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Development of scanning damage index for the damage detection of plate structures using modal strain energy method
ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 274– 287
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Development of scanning damage index for the damage detection of plate structures using modal strain energy method Huiwen Hu , Chengbo Wu Department of Vehicle Engineering, National Pingtung University of Science and Technology, 1 Hseuh-Fu Road, Neipu, Pingtung 91201, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 24 August 2007 Received in revised form 25 April 2008 Accepted 1 May 2008 Available online 10 May 2008
This work presents a novel approach of nondestructive detection of damage in plate structures by using experimental modal analysis (EMA) and modal strain energy method (MSEM). An aluminum alloy 6061 thin plate with a surface crack is investigated in this study. EMA is conducted on the plate to obtain the mode shapes before and after damage. The modal displacements of each mode shape are then used to compute the modal strain energy. For all measured mode shapes, a damage index is defined by using the ratio of modal strain energies of the plate before and after damage. In fact, small damage causes very little change in system response, but it is an essential early warning of structure damage. As the second-order derivatives, modal strain energy is much more sensitive to the small change of structural response than frequencies and mode shapes. It is therefore feasible to approach the small damage by using a damage index defined by fractional MSE of the structure before and after damage. In this study, a scanning damage index (SDI) is developed by moving damage indices obtained from the local area throughout the structure as if a scanning sensor is used to inspect the structure. The damage indices in overlap areas are added up and the summation may intensify the signals of damage in the plate. Limited by the numbers of measured point, a differential quadrature method is employed to calculate the partial differential terms in strain energy formula. Experimental results show that SDI well identifies a surface crack location by using only few measured mode shapes of the aluminum plate. This novel approach provides a flexible, cost-effective, and nondestructive damage evaluation in either local or global structure. Its applicability to different types of structures and different sizes of damage is to be experimentally validated in the future work. & 2008 Published by Elsevier Ltd.
Keywords: Scanning damage index Surface crack Modal strain energy Experimental modal analysis
1. Introduction Vibration-based methods have increasingly become an essential field of research in structural damage detection and structural health monitoring due to their flexibility of measurement, cost-effective, and nondestructive approach of damages in global structure. Basically, any significant change in the properties of baseline structure is regarded as potential damage which can be characterized in terms of modal parameters, i.e. natural frequencies, mode shapes or damping. From the change of such physical insight, a series of nondestructive detection techniques have already been developed to identify the damage, or even to predict the location and level of damage [1].
Corresponding author. Tel.: +886 8 77 03202; fax: +886 8 774 0398.
E-mail address: [email protected] (H. Hu). 0888-3270/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2008.05.001
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Cawley and Adams [2,3] simply employed the frequency shifts for different modes to detect the damage in composite structures. Tracy and Pardoen [4] indicated that the natural frequencies of composite structure were affected by the size and damage location. Shen and Grady [5] found that local delamination does not have a noticeable effect on global mode shape of vibration of composite beams, but it does cause irregularity of mode shapes. Pandey et al. [6] also showed that the irregularity of mode shape is significant for relatively large damage. Zou et al. [7] indicated that the changes of natural frequencies and mode shapes are unable to identify very small damage by modal experiments. Nevertheless, as the secondorder derivatives of mode shapes, modal strain energy (MSE) is much more sensitive to the response change than natural frequencies and mode shapes. Stubbs et al. [8,9] first applied this concept to the damage detection of structures based on the decrease in MSE of structures. Shi et al. [10–12] also used the change of MSE to define a damage index which can successfully locate structural damage and quantify the damage level. Most of their researches focused the studies on frame or truss structures such as bridge, building, and space structures. Cornwell et al. [13,14] extended MSE to define an index of damages in plate-like structure characterized by using twodimensional curvatures. In their approach, the fractional strain energy of the plate before and after damage was used to define a damage index which may successfully locate the area with stiffness reduction as low as 10%. Twelve measured modes are used to compute the damage indices which successfully identify two edge cracks with certain severities. Choi et al. [15] adopted the changes in distributed modal compliance for the detection of damage in plate structures and successfully located two penetrated cracks in a steel plate. However, it seems that current experimental achievements in both frame and plate structures are still limited in the detections of large or severe damages. The main challenge still lies in the measurement of the response changes of structure before and after damage. Especially, if the size of damage is small, a large amount of measured points in unit area of the structures are required for the further analysis of damage localization. Hu et al. [16] applied MSE to successfully identify surface crack in various composite laminates by employing a differential quadrature method (DQM) for the computation of the partial differential terms of MSE formula. Only the first six measured mode shapes are required to locate the surface crack in the laminate plates. It was reported that the numerical method is able to rapidly compute accurate solutions of partial differential equations by using only a few grid points in the respective solution domains [17]. Bert et al. [18] first applied this method to solve the structural mechanics problems. In Hu’s experiment, it was found that the summation of damage indices obtained from different tests may intensify the signal of damage location. This gives birth to the idea of scanning damage index method (SDIM), which is developed by shifting the damage indices obtained from the local area throughout the structure as if a scanning sensor is used to inspect the structure. Therefore, the objective of this paper is to develop a SDIM for the damage detection of plate structures by using MSEM. This novel approach provides a practical method of nondestructive damage evaluation in either local or global structure, especially for identifying the small damage. Its preliminary experimental validation to identify a surface crack in aluminum plate is investigated in the following.
