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A numerical technique for structural damage detection
Applied Mathematics and Computation 215 (2009) 2775–2780
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A numerical technique for structural damage detection Q.W. Yang Department of Civil Engineering, Shaoxing University, Shaoxing 312000, China
a r t i c l e
i n f o
Keywords: Damage detection Matrix disassembly Modal residual force
a b s t r a c t A computationally attractive method is proposed in this study to provide an insight to the location and extent of structural damage. The proposed method makes use of the matrix disassembly technique and approaches the damage location and extent problem in a decoupled fashion. First, a scheme is developed to determine the damage location by calculating a damage localization vector, which is derived from the modal residual force criteria. With location determined, the corresponding damage extent can be easily obtained only by the processes of division. The algorithm is applied to a numerical example and its efficiency is demonstrated through damage simulations. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Structural damage detection using measured dynamic data has emerged as a new research area in civil, mechanical and aerospace engineering communities in recent years. The basic idea of this technique is that modal parameters are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause changes in the modal properties. Recent surveys on the technical literature, such as on Doebling et al. [1] and Chang [2], among others, show that extensive efforts have been developed to find reliable and efficient numerical and experimental models to identify damage in structures. Most prior work in damage identification is based on the modification of a structural finite element model (FEM). The goal of these methods is to use test data from the damaged structure and the correlated FEM of the undamaged structure to determine changes to the stiffness and/or mass matrices. This class of methods can be divided into three main groups: One group is the sensitivity-based model update. These methods make use of the derivatives of modal parameters with respect to physical design variables. Using the eigenvalue sensitivity, Messina et al. proposed an assurance criterion for detecting single damage sites [3] and extended this method to identify the relative amount of damage at multiple sites [4]. Shi et al. extended the multiple damage location assurance criterions by using mode shape sensitivity instead of modal frequency sensitivity [5]. Yu et al. make use of eigenvalue perturbation theory and artificial neural network to detect small structural damage [6]. Yang developed a mixed sensitivity method to identify structural damage by combining the eigenvalue sensitivity with the flexibility sensitivity [7]. Recently, Yang and Liu make use of the eigenparameter decomposition of structural flexibility change to detect structural damage in a decoupled manner [8]. Another group is control-based eigenstructure assignment techniques. Andry et al. presented an excellent overview of eigenstructure assignment theory and applications [9]. Zimmerman and Kaouk utilized a symmetric eigenstructure assignment algorithm to perform the partial spectral assignment [10]. Lim and Kashangaki referred to a best achievable eigenvector as a damage indicator [11]. Kiddy and Pines used the eigenstructure assignment technique to detect damage in rotating structures [12]. The third group, known as optimal matrix update, solves a closed-form equation for the matrix perturbations that minimize the modal force error or constrain the solution to satisfy it. Chen and Garba put forward a theory of detecting potential damage by performing a minimum-norm
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.039
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solution of the residual force equation [13]. Zimmerman and Kaouk presented a minimum-rank solution of the residual force equation to identify structural damages. They firstly located the damage sites by the residual force vector and then assessed the damage extents by the minimum-rank update theory [14]. Doebling developed their method and presented a technique termed the minimum-rank elemental update by computing the minimum rank updates directly to the elemental stiffness parameters [15]. Yang and Liu studied several damage detection methods based on the residual force vector and presented the algebraic solution of the residual force equation [16]. The method proposed in this paper also belongs to the third group, which approaches the damage location and extent problem in a decoupled fashion. The damage location algorithm is derived from