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Numerical and experimental detection of concentrated damage in a parabolic arch by measured frequency variations A. Pau, A. Greco and F. Vestroni Journal of Vibration and Control 2011 17: 605 originally published online 22 November 2010 DOI: 10.1177/1077546310362861 The online version of this article can be found at: http://jvc.sagepub.com/content/17/4/605

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Numerical and experimental detection of concentrated damage in a parabolic arch by measured frequency variations

Journal of Vibration and Control 17(4) 605–614 ª The Author(s) 2010 Reprints and permission: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546310362861 jvc.sagepub.com

A. Pau1, A. Greco2 and F. Vestroni1

Abstract The dynamics of a parabolic arch is studied in its undamaged and damaged states. The damage consists of a notch that reduces the height of the cross section at a given abscissa. A damage identification technique, based on the minimization of an objective function measuring the differences between numerical and experimental variations of natural frequencies for undamaged and damaged states, is used. The uniqueness of the solution in different damage configurations is investigated using pseudo-experimental data and the reliability of the identification procedure is assessed. The identification procedure is then applied to an experimental case, where frequencies are obtained by impulsive tests on a prototype arch. The minimum number of experimental data needed to identify damage parameters is defined and the sensitivity of the identification algorithm to different possible choices of sets of data is analyzed. Keywords Arch dynamics, damage identification, inverse problems Date received: 9 August 2002; accepted: 12 January 2010

1. Introduction Curved bars represent a fundamental element in various structures, such as bridges, roofs, vaults, underground galleries etc., and have been studied extensively with respect to both their static and dynamic behaviors. The first studies on the dynamics of arches were by Lamb (1888), while later extensive contributions were made by Love (1944) and, more recently, by Henrych (1981). The literature mainly deals with the in-plane vibration problem of circular arches. A fundamental review article was published by Chidamparam and Leissa (1993). In the free in-plane vibration of curved beams, the motion consists of bending–extensional modes. Radial and tangential displacements and the rotation of the centroidal axis are the three independent variables required in the analysis. The overall problem is complicated by the curvilinear geometry, which causes coupling between displacements and rotations in the equations of motion. For even the simplest case of inextensible circular arches with negligible rotatory inertia and shear effects, the motion is governed by a homogeneous sixth-order differential equation with constant coefficients, for which closed-form solutions are rather complicated.

For every structural member, whether straight or curved, the presence of damage implies a loss of structural stiffness, inducing response variation. The problem of damaged curved beams has enjoyed less focus. Among the studies on this topic, the papers by Krawczuk and Ostachowicz (1997), ¨ z and Das (2006) are worth citing. Viola et al. (2005) and O Recently, there has been an increased interest in monitoring structural integrity and damage detection. The techniques developed are based on the well-established concept that changes in global response quantities can be related to the damage parameters. Variations in natural frequencies have been used to locate damage since the late 1970s (Cawley and Adams, 1979; Xu et al., 2007). Changes in flexibilities (Pandey and Biswas, 1994), stiffness (Zimmerman and

1

Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Universita` di Roma, Roma, Italy 2 Dipartimento di Ingegneria Civile e Ambientale, Universita` di Catania, Catania, Italy Corresponding Author: F. Vestroni, Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Universita´ di Roma, Roma, Italy Email: [email protected]

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Journal of Vibration and Control 17(4)

606

E = 2*105 N/mm2 ν = 0.3 m = 7.849* 10−9 N/mm3 b = 40 mm hU = 8 mm

y f = 200 mm x

hU

hD

L = 1000 mm

b

Figure 1. Geometrical and mechanical parameters of the arch.

