Structural damage identification based on residual force vector and response sensitivity analysis

Abstract

A two-stage approach based on residual force vector and response sensitivity analysis is proposed for structural damage identification in this study. Local damage is represented by a perturbation in the elemental stiffness parameter of the structural finite element model. Primarily, the difference between the virtual residual force vector of an intact and damaged structure is used to localize the damage. Then, a response sensitivity-based method is adopted to identify the perturbation of the stiffness parameter (i.e., the extent of the local damage) from the measured dynamic responses. Numerical simulations are conducted to identify both single and multiple structural damages under sinusoidal or impulsive excitation. The proposed approach gives satisfactory results with artificial measurement noise.

Keywords

Damage identification frequency domain model updating residual force vector response sensitivity time domain

1. Introduction

Many structures are subject to damage because of, for example, impact, earthquake, fatigue and corrosion. Because of the many requirements for high-quality and reliable structures, the vibration-based damage identification methods have become major topics of research over the past few decades. The main idea of vibration-based damage identification methods is that the modal parameters or vibration characteristics (such as natural frequency, mode shape, modal damping etc.) are related to the structural physical properties (mass, damping and stiffness). Therefore, structural damage alters the physical properties, and the changes in the physical properties have obvious influences on the modal characteristics or dynamic responses of the structure.

Modal parameters have been used extensively to identify the location and extent of local structural damage. Most of the existing damage identification methods are mainly based on the changes in natural frequencies (Chinchalkar, 2001; Kim and Stubbs, 2003; Urgessa, 2011), mode shapes (Rizos et al., 1990; Pandey et al., 1991; Ratcliffe, 1997), modal flexibility (Pandey and Biswas, 1994; Wu and Law, 2004; Yang, 2011) or frequency response functions (Maia et al., 2003; Chatterjee, 2010).

The residual force vectors, derived from the natural frequencies and mode shapes of the intact and damaged structure, have been used to identify structural damages. Ricles and Kosmatka (1992) used residual modal force vectors to locate possible damaged regions, and then determined the extents of stiffness variations using a weighted sensitivity analysis based on a first-order Taylor series expansion of mode shapes and eigenvalues of an undamped structural system. Zimmerman and Kaouk (1994) made use of a minimum rank update theory to detect structural damages. Damage locations are determined firstly by the damage vectors and the damage extents are assessed by a minimum rank update theory. Li and Smith (1995) used residual force vectors directly to solve the parameter changes of damaged elements, but again they used experimentally undamped mode shapes and natural frequencies in the damage estimation. Liu and Yang (2006) made use of an original finite element model (FEM) and a subset of measured eigenvalues and eigenvectors to determine the number of damaged elements and localized the damage elements by the damage localization matrix. After that, the damage extent could be easily obtained. In general, the residual force vector is used to locate damage, and other methods of damage quantification is used to assess the extent of the damage. However, the damage estimation scheme mentioned above is valid only for undamped structures.

Recently, the structural damage identification in the time domain has attracted the interest of many researchers (Hjelmstad et al., 1995; Ge and Soong, 1998). Choi and Stubbs (2004) utilized the damage indices, which are formulated by a mean strain energy in a specified time period, to identify the locations and corresponding extents of the possible damage. Cattarius and Inman (1997) used time histories of the vibration response of the structure for identifying the presence of damage. Lu and Law (2007) developed a response sensitivity-based FEM updating method to identify structural damages from the measured dynamic responses. Although this only needs a few measurement points, it still provides a high damage identification accuracy as it takes advantage of the plentiful time history data. However, this approach requires long computional time when the number of updating parameters is large.

To overcome the problems mentioned above, in this study an improved approach based on residual force vector and response sensitivity analysis is presented to identify structural damage. Damage location is first determined by the residual force vector. Then the extent of damage is assessed by using a response sensitivity-based model updating method. As the residual force vector localizes the damaged sites beforehand, it reduces the number of unknowns in the model updating, which increases the efficiency of calculation and also improves the accuracy. It is also helpful for selecting the measurement points that can be more close to the damaged elements. Three numerical examples are studied to show the effectiveness and accuracy of the proposed method, including a planar truss, a cantilevered plate and a multiple-span continuous beam. Both single damage and multiple damages can be identified successfully by using the proposed method and it is insensitive to artificial measurement noise.

