Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load

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This report addresses the analysis of frequency response sensitivity to crack for piezoelectric FGM beams subjected to moving load. First, a frequency domain model of a cracked FGM beam with a piezoelectric layer is conducted to derive an explicit expression of the electrical charge produced in the piezoelectric layer under the moving load.

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Nội dung Text: Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load

Vietnam Journal of Mechanics, Vol. 46, No. 3 (2024), pp. 191 – 205 DOI: https:/ /doi.org/10.15625/0866-7136/20933 FREQUENCY RESPONSE SENSITIVITY TO CRACK FOR PIEZOELECTRIC FGM BEAM SUBJECTED TO MOVING LOAD Nguyen Ngoc Huyen1,∗ , Duong Thanh Huan 2 1 Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam 2 Vietnam National University of Agriculture, Trau Quy, Gia Lam, Hanoi, Vietnam ∗ E-mail: nnhuyen@tlu.edu.vn Received: 29 March 2024 / Revised: 23 June 2024 / Accepted: 04 July 2024 Published online: 30 July 2024 Abstract. Since functionally graded material (FGM) is increasingly used in high-tech en- gineering, free and forced vibrations of FGM structures become an important issue. This report addresses the analysis of frequency response sensitivity to crack for piezoelectric FGM beams subjected to moving load. First, a frequency domain model of a cracked FGM beam with a piezoelectric layer is conducted to derive an explicit expression of the electri- cal charge produced in the piezoelectric layer under the moving load. It was shown in the previous works of the authors that the electrical charge is a reliable representation of the beam frequency response to moving load and can be efﬁciently employed as a measured diagnostic signal for structural health monitoring. Then, a damage indicator acknowl- edged as a spectral damage index (SDI) calculated from the electrical frequency response is introduced and used for sensitivity analysis of the response to crack. Under the sen- sitivity analysis the effect also of FGM and moving load parameters on the sensitivity is examined and illustrated by numerical results. Keywords: FGM beam, piezoelectric layer, frequency response, moving load, sensitivity analysis. 1. INTRODUCTION Damage detection in general and crack identiﬁcation in particular are essential prob- lem in the structural health monitoring that has been intensively studied through several latest decades and it was reviewed by numerous authors, for instance, Sohn et al. [1]; Fan and Qiao [2] and Hou and Xia [3]. Most of researchers in the ﬁeld of structural health
192 Nguyen Ngoc Huyen, Duong Thanh Huan monitoring have agreed to that dynamic behavior or vibration circumstance of a struc- ture provide the most useful tool for diagnosis of potential damages in the structure. Hence, the important issue in structural damage detection is to gather and examine the structural dynamic features that have been chosen as indicators for the detection. Con- ventional approach to assessing structure integrity is the dynamic testing technique that proposes to measure response of a structure under given external excitations. This tech- nique is bulky and expensive for huge structures because it requires a large number of sensors and actuators to obtain truthful signature of potential damages. Moreover, the conventional dynamic testing method does not allow direct identiﬁcation of damage and it is difﬁcult to perform in the real time mode. Alternately, many authors [4–7] have demonstrated that using smart material such as piezoelectric one the structural health monitoring becomes much more advantaged in its implementation and improved in the results obtained. This