2. Scanning damage index method (SDIM) The basic idea of SDIM is similar to most of the conventional nondestructive damage detection (NDD) methods, such as X-ray and ultrasonic inspections, which always proceed with a ‘‘scanning’’ movement throughout the whole or local structure as shown in Fig. 1. In the first place, the local area to be detected and the direction to be scanned are selected in terms of the numbers of grid point to be measured in the structure. EMA is conducted on the local area to obtain modal
Fig. 1. The basic idea of SDIM.
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displacements from the global mode shapes of the structure before and after damage. The modal displacements for each mode shape are then used to compute the MSE and obtain the damage index for the local area. SDIM is performed by moving the local measurement forward to the next area, with a small overlap on the adjacent one. The adjacent areas overlap and proceed ahead throughout the area of the structure to be inspected. The ambient excitation and dynamic response of the structure as shown in Fig. 1 may be replaced by any conventional instrument. Theoretically, there are three potential advantages for SDIM to be proposed in structure damage detection. First, a smaller selected area to be measured could increase the resolution for the response changes of the structure before and after damage as if the damage size is enlarged. Second, a shell structure with various curvatures could be inspected by scanning the local area, which can be regarded as a flat plate so that the plate theory used in this approach could be applied to the computation of MSE and damage indices. Third, the summation of damage indices in the overlap areas could also intensify the signals around damage location. In the following section, experimental validation for SDIM is conducted on a square thin aluminum plate with a surface crack. 3. Experimental validation 3.1. Experimental modal analysis An aluminum plate with dimension 246 246 2 mm3 is used in this study. Fig. 2 shows the experimental setup. Marked by a 13 13 parallel grid point, the plate is vertically hung by two cotton strings to simulate a completely free boundary condition. In general, a small surface crack is one of the early-stage damages that appeared in the plate structures. It is essential to detect such small damage as early as possible. Thus, a surface crack 40 mm long, 0.8 mm wide and 1 mm deep is created using a knife. EMA is performed by exciting the plate throughout all grid points using an impact hammer (KISTLER 9722A500) with a force transducer. The dynamic responses are measured by using an accelerometer (PCB 352B10) fixed at the corner. The weight of accelerometer, i.e. 0.7 g, is negligible in comparison with the weight of test plate, i.e. 330 g. A frequency response analyzer, Siglab Model 20–40, is employed to record the frequency response functions (FRFs) between measured acceleration and impact force. ME’Scope, a software for the general purpose curve fitting, is used to extract the natural frequencies and the associated mode shapes from the FRFs. The modal displacements of each mode shape before and after damage are then adopted to compute the MSE and damage index. In fact, small damage causes very little change in the system response. It is difficult to be identified by the changes of natural frequencies and modal displacements through EMA. From the theoretical point of view, MSE is the second-order derivative of modal displacement. It is much more sensitive to the change of system response than natural frequencies and modal displacement. This is the reason why MES is the most promising method proposed as follows. 3.2. MSE and damage index A plate structure as shown in Fig. 3 is subdivided into an Nx Ny sub-region and the location of each point is denoted by (xi, yj). The strain energy of the plate during elastic deformation is given by 2 !2 !2 ! ! !2 3 Z Z D b a 4 q2 w q2 w q2 w q2 w q2 w 5 dx dy (1) þ þ 2n þ 2ð1 nÞ U¼ 2 0 0 qxqy qx2 qy2 qx2 qy2 where w is transverse displacement; D is bending stiffness of the plate. Considering a completely free vibration problem, for the kth mode shape, the displacement w can be replaced by modal displacement fk, and the total strain energy of the
Fig. 2. Experimental setup. (a) Test plate and (b) test instruments.
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Fig. 3. A schematic illustration of plate.
Table 1 Natural frequencies on (Hz) Before damage
on (Hz) After damage
(%)
108 – 200 278 495 514 – 623 847 1000
0 – 0.5 0 0 0.4 – 0 0 0
Mode 1 2 3 4 5 6 7 8 9 10
108 – 201 278 495 516 – 623 847 1000
plate is expressed as 2 !2 Z Z D b a 4 q2 fk þ Uk ¼ 2 0 0 qx2
q2 fk qy2
!2 þ 2n
q2 fk qx2
!
q2 fk qy2
! þ 2ð1 nÞ
q2 fk qxqy
!2 3 5 dx dy
(2)
Cornwell et al. [13] suggested that if the damage is located at a single sub-region then the change of MSE in the subregion may become significant. Thus, the MSE associated with sub-region (i, j) for the kth mode is given by 2 !2 !2 ! ! !2 3 Z Z Dij yjþ1 xiþ1 4 q2 fk q2 fk q2 fk q2 fk q2 fk 5 dx dy (3) U k;ij ¼ þ þ 2n þ 2ð1 nÞ 2 yj qxqy qx2 qy2 qx2 qy2 xi Similarly, U k and U k;ij represent the total MSE and sub-regional MSE of the kth mode shape fk for a damaged plate. The fractional energies of the plate are defined as F k;ij ¼
U k;ij Uk
and
F k;ij ¼
U k;ij U k
(4)
Considering all measured modes, m, in the calculation, damage index in sub-region (i, j) is defined as bij ¼
Sm k¼1 F k;ij
Sm k¼1 F k;ij
(5)
A normalized damage index is given by Z ij ¼
bij bij sij
(6)
where bij and sij represent the mean and standard deviation of the damage indices, respectively. Eq. (6) is used to predict the damage location in plate structures. Theoretically, the finer the mesh of sub-region to be measured, the higher the
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Table 2 Natural frequencies and mode shapes (before damage)
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Fig. 4. Damage index (EMA). (a) Before truncation and (b) after truncation.