the Modal Residual Force criteria proposed by Yang and Liu [16]. However, greater insight of the modal residual force criteria is provided and a new damage localization vector is defined, by including the connectivity constraint directly into the modal residual force equation, to locate damage at the element level rather than at the structural degree of freedom (DOF). The main advantage to locating damaged elements rather than damaged DOFs is that the resulting localization criteria has direct physical relevance and thus can be used in the following damage quantification. The damage extent algorithm is very simple since it requires only the processes of division. The remainder of the paper is organized as follows. First, the modal residual force criteria are briefly reviewed. Second, the theoretical development of the proposed approach is presented. Next, the method is demonstrated using a numerical example and the conclusions of this work are summarized in the end. In this paper, we will assume that the dimension of the measured eigenvector is the same as the analytical eigenvector. This assumption allows us to focus our attention on the quality of the proposed damage detection algorithm. This is true (i) when all FEM DOF’s are measured (ii) after the application of an eigenvector expansion algorithm or (iii) after the application of a finite element model reduction algorithm [14]. The ideal situation would be to measure all FEM DOF’s since the eigenvector expansion process would introduce errors in the ‘‘expanded” eigenvectors and the model reduction process would introduce errors in the FEM. It should be noted that in both cases the additional errors may become significant as the ratio of measured to unmeasured DOF’s become smaller. 2. The modal residual force criteria In the context of structural damage detection and health monitoring, the perturbations to the stiffness properties are usually the most relevant. In this paper, only the perturbation of the structural stiffness properties will be considered. The basic theory of the modal residual force criteria begins with the free vibration eigenvalue problem of the damaged structure
ðK d kdj MÞ/dj ¼ 0
ð1Þ
where M and K d are the n n analytical mass and damaged stiffness matrices, kdj and /dj are the jth eigenvalue and eigenvector of the damaged structure, respectively. Let DK be the exact perturbation matrix that reflects the nature of the structural damage with the nonzero elements reflecting the state of damage, namely
K d ¼ K DK
ð2Þ
where K is the undamaged stiffness matrix. Substituting Eq. (2) into (1) yields
ðK kdj MÞ/dj ¼ DK/dj
ð3Þ
Eq. (3) can be written more compactly as
DK/dj ¼ bj
ð4Þ
where bj is defined as the jth modal force error and given by
bj ¼ ðK kdj MÞ/dj
ð5Þ
Inspection of bj in terms of Eq. (4) reveals that the ith element of bj will be zero when the ith rows of DK are zero, i.e., the FEM for the ith DOF is not directly affected by damage. Conversely, a DOF whose FEM has been affected by damage will result in a nonzero entry in bj . Thus, the damaged DOFs can be determined by inspecting the elements of bj . Vector bj as defined in Eq. (5) also reveals that only a single mode of vibration needs to be measured exactly to determine exact damage locations. This is true in even multiple member damage situations. More importantly, the vector bj can be determined from the intact FEM and the measured eigenparameters, kdj and /dj using Eq. (5). The proposed damage location algorithm is similar to the modal residual force criteria, by including the connectivity constraint directly into the modal residual force equation, but locates damage at the element level rather than at the structural DOF. The derivation of the proposed method is presented in the following section. 3. The proposed damage detection approach In this section, a new damage location algorithm is first developed and then a very simple damage extent technique is formulated which makes use of the knowledge obtained from the location algorithm.
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3.1. Damage localization The derivation of the proposed damage location algorithm begins with the parameterization of the global stiffness perturbation matrix as [8]
DK ¼ C DPC T
ð6Þ
where the ðn NÞ matrix C is defined as the stiffness connectivity matrix, which can be computed by the method proposed in Ref. [8], and the ðN NÞ diagonal matrix DP has the elemental stiffness damage parameters as its diagonal entries, N is the number of total elements. Mathematically, this is defined as
diagðDPÞ ¼ ða1 ; a2 ; . . . aN ÞT
ð7Þ
where ai ði ¼ 1 NÞ is the ith elemental damage parameter. Substituting Eq. (6) into Eq. (4) yields
C DPC T /dj ¼ bj
ð8Þ T
Perform a minimum-norm solution of Eq. (8) for DPC /dj to get