is because of the smart ma- terial could be used not only for transmitting load to structure (as actuator) but also for sensing signal of the structure response (as sensor). The smart sensors are distributed [8] and may be permanently installed as components of a structure of interest [9]. Recent progress in structural health monitoring by the use of distributed piezoelectric transduc- ers was reported in [10–14]. Particularly, Wang and Quek [15] used the sandwich beam model for modal anal- ysis of a Euler-Bernoulli beam embedded with piezoelectric layers and they found that natural frequency of the sandwich beam is function of stiffness and thickness of the piezo- electric layers. Wang and Quek [16] showed that the buckling and ﬂutter capacities of an elastic column could be enhanced by using piezoelectric patches bonded to both sides of the column as actuators with an applied voltage. Wang et al. [17] revealed an effect of a piezoelectric patch bonded to a beam on natural frequency of the beam and demonstrated an interesting fact that piezoelectric patch used as an actuator could restore the healthy condition of a cracked beam. Zhao et al. [18] proposed a procedure for crack identiﬁca- tion in beam based on the crack-induced frequency change that is ampliﬁed by applying a feedback voltage output from piezoelectric sensor through collocated actuator. The so- called Electro-Mechanical Impedance (EMI) method was developed in [19–21] for crack identiﬁcation in beam using piezoelectric transducers. The authors have concluded that the EMI is sensitive to local damage such as crack only at the high frequency range and when sensor patch is positioned near the damage location. Therefore, using a piezoelec- tric layer bonded to a beam structures as full-length distributed sensor is promising idea that is investigated in the present study for functionally graded beam with crack. Various problems in dynamics of functionally graded beams were studied in the widespread literature, some of which are, for instance, Li [22], Sina et al. [23], Larbi et al. [24], Su and Banerjee [25], Wang et al. [26]. A number of works is devoted also to
Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load 193 study vibrations of the beams with localized damages such as cracks, for example, Yang and Chen [27], Akbas [28], Aydin [29], Khiem et al. [30]. Some procedures were proposed by Yu and Chu [31]; Banerjee et al. [32] and Khiem and Huyen [33] to detect cracks in functionally graded beams with natural frequencies measured by the conventional tech- nique of modal testing. Stability of FGM Timoshenko beam embedded by the top and bottom piezoelectric layers has been investigated by Kharramabadi and Nezamabadi [34] and it is found a signiﬁcant effect of both the piezoelectric actuators and FGM parame- ters on the critical buckling loads. Li et al. [35] even proposed a model of functionally graded piezoelectric beam for its vibration analysis and revealed the increase of natural frequency and decrease of electric potential with increasing gradient index of the mate- rial. Bendine et al. [36] studied the problem for active vibration control of functionally graded beams with upper and lower surface-bonded piezoelectric layers by the ﬁnite el- ement method. Khiem et al. [37] examined the effect of piezoelectric patches on natural frequencies of undamaged functionally graded beam. The present paper addresses the analysis of frequency response sensitivity to crack for piezoelectric FGM beams subjected to moving load. First, a frequency domain model of a cracked FGM beam with a piezoelectric layer is conducted to derive an explicit ex- pression of the electrical charge produced in the piezoelectric layer under the moving load. It was shown in the