Fig. 5. SDI (7 7). (a) Scan-1 and (b) Scan-2.
resolution of damage index will be obtained. However, the size of the measured point and measured area is sometime limited in practice. A DQM is employed for the computation of partial differential terms in strain energy formula. It was reported that the numerical method is able to rapidly compute accurate solutions of partial differential equations by using only a few grid points in the respective solution domains [17]. 4. Differential quadrature method (DQM) DQM is an efficient and accurate numerical method and is frequently used to solve for non-linear PDEs. The main feature of DQM is that the partial derivatives of a function can be numerically evaluated by multiplication of a weighting function. The calculation of the function derivatives is nothing more than the linear algebra. If the functions f(xi, yj) are given, the partial derivatives of f(xi, yj) with respect to a spatial variable at any discrete point can be mathematically expressed as ðnÞ
f x ðxi ; yj Þ ¼
Nx X
C ðnÞ f ðxr ; yj Þ ir
(7)
r¼1
yðmÞ ðxi ; yj Þ ¼
Ny X
ðmÞ
C js f ðxi ; ys Þ
(8)
s¼1
ðnþmÞ
f xy
ðxi ; yj Þ ¼
Nx X r¼1
C ðnÞ ir
Ny X s¼1
ðmÞ
C js f ðxr ; ys Þ
(9)
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Fig. 6. Damage indices by using 7 7 (Scan-1).
where i ¼ 1,2, y, Nx and j ¼ 1,2, y, Ny are the grid points in the solution domain having Nx Ny discrete number of points. ðmÞ
Cir(n) and C js are the weighting coefficients associated with the nth order and the mth order partial derivatives of f(xi, yj) with respect to x and y at the discrete point (xi,yj) and n ¼ 1, 2, y, Nx1, m ¼ 1, 2, y, Ny1. The weighting coefficients can
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Fig. 7. Damage indices by using 7 7 data points (Scan-2).
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Fig. 8. Scanning damage indices by using 7 7 data points. (a) Scan-1, (b) Scan-2, and (c) Scan-1+Scan-2.
be obtained using the following recurrence formulae: ! C irðn1Þ ðn1Þ ð1Þ ¼ n C C C ðnÞ ir ii ir xi xr 0 ðmÞ
ðn1Þ
ðm1Þ
C js ¼ n@C jj
ð1Þ
C js
C js
yj ys
(10)
1 A
(11)
where i, r ¼ 1,2, y, Nx but r6¼i; n ¼ 2,3, y, Nx1; also j, s ¼ 1,2, y, Ny but s6¼j; m ¼ 2,3, y, Ny1. The weighting coefficients when r ¼ i and s ¼ j are given as Nx X
C ðnÞ ¼ ii
C ðnÞ ; ir
i ¼ 1; 2; :::; N x ;
and
n ¼ 1; 2; :::; N x 1
(12)
r¼1;rai
ðmÞ
C jj
¼
Ny X s¼1;saj
ðmÞ
C js ;
j ¼ 1; 2; :::; N y ;
and
m ¼ 1; 2; :::; Ny 1
(13)
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Fig. 9. SDI (5 5) (a) Scan-3 and (b) Scan-4.
C ð1Þ ir ¼
M ð1Þ ðxi Þ ðxi xr ÞM ð1Þ ðxr Þ P ð1Þ ðyj Þ
ð1Þ
C js ¼
ðyj ys ÞP ð1Þ ðys Þ
;
;
i; r ¼ 1; 2; :::; Nx ;
but
rai
(14)
j; s ¼ 1; 2; :::; Ny ;
but
jas
(15)
where M ð1Þ ðxi Þ ¼
Nx Y
ðxi xr Þ
(16)
ðyj ys Þ
(17)
r¼1;rai
Pð1Þ ðyj Þ ¼
Ny Y s¼1;saj
Once the kth modal displacement fk,ij ¼ fk(xi, yj) of the plate is obtained, the above equations are applied to compute the strain energy. 5. Results and discussion 5.1. Experimental modal parameters Tables 1 and 2 list the first ten natural frequencies and the associated mode shapes of the aluminum plate before and after damage, respectively. Mode 2 and mode 7 are absent in the result since the accelerometer is fixed at grid point 1, which is located at the stationary line of mode 2 and mode 7. Thus, the absent mode shapes are ignored in the calculation of MSE and damage index. Apparently, the changes in natural frequencies and mode shapes of the plate before and after damage are almost invisible, as shown in Tables 1 and 2. In fact, it is difficult to detect the small damage in plate structures based only on the changes of natural frequencies and mode shapes. Nevertheless, the change of MSE could be significant due to the damage of small crack. In the following section, the detection of surface crack in aluminum plate by using MSE and SDI is investigated. 5.2. Damage detection The first four mode shapes, i.e. 1, 3, 4 and 5, obtained from EMA are taken to compute the MSE and damage index. Fig. 4(a) shows the detection of surface crack by using the damage index. One peak of the damage indices clearly reveals the location of surface crack, with some noises appearing at undamaged areas as well. Cornwell et al. [13] suggested that damage indices with values greater than two are associated with potential damage locations. Fig. 4(b) shows the truncation of damage indices less than two. The damage index of surface crack becomes more visible than those of before truncation. In general, the damage index method based on MSE shows relatively good performance to detect the visible damage in 2D plate structures. But if the size of damage is small in comparison with the whole structure, the damage detection may result in a large amount of measured points, and an increase in experimental cost is inevitable. To solve this problem, a
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Fig. 10. Damage indices by using 5 5. data points (Scan-3) (continued).
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Fig. 11. Damage indices by using 5 5 data points (Scan-4) (continued).