DPC T /dj ¼ C þ bj
ð9Þ
where the superscript ‘‘+” denotes the pseudoinverse. As the connectivity matrix C is a full rank matrix (see Appendix A), the pseudoinverse of C can be given by
C þ ¼ C T ðCC T Þ1
ð10Þ
dj ¼ C T /dj ;
ð11Þ
ej ¼ C þ bj
ð12Þ
Let
in which vector ej can be rewritten, by substituting Eqs. (5) and (10) into (12), as
ej ¼ C T ðCC T Þ1 ðK kdj MÞ/dj
ð13Þ
Then Eq. (9) simplifies to
DPdj ¼ ej
ð14Þ
Namely,
2 6 6 6 6 4
a1 a2
8 9 3> d1 > 8 e1 9 > > > > > > > > > j > > j > 2 > > > > > d2 > > 7> = < < e 7 j = j 7 ¼ .. 7> . > > .. > 5> . > . > > > > > . > > > > > > . > > > ; : eN ; : dN > aN > j j
ð15Þ
Inspection of ej in terms of the Eq. (15) reveals that the ith element of ej will be zero when the ith element is undamaged ðai ¼ 0Þ. Conversely, a damaged element will result in a nonzero entry in ej by reason of ai – 0. Thus, vector ej given by Eq. (13) is defined as the damage localization vector and the damage locations can be determined by inspecting the elements of it. It is obvious that vector ej can be determined from the intact FEM and the measured eigenparameters, kdj and /dj using Eq. (13). Moreover, vector ej also reveals that only a single mode of vibration needs to be measured exactly to determine exact damage locations. This is true in even multiple member damage situations. In practice, the perfect zero/nonzero pattern of the damage localization vector ej rarely occurs due to errors present in the measured eigenvalues and eigenvectors. Then entries with larger absolute values in ej are associated with the damaged elements in practice. 3.2. Damage quantification It is sometimes necessary to determine the extent of structural damage. With the damaged elements determined by the above approach, the corresponding damage extents can be easily calculated only by the processes of division. Without loss of generality, assuming the ith element is damaged, from Eq. (15), one obtains
ai dij ¼ eij
ð16Þ
Rewriting Eq. (16), the damage extent can be obtained by
ai ¼ eij =dij
ð17Þ
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Q.W. Yang / Applied Mathematics and Computation 215 (2009) 2775–2780
In the end, a step-by-step summary of the whole technique is described as follows. Step 1: Calculate the modal force error bj by Eq. (5). In this step, the modes of the damaged structure can be obtained by a modal survey on it. Step 2: Compute the vectors dj and ej using Eqs. (11) and (12). The stiffness connectivity matrix C used in this step can be obtained by the procedure presented in Ref. [8]. And then the locations of damaged elements can be determined in this step by inspection of the damage localization vector ej . Step 3: For the identified damaged elements, calculate the damage extents using Eq. (17). 4. Numerical example To illustrate characteristics of the proposed damage detection algorithm, a two-dimensional truss structure is presented as shown in Fig. 1. The basic parameters of the structure are as follows: Young’s modulus E ¼ 200 GPa, density q ¼ 7:8 103 kg=m3 , and cross-sectional area A ¼ 0:004 m2 . Two damage cases are considered in the example. The first one is a single damage case that element 5 is damaged with a stiffness loss of 30%. The second case has two damages in which elements 5 and 16 have 30% and 20% reduction in stiffness, respectively. The mode shape is contaminated with 5% random noise in the study of the measurement error effect. The contaminated signal is represented as [17]
/ij ¼ /ij ð1 þ c/i q/ j/max;j jÞ
ð18Þ
where /ij and /ij are the mode shape components of the jth mode at the ith degrees of freedom with noise and without noise, respectively; c/i is the random number with a mean equal to zero and a variance equal to 1; q/ is the random noise level; and /max;j is the largest component in the jth mode shape. Table 1 presents the ratios of the first six eigenvalue changes for the 23-bar truss when damage is introduced. For damage case 1, mode 5 will be used in damage detection because it undergoes a significant shift in frequency. For damage case 2, mode 1 undergoes a largest change and so it will be used in damage identification.
Fig. 1. 23-Bar truss structure.
Table 1 The ratios of eigenvalue changes for the first six modes. Mode
Element 5 is damaged (%)
Elements 5 and 16 are damaged (%)
1 2 3 4 5 6
1.35 0.01 1.48 0.52 3.75 0.90
4.91 0.9 2.8 0.66 4.65 1.85
Damage Indicator
14 12
no noise
10
5% noise
8 6 4 2 0 1
5
9
13
17
21
Element Number Fig. 2. Damage localization result when element 5 is damaged.
Damage Indicator
Q.W. Yang / Applied Mathematics and Computation 215 (2009) 2775–2780
1. 4
no noise
1. 2
5% noise
2779
1 0. 8 0. 6 0. 4 0. 2 0
1
5
9 13 Element Number
17
21
Fig. 3. Damage localization result when elements 5 and 16 are damaged.
