Damage detection based on optimized incomplete mode shape and frequency

Acta Mechanica Solida Sinica, Vol. 28, No. 1, February, 2015 Published by AMSS Press, Wuhan, China

ISSN 0894-9166

DAMAGE DETECTION BASED ON OPTIMIZED INCOMPLETE MODE SHAPE AND FREQUENCY⋆⋆ Wei Chen⋆

Wenguang Zhao

Huizhen Yang

Xuquan Chen

(Hubei Key Laboratory of Control Structure, Huazhong University of Science and Technology, Wuhan 430074, China) Received 17 January 2013, revision received 17 June 2014

ABSTRACT For the purpose of structural health monitoring, a damage detection method combined with optimum sensor placement is proposed in this paper. The back sequential sensor placement (BSSP) algorithm is introduced to optimize the sensor locations with the aim of maximizing the 2-norm of information matrix, since the EI method is not suitable for optimum sensor placement based on eigenvector sensitivity analysis. Structural damage detection is carried out based on the respective advantages of mode shape and frequency. The optimized incomplete mode shapes yielded from the optimal sensor locations are used to localize structural damage. After the potential damage elements have been preliminarily identified, an iteration scheme is adopted to estimate the damage extent of the potential damage elements based on the changes in the frequency. The effectiveness of this method is demonstrated using a numerical example of a 31-bar truss structure.

KEY WORDS structural damage detection, optimum sensor placement, sensitivity analysis, information matrix

I. INTRODUCTION Civil structures, such as bridges, offshore platforms, and lattice structures, continuously accumulate damage and then begin to deteriorate during their service life. To ensure the safety of these structures, constructing a structural health monitoring system (SHMS) for finding early damage is necessary. It is an attractive topic to develop an effective and robust damage detection strategy in health monitoring. Reduction in stiffness property is generally associated with the modification of modal parameters of structure. During the past twenty years, lots of researchers have focused on the areaof damage detection based on modal parameters with different algorithms[1–3]. Reduction in stiffness is generally associated with the modification of mode parameters of structure. During the past twenty years, many researchers have focused on the area of damage detection based on modal parameters with different algorithms[1–3] . A common requirement of these methods is the necessity to use complete modal data. However, in engineering practice, only partial mode shapes can be obtained by a modal survey for a large flexible structure. One strategy is to expand the measured mode shapes to the total degrees-of-freedom (DOFs). Many methods belong to this strategy, such as the finite element model update based on the residual force vector[4, 5] and modal strain energy change methods[6, 7] . Unfortunately, methods of this kind are bound to induce additional errors in damage detection results as will the expansion process in the expanded mode shapes. The other strategy is simply use the frequency ⋆ ⋆⋆

Corresponding author. E-mail: [email protected] Project supported by the National Basic Research Program of China (973 Program) (No. 2011CB13804).

Vol. 28, No. 1 Wei Chen et al.: Damage Detection Based on Optimized Mode Shape and Frequency

· 75 ·

measured to detect structural damage. Messina[8] proposed a damage detection method based on the changes in natural frequencies. Krawczuk and Ostachowicz[9] investigated a detection method using merely changes in natural frequencies. However, only a truncated set of modal frequencies not sensitive to local damage can be obtained experimentally. Thus, an alternative method of directly using incomplete modal data to identify structural damage is desirable. Kim et al.[10] proposed a two-step damage detection method for large-scale structures based on incomplete mode shapes. The first step was to determine the approximate damage areas using the best matrix correlation method, and the second step was to determine damaged elements in the detected damage areas using sensitivity analysis method. Yang[11] presented a mixed sensitivity method that combined eigenvalue and flexibility sensitivity matrix to identify structural damage. However, the incomplete mode data obtained from sensors were not the most sensitive information for damage detection in these methods. Song and Jin[12] presented an optimum sensor placement strategy for structural damage detection. The optimum sensor locations were determined based on the eigenvector sensitivity analysis. The incomplete mode data obtained from these optimized sensor locations included lots of information about the damage. However, they did not use these optimized mode data to detect damage. In this paper, the main objective is to develop a damage detection method combined with an optimum sensor placement algorithm. For the purpose of damage localization, sensor locations are optimized according to their ability of localizing structural damage. Only a small subset of the total DOFs of the structure is selected for the sensor locations to measure the mode shapes. Since mode shapes are more sensitive to local damage and frequencies less contaminated by measuring noise, a multiple damage location assurance criterion (MDLAC) is presented to localize structural damage based on the optimized incomplete mode shapes, and the damage extent of the potential damage elements is assessed using the changes in the modal frequencies. A numerical analysis on a 31-bar truss structure with several damage cases is performed to demonstrate the effectiveness of this method.

