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 ...
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. Journal of Applied Mechanics, 64, (3), (1997), pp. 449–456. https://doi.org/10.1115/1.2788914. Crack identification in multiple cracked beams made of functionally graded material. . . 123 [3] L. L. Ke, J. Yang, S. Kitipornchai, and Y. Xiang. Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mechanics of Advanced Ma- terials and Structures, 16, (6), (2009), pp. 488–502. https://doi.org/10.1080/15376490902781175. [4] J. Yang, Y. Chen, Y. Xiang, and X. L. Jia. Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. Journal of Sound and Vibration, 312, (1-2), (2008), pp. 166–181. https://doi.org/10.1016/j.jsv.2007.10.034. [5] K. Aydin. Free vibration of functionally graded beams with arbitrary number of surface cracks. European Journal of Mechanics-A/Solids, 42, (2013), pp. 112–124. https://doi.org/10.1016/j.euromechsol.2013.05.002. [6] J. Yang and Y. Chen. Free vibration and buckling analyses of functionally graded beams with edge cracks. Composite Structures, 83, (1), (2008), pp. 48–60. https://doi.org/10.1016/j.compstruct.2007.03.006. [7] D. Wei, Y. Liu, and Z. Xiang. An analytical method for free vibration analysis of functionally graded beams with edge cracks. Journal of Sound and Vibration, 331, (7), (2012), pp. 1686–1700. https://doi.org/10.1016/j.jsv.2011.11.020. [8] K. Sherafatnia, G. Farrahi, and S. A. Faghidian. Analytic approach to free vibration and buck- ling analysis of functionally graded beams with edge cracks using four engineering beam theories. International Journal of Engineering-Transactions C: Aspects, 27, (6), (2013), pp. 979– 990. https://doi.org/10.5829/idosi.ije.2014.27.06c.17. [9] N. T. Khiem and D. T. Hung. A closed-form solution for free vibration of multiple cracked Timoshenko beam and application. Vietnam Journal of Mechanics, 39, (4), (2017), pp. 315–328. https://doi.org/10.15625/0866-7136/9641. [10] N. T. Khiem, L. K. Toan, and N. T. L. Khue. Change in mode shape nodes of multiple cracked bar: I. The theoretical study. Vietnam Journal of Mechanics, 35, (3), (2013), pp. 175– 188. https://doi.org/10.15625/0866-7136/35/3/2486. [11] N. T. Khiem, L. K. Toan, and N. T. L. Khue. Change in mode shape nodes of multiple cracked bar: II. The numerical analysis. Vietnam Journal of Mechanics, 35, (4), (2013), pp. 299–311. https://doi.org/10.15625/0866-7136/35/4/2487. [12] S. Kitipornchai, L. L. Ke, J. Yang, and Y. Xiang. Nonlinear vibration of edge cracked function- ally graded Timoshenko beams. Journal of Sound and Vibration, 324, (3-5), (2009), pp. 962–982. https://doi.org/10.1016/j.jsv.2009.02.023. [13] Z. Yu and F. Chu. Identification of crack in functionally graded material beams using the p- version of finite element method. Journal of Sound and Vibration, 325, (1-2), (2009), pp. 69–84. https://doi.org/10.1016/j.jsv.2009.03.010. [14] Sá. D. Akbasá. Free vibration characteristics of edge cracked functionally graded beams by using finite element method. International Journal of Engineering Trends and Technology, 4, (10), (2013), pp. 4590–4597. [15] A. Banerjee, B. Panigrahi, and G. Pohit. Crack modelling and detection in Timoshenko FGM beam under transverse vibration using frequency contour and response surface model with GA. Nondestructive Testing and Evaluation, 31, (2), (2016), pp. 142–164. https://doi.org/10.1080/10589759.2015.1071812. [16] K. V. Nguyen. Crack detection of a beam-like bridge using 3D mode shapes. Vietnam Journal of Mechanics, 36, (1), (2014), pp. 13–25. https://doi.org/10.15625/0866-7136/36/1/2965. [17] H. Su and J. R. Banerjee. Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Computers & Structures, 147, (2015), pp. 107–116. https://doi.org/10.1016/j.compstruc.2014.10.001. 