Damage Localization under Ambient Vibration Using Changes in Flexibility Yong Gao1 and B.F. Spencer, Jr.2 1
Ph.D. student, Department of Civil Engineering & Geological Science, University of Notre Dame, Notre Dame, IN 46556, USA. Tel: (574)631-3914. Fax: (574)631-9236. E-mail:
[email protected]. 2 Nathan M. Newmark Professor of Civil Engineering, Dept. of Civil & Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Matthews Ave, Urbana, IL 61801, USA. Tel: (217)333-8038. Fax: (443)646-0675. E-mail:
[email protected].
Abstract In recent years, Structural Health Monitoring (SHM) has emerged as a new research area in civil engineering. Most existing health monitoring methodologies require direct measurement of input excitation for implementation. However, in many cases, there is no easy way to measure these inputs – or alternatively, to externally excite the structure. Therefore, SHM methods based on ambient vibration have become important in civil engineering. In this paper, an approach is proposed based on the Damage Location Vector (DLV) method to handle the ambient vibration case. Here, this flexibility-matrix-based damage localization method is combined with a modal expansion technique to eliminate the need to measure the input excitation. As a by-product of this approach, in addition to determining the location of the damage, an estimate of the damage extent also can be determined. Finally, a numerical example analyzing a truss structure with limited sensors and noisy measurement is provided to verify the efficacy of the proposed approach.
Keywords Flexibility matrix, Damage Location Vector method, Modal expansion, Natural Excitation Technique, Eigenvalue Realization Algorithm.
1. Introduction Quantitative and objective condition assessment for effective infrastructure preservation has long been a subject of interest within the engineering community. To achieve this objective, methodologies focused on the health monitoring of structural systems have been under development in recent years. The first implementations of SHM appear to have taken place in offshore structures and bridges. One class of damage identification methods employed for SHM measures the change in frequencies to determine structural damage. Vandiver (1975) examined the change in resonant frequencies due to the damage in structural elements. Most recently, Cha and Tuck-lee (2000) examined the change in frequency response data; this information was then used to update the structural parameters. The change in mode shapes has also been used. West (1984) was perhaps the first to systematically use the mode shape information for localization of structural damage without employing a prior finite element model. Another such technique takes advantage of the
1
change in the dynamically-measured flexibility matrix. Pandey and Biswas (1994, 1995) presented a damage detection and localization method based on the flexibility changes in the structure. Bernal (2002a) computed a set of load vectors from the flexibility matrix change, designated as damage location vectors to localize damage. Although these methods can be effective, most require measurement of the input excitation and at least one co-located sensor actuator pair in the system to obtain mass-normalized modes (Alvin and Park, 1994). In numerous cases, the use of an impulse hammer or a rotating unbalanced vibrator as an exciter is impossible. Ambient vibration has become an important source of excitation for SHM. Bernal (2002b) presented an initial study to extend the Damage Location Vector (DLV) method (Bernal 2002a) to handle the ambient vibration case. In this paper, a new approach that extends the DLV method to tackle the ambient vibration case is proposed. For the case of ambient vibration, no inputs are known, so the flexibility matrix cannot be constructed directly using existing approaches. This problem is circumvented via a modal expansion technique with proposed modifications, allowing the flexibility matrix to be constructed under ambient vibration condition, and then the DLV method to be applied. As a byproduct of the proposed approach, in addition to the location of the damage, an estimate of the damage extent also can be obtained. Appropriate strategies for damage localization are proposed for single and multiple damage scenarios. This paper is organized as follows: First, motivation for flexibility-matrix-based approach is provided and the DLV method is introduced. A description on how to employ the Natural Excitation Technique (NExT) (James et al. 1993) in conjunction with the Eigenvalue Realization Algorithm (ERA) (Juang and Pappa, 1985) to obtain the modal parameters for the ambient vibration case is then presented (Dyke et al. 2000). The modal expansion technique (Lipkins and Vandeurzen, 1987) is then introduced and modifications to select the analytical model for this modal expansion technique are proposed. The flexibility matrix can then be constructed; and, for the single damage scenario, the DLV method can be applied without difficulty. For the multiple damage scenario, a detection approach with iterative searching is provided. Finally, a numerical example that analyzes a truss structure with limited sensors and noisy measurement is included to verify the efficacy of the proposed approach.