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SDIM is developed and validated by using the EMA results in this study. A local area is selected in terms of 7 7 measured points from the test plate as shown in Fig. 5. It is similar to increasing the damage size by decreasing the measured area. Two scans are performed and moved rightward with a small overlap between two adjacent local areas. The first scan (Scan-1), as shown in Fig. 5(a), covers the grid points from number 1 to number 12 in x-direction and from number 2 to number 8 in y-direction, while the second scan (Scan-2), as shown in Fig. 5(b), covers the grid points from number 1 to number 12 and from number 6 to number 12 in y-direction beyond the surface crack. For each scan, four local areas are selected to be detected. Fig. 6 shows the change of four local areas and the corresponding damage indices for Scan-1. The damage indices in the first three local areas slightly reveal the surface crack with some noises. The detection results of Scan-2, as shown in Fig. 7, demonstrate relatively low damage indices in the first three local areas except for a high one in the fourth area. It seems that the detection results obtained from these local areas in each scan are not so clear and even provide false signal for the location of surface crack. However, if we put the damage indices of the four local areas together for Scan-1, the values of index are summed up at the overlap areas. Consequently, the scanning damage indices successfully locate the surface crack in the plate as shown in Fig. 8(a). The same process is conducted on Scan-2 as shown in Fig. 8(b), and the scanning damage indices are relatively low in comparison with Scan-1. Furthermore, if we put the scanning damage indices of Scan-1 and Scan-2 together, the damage indices successfully locate the surface crack in the plate as shown in Fig. 8(c).
Fig. 12. Scanning damage indices by using 5 5 data points. (a) Scan-3, (b) Scan-4 and (c) Scan-3+Scan-4.
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Another experimental validation is performed by selecting a small area in terms of 5 5 measured points. The same processes are conducted on two scans, i.e. Scan-3 and Scan-4 as shown in Fig. 9; each scan covers five local areas. Figs. 10 and 11 show the damage indices of each of five local areas for Scan-3 and Scan-4, respectively. The damage indices of Scan-3 slightly reveal the location of surface crack, while the damage indices of Scan-4 show relatively low signal throughout the undamaged areas. Again, if we put the damage indices of the five local areas together, the values of index are summed up at the overlap areas. The scanning damage indices of Scan-3 successfully locate the surface crack in the plate as shown in Fig. 12(a). The same process is conducted on Scan-4 as shown in Fig. 12(b). The scanning damage indices are relatively low in comparison with Scan-3. If we put the scanning damage indices of Scan-3 and Scan-4 together, the damage indices successfully locate the surface crack in the plate as shown in Fig. 12(c). The experimental results indicate that scanning damage indices using smaller patch area (5 5) serve better detection than those using larger patch area (7 7). Nevertheless, the main challenge of this approach is the requirement of a large amount of measured points for the baseline structure. Prior to its engineering service, the baseline structure may be tested by adopting a coarse mesh of measured points in EMA in order to save the heavy work in experiment. Subsequently, it may require a fine mesh of measured points in local area to obtain the high resolution of damage index for the regular inspection of the structure during its service life. The corresponding grid points to the same fine mesh of area in baseline structure can be obtained from the interpolation methods. 6. Conclusions The SDIM scanning damage index method has been developed and experimentally validated to identify the surface crack in an aluminum plate. This novel approach only requires few mode shapes obtained from the structure before and after damage. A smaller patch area used in SDIM seems to increase the resolution of damage indices as if the damage size is enlarged. The summation of damage indices in the overlap areas may also intensity the signal of damage location. Although the equipment used in this study is time consuming and not good enough, the preliminary achievement of SDIM in the damage detection for plate structures is presented. It is possible to obtain more promising and effective detection results by using a more advanced equipment such as scanner laser vibrometer. In this study, SDIM is described and experimentally validated to well identify a surface crack, which is unable to be detected by the changes of natural frequencies and mode shapes, in an aluminum plate. This novel method also provides a flexible, cost-effective, nondestructive, and real-time observable approach to the potential damage in either local or global structure. Its limits and applicability to different types of structures and different sizes of damage are still to be experimentally validated in the future work. References [1] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage identification methods, The Shock and Vibration Digest 30 (2) (1998) 91–105. [2] P. Cawley, R.D. Adams, A vibration technique for non-destructive testing of fiber composite structures, Journal of Composite Materials 13 (1979) 161–175. [3] P. Cawley, R.D. Adams, The location of defects in structure from measurements of natural frequencies, Journal of Strain Analysis 14 (1979) 49–57. [4] J.J. Tracy, G.C. Pardoen, Effect of delamination on the natural frequencies of composite laminates, Journal of Composite Materials 23 (1989) 1200–1215. [5] M.H.H. Shen, J.E. Grady, Free vibration of delaminated beams, AIAA Journal 30 (1992) 1361–1370. [6] A.K. Pandey, M. Biswas, M.M. Samman, Damage detection form changes in curvature mode shapes, Journal of Sound and Vibration 145 (1991) 321–332. [7] Y. Zou, L. Tong, G.P. Steven, Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures—a review, Journal of Sound and Vibration 2 (2000) 357–378. [8] N. Stubbs, J.T. Kim, K. Topole, An efficient and robust algorithm for damage localization in offshore platforms, in: Proceedings of the ASCE 10th Structures Congress, 1992, pp. 543–546. [9] N. Stubbs, J.T. Kim, C.R. Farrar, Field verification of a non-destructive damage localization and severity estimation algorithm, in: Proceedings of the 13th International Modal Analysis Conference, 1995, pp. 210–218. [10] Z.Y. Shi, S.S. Law, L.M. Zhang, Structural damage localization from modal strain energy change, Journal of Sound and Vibration 218 (5) (1998) 825–844. [11] Z.Y. Shi, S.S. Law, L.M. Zhang, Structural damage detection from modal strain energy change, Journal of Engineering Mechanics 126 (12) (2000) 1216–1223. [12] Z.Y. Shi, S.S. Law, L.M. Zhang, Improved damage quantification from elemental modal strain energy change, Journal of Engineering Mechanics 128 (5) (2002) 521–529. [13] P.J. Cornwell, S.W. Dobeling, C.R. Farrar, Application of the strain energy damage detection method to plate-like structures, in: Proceedings of the 15th International Modal Analysis Conference, Orlando, FL, 1997, pp. 1312–1318. [14] P.J. Cornwell, S.W. Dobeling, C.R. Farrar, Application of the strain energy damage detection method to plate-like structures, Journal of Sound and Vibration 224 (2) (1999) 359–374. [15] S. Choi, P. Park, S. Yoon, N. Stubbs, Nondestructive damage identification in plate structures using changes in modal compliance, NDT&E International 38 (2005) 529–540. [16] H. Hu, B.T. Wang, C.H. Lee, J.S. Su, Damage detection of surface cracks in composite laminates using modal analysis and strain energy method, Composite Structure 74 (2006) 399–405. [17] R.E. Bellman, B.G. Kashef, J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equation, Journal of Computational Physics 10 (1972) 40–52. [18] C.W. Bert, S.K. Jang, A.G. Striz, Two new approximate methods for analyzing free vibration of structural components, AIAA Journal 26 (1988) 612–618.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Development of scanning damage index for the damage detection of plate structures using modal strain energy method Huiwen Hu , Chengbo Wu Department of Vehicle Engineering, National Pingtung University of Science and Technology, 1 Hseuh-Fu Road, Neipu, Pingtung 91201, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 24 August 2007 Received in revised form 25 April 2008 Accepted 1 May 2008 Available online 10 May 2008
