Identification of damage using natural frequencies and Markov parameters

Computers and Structures 74 (2000) 365±373

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Identi®cation of damage using natural frequencies and Markov parameters S. Hassiotis* Department of Civic and Environmental Engineering, University of South Florida, Tampa, FL 33620, USA Received 13 February 1998; accepted 2 October 1998

Abstract An optimization algorithm is formulated for the identi®cation of damage in structures, when such damage is manifested by localized reductions in the sti€ness of structural elements. The relationship between the sti€ness of the structure and its natural frequencies has been presented as a set of underdetermined equations to be solved for the sti€ness changes due to damage. The objective function that is introduced to solve these equations depends on the measured impulse-response of the damaged system in the form of the Markov parameters of the system. The derived algorithm was used to ®nd the magnitude and the location of damage when this occurs in any number of locations in a computer simulated frame. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Identi®cation of structural damage; Markov parameters; Inverse identi®cation; Sti€ness reductions

1. Introduction Some forms of damage (cracking, localized buckling, etc.) cause a reduction in the sti€ness of the damaged member. Such reductions of sti€ness a€ect the dynamic response of a structure. Dynamic data and system identi®cation procedures can be used to ®nd such localized damage. System identi®cation procedures can be used to ®nd a spatial model (i.e. the distribution of sti€ness, mass and damping in the system matrices) given measurements of response or modal parameters. However, the solutions are not unique. The main problem is that the measured data is incomplete because not all the degrees of freedom (DOF) in a ®nite-element model are accessible to measure. In reality, only a limited number of vibration modes can be identi®ed from ex-

* Tel.: +1-813-974-4746. E-mail address: [email protected] (S. Hassiotis)

periments. To overcome this problem the identi®ed system is assumed to be a perturbation of an original spatial model that reproduces dynamic measurements. Unique solutions are reached with the help of additional conditions such as minimizing the Frobenius norm of the changes from the initial model [1,4,5,11,14,15]. Such procedures have been adopted and developed to assess damage (or changes in the structural parameters) using dynamic data taken before and after the occurrence of damage. Natural frequencies depend on the global properties of a system and, thus, can be used with frequencydomain identi®cation procedures to ®nd the location and the magnitude of degradations in the sti€ness. However, used alone, these data cannot provide reliable results, especially because several combinations of damage in the structure can produce the same changes in the natural frequencies. Cawley and Adams [3] were among the ®rst to use an incomplete set of measured natural frequencies to identify the location and provide a rough estimate of structural damage. All

0045-7949/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 3 4 - 6

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S. Hassiotis / Computers and Structures 74 (2000) 365±373

Nomenclature A, B, C matrix coecients of state vector equations D matrix relating changes in sti€ness to changes in eigenvalues F matrix of force-input locations H2 third Markov parameter J optimality criterion K sti€ness matrix Ke element sti€ness matrix in global coordinates M mass matrix Q weighting matrix in quadratic optimization c vector in linear term of quadratic optimization

possible elements of a structure were checked individually to obtain the location of the damage, which makes its implementation to larger structures and multiple damages dicult. Another drawback is that the magnitude of the sti€ness change is given only in a qualitative sense. Hassiotis and Jeong [7,8] introduced an optimization algorithm to identify both the location and the magnitude of single or multiple damage using changes in the natural frequencies. The algorithm was used to detect damage in a 90 DOF frame, and was found to give good results if the number of damaged members is kept below three. Pabst and Hagedorn [19] used the changes in the frequencies to identify the location of a single crack on a cantilever beam. If a set of mode shapes is known in addition to the natural frequencies, the identi®cation problem becomes more robust. However, a disadvantage in such an approach is its dependence on a set of measured mode shapes, accurate measurements of which are not easy to obtain [12,16,18,21,23,24]. Several investigators approached the problem of damage identi®cation in the time domain [2,13,17]. In such work, acceleration records are used to estimate the decrease in structural sti€ness. These methods can be used to alert about the existence of damage, and give only an approximate location of the damage if a probability function of such a location can be supplied by the investigator. In summary, algorithms are constantly being developed to identify damage in structures using easily accessible measurements. Traditionally, the timedomain and the frequency-domain analyses have been seen or developed as rivals. However, neither procedure can be used alone to identify damage. Merging of time-domain and frequency-domain data to optimally complement each other in the search for changes in the sti€ness of the structure is possible. In this

