Multiple crack identification in stepped beam by measurements of natural frequencies

Volume 36 Number 2 2 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 2 (2014), pp. 119 – 132 MULTIPLE CRACK IDENTIFICATION IN STEPPED BEAM BY MEASUREMENTS OF NATURAL FREQUENCIES Nguyen Tien Khiem1,∗, Duong The Hung2, Vu Thi An Ninh3 1Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam 2Thai Nguyen University of Technology, Vietnam 3Hanoi University of Transport and Communication, Vietnam ∗E-mail: ntkhiem@imech.ac.vn Received July 26, 2013 Abstract. A new approach is proposed for calculating natural frequencies and crack detection in a stepped cantilever beam with arbitrary number of cracks. This is based an explicit expression of the natural frequencies in term of crack parameter derived in the form similar to the so-called Rayleigh quotient for vibrating beam. The obtained simple relationship between natural frequencies and crack parameters enables not only accurate calculating the natural frequencies but also to develop an efficient procedure for detecting multiple cracks from given natural frequencies. The proposed technique called crack scanning method is illustrated and validated by numerical results. Keywords : Multi stepped beam, Rayleigh quotient, multi-crack detection, frequency based method, modal analysis. 1. INTRODUCTION Early detecting damage in engineering structures such as cracks is vitally important to prevent catastrophe that may lead to loss of either material or human lives. The task helps also to extend the structure lifetime by prompt maintenance and repair. A lot of methods have been proposed to detect cracks in structures and most of them are based on the crack-induced change in the dynamic characteristics of structure under consideration. This is because of the fact that crack in a structure member certainly modify the structure dynamic properties that can be usually obtained as results of the dynamic testing. The core of the vibration-based crack detection is the change in the structure modal parameters as a function of crack parameters that is usually subject of the forward problem. Since the natural frequencies are overall parameter of structure that can be most easily and accurately measured or calculated, the change in natural frequencies due to cracks in structures and its application to crack identification still nowadays are of interest. Adams et al [1] first have shown that ratio of change in two natural frequencies of a bar with single crack represented by a translation spring is dependent only on the 120 Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh crack position so that the ratio can be used for localization of single crack in bar. Later, Liang et al [2] extended the result for beam with single crack represented by a rotational spring. By using the perturbation method Morassi [3] demonstrated that the change in eigenfrequency of a cracked beam is proportional to the curvature at the crack position and this result has been then applied to crack identification in simply supported beam from measured natural frequencies in Morassi and Rollo, [4] and Rubio, [5]. A compact form of the characteristic equation, the most important relationship between natural fre- quencies and crack parameters, has been obtained in Narkis, [6] for both the longitudinal and bending vibration of simply supported beam. For small crack size the equation allows obtaining analytical