previous works of the authors that the electrical charge is a reliable representation of the beam frequency response to moving load and can be efﬁ- ciently employed as a measured diagnostic signal for structural health monitoring. Then, a damage indicator acknowledged as a spectral damage index (SDI) calculated from the electrical frequency response is introduced and used for sensitivity analysis of the re- sponse to crack. Under the sensitivity analysis the effect also of FGM and moving load parameters on the sensitivity is examined and illustrated by numerical results. 2. GOVERNING EQUATIONS Consider an FGM beam of length L, cross sectional area Ab = b × hb bonded with a piezoelectric layer and subjected to a moving force as shown in Fig. 1 [38]. It is assumed also that the beam is made of functionally graded material with properties varying along the thickness direction by the power law R (z) = Rb + (Rt − Rb ) (z/h + 0.5)n , −hb /2 ≤ z ≤ hb /2, (1) where R stands for Young’s, shear modulus and material density E, G, ρ; subscripts t and b denote the top and bottom material respectively; n is power law exponent or mate- rial distribution index; z is ordinate of point along the beam height from the mid plane. According to thee Timoshenko beam theory with the constituting equations
literature, some of which are, for instance, Li [22], Sina et al. [23], Larbi et al. [24], Su and Banerjee [25], Wang et al. [26]. A number of works is devoted also to study vibrations of the beams with localized damages such as cracks, for example, Yang and Chen [27], Akbas [28], Aydin [29], Khiem et al. [30]. Some procedures were proposed by Yu and Chu [31]; Banerjee et al. [32] and Khiem and Huyen [33] to detect cracks in functionally graded beams with natural frequencies measured by the conventional technique of modal testing. Stability of FGM Timoshenko beam embedded by the top and bottom piezoelectric layers has been investigated by Kharramabadi and Nezamabadi [34] and it is found a significant effect of both the piezoelectric actuators and FGM parameters on the critical buckling loads. Li et al. [35] even proposed a model of functionally graded piezoelectric beam for its vibration analysis 194 and revealed the increase of natural frequency and decrease of electric Huan with increasing gradient Nguyen Ngoc Huyen, Duong Thanh potential index of the material. Bendine et al. [36] studied the problem for active vibration control of functionally graded beams with upper and lower surface-bonded piezoelectric layers by the finite element method. u x, z, t) examined t) − ( − h0 ) θ ( x, t) , Khiem(et al. [37] = u0 ( x,the effectzof piezoelectric patches onw ( x, z, t ) = w0 ( x, t ) , natural frequencies of undamaged functionally graded beam. (2) ε x = ∂u0 /∂x − (z − h0 ) ∂θ/∂x, γxz = ∂wto/∂x − θ, The present paper addresses the analysis of frequency response sensitivity 0 crack for piezoelectric FGM beams subjected to moving load. First, a frequency domain model of a cracked FGM where ubeamz, t),aw( x, z, t) are axial andtotransverse displacementselectrical charge ( x, with piezoelectric layer is conducted derive an explicit expression of the in cross-section at x; produced in the piezoelectric layer under the moving load. It was shown in the previous works of the u0 ( x, t) ,authorsx, t)the electrical charge is a reliable representation of the beam frequency response to moving the cross- w0 ( that are the displacements on the neutral plane and θ is rotation of section; ε x , and , σx , efficiently employed as a measured diagnostic signal for structural is geometry load γ can beτ are deformation and strain components; κ health monitoring. correction xz Then, a damage indicator acknowledged as a spectral damage index (SDI) calculated from