q r u x dK dM dk dl df l f o

vector of nodal displacements vector of residuals vector of forcing functions state vector change of the sti€ness matrix change of the mass matrix vector of element sti€ness changes vector of the changes in the eigenvalues change of an eigenvector vector of the eigenvalues of the structure an eigenvector of the structure natural frequency of the structure

work, we will introduce an identi®cation algorithm that uses measurements of the Markov parameters in addition to natural frequency measurements to identify damage, when this occurs in multiple locations.

2. Proposed method for the detection of damage Given the change in a set of natural frequencies caused by damage in the system, it is possible to ®nd the decrease in the sti€ness of a structure. Most developments in this problem start with the perturbation of the system eigenvalue problem. The equation of motion of an undamped mechanical system is given by MqÈ ‡ Kq ˆ F
u

…1†

where M is an n  n mass matrix, K is an n  n sti€ness matrix, q is an n-vector of nodal displacements, F is an n  l matrix of zeroes and ones to indicates the location of the force input, and u is an l  1 vector of forcing functions. The assumption of a harmonic solution for q leads to the eigenvalue problem …K ÿ li M†fi ˆ 0

…2†

where the eigenvalue li is the square of the circular natural frequency oi, and the eigenvector fi is the corresponding natural mode shape of the system. In the proposed method, Eq. (2) will be used to ®nd the eigenvalue sensitivities to changes in K. The result is a set of underdetermined equations. A new optimality criterion is introduced for their solution. The criterion depends on the measurements of the impulse responses, as given by the Markov parameters of the system.

S. Hassiotis / Computers and Structures 74 (2000) 365±373

2.1. Eigenvalue sensitivity analysis Damage of the original structure is assumed to cause changes of the sti€ness matrix by an amount dK. It is further assumed that the damage is not accompanied by a change in mass. The change in sti€ness produces changes in the eigenvalues dli, and changes in the eigenvectors dfi. The eigenvalue problem of the damaged structure is given by …K ‡ dK ÿ …li ‡ dli †M†…fi ‡ dfi † ˆ 0

…3†

By expanding Eq. (3) and neglecting higher order terms we arrive in the sensitivity of an eigenvalue [6] to changes in the sti€ness matrix given by: dli ˆ fTi dKfi

…4†

(The error introduced by neglecting the changes in the eigenvectors is used later to produce an optimality criterion.) To relate the changes in the eigenvalues to the changes in the sti€ness of the local sti€ness element, the changes in the sti€ness matrix is expressed in terms of the changes in the independent structural elements. If Kej is the contribution of the jth element to the total sti€ness matrix of the structure, dK can be written as ne X dK ˆ Kej dkj

…5†

jˆ1

where ne is the number of elements of the n-DOF structure, and dkj is the proportional change in the sti€ness of element j. This formulation preserves the symmetry and connectivity of the sti€ness matrix. Substitution of Eq. (5) into Eq. (4) gives a set of simultaneous equations Ddk ˆ dl

…6†

were D is an m  ne-matrix of elements dij ˆ fTi Kej fi , dk is the ne-vector of the unknown changes in sti€ness dkj, and dl is the m-vector of the changes in the eigenvalues dlj. The simultaneous equations relate the change of the sti€ness of each element to the changes in the frequencies of the structure. Because of the assumptions used to derive these equations, in no circumstances can they give an exact solution to the location and magnitude of the damage. However, with careful selection of optimality criteria, they can be used to arrive at locations and approximate magnitudes of damage, given the changes in some of the natural frequencies (dl ).