2. Theory

2.1. Brief introduction to the virtual residual force vector method

The equation of motion for a general FEM of a linear elastic undamaged structural dynamic system can be expressed as

M u ·· + C u \cdot + Ku = BF (t),

(1)

where K and M are the global stiffness and mass matrices of the undamaged structural model, respectively; C is the Rayleigh damping matrix, which is of the form

C = a 1 M + a 2 K

in this study.

u ··

u \cdot

and u are the acceleration, velocity, and displacement response vectors of the structure, respectively.

F (t)

is a vector of applied forces with matrix B mapping these forces to the associated degrees of freedom (d.f.) of the structure.

When neglecting the damping term and the external excitation force equals to zero, equation (1) will lead to the equation of undamped free vibration:

M u ·· + Ku = 0 .

(2)

And the solution of equation (2) can be obtained from the eigenvalue equation, as follows:

K ϕ i = λ i M ϕ i,

(3)

where

λ i

is the eigenvalue of the ith mode; and

ϕ i

is the vibration eigenvector of the ith mode.

In the case of damage detection, it usually assumes that there is a perturbation on the stiffness property as damage occurs while the mass parameter remains unchanged. Therefore, equations (1) and (2) can be expressed as follows:

M u ·· + C d u \cdot + K d u = BF (t),

(4)

K d ϕ d j = λ d j M ϕ d j,

(5)

where

K d

is the global stiffness matrix of the damaged structure;

C d

is the global damping matrix with

C d = a 1 M + a 2 K d

λ d j

is the jth eigenvalue of the damaged structure and

ϕ d j

is the corresponding eigenvector. The dynamic responses of the structure can be obtained by direct integration method.

The change of structural stiffness matrix $Δ K$ introduced by the structural damage can be modeled by a perturbation in the stiffness parameter of the structure in this study. So the global stiffness matrix $K d$ can be expressed as

K d = K - Δ K .

(6)

Substituting equation (6) into equation (5), we have

(K - λ d j M) ϕ d j = Δ K ϕ d j .

(7)

The residual force vector (RFV) (Zimmerman and Kaouk, 1994; Yang and Liu 2007) $f j$ from the jth mode of the damaged structure is defined as

f j = Δ K ϕ d j = (K - λ d j M) ϕ d j .

(8)

And equation (8) can also be expressed as

{f 1 j f 2 j : f nj} = [Δ k_{1}^{T} Δ k_{2}^{T} : Δ k_{n}^{T}] ϕ d j,

(9)

where

Δ k i (i = 1, 2, \dots, n)

is the stiffness change of the ith d.f. and n is the number of system d.f.

Δ k i

is non-zero only if the corresponding element is damaged. Thus, the residual force vector

f j

from equation (8) will have non-zero values only in the damaged elements. Furthermore, we can identify the damaged elements according to the relation between the element number and the d.f. number.

It should be pointed out that the residual force obtained from a single mode shape may miss some damaged element. To obtain a more robust damage localization result, the virtual residual force vector $f ¯$ (VRFV) is defined by making use of the RFV of the first p modes as (Yang and Liu, 2007)

f ¯ = {f 1 f 2 ¯ : f n ¯ ¯},

(10)

where

f n ¯ = \sum_{j = 1}^{p} \frac{| f nj |}{max | f j |} / p

2.2. Response sensitivity with respect to damage parameters in time domain

As the local damage is represented by a perturbation of the elemental stiffness parameter $Δ α$ , the response sensitivity with respect to this parameter is obtained from equation (11):

M \frac{\partial u ··}{\partial α i} + C d \frac{\partial u \cdot}{\partial α i} + K d \frac{\partial u}{\partial α i} = - \frac{\partial K d}{\partial α i} u - a 2 \frac{\partial K d}{\partial α i} u \cdot (i = 1, 2, \dots, m),

(11)

where

\frac{\partial u}{\partial α i}

\frac{\partial u \cdot}{\partial α i}

\frac{\partial u ··}{\partial α i}

are vectors of the displacement, velocity, and acceleration sensitivities with respect to the stiffness parameter of the ith element, and m is the number of damaged element.

Since u and $u \cdot$ have been obtained from equation (4), the right-hand-side of equation (11) can be considered as an equivalent forcing function, and equation (11) is of the same form as equation (4). Therefore, the sensitivities $\frac{\partial u}{\partial α i}$ , $\frac{\partial u \cdot}{\partial α i}$ , $\frac{\partial u ··}{\partial α i}$ can also be obtained by direct integration method.