124 Tran Van Lien, Ngo Trong Duc [18] T. V. Lien, N. T. Duc, and N. T. Khiem. Free vibration analysis of multiple cracked function- ally graded Timoshenko beams. Latin American Journal of Solids and Structures, 14, (9), (2017), pp. 1752–1766. https://doi.org/10.1590/1679-78253693. [19] T. V. Lien, N. T. Duc, and N. T. Khiem. Mode shape analysis of multiple cracked functionally graded Timoshenko beams. Latin American Journal of Solids and Structures, 14, (7), (2017), pp. 1327–1344. https://doi.org/10.1590/1679-78253496. [20] N. T. Khiem and T. V. Lien. The dynamic stiffness matrix method in forced vibration anal- ysis of multiple-cracked beam. Journal of Sound and Vibration, 254, (3), (2002), pp. 541–555. https://doi.org/10.1006/jsvi.2001.4109. [21] T. V. Lien, N. T. Duc, and N. T. Khiem. Mode shape analysis of multiple cracked functionally graded beam-like structures by using dynamic stiffness method. Vietnam Journal of Mechanics, 39, (3), (2017), pp. 215–228. https://doi.org/10.15625/0866-7136/8631. [22] F. Nazari and M. H. Abolbashari. Double cracks identification in functionally graded beams using artificial neural network. Journal of Solid Mechanics, 5, (1), (2013), pp. 14–21. [23] N. T. Khiem and N. N. Huyen. A method for crack identification in functionally graded Timoshenko beam. Nondestructive Testing and Evaluation, 32, (3), (2017), pp. 319–341. https://doi.org/10.1080/10589759.2016.1226304. [24] H. Sohn, C. R. Farrar, F. M. Hemez, and J. Czarnecki. A review of structural health monitoring literature: 1996–2001. Los Alamos National Laboratory, USA, (2003). [25] C. Surace and R. Ruotolo. Crack detection of a beam using the wavelet transform. In Proceedings-Spie The International Society For Optical Engineering, (1994), pp. 1141–1147. [26] K. M. Liew and Q. Wang. Application of wavelet theory for crack identifica- tion in structures. Journal of Engineering Mechanics, 124, (2), (1998), pp. 152–157. https://doi.org/10.1061/(asce)0733-9399(1998)124:2(152). [27] Q. Wang and X. Deng. Damage detection with spatial wavelets. International Journal of Solids and Structures, 36, (23), (1999), pp. 3443–3468. https://doi.org/10.1016/s0020-7683(98)00152- 8. [28] E. Douka, S. Loutridis, and A. Trochidis. Crack identification in beams using wavelet analysis. International Journal of Solids and Structures, 40, (13-14), (2003), pp. 3557–3569. https://doi.org/10.1016/s0020-7683(03)00147-1. [29] C. C. Chang and L. W. Chen. Detection of the location and size of cracks in the multiple cracked beam by spatial wavelet based approach. Mechanical Systems and Signal Processing, 19, (1), (2005), pp. 139–155. https://doi.org/10.1016/j.ymssp.2003.11.001. [30] W. Zhang, Z. Wang, and H. Ma. Crack identification in stepped cantilever beam combining wavelet analysis with transform matrix. Acta Mechanica Solida Sinica, 22, (4), (2009), pp. 360– 368. https://doi.org/10.1016/s0894-9166(09)60285-8. [31] S. Zhong and S. O. Oyadiji. Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform. Mechanical Systems and Signal Processing, 21, (4), (2007), pp. 1853–1884. https://doi.org/10.1016/j.ymssp.2006.07.007. [32] H. Gửkdag˘ and O. Kopmaz. A new damage detection approach for beam-type structures based on the combination of continuous and discrete wavelet transforms. Journal of Sound and Vibration, 324, (3-5), (2009), pp. 1158–1180. https://doi.org/10.1016/j.jsv.2009.02.030. [33] T. V. Lien, N. T. Khiem, and T. A. Hao. Crack identification in frame structures using the stationary wavelet transform of mode shapes. Jokull, 64, (6), (2014), pp. 251–262. [34] J. C. Goswami and A. K. Chan. Fundamentals of wavelets: theory, algorithms, and applications. John Wiley & Sons, (2011). Crack identification in multiple cracked beams made of functionally graded material. . . 125 [35] M. Misiti, Y. Misiti, G. Oppenheim, and J. M. Poggi. Matlab wavelet toolbox TM 4 user’s guide. The MathWorks, Inc. Natick, Massachusetts, (2009). [36] A. V. Ovanesova and L. E. Suarez. Applications of wavelet transforms to dam- age detection in frame structures. Engineering Structures, 26, (1), (2004), pp. 39–49. https://doi.org/10.1016/j.engstruct.2003.08.009. [37] X. H. Wang, R. S. H. Istepanian, and Y. H. Song. Microarray image enhancement by denois- ing using stationary wavelet transform. IEEE Transactions on Nanobioscience, 2, (4), (2003), pp. 184–189. https://doi.org/10.1109/tnb.2003.816225. APPENDIX 1 Frequency equation of (3) is η3 + aη2 + bη + c = 0. In case of A12 = 0, coefficients of above equation are a = ω2 [ I11 A33 + I11 A22 + I22A11 A11A22 ] , b = ω4 [ I11 I22 − I212 A11A22 + I11 A33 I11 A22 + I22A11 A11A22 ] −ω2 I11 A22 , c = ω4 [ ω2 I11 A33 I11 I22 − I212 A11A22 − I 2 11 A11A22 ] . 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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