II. SENSITIVITY ANALYSIS Suppose there is a lightly damped structure with n DsOF, the structural damage referring to changes in the stiffness properties of the structure with no changes in mass property. The equilibrium equation for structural vibration is (K − λi M )φi = 0 (1) where K and M are stiffness and mass matrices, λi and φi the ith eigenvalue and eigenvector, respectively. For a small perturbation in stiffness, Eq.(1) becomes [(K + ∆K) − (λi + ∆λi )M ](φi + ∆φi ) = 0

(2)

where ∆K, ∆λi and ∆φi are small change in the stiffness matrix, the ith eigenvalue, and the ith mode shape, respectively, due to damage. Neglecting higher order terms, Eq.(2) becomes (K − λi M )∆φi = ∆λi M φi − ∆Kφi ∆φi can be represented as ∆φi =

n X

dik φk

(3)

(4)

k=1

where dik is a scalar factor, and n is the total number of modes of the system. Substituting Eq.(4) into Eq.(3) and premultiplying φT r (r 6= i) to both sides of Eq.(3), we have n X

T T dik φT r (K − λi M )φk = ∆λi φr M φi −φr ∆Kφi

(5)

k=1

With the orthogonal relationship, Eq.(5) can be simplified as dir = −

φT r ∆Kφi λr − λi

(r 6= i)

(6)

· 76 ·

ACTA MECHANICA SOLIDA SINICA

2015

When r = i, φi + ∆φi is normalized in the unit-mass mode shape, i.e., (φi + ∆φi )T M (φi + ∆φi ) = 1

(7)

By substituting Eq.(4) into Eq.(7), using the orthogonal relationship, and ignoring the higher order terms, we have: drr = 0. Therefore Eq.(4) becomes n X −φT r ∆Kφi ∆φi = φr (8) λr − λi r=1 r 6= i ∆K can be represented as ∆K =

L X

αk K k

(−1 ≤ αk ≤ 0)

(9)

k=1

where K k and αk are the kth elemental stiffness matrix and its corresponding damage coefficient, respectively. By substituting Eq.(9) into Eq.(8), the change of the ith mode shape due to damage can be expressed as ∆φi =

L X

k=1

αk

n X −φT r K k φi φr = F (K)i δA λr − λi r=1 r 6= i

      

      n  X −φT r K 1 φi F (K)i = φr ,  λ r − λi    r = 1   r 6= i δA = {α1

α2

···

(10a)

n n X X −φT −φT r K 2 φi r K L φi φr , · · · , φr  λr − λi λr − λi    r=1 r=1   r 6= i r= 6 i

αL }T

(10b)

(10c)

where F (K)i is the matrix of sensitivity matrix of ith mode shape changes with the damage coefficient vector δA. When m modes are used together, the mode shape change vector can be represented as follows: {∆φ} = {F (K)}{δA} (11) where {∆φ} = {∆φ1 ; ∆φ2 ; · · · ; ∆φm }, {F (K)} = {F (K)1 ; F (K)2 ; · · · ; F (K)m }. Substituting Eq.(4) into Eq.(3) and premultiplying φT i to both sides of Eq.(3), we have n X T T dik φT (12) i (K − λi M )φk = ∆λi φi M φi −φi ∆Kφi k=1