2. Problem Formulation and Solution Approach Health monitoring methods based on the flexibility matrix have recently been shown promising. An inverse relationship exists between the flexibility matrix and the square of the modal frequencies; therefore, the flexibility matrix is not sensitive to high frequency modes. This unique characteristic allows the use of a small number of truncated modes to construct a reasonably accurate representation of the flexibility matrix. Toksoy and Aktan (1994) and Pandey and Biswas (1994, 1995) employed the flexibility matrix to detect damage in structures. Most recently, Bernal (2002a) proposed a flexibility-based damage localization method. 2.1 Motivation for Flexibility Matrix-Based Approach Consider the equations of motion of a linear structure M { x·· } + C { x· } + K { x } = { f } (1) The orthogonal property of mode shapes with respect to the mass and stiffness matrix leads to T
M = ϕ Mϕ
and
2
T
K = ϕ Kϕ
(2)
in which M = modal mass matrix; and K = modal stiffness matrix. The square of the modal fre2 quency ω can be expressed in a matrix sense as –1
2
ω = M K
(3)
Combining Eqs. (2) and (3) leads to T
T
2
ϕ Kϕ – ϕ Mϕω = 0
(4)
Therefore, the stiffness matrix can be obtained by T –1
2
–1
–1
2 –1 T
K = ( ϕ ) vω vϕ = Mϕv ω v ϕ M where v = diagonal matrix with the mass-normalized indices on the main diagonal given by T
v = ( ϕ Mϕ )
1⁄2
T
T
= ( ϕm Mm ϕm + ϕu Mu ϕu )
1⁄2
(5)
(6)
In Eq. (6), the subscript m designates the measured DOFs (i.e. sensor locations); and u designates the unmeasured DOFs. –1 From the relationship between the stiffness and flexibility matrix F = K , the flexibility matrix is derived from Eq. (5) –1
–1 T
–2
F = ( ϕv )ω ( ϕv ) and the flexibility matrix at the sensor locations can be written as –1
(7)
–1 T
–2
F m = ( ϕ m v )ω ( ϕ m v )
(8)
Equations (5) and (7) indicate the different influences of the various frequency modes on the stiffness and flexibility matrices, respectively. The influence of the modes on stiffness matrix K 2 –2 increases with ω , whereas for the flexibility matrix F , the influence decreases with ω . To quantitatively see this effect, consider the 40-DOF planar truss given in Fig. 1, which will be used for the numerical example later in this paper. The truncated stiffness matrix K n , which contains the contribution of the first n modes, can be derived from Eq. (5) and is written as n
Kn =
∑ Mϕj vj
–2
2 T
ω j ϕj M ,
n≤ N
(9)
j=1
10 x 3 m = 30 m 12
13
14
15
16
17
18
19
20
21
1
2
3
4
5
6
7
8
9
10
22
4.5 m
y x
Element 6: node 6–7 Element 18: node 19–20 Element 22: node 2–14 Element 32: node 2–13
11
Accelerometers in y direction
White noise excitations
Fig. 1. 44-bar planar truss
3
Accelerometer in x direction of node 11
in which N = number of DOFs of the structure. Two error norms are defined here to measure the difference between the exact and the truncated stiffness matrices. The first is the 2-norm given by the maximum singular value, designated by s , of the difference matrix K – Kn
= s { K – Kn }
2
(10)
The second norm calculated is the Frobenius norm N
K – Kn
F
∑ ∑
=
1⁄2
N
(K –
2 K n ) ij
(11)
i=1 j=1
Figure 2 shows these two error norms, which are normalized to have a value of one for n = 0 . As can be seen here, nearly all of the modes are required to obtain a reasonably accurate representation of the stiffness matrix. Because experimentally obtaining the higher modes of a structure is often quite challenging, stiffness matrix-based health monitoring strategies may be difficult to implement in practice. Similarly, the truncated flexibility matrix F n can be derived from Eq. (7) as n
Fn =
∑ ϕj vj
–2
–2 T
ω j ϕj ,
n≤ N
(12)
j=1
The counterparts to the error norms defined in Eqs. (10) and (11) are then F – Fn
2
= s { F – Fn } N
F – Fn
F
(13) 1⁄2
N
∑ ∑ ( F – Fn )ij 2