Kaouk, 1994), modal curvature (Pandey et al., 1991), modal strain energy (Stubbs and Kim, 1996), or static quantities, namely displacement induced by applied loads (Di Paola and Bilello, 2004; Caddemi and Greco, 2006), have been used to detect damage. However, it is the authors’ opinion that the most common and reliable quantities remain natural frequencies: these characteristics of the response can be obtained by means of a small number of measurement points, which can help in defining timesaving and cost-effective procedures in the monitoring of real structures. Two recent review papers by Farrar and Worden (2007) and Friswell (2007) contain a rich bibliography. Damage detection leads to a class of inverse problems which are often ill-conditioned and sometimes undetermined. Suitable solutions to inverse problems for damage detection have tended to be for straight (Vestroni and Capecchi, 1996; Cerri and Vestroni, 2000; Morassi, 2001; Cerri and Vestroni, 2003) rather than for curved beams (Cerri and Ruta, 2004; Cerri et al., 2008). In damage identification, modeling of damage is necessary. Concentrated damage may be modeled by a reduction in the rigidity of the beam at the corresponding abscissa, described by means of a rotational spring, as in Ostachowicz and Krawczuk (1991). Elsewhere, a weaker element is introduced into a finite-element model, while others present one-dimensional continuum theories (Christides and Barr, 1984; Chondros and Dimarogonas, 1998). Experimental data on the free vibration of damaged arches is generally lacking, to the best of the authors’ knowledge. Apart from a recent paper by Cerri et al. (2008), most of the contributions use numerical data. In this paper we present a numerical and experimental investigation of the dynamics of a parabolic arch, in which a notch reduces the height of the cross section at a given abscissa. The arch is studied in its undamaged and damaged states using a finite-element (FE) model. By studying the direct problem, the sensitivity of the modal properties with regards to the damage parameters, i.e. location and intensity, can be assessed. The inverse problem is solved for both the undamaged and damaged structure, by implementing a damage identification technique, described by

Vestroni and Capecchi (1996) and Cerri and Vestroni (2000). According to this technique, which pertains to the wide family of finite-element updating approaches, the model of the structure is first updated in the undamaged condition and then the location and intensity of damage are detected. Identification of damage parameters is based on the minimization of an objective function, measuring the differences between numerical and experimental natural frequencies in the damaged and undamaged states. Cases with pseudo-experimental data are also considered. The minimum number of experimental data needed to identify damage parameters is established and the effect of different possible choices of data sets analyzed.

2. Model of the cracked arch A double-hinged parabolic plane arch with the following centerline equation is considered: yðxÞ ¼ 0:004 fx

Lx ; L

ð1Þ

where f ¼ 200 mm is the mid-span height and L ¼ 1000 mm the horizontal length of the arch, as shown in Figure 1. The cross section is rectangular. The values of Young’s modulus E, Poisson’s ratio n and mass density m are also reported in the same figure. Rotary inertia and shear effects are ignored as they do not affect the first modal characteristics of this type of structure. Only the axial and bending deformability are taken into account. The direct and inverse problems are solved with a finite-element model. Hence, the damage identification procedure followed in this paper may be implemented for more complex structures. The damage is represented by a concentrated notch which reduces the height of the cross section from hU to hD . It is assumed that the width of the notch is such that it is possible to ignore the reduction in the structure’s total mass. The damage determines, at a given abscissa, a reduction in the flexural and axial rigidities. This latter reduction has considerable influence on the modal characteristics only in arches in which hU =L > 0:5, as already reported by Chidamparam and Leissa (1993). For the considered

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ω1 = 53.87 Hz

ω2 = 129.65 Hz

ω3 = 237.63 Hz

ω4 = 371.82 Hz

Figure 2. The first four mode shapes and related natural frequencies.

arch ðhU =L ¼ 0:008Þ, the reduction in the axial stiffness induced by the notch is negligible, and therefore only a rotational spring is introduced to model the damage. The notch causes a perturbation in the tension state in a damaged zone whose length LD is greater than the effective width of the notch itself. The relative rotation fD between the sections delimiting the damaged zone can be written as fD ¼ fU þ Df, where fU is the rotation between the two limit sections in the undamaged case and Df is the increase in rotation due to damage. Then Df ¼ M

LD b ; U EI ð1  bÞ

ð2Þ

where b ¼ ðEI U  EI D Þ=EI U , EI U and EI D are respectively the flexural rigidities of the undamaged and damaged cross sections. Here LD ¼ h=2 is assumed, which is obtained for a beam under the hypothesis of an exponential decay in the stiffness, as shown by Cerri and Vestroni (2003) and reported in the Appendix. In the case of localized damage, Df ¼ M=K, therefore, the nondimensional stiffness of an equivalent spring k ¼ K=ðEI U =Lc Þ can be expressed as k¼

2Lc 1  b ; hU b

ð3Þ

where Lc is the total length of the parabolic arch.