As

φT i (K

− λi M ) = 0, Eq.(12) can be simplified as ∆λi = φT i ∆Kφi

As λi =

4π 2 fi2 ,

(13)

Eq.(13) can be rewritten as

1 φT ∆Kφi (14) 8π 2 fi i Substituting Eq.(9) into Eq.(14), the change of the ith frequency due to damage can be expressed ∆fi =

as ∆fi =

L X 1 · αk φT i K k φi = G(K)i δA 8π 2 fi

(15a)

k=1

n o 1 T T T · φ K φ , φ K φ , · · · , φ K φ 1 2 L i i i i i i 8π 2 fi  T δA = α1 α2 ... αL

G(K)i =

(15b) (15c)

Vol. 28, No. 1 Wei Chen et al.: Damage Detection Based on Optimized Mode Shape and Frequency

· 77 ·

where G(K)i is the vector of sensitivity matrix of ith frequency changes with the damage coefficient vector δA. When m modes are used together, the frequency change vector can be represented as follows: {∆f } = {G(K)} {δA}

(16)

T

where {∆f } = {∆f1 , ∆f2 , · · · , ∆fm } , {G(K)} = {G(K)1 ; G(K)2 ; · · · ; G(K)m }.

III. SENSOR PLACEMENT OPTIMIZATION Based on the sensitivity analysis of mode shape, optimum sensor placement analysis is introduced in order to reduce the sensor quantity in this part. Through the perturbation equation of Eq.(10), the damage coefficient vector δA can be rewritten as −1 δA = [F (K)T F (K)T i F (K)i ] i ∆φi

(17)

where Q = F (K)T i F (K)i , which is defined as Fisher information matrix. Maximizing the Fisher information matrix Q with an efficient unbiased estimator will lead to the best estimate of damage coefficient vector δA. Different DsOF have different contributions to the Fisher information matrix. The DOFs in the candidate sensor set should be discarded if their contributions are small. In contrast, the DsOF with great contributions should be retained. Finally, a rational sensor placement can be obtained. The effective independence (EI) algorithm is one of the most influential methods frequently used in the area of optimum sensor placement[13] . By calculating the following matrix E, the contribution from each degree-of-freedom can be ranked according to the diagonal elements of the matrix E. If a DOF contributes least information to the rank of matrix E, it is redundant and can be discarded. The remaining DsOF are the optimal locations. The matrix E can be expressed as E = F (K)[F (K)T F (K)]−1 F (K)T

(18)

where F (K) can be calculated by Eqs.(10) and (11). From Eq.(18), we can know that the column vectors in the matrix F (K) must be linearly independent. By substituting Eq.(9) into Eq.(10b), F (K)i can be rewritten as      !  n  X −φT φT φT r Kφi r K 2 φi r K L φi F (K)i = φ + φ + ···+ φ ,  λr − λi r λr − λi r λr − λi r    r=1   r 6= i       n n  T T X −φ K L φ X −φ K 2 φ i i r r (19) φr , · · · , φr  λr − λi λr − λi    r=1 r=1   r 6= i r 6= i With the orthogonal relationship, Eq.(19) can be simplified as         n n  X  X φT φT   r K 2 φi r K L φi F (K)i =  φr + · · · + φr  ,    λr − λi λr − λi   r = 1   r=1   r 6= i r 6= i

      

n n X X −φT −φT r K 2 φi r K L φi φr , · · · , φr  λr − λi λr − λi    r=1 r=1   r 6= i r= 6 i

(20)

