Mode shape curvature of multiple cracked beam and its use for crack identification in beam like structures (download tai tailieutuoi com)

Vietnam Journal of Mechanics, VAST, Vol 42, No (2020), pp 123 – 132 DOI: https://doi.org/10.15625/0866-7136/14707 MODE SHAPE CURVATURE OF MULTIPLE CRACKED BEAM AND ITS USE FOR CRACK IDENTIFICATION IN BEAM-LIKE STRUCTURES Nguyen Tien Khiem1,∗ Institute of Mechanics, VAST, Hanoi, Vietnam ∗ E-mail: ntkhiem@imech.vast.vn Received: 18 December 2019 / Published online: 23 April 2020 Abstract The problem of using the modal curvature for crack detection is discussed in this paper based on an exact expression of mode shape and its curvature Using the obtained herein exact expression for the mode shape and its curvature, it is demonstrated that the mode shape curvature is really more sensitive to crack than mode shape itself Nevertheless, crack-induced change in the approximate curvature calculated from the exact mode shape by the central finite difference technique (Laplacian) is much greater in comparison with both the mode shape and curvature It is produced by the fact, shown in this study, that miscalculation of the approximate curvature is straightforwardly dependent upon crack magnitude and resolution step of the finite difference approximation Therefore, it can be confidently recommended to use the approximate curvature for multiple crack detection in beam by properly choosing the approximation mesh The theoretical development has been illustrated by numerical results Keywords: multiple-cracked beams, crack detection, mode shape curvature, Laplacian approximation INTRODUCTION Structural damage identification problem has attached enormous interest of either researchers or engineers for several decades Among a large number of techniques proposed to solve the problem, vibration-based method has proved to be the most efficient approach [1–4] This is because a damage occurred in a structure alters straightforwardly the structure’s dynamical characteristics that can be measured by the well-developed modal testing technique Natural frequencies and mode shapes of a structure are the essential characteristics for structural damage detection The frequencies are early used for the structural damage detection [5] because they can be most easily and accurately measured by the dynamic testing technique However, as a global feature of a structure, natural frequencies are slightly sensitive to local damages that should be appropriately c 2020 Vietnam Academy of Science and Technology 124 Nguyen Tien Khiem detected by using the spatial feature of structures such as the mode shapes [6, 7] Nevertheless, mode shapes are more difficult to be accurately determined, so that change in mode shape due to damage might be confused with measurement noise or modeling erroneousness [8] To overcome the drawbacks of the frequency-based and mode shape-based methods, numerous procedures have been proposed to use mutually both the modal parameters (frequency and mode shape) and their derivatives such as flexibility, strain energy for the damage detection problem Most of the developments focused on engaging more refined behavior of spatial characteristics in damaged structures such as mode shape curvature [9–14] Pandey et al [15] first revealed that change in mode shape curvature due to damage is greater than that of mode shape itself and stated that the curvature is a good indicator for damage detection in beams In the study, curvature was calculated from mode shape by using the central finite difference approximation acknowledged as Laplacian operator Then, Wahab [16] expanded the modal curvature technique and applied for damage detection in a real bridge Ratcliffe [17, 18] improved the curvature-based technique of damage detection by using the so-called