2.3. Quantifying the damage extent from response sensitivity-based model updating

The objective function for the model updating is defined as minimizing the residual between the measured and calculated structural dynamic responses, for instance, the acceleration responses

min g (α) = \frac{1}{2} \sum_{j = 1}^{β} \sum_{i = 1}^{μ t} (~ ·· u ij - u ·· ij) T W (~ ·· u ij - u ·· ij),

(12)

where β is the number of measurement locations,

μ t

is the number of time instances of the measured data.

u ·· ij

is the vector of calculated acceleration response of the structure and

~ ·· u ij

is the vector of measured acceleration response. W is the weighting matrix.

Then the penalty function method (Friswell and Mottershead, 1995) is generally used for modal sensitivity with a truncated Taylor series expansion of the dynamic responses in terms of the parameter α, which are used to derive the sensitivity-based formulation. For finding the vector $α$ such that the calculated responses best fit the measured acceleration responses, the identification problem can be expressed as follows:

Q u ·· = ~ ·· u,

(13)

where Q is a selection matrix with elements of zeros or ones, matching the d.f. corresponding to the measured response components. The vector of calculated acceleration

u ··

can be obtained from equation (1). Theoretically,

Q u ··

(eg.

u ·· cal

) should be equal to

~ ·· u

. However the difference between

u ·· cal

and

~ ·· u

always exists as damage occurs.

Let

δ u ·· = ~ ·· u - Q u ·· = ~ ·· u - u ·· cal,

(14)

where

δ u ··

is the error vector between the measured and calculated accelerations. By making use of the penalty function method, we have

δ u ·· = S Acc Δ α,

(15)

where

S Acc

is the two-dimensional acceleration response sensitivity matrix. Equation (14) can be solved by a simple least-squares method as follows:

Δ α = [S_{Acc}^{T} S Acc] - 1 S_{Acc}^{T} δ u ··,

(16)

α j + 1 = α j + [S Acc j T S Acc j] - 1 S Acc j T (~ ·· u - u ·· cal),

(17)

where the subscript j indicates the iteration number at which the sensitivity matrix is computed.

α j + 1

is the updated value of the parameter, and comparing it with the original parameter the damage extents in the damaged elements can be found.

Like many other inverse problems, equation (16) is an ill-conditioned problem. In order to provide bounds to the solution, the damped least-squares method (DLS) (Tikhonov, 1963) and singular-value decomposition is used in the pseudo-inverse calculation. Equation (16) can be written in the following form using the DLS method

Δ α = [S_{Acc}^{T} S Acc + γ I] - 1 S_{Acc}^{T} δ u ··,

(18)

α j + 1 = α j + [S Acc j T S Acc j + γ I] - 1 S Acc j T (~ ·· u - u ·· cal),

(19)

where γ is the non-negative damping coefficient governing the participation of least-squares error in the solution. The L-curve (Hansen, 1992) method is used in this paper to obtain the non-negative damping coefficient γ.

2.4. Effect of artificial measurement noise

The numerical examples will study the effect of artificial measurement noise on the identification, because the measured data are usually contaminated with the measurement noise Numerically, white noise is added to the calculated acceleration responses to simulate the noisy data with

u ·· \land = u ·· + E p * N noise * σ (u ··),

(20)

where

E p

is the percentage noise level,

N noise

is a standard normal distribution with zero mean and unit standard deviation,

σ (u ··)

is the standard deviation of the calculated acceleration response. The effect of artificial measurement noise on the identified extent results is studied.

The relative error is defined as

Res . = \frac{| p i - p t |}{p t} \times 100 %

(21)

where

p i

and

p t

are identified and true values, respectively.

3. Numerical simulations

3.1. A planar truss structure

As shown in Figure 1, a two-dimensional truss structure is studied as an example to evaluate the effectiveness of the proposed method. The physical parameters of the structure are as follows: mass density $ρ = 7.8 \times 103 kg / m 3$ , Young’s modulus $E = 200 Gpa$ , length of each rod is $L = 1 m$ with cross-sectional area $A = 0.004 m 2$ . The first four modes are used to locate the damage site and the first four natural frequencies are 298.6 Hz, 365.0 Hz, 602.0 Hz, and 748.4 Hz. Local damage is introduced as a reduction in the elemental Young’s modulus, but the other properties remain unchanged.