solution for crack position and this result has been later validated by an experiment in Sayyad and Kumar, [7]. The characteristic equation for a rotating cracked Timoshenko beam was then derived by Al-Said et al, [8], who have shown influence of crack, rotating speed and shear deformation on the natural frequencies of the beam. Ostachovicz and Krawczuk [9] accomplished an analysis of variation of natural frequencies in cantilever beam with two cracks using derived characteristic equation. Bahera et al [10] obtained fundamental frequency of a shaft with two cracks in viscous liquid. The natural frequencies of beam with arbitrary number of cracks were investigated in dependence on crack parameters by Shifrin and Ruotolo [11]; Li [12]; Zheng and Fan [13]; Khiem and Lien [14]; Aydin [15] and Caddemi and Caliò [16]. However, in the foregoing works the natural frequencies of multiple cracked beam are found from a complicated equation that may contain singularities sometimes troubling numerical solving the equation. Though the simplified characteristic equation obtained in Khiem and Lien, [14] has been used for multi-crack detection in beam by Zhang et al [17], the implicit relationship between the natural frequencies and crack parameters may cause serious problems for crack detection. Therefore, a simple and explicit expression of natural frequencies in term of crack param- eters for multiple cracked beam would be surely helpful for the crack detection problem. Liang et al [18]; Patil and Maiti [19] and Li [20] have derived approximate systems of linear algebraic equations relating the change in natural frequencies to damage magni- tudes that provide an efficient tool for damage detection in beam. Fernandez-Saez et al [21] obtained an explicit expression of fundamental frequency of beam with single crack through crack position and size based on the classical Rayleigh formulae. Objective of this paper is to develop an explicit expression of natural frequencies in term of crack position and size for a multiple cracked beam and make use of the obtained expression for multi- crack identification problem. First, a novel form of the Rayleigh quotient is derived for multiple stepped beam with a number of cracks and arbitrary boundary conditions. Then, the obtained Rayleigh quotient for multiple cracked uniform beam is applied to deter- mine quantity, position and depth of multiple cracks from given natural frequencies. The theoretical development is illustrated and validated by both numerical and experimental results. 2. THE RAYLEIGH QUOTIENT Consider a cantilever beam consisting of N spans with the material and geometrical constants: Young’s modulus Ej, mass density ρj, length Lj, cross section area and moment Multiple crack identification in stepped beam by measurements of natural frequencies 121 of inertia Fj , Ij of j-th beam segment (xj−1, xj), j = 1, . . . , n; x0 = 0; xN = 1 = L1+ . . .+ Ln. Suppose furthermore that in the segment (xj−1, xj) there exist nj cracks of depth aji at positions eji, i = 1, . . . , nj (xj−1 ≺ ej1 ≺ . . . ≺ ejnj ≺ xj). It is well known that k-th mode shape in the beam segment (xj−1, xj) denoted by φkj(x) satisfies the equation −ω2kmjφkj(x) + Sjd 4φkj(x)/dx 4 = 0, x ∈ (xj−1, xj), (1) with mj = ρjFj ; Sj = EjIj and ωk is k-th and the following conditions at cracks φkj(e − ji) = φkj(e + ji); φ ′′ kj(e − ji) = φ ′′ kj(e + ji) = φ ′′ kj(eji); φ ′′′ kj(e − ji) = φ ′′′ kj(e + ji); [φ′kj(e + ji)− φ ′ kj(e − ji)] = γjiφ ′′ kj(eji); i = 1, . . . , nj (2) where γji = Ic(aji/hj), the function of depth of the crack at position eji, denotes the crack magnitude and is determined accordingly to the Fracture Mechanics. Multiplying both sides of Eqs. (1) by φkj(x) and taking integration along the interval (xj−1, xj), one obtains ω2kmj xj−1∫ xj−1 φ2kj(x)dx = Sj xj−1∫ xj−1 d4φkj(x) dx4 φkj(x)dx; j = 1, . . . , N, k = 1, 2, . . . (3) Note that for the functions φkj , φ ′ kj, φ ′′ kj, φ ′′′ kj continuous in the segment (a, b), it would be easily to verify that b∫ a d4φkj(x) dx4 φkj(x)dx = b∫ a φ′′kj 2 (x)dx+ [Bkj(b)− Bkj(a)] , (4) where Bkj(x) = φ ′′′ kj(x)φkj(x)− φ ′′ kj(x)φ ′ kj(x). (5) Therefore, one has xj−1∫ xj−1 d4φkj(x) dx4 φkj(x)dx = xj∫ xj−1 φ′′kj 2 (x)dx+[Bkj(xj)−Bkj(xj−1)]+ nj∑ i=1 [Bkj(e − ji)− Bkj(e + ji)]. (6) Taking account of the conditions (2) the latter equation can be rewritten as xj−1∫ xj−1 d4φkj(x) dx4 φkj(x)dx = xj∫ xj−1 φ′′kj 2 (x)dx+ [Bkj(xj)−Bkj(xj−1)] + nj∑ i=1 γjiφ ′′ kj 2 (eji) (7) 122 Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh and, as consequence, one obtains N∑ j=1 Sj xj−1∫ xj−1 d4φkj(x) dx4 φkj(x)dx = N∑ j=1 Sj   xj∫ xj−1 φ′′kj 2 (x)dx+ nj∑ i=1 γjiφ ′′ kj 2 (eji)  + + N−1∑ j=1 [SjBkj(xj)− Sj+1Bk,j+1(xj)] + [BkN (L)SN −Bk1(0)S1]. (8) It can be easily verified that under the continuity conditions at step joints xj φkj(xj) = φk,j+1(xj); φ ′ kj(xj) = φ ′ k,j+1(xj); Sjφ ′′ kj(xj) = Sj+1φ ′′ k,j+1(xj); Sjφ ′′′ kj(xj) = Sj+1φ ′′′ k,j+1(xj), (9) the second sum in the right hand side of Eq. (8) would be vanished. So that one obtains ω2k = N∑ j=1 Sj { xj∫ xj−1 φ′′kj 2(x)dx+ nj∑ i=1 γjiφ ′′ kj 2(eji) } + [BkN (L)SN −Bk1(0)S1] N∑ j=1 mj xj∫ xj−1 φkj 2(x)dx . (10) This is Rayleigh’s quotient for multiple cracked and stepped beam with arbitrary boundary conditions. Furthermore, for all the homogeneous classical boundary conditions (including the simply supports, cantilever, fixed and free ends) the last term in numerator of quotient (10) is vanish. In such the case one has ω2k =   N∑ j=1 Sj   xj∫ xj−1 φ′′kj 2 (x)dx+ nj∑ i=1 γjiφ ′′ kj 2 (eji)     /   N∑ j=1 mj   xj∫ xj−1 φkj 2(x)dx     . (11) In particularity, if the beam is uniform the Eq. (14) becomes ω2k = EI ρF   L∫ 0 φ′′k 2 (x)dx+ n∑ j=1 γjφ ′′ k 2 (ej)   /   L∫ 0 φk 2(x)dx   , (12) with n = n1 + . . .+ nN . For multi-stepped undamaged beam the Rayleigh quotient is ω2k =   N∑ j=1 Sj   xj∫ xj−1 φ′′kj 2 (x)dx     /   N∑ j=1 mj   xj∫ xj−1 φkj 2(x)dx     . (13) As the classical Rayleigh quotient, the Eqs. (11)-(13) is simplest tool for calculating natural frequencies by a properly chosen shape functions φkj(x). Moreover, the obtained Rayleigh quotient can be engaged to develop a new procedure for multi-crack detection in stepped beam by measurements of natural frequencies. Multiple crack identification in stepped beam by measurements of natural frequencies 123 3. CALCULATION OF NATURAL FREQUENCIES The mode shape functions are chosen in the form φkj(x) = ϕkj0(x) +Akjx 3 +Bkjx 2 +Ckjx+Dkj + nj∑ i=1 γjiφ ′′ kj(eji)K(x− eji), (14) where ϕkj(x) is k-th mode shapes of undamaged beam in the segment (xj−1, xj), the constants Akj, Bkj, Ckj , Dkj would be determined latter from boundary and step joint conditions and K(x) = { x, x ≥ 0; 0, x ≤ 0. Obviously, substituting shape function (14) into first two conditions in (9) leads to ϕk,j+1(xj) = ϕkj(xj), ϕ ′ k,j+1(xj) = ϕ ′ kj(xj) (15) and Ak,j+1 = Akj , Bk,j+1 = Bkj , Ck,j+1 = Ckj+ nj∑ i=1 γjiφ ′′ kj(eji), Dk,j+1 = Dkj− nj∑ i=1 γjiφ ′′ kj(eji)eji or Akj = Ak, Bkj = Bk, Ckj = Ck + j−1∑ r=1 nr∑ i=1 γriφ ′′ kr(eri), Dkj = Dk − j−1∑ r=1 nr∑ i=1 γriφ ′′ kr(eri)eri. (16) The latter equations show that all the functions (14) would be completely determined for j = 1, ..., N by choosing four constants Ak, Bk, Ck, Dk and functions φkj(x) which are defined for different classical boundary conditions as follow. Namely, the boundary conditions of both the cantilever, φk(0) = φ ′ k(0) = φ ′′ k(1) = φ′′′k (1) = 0 and beam with free ends, φ ′′ k(0) = φ ′′′ k (0) = φ ′′ k(1) = φ ′′′ k (1) = 0, would be satisfied by choosing Ak = Bk = Ck = Dk = 0, (17) in combination respectively with ϕk1(0) = ϕ ′ k1(0) = ϕ ′′ kN (1) = ϕ ′′′ kN(1) = 0, (18) and ϕ′′k1(0) = ϕ ′′′ k1(0) = ϕ ′′ kN (1) = ϕ ′′′ kN(1) = 0. (19) Also, for simply supported beam, to satisfy the conditions φk(0) = φ ′′ k(0) = φk(1) = φ′′k(1) = 0, it is simply to choose Ak = Bk = Dk = 0; Ck = N∑ j=1 nj∑ i=1 γjiφ ′′ kj(eji)(eji − 1); (20) ϕk1(0) = ϕ ′′ k1(0) = ϕkN (1) = ϕ ′′ kN(1) = 0. (21) 124 Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh Finally, the clamped ends conditions would be fulfilled if Ck=Dk=0;Bk= N∑ j=1 nj∑ i=1 γjiφ ′′ kj(eji)(3eji − 2);Ak = − N∑ j=1 nj∑ i=1 γjiφ ′′ kj(eji)(2eji − 1); (22) ϕk1(0) = ϕ ′ k1(0) = ϕkN (1) = ϕ ′ kN(1) = 0. (23) Such chosen constants Ak, Bk, Ck, Dk lead the functions {ϕkj(x), j = 1, . . . , N} to be chosen as mode shape of undamaged beam satisfying the given boundary conditions for cracked beam under consideration. It can be seen that trivial values of the constants Ak = Bk = 0 including three cases of boundary conditions for simply supports, cantilevered and free-free ends lead to much simplified calculation. By the reason, only beam with the listed above boundary conditions is investigated in this paper except the case of clamped ends that requires a special study. For the cases of boundary conditions, because of φ′′kj(x) = ϕ ′′ kj(x); φ ′′′ kj(x) = ϕ ′′ kj(x), the two last conditions in (9) are equivalent to Sjϕ ′′ kj(xj) = Sj+1ϕ ′′ k,j+1(xj); Sjϕ ′′′ kj(xj) = Sj+1ϕ ′′′ k,j+1(xj). (24) Thus, the problem now remained to construct the mode shape of intact stepped beam {ϕkj(x), j = 1, . . . , N} that is accomplished by using the transfer matrix method as follows. It was well known that the mode shape of a multiple stepped beam in a uniform segment (xj−1, xj), j = 1, . . . , N and boundary conditions at the beam ends can be expressed as ϕj(x) = dj1 coshλjx+ dj2 sinhλjx+ dj3 cosλjx+ dj4 sinλjx; (25) x ∈ (xj−1, xj); λ 4 j = mjω 2/Sj; (26) [B0] · {d11, d12, d13, d14} T = 0; [B1] · {dN1, dN2, dN3, dN4} T = 0, (27) where B0,B1 are given (2 × 4)-dimensional matrices. The continuity conditions at step joints xj can be represented as [Tj] · {dj1, dj2, dj3, dj4} T = [Tj+1] · {dj+1,1, dj+1,2, dj+1,3, dj+1,4} T , or {dj+1,1, dj+1,2, dj+1,3, dj+1,4} T = [Tj+1] −1 · [Tj] · {dj1, dj2, dj3, dj4} T , (28) where Tj = T(xj, Sj, λj) = Tj(xj),Tj+1 = T(xj , Sj+1, λj+1) = Tj+1(xj) and T(x, S, λ) =   coshλx sinhλx cosλx sinλx λ sinhλx λ coshλx −λ sinλx λ cosλx λ2S coshλx λ2S sinhλx −λ2S cosλx −λ2S sinλx λ3S sinhλx λ3S coshλx λ3S sinλx −λ3S cosλx   . Based the recurrent relationship (28) one obtains {dj1, dj2, dj3, dj4} T = [Hj] · {d11, d12, d13, d14} T , (29) where [Hj] = [Tj(xj−1)] −1[Tj−1(xj−1)] . . . [T2(x1)] −1[T1(x1)] and {dN1, dN2, dN3, dN4} T = [H] · {d11, d12, d13, d14} T ;H = HN(x1, x2, . . . , xN−1). (30) Multiple crack identification in stepped beam by measurements of natural frequencies 125 Combining (27) with (30) yields [B] · {d11, d12, d13, d14} T = 0, (31) where B = [ B0 B1H ] . (32) The condition for existence of nontrivial solution of Eq. (31) is detB(ω) = 0, (33) that is the characteristic equation allowing finding the natural frequencies of multi-stepped beam (ωk0, k = 1, 2, 3, . . .). Every natural frequency ωk0 is associated with a nontrivial solution of Eq. (31) {d11, d12, d13, d14} T = dk{a11, a12, a13, a14} T that contains an arbitrary constant dk. Hence, mode shape in segment (xj−1, xj), j = 2, 3, . . . can be found through the vector {dj1, dj2, dj3, dj4} T = dk[Hj] · {a11, a12, a13, a14} by using expression (25). Thus, the undamaged mode shape functions {φkj(x), j = 1, . . . , N} are constructed. Now we can calculate the natural frequencies of multiple cracked and stepped beam by using Eq. (11) and shape function (14) reduced to φkj(x) = ϕkj(x) +Ckjx+Dkj + nj∑ i=1 γjiϕ ′′ kj(eji)K(x− eji). Note first that the numerator of quotient (11) in the cases is RN = N∑ j=1 Sj   xj∫ xj−1 ϕ′′kj 2 (x)dx+ nj∑ i=1 γjiϕ ′′ kj 2 (eji)   = N∑ j=1 Sj xj∫ xj−1 ϕ′′kj 2 (x)dx+ N∑ j=1 Sj nj∑ i=1 γjiϕ ′′ kj 2 (eji) (34) and the denominator can be calculated as follows. RD = N∑ j=1 mj xj∫ xj−1 φ2kj(x)dx = N∑ j=1 mj   xj∫ xj−1 ϕ2kj(x)dx+ xj∫ xj−1 2ϕkj(x)(Ckjx+Dkj)dx+ xj∫ xj−1 (Ckjx+Dkj) 2dx  + + N∑ j=1 mj   nj∑ i=1  2γjiϕ′′kj(eji) xj∫ eji [ϕkj(x) + Ckjx+Dkj](x− eji)dx    + 126 Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh + N∑ j=1 mj   nj∑ i,`=1  γjiγj`ϕ′′kj(eji)ϕ′′kj(ej`) xj∫ eji` (x− eji)(x− ej`)dx     = N∑ j=1 mj   xj∫ xj−1 ϕ2kj(x)dx   + 2Ψ1 + ω−2k0 Ψ2, (35) where eji` = eji for i ≥ ` or ej` if i ≤ `, Ψ1= N∑ j=1 mj   xj∫ xj−1 ϕkj(x)(Ckjx+Dkj)dx+ nj∑ i=1  γjiϕ′′kj(eji) xj∫ eji ϕkj(x)(x− eji)dx     = ω−2k0 N∑ j=1 nj∑ i=1 Sjγjiϕ ′′ kj 2 (eji); (36) Ψ2= N∑ j=1 mj { P 3j C 2 kj+2P 2 j CkjDkj+P 1 j D 2 kj+ nj∑ i=1 [Ckjαj(eji)+Dkjβj(eji)]γjiϕ ′′ kj(eji) + nj∑ i=1 nj∑ `=1 γjiϕ ′′ kj(eji)γj`ϕ ′′ kj(ej`)qj(eji, ej`) } = N∑ j=1 N∑ p=1 N∑ q=1 np∑ i=1 nq∑ `=1 mjQj(epi, eq`)γpiγq`ϕ ′′ kp(epi)ϕ ′′ kq(eq`) (37) with Pnj =(x n j−x n j−1)/n, n=1, 2, 3;αj(eji)=(xj−eji) 2(2xj+eji)/6;βj(eji)=(xj−eji) 2/2 and coefficients Qj(epi, eq`) given below for different boundary conditions under consideration. Namely, for cantilever and beam with free ends one has Qj(epi, eq`) =   1 3 { (xj − eji) 3 : i ≥ `; (xj − ej`) 3 : i ≤ `; p = q = j; [αj(eji)− eq`βj(eji)]/2 : q ≺ p = j; 0 : p, q  j; [αj(ej`)− epiβj(ej`)]/2 : p ≺ q = j; P 3j − P 2 j (epi + eq`)/2 + P 1 j epieq` : p, q ≺ j. (38) For the case of simply supported beam Qj(epi, eq`)=   P 3j (1− epi)(1− eq`)+(epi + eq`)/2+ 1 3 { (xj − eji) 3 : i ≥ `; (xj − ej`) 3 : i ≤ `; p = q = j; (P 2j − P 3 j )(1− epi)eq` + [αj(eji)− eq`βj(eji) + eq` − 1]/2 : q ≺ p = j; P 3j (1− epi)(1− eq`) : p, q  j; (P 2j − P 3 j )(1− eq`)epi + [αj(ej`)− epiβj(ej`) + epi − 1]/2 : p ≺ q = j; (P 3j − 2P 2 j + P 1 j )epieq` + P 2 j (epi + eq`)/2 : p, q ≺ j. (39) Multiple crack identification in stepped beam by measurements of natural frequencies 127 Thus, Rayleigh quotient for multiple cracked and stepped beam with simply supported, fixed-free and free-free ends is expressed as ω2k = RN/RD =   N∑ j=1 Sj xj∫ xj−1 ϕ′′kj 2 (x)dx+ ωk0 2Ψ1   /   N∑ j=1 mj xj∫ xj−1 ϕkj 2(x)dx+ 2Ψ1 +Ψ2   . (40) Obviously, natural frequencies of undamaged beam can be obtained ω2k0 =   N∑ j=1 