This work presents a novel approach of nondestructive detection of damage in plate structures by using experimental modal analysis (EMA) and modal strain energy method (MSEM). An aluminum alloy 6061 thin plate with a surface crack is investigated in this study. EMA is conducted on the plate to obtain the mode shapes before and after damage. The modal displacements of each mode shape are then used to compute the modal strain energy. For all measured mode shapes, a damage index is defined by using the ratio of modal strain energies of the plate before and after damage. In fact, small damage causes very little change in system response, but it is an essential early warning of structure damage. As the second-order derivatives, modal strain energy is much more sensitive to the small change of structural response than frequencies and mode shapes. It is therefore feasible to approach the small damage by using a damage index defined by fractional MSE of the structure before and after damage. In this study, a scanning damage index (SDI) is developed by moving damage indices obtained from the local area throughout the structure as if a scanning sensor is used to inspect the structure. The damage indices in overlap areas are added up and the summation may intensify the signals of damage in the plate. Limited by the numbers of measured point, a differential quadrature method is employed to calculate the partial differential terms in strain energy formula. Experimental results show that SDI well identifies a surface crack location by using only few measured mode shapes of the aluminum plate. This novel approach provides a flexible, cost-effective, and nondestructive damage evaluation in either local or global structure. Its applicability to different types of structures and different sizes of damage is to be experimentally validated in the future work. & 2008 Published by Elsevier Ltd.
Keywords: Scanning damage index Surface crack Modal strain energy Experimental modal analysis
1. Introduction Vibration-based methods have increasingly become an essential field of research in structural damage detection and structural health monitoring due to their flexibility of measurement, cost-effective, and nondestructive approach of damages in global structure. Basically, any significant change in the properties of baseline structure is regarded as potential damage which can be characterized in terms of modal parameters, i.e. natural frequencies, mode shapes or damping. From the change of such physical insight, a series of nondestructive detection techniques have already been developed to identify the damage, or even to predict the location and level of damage [1].
Corresponding author. Tel.: +886 8 77 03202; fax: +886 8 774 0398.
E-mail address: [email protected] (H. Hu). 0888-3270/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2008.05.001
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Cawley and Adams [2,3] simply employed the frequency shifts for different modes to detect the damage in composite structures. Tracy and Pardoen [4] indicated that the natural frequencies of composite structure were affected by the size and damage location. Shen and Grady [5] found that local delamination does not have a noticeable effect on global mode shape of vibration of composite beams, but it does cause irregularity of mode shapes. Pandey et al. [6] also showed that the irregularity of mode shape is significant for relatively large damage. Zou et al. [7] indicated that the changes of natural frequencies and mode shapes are unable to identify very small damage by modal experiments. Nevertheless, as the secondorder derivatives of mode shapes, modal strain energy (MSE) is much more sensitive to the response change than natural frequencies and mode shapes. Stubbs et al. [8,9] first applied this concept to the damage detection of structures based on the decrease in MSE of structures. Shi et al. [10–12] also used the change of MSE to define a damage index which can successfully locate structural damage and quantify the damage level. Most of their researches focused the studies on frame or truss structures such as bridge, building, and space structures. Cornwell et al. [13,14] extended MSE to define an index of damages in plate-like structure characterized by using twodimensional curvatures. In their approach, the fractional strain energy of the plate before and after damage was used to define a damage index which may successfully locate the area with stiffness reduction as low as 10%. Twelve measured modes are used to compute the damage indices which successfully identify two edge cracks with certain severities. Choi et al. [15] adopted the changes in distributed modal compliance for the detection of damage in plate structures and successfully located two penetrated cracks in a steel plate. However, it seems that current experimental achievements in both frame and plate structures are still limited in the detections of large or severe damages. The main challenge still lies in the measurement of the response changes of structure before and after damage. Especially, if the size of damage is small, a large amount of measured points in unit area of the structures are required for the further analysis of damage localization. Hu et al. [16] applied MSE to successfully identify surface crack in various composite laminates by employing a differential quadrature method (DQM) for the computation of the partial differential terms of MSE formula. Only the first six measured mode shapes are required to locate the surface crack in the laminate plates. It was reported that the numerical method is able to rapidly compute accurate solutions of partial differential equations by using only a few grid points in the respective solution domains [17]. Bert et al. [18] first applied this method to solve the structural mechanics problems. In Hu’s experiment, it was found that the summation of damage indices obtained from different tests may intensify the signal of damage location. This gives birth to the idea of scanning damage index method (SDIM), which is developed by shifting the damage indices obtained from the local area throughout the structure as if a scanning sensor is used to inspect the structure. Therefore, the objective of this paper is to develop a SDIM for the damage detection of plate structures by using MSEM. This novel approach provides a practical method of nondestructive damage evaluation in either local or global structure, especially for identifying the small damage. Its preliminary experimental validation to identify a surface crack in aluminum plate is investigated in the following.