To obtain the dynamic responses for damage quantification, a sinusoidal excitation force is used to excite the structure. The sinusoidal force is taken as $F (t) = 1500 \sin (20 π t)$ N, which is applied at node 5 along the global positive y-direction. The damping ratios of the first two modes are both assumed as equal to 0.01. The sampling rate is equal to 200 Hz. The response data of the first 10 seconds are used in the identification and the acceleration measurements are set in node 2, 3 and 4.

3.2. Identification of single damage

The first damage case assumes that a single damage occurs in the fifth element with 10% loss of Young’s modulus. For the first damage case, the VRFV values of the 18 d.f. are shown in Figure 2, where nodes 3 and 4 are the nodes of the most suspected damaged element because their VRFV values are much larger than others. According to Figure 1, the second, third, fourth, fifth, sixth, seventh, and eighth elements are the corresponding elements associated with nodes 3 and 4. Knowing the possible locations of damage make it more accurate and easier to identify the extent of damage. In the case of no artificial measurement noise and 10% noise level, the identified reduction in the fifth element is shown in Figure 3. Compared with the true value, the identified reduction in the fifth element is 10% with no artificial measurement noise, and even with 10% noise level, the relative error of the identified result is 0.03%. This shows the accuracy and efficiency of the proposed method.

3.3. Identification of multiple damages

The second damage case assumes that the fourth, fifth and sixth elements are damaged with 20% reduction in the Young’s modulus of each element. This is the same example studied by Liu and Yang (2006). Thus, a comparison can be made for the results from the proposed method and those from the residual force vector approach. In this case, the VRFV values for all d.f. are shown in Figure 4. Since each node has two d.f. and the VRFV values at the third, fifth, sixth, seventh, eighth and ninth d.f. are larger than those of others, it is found that their relative nodes 2, 3, 4 and 5 are nodes of the most probably damaged elements. According to Figure 1, elements 1 to 10 are taken as the suspected damaged elements. The extents of these damaged elements are accurately identified as shown in Figure 5. In the level of 5% noise, good identified results in Figure 5 can still be obtained by the proposed method with the identified error of 0.15%, 0.03% and 0.15% of the three damaged elements, respectively, and the identified extent of some undamaged elements are so small that they can be ignored.

The identification errors for different noise levels are listed in Table 1. And scenarios 5 and 6 are the damage identification results of Liu and Yang (2006), which are obtained only using the residual force method. Table 1 reveals that the proposed method can identify multiple damages successfully with high accuracy. The results are still satisfactory for relatively high noise levels. Even with 15% artificial measurement noise the largest relative error is below 0.5%.

Table 1.

Results for damage identification of the planar truss.

Scenarios	Noise	Element No. 4		Element No. 5		Element No. 6
Scenarios	Noise	Red. (%)	Res. (%)	Red. (%)	Res. (%)	Red. (%)	Res. (%)
1	Nil	20.00	0.00	20.00	0.00	20.00	0.00
2	5%	20.15	0.15	19.97	0.03	19.85	0.15
3	10%	20.31	0.31	19.95	0.05	19.70	0.30
4	15%	20.46	0.46	19.93	0.07	19.55	0.45
5	Nil	20.00	0.00	20.00	0.00	20.00	0.00
6	5%	20.30	0.30	20.89	0.89	19.30	0.70

Red.: E reduction; res: relative errors. Scenarios 5 and 6 are the results of Liu and Yang (2006).

3.4. A cantilevered plate

A steel cantilevered plate that has the dimensions of $500 \times 500 \times 50 mm 3$ as shown in Figure 6 is studied in this example. The physical parameters of the plate are: mass density $ρ = 7.8 \times 103 kg / m 3$ , Young’s modulus $E = 210 Gpa$ and Poisson’s ratio $ε = 0.3$ . The FEM of the plate consists of twenty five 4-node Reissner-Mindlin plate elements with three d.f. at each node. The damping ratios for the first two modes are both equal to 0.01. In calculating the dynamic response, the time increment is 0.0005 s and the time duration in the calculation of the dynamic response is taken as 1.0 s.

3.5. Identification of single damage

This case assumes that there a single damage occurs in the fifth element with 5% reduction in the Young’s modulus. An impulsive excitation is assumed to act on the 24th node of the plate, and it is expressed mathematically as

F (t) = {105 (t - 0.02) N (0.02 s \leq t \leq 0.04 s) 105 (0.06 - t) N (0.04 s < t \leq 0.06 s) 0 (t < 0.02 sor t > 0.06 s) .