Sj xj∫ xj−1 ϕ′′kj 2 (x)dx  /   N∑ j=1 mj xj∫ xj−1 ϕkj 2(x)dx   . (41) Selecting ϕkj(x) = dkϕ¯kj(x) with ϕ¯kj0(x) being the undamaged mode shape normalized by N∑ j=1 mj   xj∫ xj−1 ϕ2kj(x)dx   = 1, (42) one obtains ωk0 2 = N∑ j=1 Sj   xj∫ xj−1 ϕ¯′′ 2 kj(x)dx   (43) and ω2k ω2k0 = 1 + ωk0 −2 N∑ j=1 nj∑ i=1 Sjγjiϕ¯′′ 2 kj(eji) 1 + 2ωk0−2 N∑ j=1 nj∑ i=1 Sjγjiϕ¯′′ 2 kj+ N∑ j=1 N∑ p=1 N∑ q=1 np∑ i=1 nq∑ `=1 mjQj(epi, eq`)γpiγq`ϕ¯ ′′ kp(epi)ϕ¯ ′′ kq(eq`) . (44) This is an explicit expression of natural frequencies for a stepped beam with arbitrary number of cracks that can be used not only for modal analysis of the multiple cracked stepped beam but also provides an important tool for crack identification in the beam. 4. CRACK IDENTIFICATION PROCEDURE Assuming that m natural frequencies (ω¯1, . . . , ω¯m) of an N -stepped beam have been given, the problem is to predict position and size of cracks probably occurred in the beam. To solve the problem the so-called crack scanning method (CSM) proposed by Khiem and Tran [22] is used. Accordingly to the method a mesh (0 ≤ e1 ≺ e2 ≺ . . . . ≺ en ≺ L) of multiple crack positions is initialed for determining unknown depths (a1, . . . , an). For the evaluated crack depth vector the positions of probable cracks could be determined by the peaks on the map of estimated crack depth versus scanning crack positions. The desired depth of the cracks detected to appear at the positions may be corrected by repeating the crack depth assessment with the new mesh consisting from the detected crack positions. 128 Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh Introducing the notations γr = γji, er = eji; r = n1 + . . .+ nj−1 + i = 1, . . . , n = n1 + . . .+ nN ; j = 1, . . . , N, i= 1, . . . , nj, the quotient (47) can be rewritten as ω2k/ω 2 k0 = [ 1 + n∑ r=1 akrγr ] /  1 + 2 n∑ r=1 akrγr + n∑ r1,r2=1 bkr1r2γr1γr2   (45) where akr = ω −2 k0 Sjϕ¯ ′′ 2 kj(er); bkr1r2 = N∑ j=1 mjQj(er1, er2)ϕ¯ ′′ kp(er1)ϕ¯ ′′ kq(er2) (46) with   n1 + n2 + . . .+ nj−1 ≺ r ≤ n1 + n2 + . . .+ nj ; n1 + n2 + . . .+ np−1 ≺ r1 ≤ n1 + n2 + . . .+ np; n1 + n2 + . . .+ nq−1 ≺ r2 ≤ n1 + n2 + . . .+ nq. (47) Using the given natural frequencies (ω¯1, . . . , ω¯m) the Eq. (45) can be rewritten in the form n∑ j=1 [Akj + n∑ `=1 Bkj`γ`]γj = dk, k = 1, . . . , m or [A+B(γ)]{γ}= {d} (48) with A = [(2δk − 1)akj, k = 1, . . . , m; j = 1, . . . , n];B = [δkbkj`, k = 1, . . . , m; j, `= 1, . . . , n]; dk = 1− δk; δk = ω¯ 2 k/ω 2 k0. (49) Eqs. (47) can be solved with respect to unknowns (γ1, . . . , γn) by using the iterationmethod [A+B(γ(i−1))]{γ(i)} = {b}, {γ(0)} = {0}. (50) 5. NUMERICAL RESULTS First, for comparison, especially, with the experimental results the model studied in Ruoloto and Surace, 1997 is adopted here. That is a cantilever beam of length L = 0,8 m; cross section H × B = 0.02m × 0.02m; material constants E = 1.81 × 1011Pa; ρ = 7860 kg/m3. Double crack have been made at the positions e¯1 = e1/L = 0.3175; e¯2 = e2/L = 0.6812 with various scenarios of relative crack depth C1 (20% and 20%); C2 (20% and 30%) and C3 (30% and 20%). Tab. 1 shows ratio (damaged to undamaged) of the first five frequencies obtained by (a) solution of the characteristic equation established in Khiem and Tran, [22]; (b) the experiment accomplished by Ruotolo and Surace [23] and (c) calculation using the Rayleigh Quotient (29). The measured (b) and calculated (c) frequency ratios compared to the theoretical ones (a) give