3. The optimization problem The optimality criterion needed can be derived from

367

several approaches. In general, least squares formulations and error minimization lead to a minimization of a criterion in the form minimize

1 T dk Qdk ‡ dkT c 2

subject to

Ddk ˆ dl

and

…7†

dkR0

Here, c is a given ne-vector, and Q is a given positive de®nite ne  ne-matrix. The inequality constraint is derived from physical reasoning. In general, damage does not produce a positive change in the sti€ness. The problem is a quadratic programming problem with linear equality and inequality constraints. If Q is positive de®nite, this problem is strictly convex and a unique solution for dk can be found using ecient algorithms that have been developed in the ®eld of linear and nonlinear programming. Hassiotis and Jeong [8] introduced three criteria for the solution of the underdetermined equations. For the sake of completeness, these are summarized here. . Optimization criterion 1: minimization of the norm of dk minimize

1 T dk dk 2

…8†

This is the pseudo-inverse solution. In view of Eq. (7), Q=I, and c=0 . Optimization criterion 2: minimization of the Frobenius norm of dK. Minimization of kdKk

…9†

where k
k denotes the Frobenius norm of the expression, leads to the objective function minimize

dkT Qdk

…10†

where qrs=Sni = 1Snj = 1Ker (i, j )Kes (i, j ). . Optimization criterion 3: minimization of the eigenvalue problem residuals. The eigenvalue problem of the damaged structure can be stated as …K ‡ dK†fi ÿ …li ‡ dli †Mfi ˆ ri

…11†

where ri is a vector of residuals. Minimization of the 2 vector of residuals, Sm i = 1krik leads to another optimization criterion minimize

1 T dk Qdk ‡ dkT c 2

…12†

where Q is a symmetric ne  ne matrix of elements

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S. Hassiotis / Computers and Structures 74 (2000) 365±373

T e e qij ˆ Sm rˆ1 fr Ki Kj fr and c is an ne-vector of elements T e ci ˆÿSm dl r fr Ki Mfr . rˆ1

3.1. Measured Markov parameters used as optimality criteria The criteria introduced so far can be justi®ed only for the cases in which the structure has not changed a lot due to the damage. Therefore, the damaged structure can be assumed to be the closest perturbation to the undamaged structure. This is true when the damage is concentrated in a few elements only and it does not have a great e€ect on the global sti€ness matrix K. As reported in Hassiotis and Jeong [8], these optimality criteria can be used successfully to ®nd damage in one to three locations. However, these criteria fail to ®nd the damage when several elements have undergone changes in the sti€ness. The new optimality criterion that is proposed herein introduces into the problem additional measurements in the form of the system Markov parameters. The additional data allows for the determination of light to severe damage when such damage occurs in any number of members in the structure. . Optimization criterion 4: Markov parameters. The Markov parameters of a continuous system are simply the impulse response sequence of the system. . By de®ning a 2n-dimensional state vector x=[q q], Eq. (1) can be written as

G…s† ˆ C…sI ÿ A†ÿ1 B ˆ

Hi ˆ CAi B i ˆ 0,1,2, . . .

 Bˆ

0 Mÿ1 F

C ˆ ‰P



JŠ

Here P is an s  n-matrix of zeroes and ones to indicate the location of the measurements in displacement, and J is an s  n-matrix of zeroes and ones to indicate the location of the measurements in velocity. Taking the Laplace transform of this system yields the transfer function G(s ), which can be expanded into a power series in 1/s as [22]:

…15†

The Markov parameters can be obtained from the measured impulse-response of a continuous structure [20], or from the pulse response of a discrete system [10]. By substituting Eq. (13) into Eq. (15), we can show that the third Markov parameter, H2, is a function of the sti€ness matrix K and is given by H2 ˆ CA2 B ˆ ÿJMÿ1 KMÿ1 F

…16†

The assumption herein is that H2 for the structure can be measured after the occurrence of damage. In such cases, a possible optimality criterion can be derived to minimize the di€erence between the Markov parameter measured after damage and the one obtained from the model of the damaged structure: minimize

dK kH2

ÿ …ÿJMÿ1 ‰K ‡ dKŠMÿ1 F†k

…17†

After substitution of Eq. (5), the optimization problem can be written as minimize dk kH2

‡ JMÿ1 KMÿ1 F

‡ JMÿ1

…13†

where y is an s-dimensional output vector (s indicates the number of velocity and displacement measurements), and   0 I Aˆ ÿ1 ÿM K 0