=
(14)
i = 1j = 1
These two norms are shown in Fig. 3, again normalized to have a value of one for n = 0 . As can be seen, only a few modes are required to achieve reasonable accuracy in the flexibility matrix. This fact indicates significant potential for health monitoring and damage detection approaches
1.2
1.2
2-Norm
1 0.8
K-K n K 2
2
K-K n F K F
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
Frobenius Norm
1
0 0
10
20
30
40
0
Number of Modes
10
20
30
Number of Modes
Fig. 2. Normalized error in truncated stiffness matrix
4
40
1
1
Frobenius Norm
2-Norm 0.8
F-Fn F F F
F-Fn 2 F 2
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
10
20
30
40
0
10
Number of Modes
20
30
40
Number of Modes
Fig. 3. Normalized error in truncated flexibility matrix based on the flexibility matrix.
2.2 The DLV Method The Damage Location Vector (DLV) method, first proposed by Bernal (2002a), is based upon determination of a special set of vectors, the so-called damage location vectors (DLVs). The DLVs have the property that when they are applied to the structure as static forces at the sensor locations, no stress is induced in the damaged elements. This unique characteristic can be employed to localize structural damage. For a linear structure, the flexibility matrices at sensor locations are constructed from measured data for the case before and after damage and denoted as F u and F d , respectively (note that for conciseness, the subscript m used in Eq. (8) to indicate that the matrices correspond to the measured DOFs will be omitted in the sequel). First, all of the linear-independent load vectors L are collected, which satisfy the following relationship Fd L = Fu L
or
F ∆ L = ( F d – F u )L = 0
(15)
This equation implies that the load vectors L produce the same displacements at the sensor locations before and after damage. From the definition, the DLVs are seen to also satisfy Eq. (15); that is, because the DLVs induce no stress in the damaged structural elements, the damage of those elements does not affect the displacements at the sensor locations. Therefore, the DLVs are indeed the vectors in L . To calculate L , the singular value decomposition (SVD) is empoyed. The SVD of the matrix F ∆ leads to T
F ∆ = USV = U 1 U 0
S1 0 0 0
T
V1 V0
(16)
or, equivalently F∆ V1 F∆ V0 = U1 S1 0
(17)
F∆ V0 = 0
(18)
From Eq. (17), one obtains
5
Eqs. (15) and (18) indicate that L = V 0 , i.e., DLVs can be obtained from the SVD of the difference matrix F ∆ . In Eq. (16), because of noise and computational errors, the singular values corresponding to V 0 are generally not exactly zero. To select the DLVs from the SVD of the matrix F ∆ , an index svn was proposed by Bernal (2002a) and defined as 2
si ci ------------------------2 max ( s k c k )
svn i =
(19)
k
in which s i = the ith singular value of the matrix F ∆ ; c i = constant that is used to normalize the maximum stress in the structural element, which is induced by the static load c i V i , to have a value of one; and V i is the right singular vector of F ∆ . Each of the DLVs is then applied to an undamaged analytical model of the structure, and the stress in each structural element is calculated. If an element has zero normalized accumulative stress σ j , then this element is a possible candidate of damage. The normalized accumulative stress for the jth element is defined as σj σ j = ---------------------max ( σ k )
(20)
k
where n
σj =
σ ij
∑ ----------------------max ( σ ik )
i=1
(21)
k
In Eq. (21), σ ij = stress in the jth element induced by the ith DLV; σ j = cumulative stress in the jth element. In practice, the normalized accumulative stresses induced by DLVs in the damaged elements may not be exactly zero due to noise and uncertainties. Reasonable thresholds should be chosen to select the damaged elements.