gapped smoothing procedure to detect small damage that could not be identified by the proposed curvature technique Cao and Qiao [19] proposed a modification of the Laplacian scheme in combination with the Gaussian filter to ignore measurement noise, so that much enhanced the curvature-based technique Chandrashekhar and Ganguli [20] applied the fuzzy logic system that allows the curvature-based technique to detect small damage with noisy measured mode shape The wavelet transform is a useful tool for revealing small localized change in a signal [21] and was employed for crack detection in beam structures using mode shape [22] and curvature [23] However, it requires a large amount of input data and is strongly sensitive to noise or miscalculation of input data Most of the studies mentioned above employed the finite element method for modeling damaged structures that usually proposes rather distributed damage than the local one such as crack The error in finite element modeling damaged structures may affect results of the damage identification, especially, in detecting local damage like crack So, the present paper deals with discussion on the use of the curvature-based technique for multiple crack detection based on an explicit expression established for exact mode shape and its curvature of multiple cracked beams [24] The established expression allows one to investigate sensitivity of exact mode shape and its curvature to crack and obtain miscalculation of the Laplacian operator applied for multiple cracked beam There is demonstrated that the miscalculation increases sensitivity of the approximate curvature compared to the exact one and it is straightforwardly dependent not only on crack location and depth but also on the step of resolution mesh AN EXPRESSION FOR EXACT CURVATURE OF MULTIPLE CRACKED BEAM Let’s consider an Euler–Bernoulli beam with elasticity module E, mass density ρ, length L, cross section area F and moment of inertia I Assume that the beam is cracked at positions e j , j = 1, , n and the equivalent spring model of crack is adopted with the crack magnitude γ j calculated from the crack depth a j as [25] Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures γj = 125 EI = (5.346H/L) I δj , LK j (1) I (δ) = 1.8624δ2 − 3.95δ3 + 16.375δ4 − 37.226δ5 + 76.81δ6 − 126.9δ7 + 172δ8 − 143.97δ9 + 66.56δ10 , δj = a j /h, where h is the beam thickness As well known that modal parameters of the beam such as natural frequency and mode shape satisfy the equation φ( IV ) ( x ) − λ4 φ( x ) = 0, λ4 = L4 ρFω /EI, x ∈ (0, 1), (2) and compatibility conditions at the crack positions φ ej + = φ ej − , ej − , ej + = φ φ ej + = φ φ ej − , φ e j + = φ e j − + γ j φj e j − (3) The conventional boundary conditions for solution of Eq (2) can be expressed in general form (4) φ( p0 ) (0) = φ(q0 ) (0) = φ( p1 ) (1) = φ(q1 ) (1) = In the paper [24], it was shown that solution of Eqs (2), (3) can be represented as φ( x, λ) = CL1 ( x, λ) + DL2 ( x, λ), (5) where n n L1 ( x, λ) = L01 ( x, λ) + ∑ µ1j K ( x − e j ), L2 ( x, λ) = L20 ( x, λ) + ∑ µ2j K ( x − e j ), j =1 S (r ) ( x ), 0, K (r ) ( x ) = (6) j =1 for for x≥0 x≤0 r = 0, 1, 2, S( x ) = (sinh λx + sin λx )/2λ, (7) (8) j −1 µkj = γ j Lk0 e j + ∑ µki S e j − ei , k = 1, 2, j = 1, , n (9) i =1 and functions L10 ( x ), L20 ( x ) are two independent particular solutions of Eq (2) contin( p ,q ) ( p ,q ) uous inside the beam and satisfying boundary conditions L100 (0) = L200 (0) = Obviously, the solution (5) satisfies also first two conditions at x = in (4), so that the remained two conditions (4) at x = for the solution become ( p1 ) CL1 ( p1 ) (1, λ) + DL2 (1, λ) = 0, (q ) (q ) CL1 (1, λ) + DL2 (1, λ) = (10) The later equations have non-trivial constants C, D if ( p1 ) L1 (q ) (q ) ( p1 ) (1, λ) L2 (1, λ) − L1 (1, λ) L2 (1, λ) = (11) µ1j µ2k S pq λ, e j , ek = (12) Substitution of expressions (6) into Eq (11) leads to n n j =1 j =1 F0 (λ) + ∑ µ1j F1j λ, e j + ∑ µ2j F2j λ, e j + n ∑ j,k =1 126 Nguyen Tien Khiem where (q ) (p ) (p ) (q ) F1j = L201 (1)S( p1 ) (1 − e j ) − L201 (1)S(q1 ) (1 − e j ), F2j = L101 (1)S(q1 ) (1 − e j ) − L101 (1)S( p1 ) (1 − e j ), (p ) (q ) (q ) (13) (p ) F0 (λ) = L101 (1) L201 (1) − L101 (1) L201 (1), S pq = S( p1 ) (1 − e j )S(q1 ) (1 − ek ) − S(q1 ) (1 − e j )S( p1 ) (1 − ek ) Eq (12) gives an explicit form of the so-called characteristic equation for multiple cracked beam that could be solved straightforwardly with regard to λ under given crack positions and magnitudes e j , γ j , j = 1, , n Indeed, the recurrent relationships (9) for the parameters µ1j , µ2j , j = 1, , n can be rewritten as [ A ] { µ k } = { bk } , (14) where the following matrix and vectors are used [A] = a ji : a jj = 1, a ji = −γ j S e j − ei , i ≺ j, a ji = 0, i {bk } = γ1 Lk0 (e1 ) , , γn Lk0 (en ) T , j, j = 1, , n , {µk } = {µk1 , , µkn }T Since det[A] = it is easily to obtain {µk } = [A]−1 {bk }, k = 1, that allow completely calculating the parameters µ1j , µ2j , j = 1, , n with given the crack parameters Solution of Eq (12) in combination with Eq (14) gives rise the so-called eigenvalues λk , k = 1, 2, 3, of the multiple cracked beam that are simply related to natural frequencies of the beam by ωk = (λk /L)2 EI/ρF, k = 1, 2, 3, (15) Every eigenvalue or natural frequency associates with a mode shape determined as ( p1 ,q1 ) Φk ( x ) = φ ( x, λk ) = Ck L2 ( p1 ,q1 ) (1, λk ) L1 ( x, λk ) − L1 (1, λk ) L2 ( x, λk ) (16) where Ck is arbitrary constant that can be calculated from a chosen normalization condition, for example, ( p1 ,q1 ) Ck = max L2 ( p1 ,q1 ) (1, λk ) L1 ( x, λk ) − L1 (1, λk ) L2 ( x, λk ) , x ∈ (0, 1) −1 (17) Hence, a close form solution for exact curvature is easily calculated as ( p1 ,q1 ) Φk ( x ) = φ ( x, λk ) = Ck L2 ( p1 ,q1 ) (1, λk ) L1 ( x, λk ) − L1 (1, λk ) L2 ( x, λk ) (18) The above modal parameters have been obtained for general boundary conditions (4) represented through the functions L10 ( x ), L20 ( x ) that can be easily found for the conventional boundary conditions as following: (1) Simply supported beam: L10 ( x ) = sinh λx, L20 ( x ) = sin λx; (2) Clamped end beam: L10 ( x ) = cosh λx − cos λx; L20 ( x ) = sinh λx − sin λx; (3) Free ends beam: L10 ( x ) = cosh λx + cos λx; L20 ( x ) = (sinh λx + sin λx ) x)cosh = cosh − cos = sinh − sin Clamped beam: L1 0L (1x0)( = x −xcos x;Lx2;0L( 2x0)(=x)sinh x −xsin x ; x ; (2)(2) Clamped endend beam: Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures 127 SENSITIVITY EXACT MODE CURVATURE L ( xSHAPE ) = cosh x + cos TO x; LCRACK ( x) = (sinh x + sin x) (3) FreeOF ends beam: (3) Free ends beam: L1 0( 1x0) = cosh x + cos x; L2 0( 2x0) = (sinh x + sin x) The expressions (16) and (18) are used herein to examine deviation of mode shape and modal curvatureIII caused by multiple OF cracks in beam Namely, the deviations are III SENSITIVITY OF EXACT MODE SHAPE CURVATURE CRAC SENSITIVITY EXACT MODE SHAPE CURVATURE TOTO CRACK calculated as deviation (18) are used herein to examine of mode shape The (18) are used herein to examine deviation of mode shape andand m ∆Φ (19) ∆ΦThe ( x )expressions = φ ( x, λk(16) φ0and x,and λ ) −(16) kexpressions k ( x ) = φ ( x, λk ) − φ0 x, λk , k , caused multiple cracks in beam Namely, deviationsofare calculated as caused by by multiple cracks in beam Namely, thethe deviations calculated where λ0k is k-th eigenvalue of uncracked beam determined as solutionare the equationas ″ 0 ″ curvature F0 λ0k = (see Eq (12)),𝛥𝛷 φ0𝛥𝛷 , φ𝜙(𝑥, x, λ mode shape and of″intact = −0 𝜙 𝜆0𝑘𝛥𝛷 ); 𝛥𝛷 =″ (𝑥, 𝜙 (𝑥, −0″ (𝑥, 𝜙0″ (𝑥, (𝑥) 𝜆0k𝑘 )𝜆are − (𝑥,0 (𝑥, 𝜆0𝑘 ); =𝜙 𝜆𝑘 )𝜆− 𝜆0𝑘 ),𝜆𝑘 ), 𝜙(𝑥, 𝑘λ(𝑥) 𝑘 )𝜙 𝑘 )𝜙 k= 𝑘 (𝑥) 𝑘 x, 𝑘 (𝑥) beam determined as where is k-th eigenvalue of uncracked beam determined as solution of the equation 𝐹00𝑘( where 𝜆0𝑘 𝜆is𝑘 k-th eigenvalue of uncracked beam determined as solution of the equation 𝐹0 (𝜆 ( p1 ,q1 ) ( p1 ,q1 ) 0 0 0 L20 ″ x φ λ2 k0( x=0) C 1, λk L20 x, λk , 1, L10x ;x, λk − L10 0x𝜙;x,L𝜙 −0λsin : L1 0( x) = cosh x − cos (12)), 𝜆=0𝑘k𝜙sinh )k are mode shape curvature of intact beam determined ),0″ 𝜙 (12)), 𝜆𝑘 ), (𝑥,0 (𝑥, 𝜆0𝑘 )𝜆𝑘are mode shape andand curvature of intact beam determined as as (𝑥, (𝑥, ( p1 ,q1 ) ( p1 ,q1 ) 0 0 0 x, φ0 x, λ = Ck L x, λ(𝑝 (𝑝 ,𝑞1 ) 0 ),𝑞1L)10 01, λk L20 0 λk(𝑝1.,𝑞 k xL k 11− x0+(𝑥, )010.[𝐿 01,𝜆λ 0( x) = cosh x + cos x; L 0( x)k = (sinh ) 𝐶= 𝐶 0(𝑝[𝐿1 ,𝑞 𝜆 )𝐿(𝑥, 𝐿 11) (1, (1, 𝜆 )𝐿(𝑥, 𝜆0 )]; 𝜙 20𝜙 (𝑥, 𝜆sin )= (1,(1, 𝜆 )𝐿 𝜆0 )𝜆−)𝐿− 𝜆 )𝐿 𝜆0 )]; 10 (𝑥, 20 (𝑥, TY OF EXACT 𝑘 𝑘 𝑘 𝑘20 20 𝑘 𝑘10 𝑘 𝑘 10 10 18) are used herein to examine deviation of mode shape and0.08 modal curvature 0.03 0.03 0.08 eam Namely, the deviations are calculated as 0.06 0.06 0.0250.025 𝜙(𝑥, 𝜆𝑘 ) − 𝜙0 (𝑥, 𝜆0𝑘 ); 𝛥𝛷𝑘″0.02(𝑥) = 𝜙 ″ (𝑥, 𝜆𝑘 ) − 𝜙0″ (𝑥, 𝜆0𝑘 ),0.04 0.04 (19) 0.02 uncracked beam determined as solution of the equation 𝐹 0.02𝑘 ) = (see Eq (𝜆 0.02 0.0150.015 mode shape and curvature of intact beam determined as 0.01 0.01 (𝑝 ,𝑞1 ) 𝑘20 𝑘 𝑘 0.005 0.005𝜆 (1, 𝜆0𝑘 )𝐿10 (𝑥, 𝑘 ) − 𝐿10 (𝑝 ,𝑞1 ) (1, 𝜆0𝑘 )𝐿20 (𝑥, 𝜆𝑘 )]; 𝐶𝑘0 [𝐿201 (1, 𝜆0𝑘 )𝐿″10 (𝑥,0𝜆0𝑘0) − 𝐿101 (𝑝 ,𝑞1 ) (1, 𝜆0𝑘 )𝐿″20 (𝑥, 𝜆0𝑘 )] 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.15 0.15 0.1 0.1 0.05 0.05 0 1-0.05-0.05 0 0 -0.02 -0.02 𝐶𝑘0 [𝐿201 (𝑝 ,𝑞1 ) 𝑘 (𝑝 modes (𝑝11)with ,𝑞1 ) 90 0″ ″ (𝑝1 ,𝑞 The deviations (19) calculated first three beam 0″ a ″cantilever ) of 𝜆0for 𝐶 0(𝑝 𝜆0 ) − ″ 𝜙 ″ (𝑥, ,𝑞11),𝑞 ) = [𝐿 (1, 𝜆 )𝐿 (𝑥, 𝐿 𝜆 )𝐿(𝑥,(𝑥, 𝜆0 )] 𝜙 (𝑥, 𝜆 ) = 𝐶 [𝐿 (1, 𝜆 )𝐿 (𝑥, 𝜆 ) − 𝐿 (1,(1, 𝜆 )𝐿 𝜆0 )] 10 𝑘 𝑘 𝑘 𝑘 20 10 10 𝑘 𝑘 𝑘 𝑘 20 10 MODE SHAPE CURVATURE TO CRACK cracks (from 0.1 to 0.9) along the normalized beam length (horizontal axis) are shown𝑘 𝑘20 20 𝑘 𝑘 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 -0.04 -0.04 -0.1 -0.1 -0.06 -0.06 -0.15-0.15 (a) First mode 0.2 0.2 0.4 0.4 (b) Second mode (a)(a) (b)(b) (c) (c) 0.15 Deviation of three mode shapes first, b- second, c- third mode) crack Fig.Fig Deviation of three mode shapes (a- (afirst, b- second, c- third mode) duedue to 9tocracks a depths 10%; 30%; 50%; 60% 0.1 depths 10%; 30%; 50%; 60% 0.08 0.06 0.04 0.05 0.02 0.2 0.1 0.3 0.2 0.4 0.3 0.5 0.4 0.6 0.5 0.7 0.6 0.8 0.7 0.9 0.8 0.9 0.1 1 15 15 0 0.2 0.4 0.6 -0.02 -0.04 -0.06 0.2 0.4 0.6 -0.5 -0.5 0.8 -1 -1 -0.1 -1.5 -1.5 -0.15 -3.5 -3.5 15 10 10 10 5 0 20 20 0 0.2 0.1 0.3 0.2 0.4 0.3 0.5 0.4 0.6 0.5 0.1 0.1 0.2 0.4 0.3 0.4 0.5 0.7 0.6 0.8 0.7 0.9 0.8 0.9 0.2 0.3 0.5 0.6 1 -5 -50.1 -20 -20 -2 -2 (c) Third mode (b) -2.5 -2.5 (c)-10 -10 -40 -40 -15 atat0.1-0.9 first,Fig b-1.second, c-ofthird due to Deviation three mode) mode shapes due9tocracks -15 cracks 0.1–0.9 ofof depths 10%, 30%, 50%, 60% ode shapes (a-3 -3 depths 10%; 30%; 50%; 60% -0.05 40 40 0.8 -20 -20 -60 -60 (a)(a) (b)(b) (c) (c 40 Deviation of exact curvature of three modes first, b- second, c- third mode) Fig.Fig Deviation of exact curvature of three modes (a- (afirst, b- second, c- third mode) duedu t 0.1-0.9 with depth 10%;30%;50%;60% 0.1-0.9 with depth 10%;30%;50%;60% 20 0.03 0.025 0.025 0.02 ″ 𝑘 0.02 0.015 0.015 0.01 0.01 0.005 0.005 𝑘 0 0 0.08 0.15 0.1 0.1 0.06 in beam Namely, the deviations are calculated as 0.06 0.04 ) = 𝜙(𝑥, 𝜆𝑘 ) − 𝜙0 (𝑥, 𝜆0𝑘 ); 𝛥𝛷 (𝑥) = 𝜙 ″ (𝑥, 𝜆𝑘 ) − 𝜙0″ (𝑥, 𝜆0𝑘 ), 0.04 (19) 0.02 e of uncracked beam determined as solution of the equation 𝐹0 (𝜆 0.02 𝑘 ) = (see Eq 0.8 (𝑝 ,𝑞1 ) = 𝐶𝑘0 [𝐿201 = 00 -0.02 (𝑝1 ,𝑞1 ) -0.02 0 𝑘 20 𝑘 -0.04 10 Nguyen Tien-0.04 Khiem (𝑝1 ,𝑞1 ) 0 ″ 0.2 0.4 0.6 0.8 -0.06 20 𝑘 𝑘 10 0.2 0.4 0.6 0.8 -0.06 (1, 𝜆0𝑘 )𝐿10 (𝑥, 𝜆 ) − 𝐿 (1, 𝜆 )𝐿 (𝑥, 𝜆 )]; 𝐿 (1, 𝜆 )𝐿 (𝑥, 𝜆 )] 128 (𝑝 ,𝑞 ) 𝐶𝑘0 [𝐿201 (1, 𝜆0𝑘 )𝐿″10 (𝑥, 𝜆𝑘 ) − 0 ) are mode shape and curvature of intact beam determined as 𝑘) 𝑘) 0.05 0.05 0.2 0.4 0.2 0.6 0.4 0.8 0.6 0.8 00 -0.05 -0.05 0.2 0.4 0.2 0.6 0.4 0.6 -0.1 -0.1 -0.15 -0.15 in Figs and for mode Every box in the Fig(a) shapes and curvatures respectively (b) (c) (a) corresponding to various depth (b) of the cracks from 10% (c) ures demonstrates four curves Fig Deviation of three mode shapes (afirst, bsecond, cthird mode) due to cracks at 0.15 to 60% beamFig thickness that show monotony the deviation with due to cracks at Deviation of three modeincreasing shapes (a-offirst, b- second,magnitude c- third mode) depths 10%; 30%; 50%; 60% the crack depth depths 10%; 30%; 50%; 60% 0.1 0.08 0.06 0.04 0.05 0.02 0 -0.02 0.2 0.4 -1 -0.04 0.6 -0.5 -1.5 -0.06 -2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 00.5 0.6 0.7 0.8 0.9 15 -0.5 0.8 0.2 0.4 -0.05 -1 -0.1 -1.5 -0.15 -2 (b) -2.5 -2.5 ee mode shapes (a- first, b- second, c- third mode) due to -3 -3 depths 10%; 30%; -3.5 50%; 60% .8 0.9 -3.5 10 -5 0.1 0.2 0.3 0.4 -15 -20 0.6 0.8 10 20 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -5 00.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -5 -10 (c) -10 -15 cracks -15 at 0.1-0.9 of -20 -20 40 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 -20 -20 -40 -60 -40 -60 (b) Second mode (a) (b) (c) (a) (b) (c) Fig Deviation of 40 exact curvature of three modes (a- first, b- second, c- third mode) due to Fig Deviation of exact curvature of three modes (a- first, b- second, c- third mode) due to 0.1-0.9 with depth 10%;30%;50%;60% 0.1-0.9 with depth 10%;30%;50%;60% 20 15 -10 10 40 15 (a) First mode The deviations (19) calculated for first three modes of a cantilever beam with cracks (from The deviations0 (19) calculated for first three modes of a cantilever beam with cracks (fro 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 are shown in Fig and Fig for mode s along normalized beam0.2length (horizontal axis) 0.5 0.6 0.7the 0.9 -20 along0.8the normalized beam length (horizontal axis) are shown in Fig and Fig for mode curvatures respectively Every box in the Figures demonstrates four curves corresponding to var curvatures respectively Every box in the Figures demonstrates four curves corresponding to v -40 of the cracks from 10% to 60% beam thickness that show monotony increasing of the deviation of the cracks from 10% to 60% beam thickness that show monotony increasing of the deviatio -60 with the crack depth with the crack depth (c) Third mode (b) (c) Fig Deviation of exact curvature of threeto modes due to t curvature of three modes (a- first, b- second, c- third mode) due cracks at9 cracks at 0.1–0.9 with depth 10%, 30%, 50%, 60% 0.1-0.9 with depth 10%;30%;50%;60% Note that variation of mode shape due to cracks is visibly observed at the crack lculated for first three modes of a cantilever beam with cracks (from 0.1 to 0.9) positions (see Fig 1), but magnitude of the variation is very small (within 10%) So that m length (horizontal axis)would are shown in Fig andby Fig for mode shapes with and error of 10% Deviation cracks be difficult to detect mode shape measured of exact curvature caused by cracks is significantly magnified (see Fig 2) in comparison very box in the Figures demonstrates four curves corresponding to various depth with mode shape variation Nevertheless, the change in modal curvature is rather dis60% beam thickness that show the deviation tributed than monotony localized atincreasing the cracks of positions so thatmagnitude cracks are also not easily localized from measurement of curvature even if base-line data are available This encourages us to find another more efficient indicator for the crack detection, one of that is considered in subsequent section Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures 129 SENSITIVITY OF LAPLACIAN APPROXIMATE CURVATURE DUE TO CRACK Assume that mode shape and curvature of a beam have been measured at the mesh ( x0 , x1 , , xn+1 ) with resolution h and x0 = 0, xn+1 = 1, i.e there are given two sets of data: φ x j , φ x j , j = 0, , n + Let’s consider three subsequent points ( x j−1 , x j , x j+1 ) of the mesh and suppose that each of the segments ( x j−1 , x j ), ( x j , x j+1 )may contains only one crack at position e j−1 ∈ ( x j−1 , x j ), e j ∈ ( x j , x j+1 ), respectively Taylor’s expansion of the function φ( x ) at the points e j−1 , e j yields φ x j +1 − = φ e j + + φ e j + x j+1 − e j + (1/2)φ ej + x j +1 − e j + , φ( x j + 0) = φ(e j − 0) + φ (e j − 0)( x j − e j ) + (1/2)φ (e j − 0)( x j − e j )2 + , φ( x j − 0) = φ(e j−1 + 0) + φ (e j−1 + 0)( x j − e j−1 ) + (1/2)φ (e j−1 + 0)( x j − e j−1 )2 + , (20) φ( x j−1 + 0) = φ(e j−1 − 0) + φ (e j−1 − 0)( x j−1 − e j ) + (1/2)φ (e j−1 − 0)( x j−1 − e j )2 + Using the expressions (20) with neglected terms of order higher gives φ x j+1 − 2φ x j + φ x j−1 = φ with α j = γ j x j+1 − e j + h x¯ j − e j , x j h2 + φ ej αj + φ e j −1 α j −1 , α j−1 = γ j−1 e j−1 − x j−1 + h e j−1 − x¯ j−1 , x¯ j = ( x j+1 + x j )/2, x¯ j−1 = ( x j + x j−1 )/2 (21) Recalling the notations introduced for approximate curvature one gets finally φ xj − φ xj = β j φ xj , (22) where βj = φ φ xj xj −1 = φ ej αj + φ φ xj e j −1 α j −1 φ h2 e j γj + φ 2φ e j −1 γ j −1 xj h + O h2 (23) In case of no crack surrounding the mesh point x j , one has got φˆ x j − φ x j = O h2 , that implies negligible difference between approximate and exact curvatures at an intact section, i.e., (24) φˆ x j − φ0 x j = O h2 On the other hand, if both the crack locations coincide with x j , i.e., e j−1 = e j = x j , γ j−1 = γ j , Eq (22) gives φˆ xj − φ x j = γj φ x j /h (25) The latter equation shows that miscalculation of the Laplacian curvature at a crack position depends on the crack magnitude, value of curvature at the crack and resolution step Namely, the miscalculation gets to be increasing with reduction of the step h and grow with the crack magnitude γ j Also, crack appeared at the node of curvature (where curvature vanishes) makes no effect on the mode shape, curvature including the approximate one In general, Eqs (22), (24) allow one to obtain ∆φ xj = φ x j − φ0 x j = ∆φ xj + β j φ xj , (26) node curvature including the curvature(where (wherecurvature curvaturevanishes) vanishes)makes makesno noeffect effecton onthe themode mode shape, shape, curvature curvature including including the the nodeofofcurvature curvature (where curvature vanishes) makes no effect on the mode shape, 130 Nguyen Tien Khiem approximate one approximate one approximate one InIngeneral, general,Equations Equations(22), (22),(24) (24)allow allowone oneto toobtain obtain general, Equations (22), (24) allow one to obtain ″″″ ″ ″ ″ ″ ̂ ̂ ̂ ″ )=𝜙 ″ )−𝜙 where ″ ″ ̂ ̂ ̂ 𝛥𝜙 (𝑥 (𝑥 (𝑥 (26) ̂ ̂ ̂ 𝛥𝜙 (𝑥 (𝑥 =𝜙 𝜙 (𝑥 (𝑥 −𝜙 𝜙000(𝑥 (𝑥𝑗𝑗𝑗)))= =𝛥𝜙 𝛥𝜙″″(𝑥 (𝑥𝑗𝑗𝑗)))+ +𝛽 (𝑥𝑗𝑗𝑗), 𝑗𝑗𝑗)) = 𝑗𝑗𝑗)) − 𝛥𝜙 = 𝛥𝜙 (𝑥 + 𝛽𝛽𝑗𝑗𝑗𝜙 𝜙𝜙″″(𝑥 (𝑥 ),), (26) ∆φ x j = φ x j − φ0 x j (27) where where ″ ″″″ ″″″ 𝛥𝜙 (𝑥 = (𝑥 It can be seen from Eq (26) 𝛥𝜙that (𝑥 =𝜙𝜙 𝜙miscalculation (𝑥𝑗𝑗𝑗)))− −𝜙 𝜙00″0″(𝑥 (𝑥𝑗𝑗𝑗) 𝑗𝑗𝑗)))the 𝛥𝜙 (𝑥 = (𝑥 − 𝜙 (𝑥 ).) of the approximate curvature increases its sensitivity to crack in comparison with exact For illustration ofto ItItcan curvature sensitivity to seenfrom fromEq Eq.