· 78 ·

ACTA MECHANICA SOLIDA SINICA

2015

Equation (20) shows that the column vectors in the matrix F (K) are linearly dependent, so the EI method is not suitable for this situation. Therefore, an optimum sensor placement method based on the 2-norm of matrix Q is introduced. When the 2-norm of matrix Q takes the largest value, the best unbiased estimate of δA can be obtained. According to the definition of matrix norm, the 2-norm of matrix Q can be represented as follows:

2 kQk2 = F (K)T F (K) 2 = kF (K)k2 = λmax (Q)

(21)

A heuristic sequential sensor placement (SSP) algorithm, the backward SSP algorithm (BSSP), is proposed for making the optimal sensor configuration. According to the BSSP algorithm, the operation starts with n sensors placed at all DsOF of the structure, and then removes one sensor at a time from the position resulting in the smallest reduction in the maximum singular value of matrix Q which is equal to the 2-norm of matrix Q. The BSSP algorithm (shown in Fig.1) can be represented as follows: Algorithm: Backward sequential sensor placement (BSSP) 1. Initialize: all sensors selected, the number of sensors Ns = n and sensor configuration LNs 2. While the number of sensors Ns > Np (predefined number of sensors) does a. Consider combinations with one sensor less, Ns = Ns − 1. b. For counter i = 1 to Ns + 1 (number of possible sensors to be removed) i. Obtain new configuration LNsi by removing sensor i, ii. Evaluate 2-norm of matrix Q of new sensor configuration LNsi . c. End d. Select the sensor configuration LNsi that maximizes the 2-norm of matrix Q. 3. End 4. Obtain the optimal sensor configuration LNp and the sensor number is equal to Np .

Fig. 1. Flow chart of the BSSP algorithm.

The computations involved in the BSSP algorithm are an infinitesimal fraction of those involved in the exhaustive search method and can be done in realistic time.

Vol. 28, No. 1 Wei Chen et al.: Damage Detection Based on Optimized Mode Shape and Frequency

· 79 ·

IV. DAMAGE LOCALIZATION Mode shape measurements are distributed over the structure and more sensitive to local damage than frequency. According to the above optimum sensor placement, sensitivity matrix F (K) is modified by deleting the rows corresponding to the locations without sensors. Meanwhile, the mode shapes obtained from the optimal sensor locations consist of plenty of damage information. These optimal partial mode shapes can be directly combined with the columns of F (K) to localize damage elements by calculating the correlation parameter, the Multiple Damage Location Assurance Criterion (MDLAC), i.e.  2 {∆φ}T · F (K)j  MDLACj = 
T {∆φ} · {∆φ} · F (K)T j · F (K)j

(22)

where {∆φ} is the measured mode shape change vector before and after the occurrence of damage having a dimension equal to the product of the number of measured modes and the number of sensors. F (K)j is the jth column of F (K) which has been modified. When the jth element is the true damage site, MDLACj will be close to unity. In contrast, MDLACj will be very small. Therefore, the damage elements can be preliminarily estimated from all MDLAC values that are calculated for each element in turn, and the members with large MDLAC values should be taken as damage elements.

V. DAMAGE QUANTIFICATION Frequency is a global dynamic parameter and less contaminated by measuring noise than mode shape. After the damage elements have been preliminarily identified, the damage extent of the suspected damage elements can be assessed using the changes in the frequency. The extent of damage can be quantified by solving Eq.(16) in a least-square sense yielding  −1 {δA} = {G(K)}T {G(K)} {G(K)}T {∆f }

(23)

where ∆f is the frequency change vector.

VI. NUMERICAL ANALYSIS A 31-bar truss structure (shown in Fig.2) is taken as an example to demonstrate the performance of the proposed method. The basic parameters of the structure are listed as: E = 70 GPa, ρ = 2.77 × 103 kg/m3 , L = 1 m and A = 0.0006 m2 . In engineering practice, modal responses yielded from sensors are always polluted by measuring noise. To simulate the noise effect, mode shapes and frequencies are contaminated with 7% and 0.3% Gauss white noise, respectively. Four damage cases are adopted in the numerical analysis and shown in Table 1.