2.3 NExT and ERA To apply the DLV method to locate the damage, the flexibility matrix needs to be constructed first from the measured data. The flexibility matrix at the sensor locations can be constructed from the modal parameters. In this paper, these modal parameters are obtained using the Natural Excitation Technique (NExT) in conjunction with the Eigenvalue Realization Algorithm (ERA). For an underdamped linear structure, James et al. (1993) showed that under stationary white noise excitations, the cross-correlation function of any two outputs x i and x j has the following form N
R ij ( T ) =
∑ r=1
r
r
ϕ iA j ------------- exp ( – ζ r ω rn T ) sin ( ω rd T + θ r ) r r m ωd
(22)
Equation (22) indicates that the auto- or cross-correlation function has the same form as the free response of a linear structure. This characteristic of the correlation function allows application of time-domain system realization algorithms to estimate the modal frequencies and modal shapes, which is the basic idea behind the NExT (James, et al. 1993). Dyke et al. (2000) also pointed out
6
that because the modal shapes of the acceleration are the same as those of the displacement, acceleration measurements can be used by the NExT to obtain the displacement mode shapes. To apply the NExT approach, the first step is to measure structural responses. The auto- and cross-spectral density functions of these responses can then be calculated. Subsequently, the inverse Fourier transform is applied to obtain the auto- and cross-correlation functions, which are in turn used by the ERA to get the modal parameters. To obtain the desired modal parameters, the reference output selected in calculating correlation functions should include information on all modes of interest (Dyke et al. 2000).
2.4 Modal Expansion for Single Damage Case In practice, the number of measured DOFs are usually smaller than the total number of DOFs of the structure; and not all modes are identified. As can be seen from Eqs. (6) and (7), the matrix v needs to be determined to construct the flexibility matrix. Because calculation of the matrix v requires information regarding both the measured and unmeasured DOFs, a modal expansion technique is required to obtain the mode shape information for the unmeasured DOFs. Bernal (2000) has shown that when the input excitation is measured, and there is at least one co-located sensor and actuator pair, the flexibility matrix at sensor locations can be determined solely from modal information at sensor locations. However, this is not the case when inputs are unknown (e.g., the ambient vibration case). The modal expansion method suggested by Lipkins and Vandeurzen (1987) is applied herein. In this approach, an analytical model representing the structure is first developed. The equation employed for modal expansion then can be written as A
[ ϕm ]n × l
E
[ ϕm ]n × p
=
A [ ϕu ]( N – n ) × l
E [ ϕu ]( N – n ) × p
λl × p
(23)
In Eq. (23), subscripts outside of the brackets denote the dimension of the matrices. For conciseness, these subscripts will not be shown again in the sequel. Equivalent to Eq. (23), we have E
A
E
[ ϕ m ] = [ ϕ m ]λ
A
[ ϕ u ] = [ ϕ u ]λ
A
(24) E
in which ϕ = mode shapes obtained from the analytical model; ϕ = experimental mode shapes; N = number of DOFs of the structure; n = number of measured DOFs (i.e., number of sensors); p = number of identified mode shapes; l = number of analytical mode shapes for modal expansion; and λ = coefficient matrix. As long as n ≥ l , the coefficient matrix λ can be obtained from Eq. (24) in a least-squares sense as A T
A
–1
A T
E
λ = ( [ ϕm ] [ ϕm ] ) [ ϕm ] [ ϕm ] E
(25)