(26) (26)that thatthe themiscalculation miscalculationof ofthe theapproximate approximatecurvature curvature increases increases itssensitivity sensitivity canbebeseen seen from Eq (26) that the miscalculation of the approximate its the fact, deviation of the Laplacian curvature due to multiple cracks is calculated by using crack Laplacian curvature comparisonwith withexact exactcurvature curvature.For Forillustration illustrationof ofthe thefact, fact,deviation deviation of of the the Laplacian Laplacian curvature crackinincomparison comparison with exact curvature For illustration of the fact, expression cracks (16) for three lowest modes of cantilever beamlowest and results are demonstrated due cantilever beam and and multiple cracks cracksisis iscalculated calculatedby byusing usingexpression expression(16) (16)for forthree three lowestmodes modesof ofcantilever cantileverbeam duetotomultiple multiple calculated by using expression (16) for three in Fig results demonstratedinin inFig Fig.3.3 resultsare aredemonstrated demonstrated Fig 2525 25 150 150 150 2020 20 100 100 100 15 1515 50 5050 10 1010 00 000 -50 -50 -50 400 400 400 300 300 300 200 200 200 55 00 00 -5 -5-5 100 100 100 00 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 111 -10000 -100 -100 0.2 0.2 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 11 -200 -200 -200 -100 -100 -100 0.2 0.40.4 0.4 0.20.2 (a) 0.6 0.60.6 0.8 0.8 0.8 11 -300 -300 -300 -400 -400 -400 -150 -150 -150 (b) (c) Fig.3.3 Deviation approximate curvature of first three modes 3.Deviation Deviationofof ofapproximate approximatecurvature curvatureof offirst firstthree threemodes modes due due to to 99 cracks cracks at at 0.1-0.9 0.1-0.9with withequal equal Fig 0.1-0.9 with depth 10%; 30%; 50%; 60% depth10%; 10%;30%; 30%;50%; 50%;60% 60% depth Fig Deviation of approximate curvature of first three modes due to cracks at 0.1–0.9 with equal depth 10%, 30%, 50%, 60% curvature Graphs shown in Fig demonstrate strong sensitivity of approximate curvature Graphs shown in Fig demonstrate strong sensitivity of approximate curvatureto to either eithermagnitude magnitudeor or Graphs shown in Fig demonstrate strong sensitivity of approximate to either magnitude or position of cracks that confirms theoretically once more the usefulness of the approximate curvature in cracksthat thatconfirms confirmstheoretically theoreticallyonce oncemore morethe the usefulness usefulness of of the the approximate approximate curvature curvature in in position of cracks Graphs shown in Fig demonstrate strong sensitivity of approximate curvature to crack localization for beam that was only numerically acknowledged in a number of previous studies localization for beam that was only numerically acknowledged in a number of previous studies crack localization for beam that was only numerically acknowledged in a number of previous studies either magnitude or position of cracks that confirms theoretically once more the usefulV CONCLUDING REMARKS ness of the approximate curvature in crack localization for beam that was only numeriV CONCLUDING CONCLUDING REMARKS V REMARKS cally acknowledged in a number of previous studies Themain main results this study can be summarized as follow: mainresults resultsofof ofthis thisstudy studycan canbe besummarized summarizedas asfollow: follow: The Anexpression expression for exact mode shapes and mode shape curvatures expressionfor forexact exactmode modeshapes shapesand andmode mode shape shape curvatures curvatures have have been been obtained obtained for for multiple multiple 1.1 An CONCLUDING REMARKS have been obtained for multiple cracked beams that provides an efficient tool for analysis and identification of the beam structures cracked beams that provides an efficient tool for analysis and identification of the beam structures cracked beams that provides an efficient tool for analysis and identification of the beam structures Thethe main results of this study can be summarized as follow: Using the obtained expression, was shown that mode shape curvature isis really the obtained expression, was shown thatmode mode shape shape curvature is really