rise to discrepancies presented in the adjacent columns. Obviously, the discrepancies are all insignificant (less than one Multiple crack identification in stepped beam by measurements of natural frequencies 129 percent) so that usefulness of the Rayleigh Quotient derived above for calculating natural frequencies of multiple cracked beam is thus validated. Table 1. Comparison with the theoretical and experimental results Frequency Characteristic(a) Experiment (b) Rayleigh Quotient (c) Number Equation Measured Deviation (%) Calculated Deviation (%) Scenario C1: a1/h = 0.2, a2/h = 0.2 (e¯1 = e1/L = 0.3175 ; e¯2 = e2/L = 0.6812) First 0.9929 0.9924 0.05 0.9945 0.16 Second 0.9908 0.9907 0.01 0.9927 0.19 Third 0.9804 0.9814 0.10 0.9846 0.36 Fourth 0.9965 0.9966 0.01 0.9970 0.42 Fifth 0.9942 N/A N/A 0.9950 0.08 Scenario C2: a1/h = 0.2, a2/h = 0.3(e¯1 = e1/L = 0.3175 ; e¯2 = e2/L = 0.6812) First 0.9901 0.9945 0.40 0.9940 0.39 Second 0.9810 0.9813 0.03 0.9864 0.55 Third 0.9668 0.9642 0.25 0.9746 0.80 Fourth 0.9942 0.9926 0.16 0.9950 0.08 Fifth 0.9907 N/A N/A 0.9921 0,15 Scenario C3: a1/h = 0.3, a2/h = 0.2 (e¯1 = e1/L = 0.3175 ; e¯2 = e2/L = 0.6812) First 0.9844 0.9895 0.52 0.9880 0.36 Second 0.9873 0.9893 0.20 0.9897 0.24 Third 0.9698 0.9692 0.06 0.9755 0.58 Fourth 0.9945 0.9952 0.07 0.9947 0.02 Fifth 0.9907 N/A N/A 0.9906 0.01 To demonstrate also applicability of the Rayleigh Quotient to the multi-crack de- tection problem the crack detection procedure proposed above is running by using the measured natural frequencies given by Ruotolo and Surace [23] (the second column in Tab. 1). Results of the crack magnitude estimation obtained as solution of the Eq. (34) with number of scanning crack points N = 25 from the clamped end to the free one are plotted and shown in Figs. 1-3. The three plots give rise consistently the same peaks at the positions eˆ1 = 0.04; eˆ2 = 0.32; eˆ3 = 0.68 where cracks may occur. Following the crack scanning method, the first estimated crack positions would be taken as a new scanning mesh for correcting crack size. This correction has been accomplished and one obtains in result the following corrected crack magnitude Scenario C1: γˆ1 = 0.0; γˆ2 = 0.0105; γˆ3 = 0.00935; Scenario C2: γˆ1 = 0.0; γˆ2 = 0.0124; γˆ3 = 0.0289; Scenario C3: γˆ1 = 0.0; γˆ2 = 0.0252; γˆ3 = 0.0103. 130 Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh E s ti m a te d c ra c k m a g n it u d e Nondimensional scanning crack position Fig. 1. Estimated crack magnitude from the natural frequencies measured by Ruotolo and Surace in the case of actual cracks are at 0.3175 (depth 20%) and 0.6812 (depth 30%) E s ti m a te d c ra c k m a g n it u d e Nondimensional scanning crack position Fig. 2. Estimated crack magnitude from the natural frequencies measured by Ruotolo and Surace in the case of actual cracks are at 0.3175 (depth 20%) and 0.6812 (depth 20%) Nondimensional scanning crack position E s ti m a te d c ra c k m a g n it u d e Fig. 3. Estimated crack magnitude from the natural frequencies measured by Ruotolo and Surace in the case of actual cracks are at 0.3175 (depth 30%) and 0.6812 (depth 20%) The latter allow us to make a conclusion that the number of cracks detected equal to two and the crack positions are e¯1 = 0.32; e¯2 = 0.68. Final results of crack identification and its accuracy are presented in Tab. 2. Table 2. Results of crack detection Actual crack scenarios Identified crack position