A sinusoidal excitation is also used to give a comparison on the results from impulsive excitation in damage identification. The sinusoidal force is taken as $F (t) = 15000 \sin (20 π t)$ N, which acts on the same node as the impulsive excitation.

In the first step, differences of the VRFV value of each d.f. are plotted in Figure 7. There are peak values at d.f. of 10, 11, 12, 15, 25, 26, 27, 28, 29, and 30. Each node has three d.f., and d.f. of nodes 1, 7, 13, 19, 25, and 32 are neglected because they lie on the fixed end. It reveals that nodes 5, 6 11 and 12 are the nodes of the most suspected damaged elements. According to Figure 6, the fourth, fifth, ninth and tenth elements are the corresponding damaged elements. Then, the response sensitivities approach is used to identify the perturbation in the Young’s modulus (i.e., the damage extent). Figure 8 shows the identified extent of damage with no artificial measurement noise and 10% artificial measurement noise, respectively. The damage extent has been identified accurately.

Table 2 gives the comparison of the identified results using different excitation forces. It shows that the damage can be identified successfully from both impulsive and sinusoidal excitation. When artificial measurement noise is nil, there is no error of identification for both excitations. It can also be observed that impulsive force can give better identification results than does sinusoidal force when the measurements are contaminated with high level noise. For impulsive excitation, even with 15% artificial measurement noise, the largest relative error is below 0.7%. Increasing measurement points can improve the accuracy of the identification results when the sinusoidal excitation is used.

Table 2.

Results for single damage identification of the cantilevered plate.

	Noise	Element No. 5		Measurement (node number)	Excitation
Scenarios	Noise	Red. (%)	Res. (%)	Measurement (node number)	Excitation
1	Nil	5.00	0.00	11, 12	Impulsive
2	5%	5.23	0.23	11, 12	Impulsive
3	10%	5.45	0.45	11, 12	Impulsive
4	15%	5.67	0.67	11, 12	Impulsive
5	Nil	5.00	0.00	11, 12	Sinusoidal
6	5%	4.65	0.35	11, 12	Sinusoidal
7	10%	4.31	0.69	11, 12	Sinusoidal
8	5%	4.91	0.09	5, 11, 12	Sinusoidal
9	10%	4.81	0.19	5, 11, 12	Sinusoidal

Red.: E reduction; res: relative errors.

3.6. Identification of multiple damages

Three damages are assumed to occur at the fifth, 13th and 25th elements, respectively. The relative damage severities are arbitrarily assumed to be 10%, 5%, and 15% loss in the Young’s modulus, respectively. And the same impulsive excitation as the above case is used. The calculated VRFV values are shown in Figure 9. There are peak values at d.f. of 10∼12, 15, 25∼27, 28∼30, 34∼39, 49∼54 70∼75 and 85∼90. And each node has three d.f. as mentioned above. According to the VRFV values, nodes 5, 6, 11, 12, 15, 16, 21, 22, 29, 30, 35 and 36 are the nodes of the most probably damaged element. And according to Figure 6, the suspected damaged elements are the members related to the nodes mentioned above, which are the 4th, 5th, 7th∼10th, 12th∼14th, 17th∼20th and 24th∼25th elements. After localizing the damaged elements, the damage extents of each element are identified accurately by using the response sensitivity method as shown in Figure 10. Figure 11 shows the identified results of damage extent without localization first by the VRFV method, including two situations (no artificial measurement noise and 10% noise level). It reveals that the identified results are just slightly different from the true values when there is no artificial measurement noise. However, the maximum identification error is around 1.1% at the 16th element when the artificial measurement noise is 10%. It may be mistaken for a damaged element.

In order to illustrate the improvement of the accuracy of identified results using the proposed method, a comparison between the results from the proposed method and those from the response sensitivity method by Lu and Law (2007) is made as shown in Table 3. It is found that more accurate identified results can be obtained using the proposed method. Even under 15% noise level, it can achieve accurate identified results with the relative errors that are less than 1%.

Table 3.

Results for multiple damages identification of the cantilevered plate.

