aerica, 9(2): 457-464. http://dx.doi.org/.2/.3529 Li, J., Rose, J.L., 2b. Ipleenting guided wave ode control by use of a phased transducer array. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 48(3):76-768. http://dx.doi.org/.9/58.9278 Li, J., Rose, J.L., 22. Angular-profile tuning of guided waves in hollow cylinders using a circuferential phased array. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 49(2):72-729. http://dx.doi.org/.9/tuffc.22.59849 Luo, W., 25. Ultrasonic Guided Waves and Wave Scattering in Viscoelastic Coated Hollow Cylinder. PhD Thesis, The Pennsylvania State University, State College, USA. Marty, P.., 22. Modelling of Ultrasonic Guided Wave Field Generated by Pieoelectric Transducers. PhD Thesis, Iperial College, London, UK. Marani, A., 28. Tie-transient response for ultrasonic guided waves propagating in daped cylinders. International Journal of Solids and Structures, 45(25-26):6347-6368. http://dx.doi.org/.6/j.ijsolstr.28.7.28 Moreau, L., Velichko, A., Wilcox, P.D., 22. Accurate finite eleent odelling of guided wave scattering fro
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