and depth (error in %) C1: a1= a2= 0.02 a1= 0.2218 (10.9%) a2= 0.2105 (5.25%) C2: a1= 0.2; a2= 0.3 a1= 0.2427 (21.3%) a2= 0.3588 (19.6%) C3: a1= 0.3; a2= 0.2 a1= 0.3416 (13.8%) a2= 0.2218 (10.9%) e1= 0.3175; e2= 0.6812 e1= 0.32 (0.78%) e2= 0.68 (0.17%) Tab. 2 shows that the proposed herein procedure enables exact crack localization (within accuracy of frequency measurement error) and error in crack extent estimation can be within 10% for the cracks depth less than 20% beam thickness. REFERENCES 131 6. CONCLUSION In the present study the well known Rayleigh Quotient has been developed for multi-stepped beam with arbitrary number of cracks and boundary conditions. This is an explicit expression of natural frequencies in term of crack parameters that provides a simplified method for calculating natural frequencies of the beam. Moreover, the Rayleigh Quotient obtained in more generalized form can be usefully applied also for identifica- tion of multi-stepped beam and boundary condition evaluation from measured natural frequencies. Specifically, based on the explicit expression there has been conducted a sim- ple procedure for multiple crack detection of uniform Euler-Bernoulli beam that allows determining not only the crack parameters but also the amount of cracks in a beam from measured natural frequencies. The procedure is a further development of the so-called crack scanning method for the case of available natural frequencies. The theoretical de- velopment has been illustrated and validated by either numerical or experimental results. Namely, the natural frequencies calculated by the Rayleigh Quotient very well agreed with those computed from the characteristic equation and measured from an experiment. 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Journal of Sound and Vibration, 234(1), (2000), pp. 1–19. [21] J. Fernandez-Saez, L. Rubio, and C. Navarro. Approximate calculation of the fun- damental frequency for bending vibrations of cracked beams. Journal of Sound and Vibration, 225(2), (1999), pp. 345–352. [22] N. T. Khiem and H. T. Tran. A procedure for multiple crack identification in beam- like structure from natural vibration mode. Journal of Vibration and Control, (2013). DOI: 10.1177/1077546312470478. [23] R. Ruotolo and C. Surace. Damage assessment of multiple cracked beams: numerical results and experimental validation. Journal of Sound and Vibration, 206(4), (1997), pp. 567–588. VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 36, N. 2, 2014 CONTENTS Pages 1. Dao Huy Bich, Nguyen Dang Bich, A coupling successive approximation method for solving Duffing equation and its application. 77 2. Nguyen Thai Chung, Hoang Xuan Luong, Nguyen Thi Thanh Xuan, Dynamic stability analysis of laminated composite plate with piezoelectric layers. 95 3. Vu Le Huy, Shoji Kamiya, A direct evidence of fatigue damage growth inside silicon MEMS structures obtained with EBIC technique. 109 4. Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh, Multiple crack identification in stepped beam by measurements of natural frequencies. 119 5. Nguyen Hong Son, Hoang Thi Bich Ngoc, Dinh Van Phong, Nguyen Manh Hung, Experiments and numerical calculation to determine aerodynamic char- acteristics of flows around 3D wings. 133 6. Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha, Free vi- bration analysis of four parameter functionally graded plate accounting for realistic transverse shear mode. 145