more more sensitive sensitive to to Using obtained expression, itititwas shown that curvature really more sensitive to - Anthan expression for exact mode shapes and mode shape curvatures have been obcracks the mode shape itself, however, the exact curvature sensitivity to crack is much less than thanthe themode modeshape shapeitself, itself,however, however,the theexact exactcurvature curvature sensitivity sensitivity to to crack crack is is much much less less than than cracks than tained for multiple cracked beams that provides an efficient tool for analysis and identithat of approximate curvature calculated by the finite difference approximation approximate curvature calculated by the finite difference approximation that of approximate curvature calculated by the finite difference approximation of the can beam structures 3.fication The paradox be explained by the fact that sensitivity of the approximate paradox canbe beexplained explainedby bythe thefact factthat that sensitivity sensitivity of of the the approximate approximate curvature curvature to to crack crack isis is The paradox can curvature to crack Using the obtained expression, it was shown that mode shape curvature is really magnified by its miscalculation, that is also strongly depended upon crack magnitude and resolution magnified by its miscalculation, that is also strongly depended upon crack magnitude and resolution magnified by itsto miscalculation, also strongly depended upon crack and resolution more sensitive cracks than that the ismode shape itself, however, the magnitude exact curvature senstep step to crack is much less than that of approximate curvature calculated by the finite sitivity Finally, the approximate Laplacian curvature would be useful indicator for Finally,the the approximateLaplacian Laplaciancurvature curvaturewould wouldbe beaaauseful usefulindicator indicator for for multiple-crack multiple-crackdetection, detection, approximation difference Finally, approximate multiple-crack detection, if the base-line mode shape has been with sufficient accuracy base-line mode shape has been measured measured with sufficient accuracy.of the approximate cur- The paradox can behas explained by the factsufficient that sensitivity if the base-line mode shape been measured with accuracy 5.vature The effect of noise in measurement of mode shape on the sensitivity of the curvature effect of noise in measurement of mode shape on the sensitivity of the approximate approximate curvature to to crack is magnified by its miscalculation, is alsoof strongly depended uponto The effect of noise in measurement of mode shape on thethat sensitivity the approximate curvature to crack is in paper, is not not yet yet considered considered in the the present present paper, itit would would be be aa topic topic for for further further study study of of the theauthor author crack and resolution step.paper, crackmagnitude is not yet considered in the present it would be a topic for further study of the author - Finally, the approximate Laplacian curvature would be a useful indicator for multiple-crack detection, if the base-line mode shape has been measured with sufficient accuracy Mode shape curvature of multiple cracked beam and its use for crack identification in 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multiple multiple 1.1 An CONCLUDING REMARKS have been obtained for multiple. .. shape curvature of multiple cracked beam and its use for crack identification in beam- like structures 129 SENSITIVITY OF LAPLACIAN APPROXIMATE CURVATURE DUE TO CRACK Assume that mode shape and curvature. .. curvature of multiple cracked beam and its use for crack identification in beam- like structures 131 - The effect of noise in measurement of mode shape on the sensitivity of the approximate curvature