Fig. 2 shows the damage localization result for the first damage case using Eq. (13) with the 5th mode of vibration data. It is clear that the location algorithm is able to accurately locate the correct damage even if the mode shape is contaminated with 5% random noise. For the error-free case, the calculated damage parameter of element 5 using Eq. (17) is 0.3, which is exactly equal to the assumed value (0.3). When 5% noise is introduced, the corresponding damage parameter obtained by Eq. (17) is 0.2858, which has 4.7% relative error as compared to the assumed value. Fig. 3 presents the results of using Eq. (13) with the first mode eigenvalue/eigenvector information. Again, the location algorithm is able to cleanly detect damage even when presented with eigenvector information corrupted with 5% noise. Using Eq. (17), for the error-free case, the damage extents can be readily calculated as a5 ¼ 0:3 and a16 ¼ 0:2, which exactly equal the assumed values (0.3, 0.2). For the case with 5% error, the calculated damage parameters in elements 5 and 16 are 0.2874 and 0.1802, respectively. From the above results, it is obvious that the predicted damage location is exactly correct, and the predicted damage extent has slight deviation from the true value because of the error in the measured data. 5. Conclusions A new optimal matrix update method is developed in this paper. The algorithm makes use of an existing finite element model of the ‘‘healthy” structure and a subset of experimentally measured modal properties of the ‘‘damaged” structure. The procedure includes two parts or stages. The first stage is to locate structural damage using a localization vector derived from the modal residual force vector. With damaged element determined, estimation of its severity is performed in the second stage. The proposed approach is shown to be computationally attractive because the damage location only requires matrix–vector and matrix–scalar multiplications and the damage extent only requires the processes of division. The illustrative example shows the good efficiency and stability of the presented method on the identification of single damage or multiple damages. It has been shown that the proposed procedure may be useful for structural damage identification. Further research on the technique can be carried out to tackle the problem of the test/analysis DOF mismatch, to compare the method with other damage detection techniques, and to demonstrate the procedure using experimentally measured data. Acknowledgments The author thanks the reviewers who gave a thorough and careful reading to the original manuscript. Their comments are greatly appreciated and have helped to improve the quality of the paper. This work is supported by the scientific research project of education of Zhejiang Province (20070503). Appendix A.
The statement after Eq. (9) ‘‘As the connectivity matrix C is a full rank matrix, . . .” can be proven as follows:
For C is of dimension n N, the rank of C must satisfy
rankðCÞ 6 minðn; NÞ
ðA:1Þ
On the other hand, according to Eq. (7), one obtains
rankðCÞ P rankðKÞ ¼ n
ðA:2Þ
From Eqs. (A.1) and (A.2), one has
rankðCÞ ¼ n Eq. (A.3) implies that the connectivity matrix C is a full rank matrix.
ðA:3Þ
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References [1] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in their Vibration Characteristics: A Literature Review, Los Alamos National Laboratory, LA-13070-MS, 1996. [2] F.K. Chang, Structural Health Monitoring Current Status and Perspectives, Technomic Publishing Co., Inc., Lancaster, PA, 1997. [3] A. Messina, I.A. Jones, J.E. Williams. Damage Detection and Localization Using Natural Frequency Changes, in: Proceedings of Conference on Identification in Engineering Systems, Swansea, UK, 1996, pp. 67–76. [4] A. Messina, J.E. Williams, T. Contursi, Structural damage detection by a sensitivity and statistical-based method, Journal of Sound and Vibration 216 (5) (1996) 791–808. [5] Z.Y. Shi, S.S. Law, L.M. Zhang, Damage localization by directly using incomplete mode shapes, Journal of Engineering Mechanics 126 (6) (2000) 656– 660. [6] L. Yu, L. Cheng, L.H. Yam, Y.J. Yan, Application of eigenvalue perturbation theory for detecting small structural damage using dynamic responses, Composite Structures 78 (2007) 402–409. [7] Q.W. Yang, A mixed sensitivity method for structural damage detection, Communications in Numerical Methods in Engineering 25 (4) (2009) 381–389. [8] Q.W. Yang, J.K. Liu, Damage identification by the eigenparameter decomposition of structural flexibility change, International Journal for Numerical Methods in Engineering 78 (4) (2009) 444–459. [9] A.N. Andry, E.Y. Shapiro, J.C. Chung, Eigenstructure assignment for linear systems, IEEE Transactions on Aerospace and Electronic Systems 19 (5) (1983) 711–729. [10] D.C. Zimmerman, M. Kaouk, Eigenstructure