2. Scanning damage index method (SDIM) The basic idea of SDIM is similar to most of the conventional nondestructive damage detection (NDD) methods, such as X-ray and ultrasonic inspections, which always proceed with a ‘‘scanning’’ movement throughout the whole or local structure as shown in Fig. 1. In the first place, the local area to be detected and the direction to be scanned are selected in terms of the numbers of grid point to be measured in the structure. EMA is conducted on the local area to obtain modal
Fig. 1. The basic idea of SDIM.
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displacements from the global mode shapes of the structure before and after damage. The modal displacements for each mode shape are then used to compute the MSE and obtain the damage index for the local area. SDIM is performed by moving the local measurement forward to the next area, with a small overlap on the adjacent one. The adjacent areas overlap and proceed ahead throughout the area of the structure to be inspected. The ambient excitation and dynamic response of the structure as shown in Fig. 1 may be replaced by any conventional instrument. Theoretically, there are three potential advantages for SDIM to be proposed in structure damage detection. First, a smaller selected area to be measured could increase the resolution for the response changes of the structure before and after damage as if the damage size is enlarged. Second, a shell structure with various curvatures could be inspected by scanning the local area, which can be regarded as a flat plate so that the plate theory used in this approach could be applied to the computation of MSE and damage indices. Third, the summation of damage indices in the overlap areas could also intensify the signals around damage location. In the following section, experimental validation for SDIM is conducted on a square thin aluminum plate with a surface crack. 3. Experimental validation 3.1. Experimental modal analysis An aluminum plate with dimension 246 246 2 mm3 is used in this study. Fig. 2 shows the experimental setup. Marked by a 13 13 parallel grid point, the plate is vertically hung by two cotton strings to simulate a completely free boundary condition. In general, a small surface crack is one of the early-stage damages that appeared in the plate structures. It is essential to detect such small damage as early as possible. Thus, a surface crack 40 mm long, 0.8 mm wide and 1 mm deep is created using a knife. EMA is performed by exciting the plate throughout all grid points using an impact hammer (KISTLER 9722A500) with a force transducer. The dynamic responses are measured by using an accelerometer (PCB 352B10) fixed at the corner. The weight of accelerometer, i.e. 0.7 g, is negligible in comparison with the weight of test plate, i.e. 330 g. A frequency response analyzer, Siglab Model 20–40, is employed to record the frequency response functions (FRFs) between measured acceleration and impact force. ME’Scope, a software for the general purpose curve fitting, is used to extract the natural frequencies and the associated mode shapes from the FRFs. The modal displacements of each mode shape before and after damage are then adopted to compute the MSE and damage index. In fact, small damage causes very little change in the system response. It is difficult to be identified by the changes of natural frequencies and modal displacements through EMA. From the theoretical point of view, MSE is the second-order derivative of modal displacement. It is much more sensitive to the change of system response than natural frequencies and modal displacement. This is the reason why MES is the most promising method proposed as follows. 3.2. MSE and damage index A plate structure as shown in Fig. 3 is subdivided into an Nx Ny sub-region and the location of each point is denoted by (xi, yj). The strain energy of the plate during elastic deformation is given by 2 !2 !2 ! ! !2 3 Z Z D b a 4 q2 w q2 w q2 w q2 w q2 w 5 dx dy (1) þ þ 2n þ 2ð1 nÞ U¼ 2 0 0 qxqy qx2 qy2 qx2 qy2 where w is transverse displacement; D is bending stiffness of the plate. Considering a completely free vibration problem, for the kth mode shape, the displacement w can be replaced by modal displacement fk, and the total strain energy of the
Fig. 2. Experimental setup. (a) Test plate and (b) test instruments.
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Fig. 3. A schematic illustration of plate.