Mode shapes at unmeasured DOFs, i.e., [ ϕ u ] , can then easily be computed from Eq. (24). Subsequently, the flexibility matrix at the sensor locations can be constructed by Eq. (8). A In this paper, the number of analytical mode shapes [ ϕ ] is set equal to the number of identiE fied mode shapes [ ϕ ] ; that is to say, l = p . From Eq. (23), we see that experimental mode shapes are presumed as a linear combination of analytical mode shapes. One way to select these analytical mode shapes for a given analytical model is based on the Modal Assurance Criterion (MAC), suggested by Ewins (1985), which is defined as
7
E T
A
2
{ ϕ } i{ ϕ } j MAC ( { ϕ } i, { ϕ } j ) = -----------------------------------------------------------------------E T E A T A { ϕ } i {ϕ } i { ϕ } j { ϕ } j E
A
(26)
Analytical mode shapes with a MAC value close to 1.0 will be selected. Then, the Total Modal Assurance Criterion (TMAC) can be used to determine the analytical model, which yields the analytical mode shapes in Eq. (23) p
TMAC =
∏ MAC ( { ϕm } i, { ϕm } i ) E
A
(27)
i=1
The key question then lies in determining exactly how to employ the TMAC to select the analytical model, denoted as the damaged analytical model (DAM). First, the undamaged analytical model is constructed. Then the undamaged analytical model with one element damaged will be selected as the DAM. There are two steps to follow in finally choosing the DAM. The first step attempts to determine which element, if damaged, may generate the DAM. In this step, each element is damaged to a few different damage extents (for example, 8 evenly-distributed damage extents). Results are compared, and the analytical model (e.g., the ith element is damaged) with the highest TMAC value is chosen. In the second step, a wide range of damage extents (for example, 100 damage extents) is evaluated for the ith element alone to choose the damage extent corresponding to the highest TMAC. After these two steps, i.e., selecting the damage element and then the damage extent, the DAM is fixed. Mode shapes at unmeasured DOFs then can be calculated by Eq. (24). The flexibility matrix at the sensor locations can be obtained easily from Eq. (8). After the pre- and post-damage flexibility matrices are obtained, the DLV method can be applied to locate the damaged element in the structure. Note, however, that if the damaged elements indicated by the DLV method do not include the damage element selected in the DAM, the DAM corresponding to the next highest TMAC value then must be selected. This step is important, as the DAM that carries the correct damaged element does not necessarily produce the highest TMAC value because of noise and uncertainties. Additionally, TMAC values reflect only changes in mode shapes, while the DLV method includes both frequency and mode shape information through the flexibility matrix. Also, assuming the analytical model is reasonably accurate, if the DAM includes the correct damaged element, then the DLV method will indicate the correct damaged element. In summary, the damage element selected in the DAM is detected as the real damage location only when it is also indicated by the DLV method.
2.5 Multiple Damage Case At this point, the proposed approach identifies only one damaged element in the structure. For the multiple damage case, iterative searching is necessary. In this case, one of the damaged elements is identified based on the approach outlined above. Then the identified DAM is denoted as the new baseline model (undamaged analytical model), and the above approach is repeated to detect the next damaged element. The remaining problem lies in determining when to stop this iterative searching. An idea proposed here is based on the fact that the DAM with the same number of damaged elements as the damaged structure should have more information than others. An index, so-called Averaged Highest TMAC value (AHTMAC) is proposed here as the flag
8
NE
1 AHTMAC = -----NE
∑ HTMACj
(28)
j=1
in which HTMAC j = the highest TMAC value observed when the jth element of DAM is damaged; and N E = total number of elements in the DAM. If the index AHTMAC for a new search is smaller than the previous one, the iterative searching should stop.