the electrical factor; hfrequency response is introduced exact position ofanalysis of the response to crack. Under the the beam 0 is acknowledged as and used for sensitivity neutral plane measured from midplane. illustrated by numerical results. FGM and moving load parameters on the sensitivity is examined sensitivity analysis the effect also of and 2. Governing equations Fig.1. Model of piezoelectric FGM beam under moving force Fig. 1. Model of piezoelectric FGM beam under moving force Let the piezoelectric layer be considered as a homogeneous Timoshenko beam ele- ment, so that constitutive equations can be expressed as u p ( x, z, t) = u p0 ( x, t) − zθ p ( x, t) , w p ( x, z, t) = w p0 ( x, t) , ε px = u p0 − zθ p , γ p = w p0 − θ p , (3) p p p σpx = C11 ε px − h13 D, τp = C55 γ p , ∈ = −h13 ε px + β 33 D, p p where C11 , h13 , β 33 are elastic modulus, piezoelectric and dielectric constants respectively, ∈ and D are electric ﬁeld and displacement of the piezoelectric layer. Hence, conditions of perfect bonding between the base beam and piezoelectric layer can be represented as hb hp u x, ,t = u p x, − ,t , w ( x, hb /2, t) = w p x, −h p /2, t , 2 2 that yield u p0 = u0 − θh/2, h = hb + h p , w p0 = w0 , θ = θp, h (4) ε px = u0 − z + θ, γ p = w0 − θ. 2
Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load 195 Using the constituting equations (2)–(3)–(4) and Hamilton’s principle, one gets the following equations of motion [38] ∗ ∗ ∗ ¨ ∗ I11 u0 − B11 u0 + I12 θ − B12 θ = 0, ¨ ∗ ∗ ∗ ¨ ∗ ∗ I12 u0 − B12 u0 + I22 θ − B22 θ ¨ − A33 w0 − θ = 0, (5) ∗ ∗ I11 w0 − A33 w0 − θ ¨ = P (t) δ ( x − vt) , and p D ( x, t) = h13 u0 + hθ /β 33 , (6) where ∗ ∗ ∗ p p p B11 = A11 + E p A p , B12 = E p A p h, B22 = A22 + C11 I p + E p A p h2 , E p = C11 − h2 /β 33 , 13 ∗ ∗ ∗ ∗ p I11 = I11 + ρ p A p , I12 = I12 + ρ p A p h, I22 = I22 + ρ p I p + ρ p A p h2 , A33 = κA33 + C55 A p , A11 = bhb Eb ϕ1 (re , n) , A22 = bh3 Eb ϕ3 (re , n) , b A33 = bhb Gb ϕ1 r g , n , I11 = bhb ρb ϕ1 rρ , n , I12 = bh3 ρb ϕ2 b rρ , n , I22 = bh3 ρb ϕ3 rρ , n , b ϕ1 (r, n) = (r + n) / (1 + n) , ϕ2 (r, n) = (2r + n) /2 (2 + n) − α (r + n) / (1 + n) , ϕ3 (r, n) = (3r + n) /3 (3 + n) − α (2r + n) / (2 + n) − α2 (r + n) / (1 + n) , α = 1/2 + h0 /hb , re = Et /Eb , rρ = ρt /ρb , r g = Gt /Gb . (7) Transferring equations (5) and (6) into the frequency domain, one gets [A] Z ( x, ω ) + [B] Z ( x, ω ) + [Ω] { Z ( x, ω )} = − {P ( x, ω )} , (8) ∞ { Z ( x, ω )} = {u0 ( x, t) , θ ( x, t) , w0 ( x, t)} e−iωt dt, Z = dZ/dx, Z = d2 Z/dx2 , −∞ P ( x, ω ) = {0, 0, Q ( x, ω )} T , Q ( x, ω ) = P ( x/v) exp {−iωx/v} , (9) with the matrices ∗ ∗     B11 B12 0 0 0 0 ∗ ∗  B12 B22 ∗ [A] = 0 , [B] =  0 0 A33  , ∗ ∗ 0 0 A33 0 − A33 0 ∗ ∗ (10) ω 2 I11 ω 2 I12   0 2 ∗ ∗ ∗ [Ω] =  ω I12 ω 2 I22 − A33 0 , ∗ 0 0 ω 2 I11 and ˆ p p D ( x, ω ) = h13 Z1 ( x, ω ) + hZ2 ( x, ω ) /β 33 = h13 U ( x, ω ) + hΘ ( x, ω ) /β 33 . (11) If the piezoelectric layer is employed as a distributed sensor, the frequency domain output charge of which can be calculated as L L ˆ p Q (ω ) = b Ddx = bh13 /β 33 U ( x, ω ) − hΘ ( x, ω ) /2 dx. (12) 0 0
196 Nguyen Ngoc Huyen, Duong Thanh Huan Furthermore, assume that the host FGM beam has been cracked at the position e measured from its left end and crack is modeled by a pair of equivalent springs of stiff- ness T for transnational spring and R for rotational one [39]. In this case, conditions that must be satisﬁed at the crack are U (e + 0) = U (e − 0) + N (e)/T, Θ(e + 0) = Θ(e − 0) + M (e)/R, W ( e + 0) = W ( e − 0), Ux ( e + 0 ) = Ux ( e − 0 ) , (13) Θ x ( e + 0) = Θ x ( e − 0) , Wx (e + 0) = Wx (e − 0) + M (e) /R, where N ( x ) = A11 Ux ( x ) , M ( x ) = A22 Θ x ( x ) are respectively internal axial force and bending moment at section x. Substituting the expressions