Fig. 2. A 31-bar truss structure.

6.1. Sensor Placement Optimization According to the method illustrated in §III, the optimal sensor configuration can be obtained using the BSSP algorithm based on the first three modes. 12 DsOF are chosen as the sensor locations, with the optimal sensor configuration reported in Table 2.

· 80 ·

ACTA MECHANICA SOLIDA SINICA

2015

Table 1. Damage cases in numerical analysis

Damage cases Case 1 Case 2 Case 3 Case 4

Damaged elements 2 24 6, 8 3, 16, 25

Damage extent 20% 20% 10%, 15% 30%, 30%, 15%

Sensor quantity 12 12 12 12

Table 2. Optimal sensor configuration

DOF 1 3 4 6 7 14

Node 2 3 3 4 5 8

Direction X X Y Y X Y

DOF 15 17 19 21 23 25

Node 9 10 11 12 13 14

Direction X X X X X X

6.2. Damage Localization Based on the optimal sensor configuration, incomplete measured mode shapes are yielded from the 12 installed sensors. The possible damage elements can be identified based on these optimized incomplete mode shapes using MDLAC method. The MDLAC value of each element which indicates the correlation degree between {∆φ} and each column of sensitivity matrix F (K) is calculated by Eq.(22) and then plotted. The damage elements are identified preliminarily as those having a high MDLAC value. In this paper, we assume that the MDLAC for a group of damage elements could be approximately represented by a group of MDLAC for individual damage element. Based on the first three optimized incomplete mode shapes, damage localization is carried out in cases 1-4 with the results shown in Figs.3-6.

Fig. 3. Damage location for case 1.

Fig. 4. Damage location for case 2.

A higher MDLAC value means that the damaged element is closely correlated to the true damage state, and this element is the more probable damage element. According to Figs.3-6, the suspected damage elements identified by damage localization in cases 1-4 are listed in Table 3.

Vol. 28, No. 1 Wei Chen et al.: Damage Detection Based on Optimized Mode Shape and Frequency

· 81 ·

Fig. 5. Damage location for case 3.

Fig. 6. Damage location for case 4. Table 3. Damage localization results

Damage cases Case 1 Case 2 Case 3 Case 4

True damaged elements 2 24 6,8 3,16,25

Detected damaged elements 2,20 7,24,25 6,8,10 1,3,16,25

Misjudged elements 20 7,25 10 1

6.3. Damage Quantification After the potential damage elements have been identified, the damage extent of the suspected damage elements can be calculated by Eq.(23) based on the changes in the frequencies. With the effect of the measuring noise and the increase of damage extent and the number of damage elements, the effect of nonlinear relationship between the frequency change and the damage can induce significant errors in the damage extent estimate. An iteration scheme is adopted to estimate the true damage extent and the locations. Vector δA is estimated from the original ∆f by Eq.(23). Then, the stiffness matrix is updated, new sets of vector ∆f and matrix G(K) are computed, and vector δA is computed again in a subsequent iteration. Damage quantification is carried out in cases 1-4 with the results listed in Table 4. According to the above numerical analysis in damage cases 1-4, several conclusions can be summarized as follows: 1) The method proposed in this paper has excellent damage detecting capability for both single and multiple damage situations. 2) The proposed method has a good anti-noise pollution capability. 3) Several elements are incorrectly predicted during the damage localization process. The probable reasons are as follows: a. the effects of nonlinear relationship between the mode shape change and the damage affect the accuracy of damage localization as a result of neglecting the higher orders in sensitivity analysis; b. the measured incomplete mode shapes obtained from sensors are always polluted by noise, which has a bad effect on damage localization. 4) By using the iteration scheme, the damage extent of the suspected damaged elements can be accurately estimated in the damage quantification process, and then the misjudged elements can be eliminated since the quantified values of the misjudged elements are far less than the true damaged elements. Finally, the true damage elements and extent can be identified in the structure.