3. Numerical Example The proposed detection approach is demonstrated using a planar truss structure shown in Fig. 1 (note that a similar truss was considered by Bernal (2002a)). In this numerical example, limited sensors are employed (9 sensors compared with 40 DOFs), and a 5% RMS noise is added to the measured outputs. A Matlab finite element model consisting of 44 bars, 22 nodes, and 40 DOFs is developed. Elements are connected at pinned joints, each having two DOFs. The simulated excitations are independent band-limited white noises applied in the y direction at nodes 5 and 7. Outputs (9 accelerations) are collected in the y direction at nodes [2, 3, 4, 5, 7, 8, 9, 10] and in the x direction at node 11. The acceleration in the y direction of node 4 is selected as a reference output to obtain the spectral density functions when employing the NExT. To simulate the ambient vibration case, inputs are not collected. Two categories of damages are considered: • Case 1: Single damage scenario – 15% stiffness reduction at a single element. • Case 2: Multiple damage scenario – 30% stiffness reduction in elements 18 and 22. First, the modal parameters are obtained using the NExT and the ERA. Here, the first 10 natural frequencies are identified when element 6 has 0, 15%, and 30% damage. These results are compared with the exact solution from the finite element model in Table 1, which shows that this Table 1: Comparison of identified natural frequencies when element 6 is damaged Exact Results Undamaged 12.4865 24.8554 29.2125 44.9624 68.0143 73.2656 80.6575 90.1743 96.7830 105.5592
15%a 12.4321 24.7833 29.1837 44.9170 67.5132 73.1240 80.3934 90.1743 96.5721 105.5525
Results from NExT & ERA 30%b 12.3556 24.6824 29.1451 44.8527 66.7746 72.9577 80.0775 90.1742 96.3080 105.5438
Undamaged 12.4846 *c 29.1959 44.9484 67.9838 73.2495 80.6690 90.1395 96.8070 105.5830
a 15% stiffness reduction. b 30% stiffness reduction. c Second mode is missed in identification.
9
15%a 12.4284 *c 29.1655 44.9020 67.5134 73.0887 80.4158 90.1394 96.5800 105.5749
30%b 12.3576 *c 29.1273 44.8381 66.7963 72.9216 80.0799 90.1392 96.2978 105.5651
0
2nd Mode Magnitude (dB)
-50
-100
-150
Acceleration in x direction of node 11 Acceleration in y direction of node 5
-200 0
20
40
60
80
100
120
Frequency (Hz)
Fig. 4. Power spectral density functions identification approach is quite accurate. As noted in Table 1, the second mode is not identified. This result is because the contribution of the second mode to the y direction response is too small to be captured. Figure 4 clearly illustrates this phenomenon, in which the power spectral density function of the acceleration in the x direction (without noise) has a clear second mode (24.8554 Hz), while the result for the acceleration in the y direction does not.
3.1 Single Damage Scenario: 15% Stiffness Reduction Once the modal parameters are obtained, the TMAC is applied to find the DAM for use in modal expansion. The damage detection results when using the highest TMAC value are listed in Table 2 for damage in various element. Due to the limited space, only results for damage of selected elements are listed. Results for this damage level in other elements were similar to these. The table shows that, in this single damage case, most of the selected DAMs corresponding to the Table 2: Single damage case: detection results for selected damaged elements (using the highest TMAC value) The DAM Real Damaged Damage Element Element # Extent 1 1 0.15 2 17 0.06 3 3 0.19 4 4 0.24 5 17 0.06 6 6 0.13 7 7 0.13 8 8 0.10 9 9 0.16 10 10 0.17
The DAM Real The DLV Damaged Damage methoda Element Element # Extent yes 31 31 0.17 no 32 17 0.06 yes 33 33 0.23 yes 34 34 0.15 no 35 35 0.16 yes 36 36 0.20 yes 37 37 0.12 yes 38 38 0.17 yes 39 39 0.14 yes 40 40 0.14
The DLV methoda yes no yes yes yes yes yes yes yes yes
a Do the elements indicated as possible damage locations by the DLV method include the damaged element selected in the DAM?