for axial force and bending moment into (13) that can be rewritten as U (e + 0) = U (e − 0) + γ1 Ux (e) , Θ (e + 0) = Θ (e − 0) + γ2 Θ x (e) , W ( e + 0) = W ( e − 0) , Ux ( e + 0 ) = Ux ( e − 0 ) , (14) Θ x ( e + 0) = Θ x ( e − 0) , Wx (e + 0) = Wx (e − 0) + γ2 Θ x (e) , γ1 = A11 /T, γ2 = A22 /R. The so-called crack magnitudes γ1 , γ2 are functions of the material parameters such as elastic modulus and they should be those of homogeneous beam when Et = Eb = E0 . Using expressions (7) and the latter conditions the crack magnitudes can be rewritten as γ1 = γa ϕ1 (re , n) , γ2 = 12γb ϕ3 (re , n) , (15) where [39] 2 2 γa = E0 A/T = 2π 1 − ν0 h f 1 (z) , γb = E0 I0 /R = 6π 1 − ν0 h f 2 (z) , z = a/h, (16) f 1 (z) = z2 0.6272 − 0.17248z + 5.92134z2 − 10.7054z3 + 31.5685z4 − 67.47z5 + 139.123z6 − 146.682z7 + 92.3552z8 , (17) f 2 (z) = z2 0.6272 − 1.04533z + 4.5948z2 − 9.9736z3 + 20.2948z4 − 33.0351z5 + 47.1063z6 − 40.7556z7 + 19.6z8 . Obviously, for uncraled beam when Et = Eb = E0 or re = 1, Eqs. (15) yield γ1 = γa ϕ1 (1, n) = γa , γ2 = 12γb ϕ3 (1, n) = γb . (18) 3. FREQUENCY RESPONSE CRACKED PIEZOELECTRIC FGM BEAM SUBJECTED TO MOVING HARMONIC LOAD First, seeking solutions of homogeneous equation (8) in the form: Z 0 = deλx one gets general solution for free vibration of the beam in the form { Z 0 ( x, ω )} = [G0 ( x, ω )] {C } , (19)
Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load 197 where {C } = (C1 , . . . , C6 )T is constant vector and G0 ( x, ω ) is matrix α1 e k 1 x α2 e k 2 x α3 e k 3 x α1 e − k 1 x α2 e − k 2 x α3 e − k 3 x   [ G0 ( x, ω )] =  ek1 x ek2 x ek3 x e−k1 x e−k2 x e−k3 x  , β 1 ek1 x β 2 ek2 x β 3 ek3 x − β 1 e−k1 x − β 2 e−k2 x − β 3 e−k3 x (20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ α j = ω 2 I11 + k2 B11 / ω 2 I12 + k2 B12 , β j = k j A33/ ω 2 I11 + k2 A33 , j = 1, 2, 3 j j j and k j ( j = 1, 2, 3) are wave numbers obtained from the roots λ1,4 = ±k1 , λ2,5 = ±k2 , λ3,6 = ±k3 of the characteristic equation det λ2 A + λB + Ω = 0. Using expression (18) we can ﬁnd a particular solution Z c ( x, ω ) in the form { Z c ( x, ω )} = [Gc ( x, ω )] Z 0 (e, ω ) , (21) where G( x, ω ) is 3×3-matrix of the form 3 3    γa ∑ αi δi1 coshk i x γb ∑ αi (δi2 + δi3 ) coshk i x 0   i =1 i =1  3 3    γa ∑ δi1 coshk i x γb ∑ (δi2 + δi3 ) coshk i x 0  ,   [Gc ( x, ω )] =   (22)  i =1 i =1   3 3  γa ∑ β i δi1 sinhk i x γb ∑ β i (δi2 + δi3 ) sinhk2 x 0   i =1 i =1 and δ11 = (k3 β 3 − k2 β 2 ) /∆, δ12 = (α3 k2 β 2 − α2 k3 β 3 ) /∆, δ13 = (α2 − α3 ) /∆, δ21 = (k1 β 1 − k3 β 3 ) /∆, δ22 = (α1 k3 β 3 − α3 k1 β 1 ) /∆, δ23 = (α3 − α1 ) /∆, δ31 = (k2 β 2 − k1 β 1 ) /∆, δ32 = (α2 k1 β 1 − α1 k2 β 2 ) /∆, δ33 = (α1 − α2 ) /∆, ∆ = k 1 β 1 ( α2 − α3 ) + k 2 β 2 ( α3 − α1 ) + k 3 β 3 ( α1 − α2 ) , that satisﬁes the conditions T T { Z c (0)} = γa U0 (e) , γb Θ0 (e) , 0 , Z c (0) = 0, 0, γb Θ0 (e) . (23) So, it is easily to verify that solution (18) for free vibration of integrated piezoelectric FGM beam satisfying the conditions at crack (14) can be represented as { Z 0 ( x, ω )} : for x < e, { Z ( x, ω )} = { Z 0 ( x, ω )} + { Z c ( x − e, ω )} : for e ≤ x, that is rewritten in the form { Z ( x, ω )} = { Z 0 ( x, ω )} + [K ( x − e)] Z 0 (e, ω ) = [Φ ( x, ω )] {C } , (24) with the matrices introduced [Φ ( x, ω )] = G0 ( x, ω ) + K ( x − e) G0 ( x, ω ) , [Gc ( x )] : x > 0, Gc ( x ) : x > 0, (25) [K ( x )] = K (x) = [0] : x ≤ 0, [0] : x ≤ 0.