Scenarios	Noise	Element No. 5		Element No. 13		Element No. 25		Methods
Scenarios	Noise	Red. (%)	Res. (%)	Red. (%)	Res. (%)	Red. (%)	Res. (%)	Methods
1	Nil	10.00	0.00	5.00	0.00	15.00	0.00	Proposed
2	Nil	10.00	0.00	5.00	0.00	15.00	0.00	Lu and Law (2007)
3	5%	10.32	0.32	5.71	0.71	15.27	0.27	Proposed
4	5%	10.61	0.61	4.02	0.98	15.33	0.33	Lu and Law (2007)
5	10%	10.28	0.28	5.40	0.40	15.46	0.46	Proposed
6	10%	11.14	1.14	3.40	1.60	15.68	0.68	Lu and Law (2007)
7	15%	10.42	0.42	5.53	0.53	15.67	0.67	Proposed
8	15%	11.67	1.67	2.62	2.38	16.01	1.01	Lu and Law (2007)

Red.: E reduction; res: relative errors.

3.7. Identification of multiple damages in a continuous beam

In this numerical example, identification of multiple damages in a two-span continuous beam is studied to further illustrate the effectiveness of the proposed method. Figure 12 shows the continuous beam that is under study. The physical parameters of the beam are: mass density $ρ = 2.8 \times 103 kg / m 3$ , Young’s modulus $E = 34 Gpa$ , total length $L = 30 m$ , width $b = 0.5 m$ and height $h = 1.0 m$ . The FEM of the beam consists of 20 Euler-Bernoulli beam elements with two d.f. at each node. The sampling rate is 500 Hz and the time duration in the calculation of dynamic responses of the structure is 2 seconds.

To obtain the dynamic responses of the beam, an impulsive force is assumed to act at 7.5 metres from the left support at 0.05 seconds and last for 0.05 seconds. It is expressed mathematically as

F (t) = {3.0 \times 104 N (0.05 s \leq t \leq 0.1 s) 0 (0 \leq t < 0.05 sor t > 0.1 s) .

In this case, a two-span continuous supported beam with multiple damages is studied under different artificial measurement noise circumstances. It assumes that three damages occur in the fifth, 10th and 19th elements with a Young’s modulus loss of 10%, 5% and 8%, respectively.

As shown in Figure 13, it is easy to identify that nodes 5, 6, 10,11, 19, and 20 are nodes of the most suspected damaged elements. Figure 14 shows RFV values when only the first mode is used . Comparing Figure 13 with Figure 14, it can be seen that the VRFV method distinguishes the damaged and undamaged elements more clearly than does the RFV method. The VRFV method highlights the damaged elements. In addition, the extents of these damaged elements are assessed as the reductions in the elemental Young’s modulus shown in Figure 15. When the noise is free, the identified results are exactly the same as the assumed damage extents in each elements. In addition, the identified results are 9.82%, 4.91%, and 7.45% in the case of 5% noise level, which are just slightly different from the true values.

The relative errors in identified results for different scenarios of measurement locations and noise level are shown in Table 4, where it can be seen that when measurement points are chosen closer to the damaged elements, more accurate damage identification results can be achieved. This can be accomplished by making use of the VRFV to localize the damage first, and the locations of the measurement can be determined accordingly. This table also reveals that increasing the number of measurement points can improve the accuracy of the identification results when noisy measurements are used in the damage identification.

Table 4.

Results for the multiple damages identification using different number of measurements.

Scenarios	Noise	Element No.5		Element No.10		Element No.19		Measurement (node number)
Scenarios	Noise	Red. (%)	Res. (%)	Red. (%)	Res. (%)	Red. (%)	Res. (%)	Measurement (node number)
1	Nil	10.00	0.00	5.00	0.00	8.00	0.00
2	5%	9.18	0.82	3.32	1.68	8.56	0.56	3, 8 18
3	5%	9.82	0.18	4.91	0.09	7.45	0.55	5, 10, 19
4	5%	10.04	0.04	5.29	0.29	7.81	0.19	5, 6, 10, 19, 20
5	10%	8.35	1.65	3.04	1.96	9.88	1.88	3, 8 18
6	10%	9.64	0.36	4.81	0.19	6.81	1.19	5, 10, 19
7	10%	9.85	0.15	5.03	0.03	7.71	0.29	5, 6, 10, 19, 20
8	15%	10.22	0.22	5.57	0.57	5.57	2.43	5, 10, 19
9	15%	9.78	0.22	5.05	0.05	7.55	0.45	5, 6, 10, 19, 20

Red.: E reduction; res: relative errors.