10
Normalized Accumulative Stress σj
1 0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
35
40
45
35
40
45
35
40
45
Element Number
Normalized Accumulative Stress σj
Normalized Accumulative Stress σj
Fig. 5. Case 1: element 6 damaged Using the highest TMAC value 1
(a)
0.8 0.6 0.4 0.2 0
0
5
10
15
20
25
30
Element Number
Using the third highest TMAC value 1
(b) 0.8 0.6 0.4 0.2 0
0
5
10
15
20
25
30
Element Number
Fig. 6. Case 1: element 32 damaged highest TMAC value have a damaged element that is consistent with the candidate damage locations indicated by the DLV method; that is, the method correctly identifies the damage location. However, some of the selected DAMs are not consistent with damage locations predicted by the DLV method. An iterative procedure, i.e., re-selecting the DAM that corresponds to the next highest TMAC value and repeating the comparison, is necessary. This procedure is continued until the damage element in the DAM corresponds to the damage element indicated by the DLV method. After a few iterations, the correct damaged element is identified. Figure 5 provides the normalized accumulative stress indicated by the DLVs in each element for the case when element 6 is damaged. These results clearly indicates that element 6 is the dam-
11
Normalized Accumulative Stress σj Normalized Accumulative Stress σj
First iteration: damaged element is #18 1
(a) 0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
35
40
45
35
40
45
Element Number Second iteration: damaged element is #22 1
(b) 0.8 0.6 0.4 0.2 0
0
5
10
15
20
25
30
Element Number
Fig. 7. Case 2: elements 18 and 22 damaged aged location. In this case, the DAM corresponding to the highest TMAC value is consistent with the candidate damage locations indicated by the DLV method. In Table 2, the case with damage in elements 2, 5 or 32 represents the more general situation; that is, the damaged element selected in the DAM corresponding to the highest TMAC value is not consistent with the damage locations indicated by the DLV method. However, following the iterative procedure described previously, the damaged element in the structure, in the DAM, and indicated by the DLV method coincide. Results for the case with damage in element 32 are shown in Fig. 6. Figure 6a displays the results with the DAM corresponding to the highest TMAC value. The DAM indicates a 6% stiffness reduction in element 17. All normalized accumulative stresses are relatively large, and there is no definitive indication on which element is damaged. Figure 6b displays the converged results. In this case, the DAM corresponding the third-highest TMAC value has a 19% stiffness reduction in element 32. Figure 6b shows that element 32 has a relatively small normalized accumulative stress. The damaged element is predicted correctly.
3.2 Case 2: Multiple Damage Scenario In this case, elements 18 and 22 are both given 30% stiffness reduction. As discussed previously, for the multiple damage scenario, in each step one damaged element will be determined. First, element 18 is identified as shown in Fig. 7a. The DAM indicates element 18 has 31% stiffness reduction. After element 18 is identified, the model with element 18 damaged is designated as the new baseline model. Then, this new baseline model can be used to detect the damage in element 22. The results for element 22 are shown in Fig. 7b, in which the normalized accumulative
12
stress of element 22 is reasonably small. In this case, the DAM indicates a 31% stiffness reduction of element 22. Therefore, the multiple damage case is also able to be handled by the proposed approach. Note that, although element 33 in Fig. 7 also has a small stress, it is only the possible damage location indicated by the DLV method; this is because the damaged element selected in the DAM is element 22 but not element 33.