198 Nguyen Ngoc Huyen, Duong Thanh Huan Thus, expression (24) is general solution for free vibration of cracked FGM piezoelec- tric beam that would be determined seeking constant vector {C } = (C1 , . . . , C6 )T from speciﬁcally given boundary conditions. For example, in case of simply supported beam with boundary conditions U (0) = W (0) = M (0) = U ( L) = W ( L) = M ( L) = 0, ∗ ∗ where M ( x ) = B12 ∂ x U ( x ) − B22 ∂ x Θ ( x ), one gets [G (ω )] {C } = 0, (26) where   α1 α2 α3 α1 α2 α3   β1 β2 β3 − β1 − β2 − β3    m1 m2 m3 − m1 − m2 − m3  [G(ω )] = [BSS (ω )] =  ,   φ11 ( L) φ12 ( L) φ13 ( L) φ14 ( L) φ15 ( L) φ16 ( L)    φ31 ( L) φ32 ( L) φ33 ( L) φ34 ( L) φ35 ( L) φ36 ( L)  M1 ( L) M2 ( L) M3 ( L) M4 ( L) M5 ( L) M6 ( L) ∗ ∗ ∗ ∗ m j = B12 α j − B22 k j , j = 1, 2, 3, M j ( L) = B12 φ1j ( L) − B22 φ2j ( L) , j = 1, 2, . . . , 6, φij ( x ) , φij ( x ) , i = 1, 2, 3; j = 1, 2, . . . , 6 are elements of matrices [Φ ( x, ω )] and Φ ( x, ω ) deﬁned in (26). Therefore, frequency equation of the beam is det [G (ω )] = 0, (27) positive roots of which give rise desired natural frequencies ω1 , ω2 , ω3 , . . . of simply sup- ported FGM beam with piezoelectric layer and cracks. As a consequence, the natural frequencies are dependent On crack parameters such as crack location e and depth a as well as material properties and piezoelectric layer thickness. The effect of latter factors on natural frequencies was studied in [39] and [37]. In this study, the natural frequencies are computed as function of crack parameters (e, a): ωk = ωk (e, a) , k = 1, 2, . . ., that would be employed as a database for crack detection from measured natural frequencies. Now, we are going to ﬁnd the solution of the inhomogeneous equation (8) that is acknowledged as the frequency response of the integrated beam subjected to a mov- ing force. Let’s consider the case of moving harmonic force P (t) = P0 exp {iΩm t} that gives rise Q ( x, ω ) = ( P0 /v) exp {−iΩx } , Ω = (ω − Ωm ) /v. (28) It is not difﬁcult to ﬁnd a particular solution Z q ( x, ω ) of Eq. (8)–(9)–(28) in the form T Z q ( x, ω ) = Uq (ω ) , Θ0 (ω ) , Wq (ω ) 0 q 0 exp {−iΩx } , (29) where ∗ ∗ ∗ ∗ ∗ ∗ Uq (ω ) = (iΩ) P0 A33 Ω2 B12 − ω 2 I12 /v∆, 0 Θ0 (ω ) = (iΩ) P0 A33 ω 2 I11 − Ω2 B11 /v∆, q
Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load 199 ∗ ∗ ∗2 ∗ ∗ 0 Wq (ω ) = P0 D/∆, ∆ = ω 2 I11 − Ω2 A33 D + iΩA33 ω 2 I11 − Ω2 B11 , ∗ ∗ ∗2 ∗ ∗ ∗2 ∗ ∗ ∗ D = ω 4 I11 I22 − I12 + Ω4 B11 B22 − B12 + A33 Ω2 B11 − ω 2 I11 ∗ ∗ ∗ ∗ ∗ ∗ + ω 2 Ω2 (2I12 B12 − I11 B22 − I22 B11 ) . Therefore, general solution of Eq. (8) can be expressed as Z ( x, ω ) = {U ( x, ω ) , Θ ( x, ω ) , W ( x, ω )} T = [Φ ( x, ω )] {C } + Z q ( x, ω ) , (30) where matrix Φ ( x, ω ) is deﬁned in Eq. (25) and constant vector C is sought by given boundary conditions. Namely, for simply supported beam one can ﬁnd {C } = − [G (ω )]−1 P (ω ) , ˆ (31) T ˆ ˆ ˆ with matrix G (ω ) deﬁned in Eq. (27) and vector P (ω ) = P1 (ω ) , . . . , P6 (ω ) where ∗ ∗ ˆ 0 P1 (ω ) = Uq (ω ) , P2 (ω ) = −iΩ B12 Uq (ω ) − B22 Θ0 (ω ) , ˆ 0 q ˆ 0 P3 (ω ) = Wq (ω ) , ∗ ∗ P4 (ω ) = −iΩ B11 Uq (ω ) − B12 Θ0 (ω ) exp {−iΩL} , ˆ 0 q ∗ ∗ P5 (ω ) = −iΩ B12 Uq (ω ) − B22 Θ0 (ω ) exp {−iΩL} , ˆ 0 q ˆ 0 P6 (ω ) = Wq (ω ) exp {−iΩL} . Thus, mechanical frequency response (30) for simply supported beams gets the form Z ( x, ω ) = Z q ( x, ω ) − [Φ ( x, ω )] [G (ω )]−1 P (ω ) . ˆ (32) ˆ Owning mechanical frequency response Z ( x, ω ) , the sensor output charge Q (ω ) can be calculated by L ˆ p Q (ω ) = bh13 /β 33 U ( x, ω ) − hΘ ( x, ω ) /2 dx 0 bh13 (33) = p U ( L, ω ) − U (0, ω ) − γ1 Ux (e, ω ) β 33 − (h/2) Θ ( L, ω ) − Θ (0, ω ) − γ2 Θ x (e, ω ) , where U ( x, ω ) , Θ ( x, ω ) are components of solution (30) acknowledged as mechanical frequency response of cracked FGM piezoelectric beam subjected to moving load. As consequence, the frequency domain charge generated in the piezoelectric layer is called herein electrical frequency response of the beam. This is the basics for using the piezo- electric layer as a distibuted sensor and its output charge as a diagnostic signal for crack detection in FGM beam subjected to moving load.