4. Conclusions

A combined approach based on residual force vector and response sensitivity analysis is proposed for structural damage identification. Although both the residual force method and the response sensitivity analysis are not new, the novelty of the combination of the two methods lies in the improvement of the accuracy and efficiency of the damage identification. With the help of virtual residual force vector, it is convenient to find the location of the local damage. Consequently, it reduces the number of updating parameters in response to the sensitivity-based method and increases the efficiency of damage quantification so that it does not need a lot of measurement points. Both impulsive excitation and sinusoidal excitation can be used for damage identification. But the impulsive excitation seems to give better identified results when the data is contaminated by noise. Both single damage and multiple damages can be identified successfully from the proposed method and it is insensitive to artificial measurement noise. It is also found that when more measurements are used around the suspected elements, better identified results can be obtained.

Footnotes

Funding

This work was supported in part by the National Natural Science Foundation of China (grant numbers 11172333 and 11272361), the Fundamental Research Funds for the Central Universities (grant number 13lgzd06), the Doctoral Program Foundation of the Ministry of Education of China (grant number 20130171110039), and the Guangdong Province Science and Technology Program (grant number 2012A030200011). These financial aids are gratefully acknowledged.

References

Cattarius

Inman

(1997) Time domain analysis for damage detection in smart structures. Mechanical Systems and Signal Processing 11(3): 409–423.

Chatterjee

(2010) Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions. Journal of Sound and Vibration 329(16): 3324–3335.

Chinchalkar

(2001) Determination of crack location in beams using natural frequencies. Journal of Sound and Vibration 247(3): 417–429.

Choi

Stubbs

(2004) Damage identification in structures using the time-domain response. Journal of Sound and Vibration 275(3–5): 577–590.

Friswell

Mottershead

(1995) Finite Element Model Updating in Structural Dynamics, Dordrecht, The Netherlands: Kluwer Academic Publishers.

Soong

(1998) Damage identification through regularization method I: theory. Journal of Engineering Mechanics 124(1): 103–108.

Hansen

(1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review 34(4): 561–580.

Hjelmstad

Banan

(1995) Time-domain parameter estimation algorithm for structures I: computational aspects. Journal of Engineering Mechanics 121(3): 424–434.

Kim

Stubbs

(2003) Crack detection in beam-type structures using frequency data. Journal of Sound and Vibration 259(1): 145–160.

10.

Smith

(1995) Hybrid approach for damage detection in flexible structures. Journal of Guidance, Control, and Dynamics 18(3): 419–425.

11.

Liu

Yang

(2006) A new structural damage identification method. Journal of Sound and Vibration 297(3): 694–703.

12.

Law

(2007) Features of dynamic response sensitivity and its application in damage detection. Journal of Sound and Vibration 303(1): 305–329.

13.

Maia

NMM

Silva

JMM

Almas

EAM

Sampaio

RPC

(2003) Damage detection in structures: from mode shape to frequency response function methods. Mechanical Systems and Signal Processing 17(3): 489–498.

14.

Pandey

Biswas

(1994) Damage detection in structures using changes in flexibility. Journal of Sound and Vibration 169(1): 3–17.

15.

Pandey

Biswas

Samman

(1991) Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration 145(2): 321–332.

16.

Ratcliffe

(1997) Damage detection using a modified Laplacian operator on mode shape data. Journal of Sound and Vibration 204(3): 505–517.

17.

Ricles

Kosmatka

(1992) Damage detection in elastic structures using vibratory residual forces and weighted sensitivity. AIAA Journal 30(9): 2310–2316.

18.

Rizos

Aspragathos

Dimarogonas

(1990) Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration 138(3): 381–388.

19.

Tikhonov

(1963) On the solution of ill-posed problems and the method of regularization. Soviet Mathematics 4: 1035–1038.

20.

Urgessa

(2011) Vibration properties of beams using frequency-domain system identification methods. Journal of Vibration and Control 17(9): 1287–1294.

21.

Law

(2004) Model error correction from truncated modal flexibility sensitivity and generic parameters: Part I: Simulation. Mechanical Systems and Signal Processing 18(6): 1381–1399.

22.

Yang

(2011) A new damage identification method based on structural flexibility disassembly. Journal of Vibration and Control 17(7): 1000–1008.

23.

Yang

Liu

(2007) Structural damage identification based on residual force vector. Journal of Sound and Vibration 305(1): 298–307.

24.

Zimmerman

Kaouk

(1994) Structural damage detection using a minimum rank update theory. Journal of Vibration and Acoustics 116(2): 222–231.