4. Conclusions An approach that extends the DLV method to include the ambient vibration case has been presented in this paper. By applying the modal expansion method with the proposed approach for selecting the analytical model, the need to measure the input excitation to construct the flexibility matrix has been eliminated. The DLV method could then be applied to locate the damage in the structure. Appropriate strategies were proposed for both single and multiple damage scenarios. As a by-product of the proposed approach, the possible damage extent of the damage element were also estimated. The numerical examples indicate that the results are quite reasonable for damage as small as a 15% stiffness reduction on a local element. In these numerical studies, the approach works well for both single and multiple damage cases. Experimental verification of this proposed approach is currently underway. References Alvin, K.F. and Park, K.C. (1994), “Second-order Structural Identification Procedure via StateSpace-Based System Identification,” AIAA Journal, 32(2), pp. 397–406. Bernal, D. (2000), “Extracting Flexibility Matrices from State-Space Realizations,” COST F3 Conference, Madrid, Spain, pp. 127–135. Bernal, D. (2002a), “Load Vectors for Damage Localization,” Journal of Engineering Mechanics, 128(1), pp. 7–14. Bernal, D. and Gunes, B. (2002b), “Damage localization in output-only systems: A flexibility based approach,” IMAC-XX, Los Angeles, California, pp. 1185–1191. Cha, P.D. and Tuck-Lee, J.P. (2000), “Updating Structural System Parameters Using Frequency Response Data,” Journal of Engineering Mechanics, 126(12), pp. 1240–1246. Dyke, S.J., Caicedo, J.M. and Johnson, E.A. (2000), “Monitoring a Benchmark Structure for Damage Identification,” Proceedings of the Engineering Mechanics Specialty Conference, Austin, Texas, May 21-24. Ewins, D.J. (1985), Modal Testing: Theory and Practice, New York, John Wiley. James, G.H. III, Carne, T.G., Lauffer, J.P. (1993), “The Natural Excitation Technique (NExT) for Modal Parameter Extraction from Operating Wind Turbines,” SAND92-1666, UC-261, Sandia National Laboratories. Juang, J.N. and Pappa, R.S. (1985), “An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction,” Journal of Guidance Control and Dynamics, 8, pp. 620–627. Lipkins, J. and Vandeurzen, U. (1987), “The Use of Smoothing Techniques for Structural Modification Applications,” Proceedings of 12 International Seminar on Modal Analysis, S1–3.
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Pandey, A.K. and Biswas, M. (1994), “Damage Detection in Structures Using Changes in Flexibility,” Journal of Sound and Vibration, 169(1), pp. 3–17. Pandey, A.K. and Biswas, M. (1995), “Damage Diagnosis of Truss Structures by Estimation of Flexibility Change,” The International Journal of Analytical and Experimental Modal Analysis, 10(2), pp. 104–117. Toksoy, T. and Aktan, A.E. (1994), “Bridge-Condition Assessment by Modal Flexibility,” Experimental Mechanics, 34, pp. 271–278. Vandiver, J.K. (1975), “Detection of Structural Failure on Fixed Platforms by Measurement of Dynamic Response,” Proceedings of the 7th Annual Offshore Technology Conference, pp. 243–252. West, W.M. (1984), “Illustration of the Use of Modal Assurance Criterion to Detect Structural Changes in an Orbiter Test Specimen,” Proceedings Air Force Conference on Aircraft Structural Integrity, pp. 1–6.
Vitae Yong Gao was born in Zhejiang province, China. He obtained the Bachelor degree in Civil Engineering from Zhejiang University in 1996. Mr. Gao was then enrolled in University of Macau, and received the Master degree in Civil Engineering in 1999. Currently, he is a Ph.D. student in structural engineering at the University of Notre Dame. B.F. Spencer, Jr. received his B.S. (1981) in mechanical engineering from the University of Missouri-Rolla and his M.S. (1983) and Ph.D. (1985) in theoretical and applied mechanics from the University of Illinois at Urbana-Champaign. He was a member of the faculty at the University of Notre Dame for 17 years, before taking a position as the Nathan M. Newmark Professor of Civil Engineering at the University of Illinois at Urbana-Champaign in 2002. His research has been primarily in the areas of stochastic fatigue, computational stochastic mechanics, civil engineering applications of structural control and health monitoring. He has published numerous technical papers/reports, including two books. His research was cited by the Science Coalition in 1998 Great Advances in Scientific Discovery and he is the co-recipient of the American Society of Civil Engineers 1999 Norman Medal.
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