· 82 ·

ACTA MECHANICA SOLIDA SINICA

2015

Table 4. Damage quantification results

Damage cases Case 1 Case 2

Case 3

Case 4

Detected damaged element 2 20 7 24 25 6 8 10 1 3 16 25

True damage extent 10% 0 0 20% 0 10% 15% 0 0 30% 30% 15%

Detected damaged extent Iteration 1 Iteration 2 Iteration 3 24.83% 20.35% — -0.17% — — 0.077% — — 21.99% 19.46% — -0.11% — — 12.17% 10.03% — 17.50% 14.78% — -2.78% 0.32% — 4.59% 0.06% — 22.51% 32.57% 30.47% 39.64% 24.26% 29.09% 20.79% 13.50% 14.96%

Relative error (abs) 1.77% — — 3.66% — 0.288% 1.48% — — 1.55% 3.04% 0.29%

VII. CONCLUSIONS A sensitivity method for structural damage detection has been developed in this paper. With the method developed, the damaged elements and extent have been identified for a numerical example. Both in the single and multiple damage situations, damaged elements are correctly identified by directly using the optimized incomplete mode shapes, and the accuracy of the quantification of the damage extent which is identified by frequencies is excellent. Meanwhile, the elements misjudged from the damage localization process can be eliminated through the damage quantification process. Finally, the true damage locations and extent can be accurately identified. Also, this method has excellent anti-noise pollution ability.

References [1] Ismail,Z., Razak,H.A. and Rahman,A.G.A., Determination of damage location in RC beams using mode shape derivatives. Engineering Structures, 2006, 28(11): 1566-1573. [2] Sheng,E.F. and Ricardo,P., Power mode shapes for early damage detection in linear structures. Journal of Sound and Vibration, 2009, 324(1-2): 45-56. [3] Zhu,H.P., Li,L. and He,X.Q., Damage detection method for shear building using the changes in the first mode shape slops. Computers and Structures, 2011, 89(9-10): 733-743. [4] Liu,J.K. and Yang,Q.W., A new structural damage identification method. Journal of Sound and Vibration, 2006, 297(3-5): 694-703. [5] Yang,Q.W. and Liu,J.K., Structural damage identification based on residual force vector. Journal of Sound and Vibration, 2007, 305(1-2): 298-307. [6] Law,S.S., Shi,Z.Y. and Zhang,L.M., Structural damage detection from incomplete and noisy modal test data. Journal of Engineering Mechanics, 1998, 124(11): 1280-1288. [7] Shi,Z.Y., Law,S.S. and Zhang,L.M., Structural damage detection from modal strain energy change. Journal of Engineering Mechanics, 2000, 126(6): 656-660. [8] Messina,A., Williams,E.J. and Contursi,T., Structural damage detection by a sensitivity and statisticalbased method. Journal of Sound and Vibration, 1998, 216(5): 791-808. [9] Ostachowicz,W. and Krawczuk,M., Identification of delamination in composite beams by genetic algorithm. Science and Engineering of Composite Materials, 2002, 10(2): 147-155. [10] Kim,H.M. and Membertkowicz,T.J., An experimental study for damage detection using a hexagonal truss. Computers and Structures, 2001, 79(2): 173-182. [11] Yang,Q.W., A mixed sensitivity method for structural damage detection. Communications in Numerical Methods in Engineering, 2009, 25(4): 381-389. [12] Song,Y. and Jin,H., A sensitivity based method for sensor placement optimization of bridges. In: Proceedings of Sensors and Smart Structures Technologies for Civil, Mechanical and Aerospace Systems, San Diego, USA, 2008: 1137-1144. [13] Li,D.S., Li,H.N. and Fritzen,C.P., The connection between effective independence and modal kinetic energy methods for sensor placement. Journal of Sound and Vibration, 2007, 305(4-5): 945-955.