Table 1 Natural frequencies on (Hz) Before damage
on (Hz) After damage
(%)
108 – 200 278 495 514 – 623 847 1000
0 – 0.5 0 0 0.4 – 0 0 0
Mode 1 2 3 4 5 6 7 8 9 10
108 – 201 278 495 516 – 623 847 1000
plate is expressed as 2 !2 Z Z D b a 4 q2 fk þ Uk ¼ 2 0 0 qx2
q2 fk qy2
!2 þ 2n
q2 fk qx2
!
q2 fk qy2
! þ 2ð1 nÞ
q2 fk qxqy
!2 3 5 dx dy
(2)
Cornwell et al. [13] suggested that if the damage is located at a single sub-region then the change of MSE in the subregion may become significant. Thus, the MSE associated with sub-region (i, j) for the kth mode is given by 2 !2 !2 ! ! !2 3 Z Z Dij yjþ1 xiþ1 4 q2 fk q2 fk q2 fk q2 fk q2 fk 5 dx dy (3) U k;ij ¼ þ þ 2n þ 2ð1 nÞ 2 yj qxqy qx2 qy2 qx2 qy2 xi Similarly, U k and U k;ij represent the total MSE and sub-regional MSE of the kth mode shape fk for a damaged plate. The fractional energies of the plate are defined as F k;ij ¼
U k;ij Uk
and
F k;ij ¼
U k;ij U k
(4)
Considering all measured modes, m, in the calculation, damage index in sub-region (i, j) is defined as bij ¼
Sm k¼1 F k;ij
Sm k¼1 F k;ij
(5)
A normalized damage index is given by Z ij ¼
bij bij sij
(6)
where bij and sij represent the mean and standard deviation of the damage indices, respectively. Eq. (6) is used to predict the damage location in plate structures. Theoretically, the finer the mesh of sub-region to be measured, the higher the
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Table 2 Natural frequencies and mode shapes (before damage)
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Fig. 4. Damage index (EMA). (a) Before truncation and (b) after truncation.
Fig. 5. SDI (7 7). (a) Scan-1 and (b) Scan-2.
resolution of damage index will be obtained. However, the size of the measured point and measured area is sometime limited in practice. A DQM is employed for the computation of partial differential terms in strain energy formula. It was reported that the numerical method is able to rapidly compute accurate solutions of partial differential equations by using only a few grid points in the respective solution domains [17]. 4. Differential quadrature method (DQM) DQM is an efficient and accurate numerical method and is frequently used to solve for non-linear PDEs. The main feature of DQM is that the partial derivatives of a function can be numerically evaluated by multiplication of a weighting function. The calculation of the function derivatives is nothing more than the linear algebra. If the functions f(xi, yj) are given, the partial derivatives of f(xi, yj) with respect to a spatial variable at any discrete point can be mathematically expressed as ðnÞ
f x ðxi ; yj Þ ¼
Nx X
C ðnÞ f ðxr ; yj Þ ir
(7)
r¼1
yðmÞ ðxi ; yj Þ ¼
Ny X
ðmÞ
C js f ðxi ; ys Þ
(8)
s¼1
ðnþmÞ
f xy
ðxi ; yj Þ ¼
Nx X r¼1
C ðnÞ ir
Ny X s¼1
ðmÞ
C js f ðxr ; ys Þ
(9)
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Fig. 6. Damage indices by using 7 7 (Scan-1).
where i ¼ 1,2, y, Nx and j ¼ 1,2, y, Ny are the grid points in the solution domain having Nx Ny discrete number of points. ðmÞ
Cir(n) and C js are the weighting coefficients associated with the nth order and the mth order partial derivatives of f(xi, yj) with respect to x and y at the discrete point (xi,yj) and n ¼ 1, 2, y, Nx1, m ¼ 1, 2, y, Ny1. The weighting coefficients can
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Fig. 7. Damage indices by using 7 7 data points (Scan-2).
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Fig. 8. Scanning damage indices by using 7 7 data points. (a) Scan-1, (b) Scan-2, and (c) Scan-1+Scan-2.
be obtained using the following recurrence formulae: ! C irðn1Þ ðn1Þ ð1Þ ¼ n C C C ðnÞ ir ii ir xi xr 0 ðmÞ
ðn1Þ
ðm1Þ
C js ¼ n@C jj
ð1Þ
C js
C js
yj ys
(10)
1 A
(11)
where i, r ¼ 1,2, y, Nx but r6¼i; n ¼ 2,3, y, Nx1; also j, s ¼ 1,2, y, Ny but s6¼j; m ¼ 2,3, y, Ny1. The weighting coefficients when r ¼ i and s ¼ j are given as Nx X
C ðnÞ ¼ ii
C ðnÞ ; ir
i ¼ 1; 2; :::; N x ;
and
n ¼ 1; 2; :::; N x 1
(12)
r¼1;rai
ðmÞ
C jj
¼
Ny X s¼1;saj
ðmÞ
C js ;
j ¼ 1; 2; :::; N y ;
and
m ¼ 1; 2; :::; Ny 1
(13)
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Fig. 9. SDI (5 5) (a) Scan-3 and (b) Scan-4.