assignment approach for structural damage detection, AIAA Journal 30 (7) (1992) 1848–1855. [11] T.W. Lim, T.A.L. Kashangaki, Structural damage detection of space truss structures using best achievable eigenvectors, AIAA Journal 32 (5) (1994) 1049–1057. [12] J. Kiddy, D. Pines, Eigenstructure assignment technique for damage detection in rotating structures, AIAA Journal 36 (9) (1998) 1680–1685. [13] J.C. Chen, J.A. Garba, On-orbit damage assessment for large space structures, AIAA Journal 26 (9) (1988) 1119–1126. [14] D.C. Zimmerman, M. Kaouk, Structural damage detection using a minimum rank update theory, Journal of Vibration and Acoustics 116 (2) (1994) 222– 231. [15] S.W. Doebling, Minimum-rank optimal update of elemental stiffness parameters for structural damage identification, AIAA Journal 34 (12) (1996) 2615–2621. [16] Q.W. Yang, J.K. Liu, Structural damage identification based on residual force vector, Journal of Sound and Vibration 305 (2007) 298–307. [17] Z.Y. Shi, S.S. Law, L.M. Zhang, Structural damage detection from modal strain energy change, Journal of Engineering Mechanics ASCE 126 (12) (2000) 1216–1223.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A numerical technique for structural damage detection Q.W. Yang Department of Civil Engineering, Shaoxing University, Shaoxing 312000, China
a r t i c l e
i n f o
Keywords: Damage detection Matrix disassembly Modal residual force
a b s t r a c t A computationally attractive method is proposed in this study to provide an insight to the location and extent of structural damage. The proposed method makes use of the matrix disassembly technique and approaches the damage location and extent problem in a decoupled fashion. First, a scheme is developed to determine the damage location by calculating a damage localization vector, which is derived from the modal residual force criteria. With location determined, the corresponding damage extent can be easily obtained only by the processes of division. The algorithm is applied to a numerical example and its efficiency is demonstrated through damage simulations. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Structural damage detection using measured dynamic data has emerged as a new research area in civil, mechanical and aerospace engineering communities in recent years. The basic idea of this technique is that modal parameters are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause changes in the modal properties. Recent surveys on the technical literature, such as on Doebling et al. [1] and Chang [2], among others, show that extensive efforts have been developed to find reliable and efficient numerical and experimental models to identify damage in structures. Most prior work in damage identification is based on the modification of a structural finite element model (FEM). The goal of these methods is to use test data from the damaged structure and the correlated FEM of the undamaged structure to determine changes to the stiffness and/or mass matrices. This class of methods can be divided into three main groups: One group is the sensitivity-based model update. These methods make use of the derivatives of modal parameters with respect to physical design variables. Using the eigenvalue sensitivity, Messina et al. proposed an assurance criterion for detecting single damage sites [3] and extended this method to identify the relative amount of damage at multiple sites [4]. Shi et al. extended the multiple damage location assurance criterions by using mode shape sensitivity instead of modal frequency sensitivity [5]. Yu et al. make use of eigenvalue perturbation theory and artificial neural network to detect small structural damage [6]. Yang developed a mixed sensitivity method to identify structural damage by combining the eigenvalue sensitivity with the flexibility sensitivity [7]. Recently, Yang and Liu make use of the eigenparameter decomposition of structural flexibility change to detect structural damage in a decoupled manner [8]. Another group is control-based eigenstructure assignment techniques. Andry et al. presented an excellent overview of eigenstructure assignment theory and applications [9]. Zimmerman and Kaouk utilized a symmetric eigenstructure assignment algorithm to perform the partial spectral assignment [10]. Lim and Kashangaki referred to a best achievable eigenvector as a damage indicator [11]. Kiddy and Pines used the eigenstructure assignment technique to detect damage in rotating structures [12]. The third group, known as optimal matrix update, solves a closed-form equation for the matrix perturbations that minimize the modal force error or constrain the solution to satisfy it. Chen and Garba put forward a theory of detecting potential damage by performing a minimum-norm
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.039