…14†

The matrices Hi are called the Markov parameters of the continuous system and have the value

xÇ ˆ Ax ‡ Bu y ˆ Cx

inf X 1 H i‡1 i s iˆ0

! ne X Kek dkk Mÿ1 Fk

…18†

kˆ1

To simplify manipulation of this expression, we de®ne an s  l matrix S=JMÿ1KMÿ1F and an s  l matrix Tk=JMÿ1KekMÿ1F. The Frobenius norm can thus be written as ne s X l X X fH2 …i,j † ‡ S…i,j † ‡ Tk …i,j †dkk g2 iˆ1 jˆ1

…19†

kˆ1

This norm includes constant terms which do not in¯uence the optimization and can be dropped. After expansion and simpli®cation, the norm that is minimized can be written as ne X ne s X l X X Tk …i,j †dkk Tp …i,j †dkp iˆ1 jˆ1 kˆ1 pˆ1

‡

ne s X l X X 2fH2 …i,j †Tk …i,j † ‡ S…i,j †Tk …i,j †gdkk iˆ1 jˆ1 kˆ1

…20†

S. Hassiotis / Computers and Structures 74 (2000) 365±373

369

Fig. 1. Ten-story, 90 DOF frame.

We can present Eq. (20) in a quadratic form as minimize

1 T dk Qdk ‡ dkT c 2

…21†

where Q(p,k)=Ssi = 1Slj = 1Tk (i,j )Tp (i,j ), and s l ck=Si = 1Sj = 1H2(i, j )Tk (i, j )+S(i, j )Tk (i, j ). This optimization criterion can be used for the solution of optimization problem 7. It can be shown that Q is a positive de®nite matrix, hence, the optimization problem is a convex quadratic optimization. 4. Assessment of damage in frame The proposed algorithm is applied here to locate the

damage in the 10-story, two-bay steel frame shown in Fig. 1. The structure is divided into 50 frame elements. Nodes are assigned at every joint, and each node provides three degrees of freedom for a total of 90 DOF Elements 1±30 make up the columns, and elements 31±50 make up the ¯oors of the frame. In assessing the damage, ®rst we assume that the ®nite element model of the frame describes the system before damage. This model provides the natural frequencies of the undamaged structure. Then, we induce a known decrease in the sti€ness of one or more elements which is referred herein as the `actual' damage. The natural frequencies and Markov parameters calculated using the damaged structure serve as the input data of the identi®cation algorithm. The identi®ed

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Table 1 Eigenvalues of a steel frame Change due to damage Mode

Undamaged eigenvalues

Exact data

Noise-polluted data

1 2 3 4 5 6 7 8 9 10 11 12

0.612  100 0.427  101 0.145  102 0.340  102 0.686  102 0.130  103 0.151  103 0.192  103 0.217  103 0.255  103 0.338  103 0.392  103

ÿ0.413  10ÿ1 ÿ0.188  100 ÿ0.916  100 ÿ0.140  101 ÿ0.290  101 ÿ0.407  101 ÿ0.157  101 ÿ0.427  101 ÿ0.850  101 ÿ0.120  102 ÿ0.832  101 ÿ0.177  102

ÿ0.453  10ÿ1 ÿ0.188  100 ÿ0.916  100 ÿ0.139  101 ÿ0.291  101 ÿ0.407  101 ÿ0.157  101 ÿ0.427  101 ÿ0.850  101 ÿ0.120  102 ÿ0.106  102 ÿ0.151  102

damage is referred herein as the `predicted' damage. The International Mathematical and Statistical Libraries [9] were used to solve the eigenvalue and the optimization problems. The eigenvalues calculated for the frame of Fig. 1 are listed in Table 1. Data on the ®rst 12 eigenvalues are considered. The eigenvalues of the undamaged structure are listed in the second column. The changes in the data due to damage are listed in the third column for clean data, and in the fourth column for noise-polluted data. Fig. 2 shows the successful identi®cation of damage in elements 24, 40 and 49 of the steel frame using 12 natural frequency measurements. The optimality criterion minimizes the norm of dk (Eq. (8)). This algorithm fails to identify the damage on elements 7, 24, 33, 38, 40, 48 and 49 as shown in Fig. 3, where the predicted damage is spread throughout the structure. In