200 Nguyen Ngoc Huyen, Duong Thanh Huan 4. SENSITIVITY OF ELECTRICAL FREQUENCY RESPONSE TO CRACK - NUMERICAL RESULTS For sensitivity analysis of electrical frequency response, the sensor output charge ˆ Q (ω ), by using so-called spectral damage index deﬁned for the responses of intact and ˆ ˆ cracked beams Q (ω, e, a), Q0 (ω ) as [40] 1/2 N N N SDI (e, a) = ∑ ˆ ˆ Q (ωk , e, a) Q0 (ω ) / ∑ ˆ Q2 (ωk , e, a) ∑ ˆ Q02 (ωk ) . (34) k =1 k =1 k =1 The introduced above damage index lies between 0 and 1, which equals 1 only if the two frequency-dependent functions are fully similar. Hence, its deviation of unique represents a measure of effect of crack on the index and as usual it is acknowledged as sensitivity of the frequency response to crack. Note, the sensitivity represented by the spectral damage index depends also on the material and load parameters such as the material distribution index n, frequency and speed of the moving load Ωm , v that are all useful for us to control the success of crack detection. Thus, the spectral damage index is numerically examined herein with the following geometry and material constants: Nguyen Lb =Huyen, Duong ThanhThanh0.1 m; hbNguyen Tien Khiem Nguyen Ngoc = L = 1 m; b = Huan and = L/10; Ngoc L p Huyen, Duong Huan and Nguyen Tien Khiem Et = 390 MPa; ρt = 3960 kg/m3 ; µt = 0.25; Eb = 210 MPa; ρb = 7800 kg/m3 ; µb = 0.31; p In the Figs. Figs. there theredepicted spectral damage indexindex as function of crack location in In the 2-5 2-5 are are depicted spectral damage as function of crack location in p C11 dependnce crack depthdepth21.0526 frequency = 7750load (Fig. h13 = −7.70394 × 108 V/m. = 69.0084 GPa; C and and load GPa; ρ p (Fig. 2), kg/m3 ; (Fig. 3), material distribution dependnce upon upon crack55 =load frequency (Fig. 2), load speedspeed 3), material distribution index index n (Fig. 4) and thickness of theof the piezoelectric layer 5). 5). n (Fig. 4) and thickness piezoelectric layer (Fig. (Fig. Fig.2. Fig.2. Spectral damage index versus location Spectral damage index versus crack crack location Fig.3. Fig.3. Spectral damage versusversus crack Spectral damage index index crack Fig. in dependence damageand moving load frequency 2. Spectral on crack index versus load frequency Fig.location in dependence on index speed speed (v) in dependence on crack depth depth and movingcrack 3. dependence on moving load load (v) location in Spectral damage movingversus crack location in dependence on crack depth location in dependence on moving load and moving load frequency speed (v)
Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load 201 In Figs. 2–5, there are depicted spectral damage index as function of crack location in dependence on crack depth and load frequency (Fig. 2), load speed (Fig. 3), material distribution Spectral n (Fig.index versus crack locationthe Fig.3.Fig.3. Spectral damage index versus crack Fig.2.Fig.2. index damage versus crack location of Spectral damage index 4) and thickness piezoelectric layer (Fig. 5). crack Spectral damage index versus in dependence on crack crack depth and movingfrequency in dependence on depth and moving load load frequency location in dependence on moving load speed speed (v) location in dependence on moving load (v) Fig.4.Fig.4. Spectral damage index versus crack location Fig.5.Fig.5. Spectral damage index versus crack location Spectral damage index versus crack location Spectral damage index versus crack location Fig. 4. in dependence on material distribution crack(n) Spectral damage index versus (n) in dependence on material distribution index index Fig. in dependence damage index versus (hp) 5. Spectral on thickness of sensor layer in dependence on thickness of sensor layer (hp) crack location in dependence on material location in dependence on thickness of Observing graphs presented in thein the Figues one can make the following discussion: (1) Change in the Observing graphs presented Figues one can make the following discussion: (1) Change in the distribution index (n) sensor layer (hp) spectral damage indexindex due to crack position and depth is similar to the crack-induced change in fundametal spectral damage due to crack position and depth is similar to the crack-induced change in fundametal frequency with an exception that magnitude of the change in spectral damage indexindex is larger that of of frequency with an exception that magnitude of the change in spectral damage is larger than than that natural fundamental frequency. In the other other words, itbe one acknowledgedspectral damage indexindex is Observing graphs presented in the ﬁgures acknowledged thatthe following discussion: natural fundamental frequency. In the words, it can can be can make that spectral damage is significantly moremore senstitive to crackthe natural frequencies; (2) The effecteffect depth isparameters the significantly senstitive to crack than index due to crack position of moving load similar to (i) Change in the spectral damagethan the natural frequencies; (2) The and of moving load parameters such such as frequency and speedspeed on the sensitivity of spectral damage index to crack slightslight that they as their their and on the sensitivity of spectral damage index to crack is so is so that they crack-induced frequencyin crack detection asfrequencyfor homogeneous beamsthat