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solution of the residual force equation [13]. Zimmerman and Kaouk presented a minimum-rank solution of the residual force equation to identify structural damages. They firstly located the damage sites by the residual force vector and then assessed the damage extents by the minimum-rank update theory [14]. Doebling developed their method and presented a technique termed the minimum-rank elemental update by computing the minimum rank updates directly to the elemental stiffness parameters [15]. Yang and Liu studied several damage detection methods based on the residual force vector and presented the algebraic solution of the residual force equation [16]. The method proposed in this paper also belongs to the third group, which approaches the damage location and extent problem in a decoupled fashion. The damage location algorithm is derived from the Modal Residual Force criteria proposed by Yang and Liu [16]. However, greater insight of the modal residual force criteria is provided and a new damage localization vector is defined, by including the connectivity constraint directly into the modal residual force equation, to locate damage at the element level rather than at the structural degree of freedom (DOF). The main advantage to locating damaged elements rather than damaged DOFs is that the resulting localization criteria has direct physical relevance and thus can be used in the following damage quantification. The damage extent algorithm is very simple since it requires only the processes of division. The remainder of the paper is organized as follows. First, the modal residual force criteria are briefly reviewed. Second, the theoretical development of the proposed approach is presented. Next, the method is demonstrated using a numerical example and the conclusions of this work are summarized in the end. In this paper, we will assume that the dimension of the measured eigenvector is the same as the analytical eigenvector. This assumption allows us to focus our attention on the quality of the proposed damage detection algorithm. This is true (i) when all FEM DOF’s are measured (ii) after the application of an eigenvector expansion algorithm or (iii) after the application of a finite element model reduction algorithm [14]. The ideal situation would be to measure all FEM DOF’s since the eigenvector expansion process would introduce errors in the ‘‘expanded” eigenvectors and the model reduction process would introduce errors in the FEM. It should be noted that in both cases the additional errors may become significant as the ratio of measured to unmeasured DOF’s become smaller. 2. The modal residual force criteria In the context of structural damage detection and health monitoring, the perturbations to the stiffness properties are usually the most relevant. In this paper, only the perturbation of the structural stiffness properties will be considered. The basic theory of the modal residual force criteria begins with the free vibration eigenvalue problem of the damaged structure
ðK d kdj MÞ/dj ¼ 0
ð1Þ
where M and K d are the n n analytical mass and damaged stiffness matrices, kdj and /dj are the jth eigenvalue and eigenvector of the damaged structure, respectively. Let DK be the exact perturbation matrix that reflects the nature of the structural damage with the nonzero elements reflecting the state of damage, namely
K d ¼ K DK
ð2Þ
where K is the undamaged stiffness matrix. Substituting Eq. (2) into (1) yields
ðK kdj MÞ/dj ¼ DK/dj
ð3Þ
Eq. (3) can be written more compactly as
DK/dj ¼ bj
ð4Þ
where bj is defined as the jth modal force error and given by
bj ¼ ðK kdj MÞ/dj
ð5Þ
Inspection of bj in terms of Eq. (4) reveals that the ith element of bj will be zero when the ith rows of DK are zero, i.e., the FEM for the ith DOF is not directly affected by damage. Conversely, a DOF whose FEM has been affected by damage will result in a nonzero entry in bj . Thus, the damaged DOFs can be determined by inspecting the elements of bj . Vector bj as defined in Eq. (5) also reveals that only a single mode of vibration needs to be measured exactly to determine exact damage locations. This is true in even multiple member damage situations. More importantly, the vector bj can be determined from the intact FEM and the measured eigenparameters, kdj and /dj using Eq. (5). The proposed damage location algorithm is similar to the modal residual force criteria, by including the connectivity constraint directly into the modal residual force equation, but locates damage at the element level rather than at the structural DOF. The derivation of the proposed method is presented in the following section. 3. The proposed damage detection approach In this section, a new damage location algorithm is first developed and then a very simple damage extent technique is formulated which makes use of the knowledge obtained from the location algorithm.
