Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes
Tóm tắt Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes: ...)] {CL} . (21) Substituting (21) into (16) we obtain {zc(x)} = ( [G0(x)] + n ∑ j=1 [ G¯(x− ej) ] . [ χj ]) . {CL} = [GL (x,ω)] . {CL} , (22) Crack identification in multiple cracked beams made of functionally graded material. . . 111 where [GL (x,ω)] = [G0(x)] + n ∑ j=1 [ G¯(...e Nunber of measured points:200/span 10 % 20% 30% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 -4 SWT detail coefficients: Mode 2 L(m) A m p lit u d e Nunber of measured points:200/span 10 % 20% 30% (f) 116 Tran Van Lien, Ngo Trong Duc 11 ...nd third mode shapes if the SNR exceeds 75dB (Figs. 8d, 9d). a) b) c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x 10 -4 SWT of eigenmodes: 1 Depth Cr= 30% SNR= 75db L(m) D et ai l d iff er en ce dB4 0 0.2 0.4 0.6 0.8 1 -1 0 ...
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odes: 2 Depth Cr= 30% SNR= 80db L(m) D e ta il d if fe re n c e dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 -4 SWT of eigenmodes: 2 Depth Cr= 30% SNR= 90db L(m) D e ta il d if fe re n c e dB4 (f) 18 g) h) i) Fig. 8: Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 crack at 0.2m from the left node with the depth of 30% and noise level 75, 80 và 90dB 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 75db L(m) D et ai l d iff er en ce dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 80db L(m) D et ai l d iff er en ce dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 90db L(m) D et ai l d iff er en ce dB4 (g) 18 g) h) i) Fig. 8: Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 crack at 0.2m from the left node with the depth of 30% and noise level 75, 80 và 90dB 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 75db L(m) D e ta il d iff e re n ce dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 80db L(m) D e ta il d if fe re n c e dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 90db L(m) D et ai l d iff er en ce dB4 (h) 18 g) h) i) Fig. 8: Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 crack at 0.2m from the left node with the depth of 30% and noise level 75, 80 và 90dB 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 75db L(m) D e ta il d if fe re n c e dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 80db L(m) D e ta il d if fe re n c e dB4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 90db L(m) D et ai l d iff er en ce dB4 (i) Fig. 7. Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 crack at 0.2 m from the left node with the depth of 30% and noise level 75, 80 and 90 dB shape if the SNR exceeds 80 dB (Figs. 7(b) and 8(b)), using the second and third mode shapes if the SNR exceeds 75 dB (Figs. 7(d) and 8(d)). Crack identification in multiple cracked beams made of functionally graded material. . . 121 19 a) b) c) (a) 19 a) b) c) (b) 19 a) b c) (c) 20 d) e) f) (d) 20 d) e) f) (e) 20 d) e) f) (f) 21 g) h) i) Fig. 9: Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM beam that has 4 equivalent crack at 0.2m 0.4m 0.6m 0.8m from the left node with the depth of 30% and noise level 75, 80 and 90dB (g) 21 g) h) i) Fig. 9: Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM beam that has 4 equivalent crack at 0.2m 0.4m 0.6m 0.8m from the left node with the depth of 30% and noise level 75, 80 and 90dB (h) 122 Tran Van Lien, Ngo Trong Duc 21 g) h) i) Fig. 9: Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM beam that has 4 equivalent crack at 0.2m 0.4m 0.6m 0.8m from the left node with the depth of 30% and noise level 75, 80 and 90dB (i) Fig. 8. Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM beam that has 4 equivalent cracks at 0.2 m, 0.4 m, 0.6 m, 0.8 m from the left node with the depth of 30% and noise level 75, 80 and 90 dB 4. CONCLUSIONS In this paper, crack identification in a multiple cracked FGM beam by using SWT of mode shapes and taking into account influence of Gaussian noise is addressed. Mode shapes are obtained from the multiple cracked FGM beam element model using DSM and spring model of cracks. Hence, SWT is applied for crack identification of the multiple cracked FGM beam. Numerical analysis was carried out to validate the proposed method and to investi- gate effect of cracks, material properties and Gaussian noise on crack identification of a multiple cracked FGM beam. The illustrating numerical results verify that the multiple cracked FGM beam element model combined with the SWT can be reliably employed for localization of cracks in beam-like structures with data contaminated with noise of SNR from 75 dB. Moreover, crack position can be more clearly detected using only one mode shape of the cracked beam. The investigated results show that proposed method is efficient and realizable. ACKNOWLEDGEMENT This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.301. REFERENCES [1] Z. H. Jin and R. C. Batra. Some basic fracture mechanics concepts in functionally graded materials. Journal of the Mechanics and Physics of Solids, 44, (8), (1996), pp. 1221–1235. https://doi.org/10.1016/0022-5096(96)00041-5. [2] F. Erdogan and B. H. Wu. The surface crack problem for a plate with function- ally graded properties. 