magnitude of the may be difficult tochangefor fundamental it hasit has done for homogeneous beams [40]; (3) Sensitivity with an exception [40]; (3) Sensitivity may be difficult to employ for crack detection as done employ changeSDI spectralisdamage index is larger than that ofindex (n) from 0𝑛 to 0 = 1.0, then the the crack is first first increasing material distribution index (n) from 𝑛 = = 𝑛 to 𝑛 frequency. of SDI toin to crack increasing with with material distribution natural fundamental = 1.0, then In of the other decreases it can𝑛 be 5, = 5, fromit becomes againagain increasing.index that dependence ofmore sensitivity decreases = 𝑛 from that that it becomes increasing. This This means that dependence of sensitivity words, until until acknowledged that spectral damage means is signiﬁcantly the SDI sensitivity to crackcrack on the FGM property is not monotonous, but it reaches maximum at the value the SDI crack thanon the FGM property is not monotonous, but it reaches maximum at the value sensitive tosensitivity to the natural frequencies; (ii) The effect of moving load parameters 𝑛 = 1.0; (4) Sensitivity of SDI to crackcrack is increasing piezoelectric layer layer thickness the thickness 𝑛 = 1.0; (4) Sensitivity of SDI to is increasing with with piezoelectric thickness until until the thickness suchreaches critical value ℎ and. speed on the sensitivity thickness fromdamage index to crack reaches critical value ℎ' = 0.2ℎ! Further increase of the thickness spectral critical valuevalue leads to as their frequency = 0.2ℎ . Further increase of the of from the the critical leads to ' ! is sodecrease of sensitivity may to crack. discussionemploy for designingdetection detection procedure for decrease of SDI SDI sensitivity be This This discussion is useful for designing crack as it has done in slight that they to crack. difﬁcult to is useful for crack crack detection procedure in homogeneous beams [40];and piezoelectric distributed sensor. is ﬁrst increasing with material FGMFGM beams using moving load and piezoelectric distributed sensor. beams using moving load (iii) Sensitivity of SDI to crack distribution index (n) from n = 0 to n = 1.0, then the sensitivity decreases until n = 5, 5. Conclusions 5. Conclusions from that it becomes again increasing.summarized as follow: The mainmain results obtained instudystudybe This means thatfollow: dependence of the SDI sensitivity The results obtained in this this can can be summarized as to crack on the FGM property is not monotonous, but it reaches maximum at the value n = 1.0; (iv) Sensitivity of SDI to crack is increasing with piezoelectric layer thickness until the thickness reaches critical value h p = 0.2hb . Further increase of the thickness from the critical value leads to decrease of SDI sensitivity to crack. This discussion is useful for designing crack detection procedure in FGM beams using moving load and piezoelectric distributed sensor. 5. CONCLUSIONS The main results obtained in this study can be summarized as follow:
202 Nguyen Ngoc Huyen, Duong Thanh Huan - A frequency domain model of cracked functionally graded beams bonded with a piezoelectric layer and subjected to a harmonic moving force has been established and it could be usefully employed for both analysis and identiﬁcation of the composite struc- tures. - There has been introduced a novel damage index called spectral damage index for cracked functionally graded beams with piezoelectric layer. The index is easily calculated from the output charge produced in the piezoelectric layer acknowledged as an electrical frequency response to the moving load. - Sensitivity of the electrical frequency response to crack has been examined in de- pendence On load parameters, FGM properties, and thickness of the piezoelectric layer used as a distributed sensor for measuring the frequency response of FGM beam sub- jected to moving load. - The completed above theoretical development and numerical analysis provide use- ful information and instruction for solving the crack detection problem that is the subject of the next study for the authors. DECLARATION OF COMPETING INTEREST The authors declare that they have no known competing ﬁnancial interests or per- sonal relationships that could have appeared to inﬂuence the work reported in this paper. FUNDING This research received no speciﬁc grant from any funding agency in the public, com- mercial, or not-for-proﬁt sectors. REFERENCES [1] H. Sohn, C. R. Farrar, F. M. Hemez, and J. J. Czarnecki. A review of structural health review of structural health monitoring literature 1996-2001. Los Alamos National Laboratory Report, (2002). [2] W. Fan and P. Qiao. Vibration-based damage identiﬁcation methods: A re- view and comparative study. Structural Health Monitoring, 10, (2010), pp. 83–111. https://doi.org/10.1177/1475921710365419. [3] R. Hou and Y. Xia. Review on the new development of vibration-based damage identiﬁca- tion for civil engineering structures: 2010–2019. Journal of Sound and Vibration, 491, (2021). https://doi.org/10.1016/j.jsv.2020.115741. [4] S. S. Rao and M. Sunar. Piezoelectricity and its use in disturbance sensing and con- trol of ﬂexible structures: A survey. Applied Mechanics Reviews, 47, (1994), pp. 113–123. https://doi.org/10.1115/1.3111074.
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