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3.1. Damage localization The derivation of the proposed damage location algorithm begins with the parameterization of the global stiffness perturbation matrix as [8]
DK ¼ C DPC T
ð6Þ
where the ðn NÞ matrix C is defined as the stiffness connectivity matrix, which can be computed by the method proposed in Ref. [8], and the ðN NÞ diagonal matrix DP has the elemental stiffness damage parameters as its diagonal entries, N is the number of total elements. Mathematically, this is defined as
diagðDPÞ ¼ ða1 ; a2 ; . . . aN ÞT
ð7Þ
where ai ði ¼ 1 NÞ is the ith elemental damage parameter. Substituting Eq. (6) into Eq. (4) yields
C DPC T /dj ¼ bj
ð8Þ T
Perform a minimum-norm solution of Eq. (8) for DPC /dj to get
DPC T /dj ¼ C þ bj
ð9Þ
where the superscript ‘‘+” denotes the pseudoinverse. As the connectivity matrix C is a full rank matrix (see Appendix A), the pseudoinverse of C can be given by
C þ ¼ C T ðCC T Þ1
ð10Þ
dj ¼ C T /dj ;
ð11Þ
ej ¼ C þ bj
ð12Þ
Let
in which vector ej can be rewritten, by substituting Eqs. (5) and (10) into (12), as
ej ¼ C T ðCC T Þ1 ðK kdj MÞ/dj
ð13Þ
Then Eq. (9) simplifies to
DPdj ¼ ej
ð14Þ
Namely,
2 6 6 6 6 4
a1 a2
8 9 3> d1 > 8 e1 9 > > > > > > > > > j > > j > 2 > > > > > d2 > > 7> = < < e 7 j = j 7 ¼ .. 7> . > > .. > 5> . > . > > > > > . > > > > > > . > > > ; : eN ; : dN > aN > j j
ð15Þ
Inspection of ej in terms of the Eq. (15) reveals that the ith element of ej will be zero when the ith element is undamaged ðai ¼ 0Þ. Conversely, a damaged element will result in a nonzero entry in ej by reason of ai – 0. Thus, vector ej given by Eq. (13) is defined as the damage localization vector and the damage locations can be determined by inspecting the elements of it. It is obvious that vector ej can be determined from the intact FEM and the measured eigenparameters, kdj and /dj using Eq. (13). Moreover, vector ej also reveals that only a single mode of vibration needs to be measured exactly to determine exact damage locations. This is true in even multiple member damage situations. In practice, the perfect zero/nonzero pattern of the damage localization vector ej rarely occurs due to errors present in the measured eigenvalues and eigenvectors. Then entries with larger absolute values in ej are associated with the damaged elements in practice. 3.2. Damage quantification It is sometimes necessary to determine the extent of structural damage. With the damaged elements determined by the above approach, the corresponding damage extents can be easily calculated only by the processes of division. Without loss of generality, assuming the ith element is damaged, from Eq. (15), one obtains
ai dij ¼ eij
ð16Þ
Rewriting Eq. (16), the damage extent can be obtained by
ai ¼ eij =dij
ð17Þ
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In the end, a step-by-step summary of the whole technique is described as follows. Step 1: Calculate the modal force error bj by Eq. (5). In this step, the modes of the damaged structure can be obtained by a modal survey on it. Step 2: Compute the vectors dj and ej using Eqs. (11) and (12). The stiffness connectivity matrix C used in this step can be obtained by the procedure presented in Ref. [8]. And then the locations of damaged elements can be determined in this step by inspection of the damage localization vector ej . Step 3: For the identified damaged elements, calculate the damage extents using Eq. (17). 4. Numerical example To illustrate characteristics of the proposed damage detection algorithm, a two-dimensional truss structure is presented as shown in Fig. 1. The basic parameters of the structure are as follows: Young’s modulus E ¼ 200 GPa, density q ¼ 7:8 103 kg=m3 , and cross-sectional area A ¼ 0:004 m2 . Two damage cases are considered in the example. The first one is a single damage case that element 5 is damaged with a stiffness loss of 30%. The second case has two damages in which elements 5 and 16 have 30% and 20% reduction in stiffness, respectively. The mode shape is contaminated with 5% random noise in the study of the measurement error effect. The contaminated signal is represented as [17]
/ij ¼ /ij ð1 þ c/i q/ j/max;j jÞ
ð18Þ
where /ij and /ij are the mode shape components of the jth mode at the ith degrees of freedom with noise and without noise, respectively; c/i is the random number with a mean equal to zero and a variance equal to 1; q/ is the random noise level; and /max;j is the largest component in the jth mode shape. Table 1 presents the ratios of the first six eigenvalue changes for the 23-bar truss when damage is introduced. For damage case 1, mode 5 will be used in damage detection because it undergoes a significant shift in frequency. For damage case 2, mode 1 undergoes a largest change and so it will be used in damage identification.