C ð1Þ ir ¼
M ð1Þ ðxi Þ ðxi xr ÞM ð1Þ ðxr Þ P ð1Þ ðyj Þ
ð1Þ
C js ¼
ðyj ys ÞP ð1Þ ðys Þ
;
;
i; r ¼ 1; 2; :::; Nx ;
but
rai
(14)
j; s ¼ 1; 2; :::; Ny ;
but
jas
(15)
where M ð1Þ ðxi Þ ¼
Nx Y
ðxi xr Þ
(16)
ðyj ys Þ
(17)
r¼1;rai
Pð1Þ ðyj Þ ¼
Ny Y s¼1;saj
Once the kth modal displacement fk,ij ¼ fk(xi, yj) of the plate is obtained, the above equations are applied to compute the strain energy. 5. Results and discussion 5.1. Experimental modal parameters Tables 1 and 2 list the first ten natural frequencies and the associated mode shapes of the aluminum plate before and after damage, respectively. Mode 2 and mode 7 are absent in the result since the accelerometer is fixed at grid point 1, which is located at the stationary line of mode 2 and mode 7. Thus, the absent mode shapes are ignored in the calculation of MSE and damage index. Apparently, the changes in natural frequencies and mode shapes of the plate before and after damage are almost invisible, as shown in Tables 1 and 2. In fact, it is difficult to detect the small damage in plate structures based only on the changes of natural frequencies and mode shapes. Nevertheless, the change of MSE could be significant due to the damage of small crack. In the following section, the detection of surface crack in aluminum plate by using MSE and SDI is investigated. 5.2. Damage detection The first four mode shapes, i.e. 1, 3, 4 and 5, obtained from EMA are taken to compute the MSE and damage index. Fig. 4(a) shows the detection of surface crack by using the damage index. One peak of the damage indices clearly reveals the location of surface crack, with some noises appearing at undamaged areas as well. Cornwell et al. [13] suggested that damage indices with values greater than two are associated with potential damage locations. Fig. 4(b) shows the truncation of damage indices less than two. The damage index of surface crack becomes more visible than those of before truncation. In general, the damage index method based on MSE shows relatively good performance to detect the visible damage in 2D plate structures. But if the size of damage is small in comparison with the whole structure, the damage detection may result in a large amount of measured points, and an increase in experimental cost is inevitable. To solve this problem, a
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Fig. 10. Damage indices by using 5 5. data points (Scan-3) (continued).
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Fig. 11. Damage indices by using 5 5 data points (Scan-4) (continued).
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SDIM is developed and validated by using the EMA results in this study. A local area is selected in terms of 7 7 measured points from the test plate as shown in Fig. 5. It is similar to increasing the damage size by decreasing the measured area. Two scans are performed and moved rightward with a small overlap between two adjacent local areas. The first scan (Scan-1), as shown in Fig. 5(a), covers the grid points from number 1 to number 12 in x-direction and from number 2 to number 8 in y-direction, while the second scan (Scan-2), as shown in Fig. 5(b), covers the grid points from number 1 to number 12 and from number 6 to number 12 in y-direction beyond the surface crack. For each scan, four local areas are selected to be detected. Fig. 6 shows the change of four local areas and the corresponding damage indices for Scan-1. The damage indices in the first three local areas slightly reveal the surface crack with some noises. The detection results of Scan-2, as shown in Fig. 7, demonstrate relatively low damage indices in the first three local areas except for a high one in the fourth area. It seems that the detection results obtained from these local areas in each scan are not so clear and even provide false signal for the location of surface crack. However, if we put the damage indices of the four local areas together for Scan-1, the values of index are summed up at the overlap areas. Consequently, the scanning damage indices successfully locate the surface crack in the plate as shown in Fig. 8(a). The same process is conducted on Scan-2 as shown in Fig. 8(b), and the scanning damage indices are relatively low in comparison with Scan-1. Furthermore, if we put the scanning damage indices of Scan-1 and Scan-2 together, the damage indices successfully locate the surface crack in the plate as shown in Fig. 8(c).
Fig. 12. Scanning damage indices by using 5 5 data points. (a) Scan-3, (b) Scan-4 and (c) Scan-3+Scan-4.
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Another experimental validation is performed by selecting a small area in terms of 5 5 measured points. The same processes are conducted on two scans, i.e. Scan-3 and Scan-4 as shown in Fig. 9; each scan covers five local areas. Figs. 10 and 11 show the damage indices of each of five local areas for Scan-3 and Scan-4, respectively. The damage indices of Scan-3 slightly reveal the location of surface crack, while the damage indices of Scan-4 show relatively low signal throughout the undamaged areas. Again, if we put the damage indices of the five local areas together, the values of index are summed up at the overlap areas. The scanning damage indices of Scan-3 successfully locate the surface crack in the plate as shown in Fig. 12(a). The same process is conducted on Scan-4 as shown in Fig. 12(b). The scanning damage indices are relatively low in comparison with Scan-3. If we put the scanning damage indices of Scan-3 and Scan-4 together, the damage indices successfully locate the surface crack in the plate as shown in Fig. 12(c). The experimental results indicate that scanning damage indices using smaller patch area (5 5) serve better detection than those using larger patch area (7 7). Nevertheless, the main challenge of this approach is the requirement of a large amount of measured points for the baseline structure. Prior to its engineering service, the baseline structure may be tested by adopting a coarse mesh of measured points in EMA in order to save the heavy work in experiment. Subsequently, it may require a fine mesh of measured points in local area to obtain the high resolution of damage index for the regular inspection of the structure during its service life. The corresponding grid points to the same fine mesh of area in baseline structure can be obtained from the interpolation methods. 6. Conclusions The SDIM scanning damage index method has been developed and experimentally validated to identify the surface crack in an aluminum plate. This novel approach only requires few mode shapes obtained from the structure before and after damage. A smaller patch area used in SDIM seems to increase the resolution of damage indices as if the damage size is enlarged. The summation of damage indices in the overlap areas may also intensity the signal of damage location. Although the equipment used in this study is time consuming and not good enough, the preliminary achievement of SDIM in the damage detection for plate structures is presented. It is possible to obtain more promising and effective detection results by using a more advanced equipment such as scanner laser vibrometer. In this study, SDIM is described and experimentally validated to well identify a surface crack, which is unable to be detected by the changes of natural frequencies and mode shapes, in an aluminum plate. 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