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Roots of cub algebraic equation are η1(ω), η2(ω), η3(ω) η1 = −a/3+ u− b1/u, η2,3 = −a/3− (u− b1/u)/2± i √ 3(u + b1/u)/2, where u = (a1+ √ b31 + c 2 1− a3/27)1/3, a1 = ab/6− c/2, b1 = b/3− a2/9, c1 = a3/27− a1. Constants in formula (6) are λ1,4 = ±k1, λ2,5 = ±k2, λ3,6 = ±k3, k j = √ηj, j = 1, 2, 3, αj = ω2 I12 ω2 I11 + λ2j A11 , β j = λj A33 (ω2 I11 + λ2j A33) , j = 1, 2, . . . , 6. APPENDIX 2 Differential matrix operators - Simply supported (S): u(x, t) = M(x, t) = w(x, t) = 0 24 2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−= where 1 3 1 2 11 3/132 1 3 11 27/;9/3/;2/6/;)27/( aacabbcabacbau −=−=−=−++= Constants in formula (6) are 3,2,1,;;; 36,325,214,1 ===== jkkkk jj 6,...,2,1; )( ; 33 2 11 2 33 11 2 11 2 12 2 = + = + = j AI A AI I j j j j j A2. Differential matrix operators - Simply supported (S): 0),(),(),( === txwtxMtxu = 100 0 001 2212 xxS AAB - Pined (P): 0),(),(),( === txwtxMtxN − = 100 0 0 2212 1211 xx xx P AA AA B - Clamped (C): 0),(),(),( === txwtxtxu = 100 010 001 CB - Free (F): 0),(),(),( === txQtxMtxN − − = x xx xx AA AA AA 3333 2212 1211 0 0 0 FB • Simply supported beam ends (SS): SBBB L ==0 • Pined ends (PP): PBBB L ==0 • Simple beam (SP): PBBBB LS == ;0 • Cantilevered beam (CF): FC BBBB L == ;0 • Clamped beam (CC): CBBB L ==0 [BS] = 1 0 0A12∂x A22∂x 0 0 0 1 . - Pined (P): N(x, t) = M(x, t) = w(x, t) = 0 126 Tran Van Lien, Ngo Trong Duc 24 2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−= where 1 3 1 2 11 3/132 1 3 11 27/;9/3/;2/6/;)27/( aacabbcabaacbau −=−=−=−++= Constants in formula (6) are 3,2,1,;;; 36,325,214,1 ===== jkkkk jj 6,...,2,1; )( ; 33 2 11 2 33 11 2 11 2 12 2 = + = + = j AI A AI I j j j j j A2. Differential matrix operators - Simply supported (S): 0),(),(),( === txwtxMtxu = 100 0 001 2212 xxS AAB - Pined (P): 0),(),(),( === txwtxMtxN − = 100 0 0 2212 1211 xx xx P AA AA B - Clamped (C): 0),(),(),( === txwtxtxu = 100 010 001 CB - Free (F): 0),(),(),( === txQtxMtxN − − = x xx xx AA AA AA 3333 2212 1211 0 0 0 FB • Simply supported beam ends (SS): SBBB L ==0 • Pined ends (PP): PBBB L ==0 • Simple beam (SP): PBBBB LS == ;0 • Cantilevered beam (CF): FC BBBB L == ;0 • Clamped beam (CC): CBBB L ==0 [BP] = A11∂x −A12∂x 0A12∂x A22∂x 0 0 0 1 . - Clamped (C): u(x, t) = θ(x, t) = w(x, t) = 0 24 2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−= where 1 3 1 2 11 3/132 1 3 11 27/;9/3/;2/6/;)27/( aacabbcabaacbau −=−=−=−++= Constants in formula (6) are 3,2,1,;;; 36,325,214,1 ===== jkkkk jj 6,...,2,1; )( 33 2 11 2 33 11 2 11 2 12 2 = + = + = j AI A AI I j j j j j A2. Differential matrix operators - Simply supported (S): 0),(),(),( === txwtxMtxu = 100 0 001 2212 xxS AAB - Pined (P): 0),(),(),( === txwtxMtxN − = 100 0 0 2212 1211 xx xx P AA AA B - Clamped (C): 0),(),(),( === txwtxtxu = 100 010 001 CB - Free (F): 0),(),(),( === txQtxtxN − − = x xx xx AA AA AA 3333 2212 1211 0 0 0 FB • Simply supported beam ends (SS): SBBB L ==0 • Pined ends (PP): PBBB L ==0 • Simple beam (SP): PBBBB LS == ;0 • Cantilevered beam (CF): FC BBBB L == ;0 • Clamped beam (CC): CBBB L ==0 [BC] = 1 0 00 1 0 0 0 1 . - Free (F): N(x, t) = M(x, t) = Q(x, t) = 0 24 2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−= where 1 3 1 2 11 3/132 1 3 11 27/;9/3/;2/6/;)27/( aacabbcabaacbau −=−=−=−++= Constants in formula (6) are 3,2,1,;;; 36,325,214,1 ===== jkkkk jj 6,...,2,1; )( ; 33 2 11 2 33 11 2 1 2 12 2 = + = + = j AI A AI I j j j j j A2. Differential matrix operators - Simply supported (S): 0),(),(),( === txwtxMtxu = 100 0 001 2212 xxS AAB - Pined (P): 0),(),(),( === txwtxMtxN − = 100 0 0 2212 1211 xx xx P AA AA B - Clamped (C): 0),(),(),( === txwtxtxu = 100 010 001 CB - Free (F): 0),(),(),( === txQtxMtxN − − = x xx xx AA AA AA 3333 2212 1211 0 0 0 FB • Simply supported bea ends (SS): SBBB L ==0 • Pined ends (P ): PBB =0 • Simple beam (SP): PBBBB LS == ;0 • Cantilevered beam (CF): FC BBBB L == ;0 • Clamped beam (CC): CBBB L ==0 [BF] = A11∂x −A12∂x 0A12∂x 2∂ 0 0 −A33 A33∂x . - Simply supported beam ends (SS): [B0] = [BL] = [BS]. - Pined ends (PP): [B0] = [BL] = [BP]. - Simple a (SP): [B0] = [ S] , [BL] = [BP]. - Cantilevered beam (CF): [B0] = [BC] , [BL] = [BF]. - Clamped beam (CC): [B0] = [BL] = [BC].
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