Fig. 1. 23-Bar truss structure.
Table 1 The ratios of eigenvalue changes for the first six modes. Mode
Element 5 is damaged (%)
Elements 5 and 16 are damaged (%)
1 2 3 4 5 6
1.35 0.01 1.48 0.52 3.75 0.90
4.91 0.9 2.8 0.66 4.65 1.85
Damage Indicator
14 12
no noise
10
5% noise
8 6 4 2 0 1
5
9
13
17
21
Element Number Fig. 2. Damage localization result when element 5 is damaged.
Damage Indicator
Q.W. Yang / Applied Mathematics and Computation 215 (2009) 2775–2780
1. 4
no noise
1. 2
5% noise
2779
1 0. 8 0. 6 0. 4 0. 2 0
1
5
9 13 Element Number
17
21
Fig. 3. Damage localization result when elements 5 and 16 are damaged.
Fig. 2 shows the damage localization result for the first damage case using Eq. (13) with the 5th mode of vibration data. It is clear that the location algorithm is able to accurately locate the correct damage even if the mode shape is contaminated with 5% random noise. For the error-free case, the calculated damage parameter of element 5 using Eq. (17) is 0.3, which is exactly equal to the assumed value (0.3). When 5% noise is introduced, the corresponding damage parameter obtained by Eq. (17) is 0.2858, which has 4.7% relative error as compared to the assumed value. Fig. 3 presents the results of using Eq. (13) with the first mode eigenvalue/eigenvector information. Again, the location algorithm is able to cleanly detect damage even when presented with eigenvector information corrupted with 5% noise. Using Eq. (17), for the error-free case, the damage extents can be readily calculated as a5 ¼ 0:3 and a16 ¼ 0:2, which exactly equal the assumed values (0.3, 0.2). For the case with 5% error, the calculated damage parameters in elements 5 and 16 are 0.2874 and 0.1802, respectively. From the above results, it is obvious that the predicted damage location is exactly correct, and the predicted damage extent has slight deviation from the true value because of the error in the measured data. 5. Conclusions A new optimal matrix update method is developed in this paper. The algorithm makes use of an existing finite element model of the ‘‘healthy” structure and a subset of experimentally measured modal properties of the ‘‘damaged” structure. The procedure includes two parts or stages. The first stage is to locate structural damage using a localization vector derived from the modal residual force vector. With damaged element determined, estimation of its severity is performed in the second stage. The proposed approach is shown to be computationally attractive because the damage location only requires matrix–vector and matrix–scalar multiplications and the damage extent only requires the processes of division. The illustrative example shows the good efficiency and stability of the presented method on the identification of single damage or multiple damages. It has been shown that the proposed procedure may be useful for structural damage identification. Further research on the technique can be carried out to tackle the problem of the test/analysis DOF mismatch, to compare the method with other damage detection techniques, and to demonstrate the procedure using experimentally measured data. Acknowledgments The author thanks the reviewers who gave a thorough and careful reading to the original manuscript. Their comments are greatly appreciated and have helped to improve the quality of the paper. This work is supported by the scientific research project of education of Zhejiang Province (20070503). Appendix A.
The statement after Eq. (9) ‘‘As the connectivity matrix C is a full rank matrix, . . .” can be proven as follows:
For C is of dimension n N, the rank of C must satisfy
rankðCÞ 6 minðn; NÞ
ðA:1Þ
On the other hand, according to Eq. (7), one obtains
rankðCÞ P rankðKÞ ¼ n
ðA:2Þ
From Eqs. (A.1) and (A.2), one has
rankðCÞ ¼ n Eq. (A.3) implies that the connectivity matrix C is a full rank matrix.
ðA:3Þ
2780
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