this case, no amount of data was successfull in identifying the damage. It is apparent that natural frequencies alone cannot be used to identify damage when this occurs in many elements. In addition to the measurements of 12 natural frequencies, the data proposed herein consists of the measurements of the third Markov parameter, H2, of the damaged structure. In constructing the Markov parameter for our analytical experiment, we assumed that J in Eq. (16) indicates velocity measurements in the translational direction of each node in elements 33, 34, 41, 42, 49 and 50 of the damaged structure (i.e. nine degrees of freedom are supplied with sensors: DOF 10, 13, 16; 46, 49, 52; 82, 85, 88). Speci®cally, matrix J is a 9  90 matrix of mainly zeroes. A one is placed at positions (1, 10), (2, 13), . . . , (9, 88) to indicate that a velocity measurement is available there.

Fig. 2. Identi®cation of damage in three locations using natural frequencies only.

S. Hassiotis / Computers and Structures 74 (2000) 365±373

371

Fig. 3. Identi®cation of damage in seven locations using natural frequencies only.

Matrix F of Eq. (16) assumes impulse input in the translational direction at the nodes of elements 33, 34, 35, 36, 39, 40, and 45. (i.e. 12 DOF are used for an impulse input: DOF 10, 13, 16; 19, 22, 25; 37, 40, 43; 64, 67, 70). Thus, matrix F is a 90  12-matrix of mainly zeroes. A one is placed at positions (10, 1), (13, 2), . . ., (70, 12) to indicate that an impulse loading is input there. This data was used in constructing the optimization criterion of Eq. (21). The results of the new algorithm are shown in Fig. 4. It is evident from this ®gure that the new algorithm is capable of identifying the location and the magnitude of multiple damages. Additional parametric studies in similar cases showed that the location of the impulse input and measured output play a signi®cant role in the identi®cation. Optimal placement of sensors and actuators should be sought. This is beyond the scope of the present work, however it is under investigation.

Fig. 5 shows that if a data can be obtained at every structural element (velocity measurements were assumed to be available on all 50 elements), any amount and degree of damage can be identi®ed. In this case, all structural elements were assumed to have experienced damage at the same time. In order to identify the damage when this occurs in all (or a large number) of the elements in the structure, a large number of data must be available. The number of response data that are needed for correct identi®cation depends on the number of damage locations, the degree of damage, the number of natural frequencies that are assumed to be available, and the location of the actuators and sensors. Damage in one location was identi®ed quite easily with very few measurements. To test the sensitivity of the algorithm to noise in the data, a white-noise sequence was superimposed on the calculated eigenvalues of the damaged structure. The dependence of the identi®cation on the variance of

Fig. 4. Identi®cation of damage in seven locations using natural frequencies and Markov parameters.

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S. Hassiotis / Computers and Structures 74 (2000) 365±373

Fig. 5. Identi®cation of damage using one natural frequency and Markov parameters.

the noise was then explored. It was found that a coecient of variation above 0.8% contributes to erroneous identi®cation of the location and magnitude of the damage. The identi®cation of the damage was successful for coecients of variation of less than 0.7%, as shown in Fig. 6. Overall, the identi®cation of the correct location and magnitude of the damage is sensitive to the accuracy of the data. 5. Conclusions The sensitivities of the natural frequencies to localized changes in the sti€ness of a structure were formulated as a set of simultaneous underdetermined equations. We assumed that an incomplete set of natural frequency measurements taken before and after

damage constitute the data. Measurements of (1) the changes in the natural frequencies and (2) the Markov parameters are used to identify the location and severity of the damage. An identi®cation algorithm was developed to ®nd the damage in a 90 DOF frame. Parametric studies conducted with the new algorithm lead to the conclusions that (1) the combination in measurements of natural frequencies and Markov paramenters has contributed to an improved algorithm for the identi®cation of damage; (2) the number of measurements needed to identify damage depends on the number of damaged locations; (3) identi®cation of damage that occurs simultaneously in all the elements in a structure requires that measurements exist at all elements; (4) if damage occurs in a single location, very little data is needed to locate and quantify it; and (5) the identi®-

Fig. 6. Identi®cation of damage using noise-polluted data (c = 0.7%).

S. Hassiotis / Computers and Structures 74 (2000) 365±373

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