3. Direct problem For the undamaged arch reported in Figure 1, the first four in-plane frequencies and vibration modes have been evaluated by means of a one-dimensional finite-element model (Figure 2). The system does not exhibit close frequencies. Moreover, in contrast to the flexural behavior of beams, odd modes are skew-symmetric while even modes are symmetric. The predominant component of displacement in the first modes shown in Figure 2 is orthogonal to the axis of the arch. Damage implies a loss of mechanical stiffness and modifies the dynamic response of a structure. Figure 3 shows, for different values of the normalized stiffness of the rotational spring, the influence of the locations of the damage on the variation between the undamaged and

damaged state of the first four frequencies. The location is represented through its normalized abscissa s ¼ x=L. Since the changes in frequencies at two symmetric locations are the same, as the structure is symmetric, only half of the axis is reported. Differences in natural frequencies increase with the intensity of the damage and depend on its location. For each frequency, differences reach their maximum when the damage is located at a peak value of the corresponding bending curvature (Figure 4), while equal to zero when the damage is located at a node of the bending curvature, as observed in beams and circular arches (Cerri and Ruta, 2004). However, since the mode shapes of the parabolic arch are not simply harmonic functions, their peaks and valleys do not occur precisely at submultiples of L, as is the case with simply supported straight beams. This analysis is useful in developing an inverse procedure for the characterization of damage when using experimental dynamic data. In the direct problem, for assumed values of k and s, the finite-element model provides the values of the ith natural frequency oi . In contrast, when oi is known for the damaged arch, for each possible damage position sj , a stiffness ki ðsj Þ exists which corresponds to a value of the ith natural frequency equal to oi . Therefore, by taking into account all of the possible positions of the damage in the FE model, a curve ki ðsj Þ can be obtained. In fact, the curve is built by a series of segments since only a finite number of damage positions can be considered in the model. Figure 5a represents the curves ki ðsj Þ for the first five frequencies of vibration, obtained by using s ¼ 0:3 and k ¼ 20 as damage parameters. This analysis can be used to examine the uniqueness of the solution to the inverse problem. In fact, curves ki ðsj Þ obtained for different oi cross at the abscissa where damage is localized, providing the solution to the inverse problem. In the case of simply supported straight beams, the curves corresponding to the first and the second frequencies, k1 ðsj Þ and k2 ðsj Þ, cross only once, providing the solution to the inverse problem using only two frequencies (Figure 5b). Regarding the arch presented here, however, each pair of curves shows more than one intersection, and thus the solution is undetermined when only two frequencies are known (Figure 5a). This is due to the wavelengths of curves ki ðsj Þ, which present at least one node along the length of the arch, one more than with respect to the simply supported straight beam. Thus, at least three different frequencies are necessary to uniquely determine the two damage parameters, as is shown in the case presented in Figure 5a.

4. Inverse problem The identification procedure is based on the response comparison. An optimal estimate of the damage parameters k; s is obtained by minimizing the objective function:

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Journal of Vibration and Control 17(4)

608

Figure 3. Frequency variation ratios as a function of the damage parameters k and s.

Since the numerical values Doi ðk; sÞ are provided by a finite-element model of the arch, it is convenient to obtain the damage parameters k; s in two phases, by successively seeking two distinct minima. For each possible discrete damage position s in the finite-element model, the minimization of the function (equation 3) with respect to k provides the function:

200

1/R

100

0

1 2 3 4

−100

e ¼ min Gðk; sÞ: GðsÞ k

−200 0

0.1

0.20

0.3

0.4

0.5

s

Figure 4. Bending curvatures of the first four mode shapes.

Gðk; sÞ ¼

XDoi ðk; sÞ i

oU i



Doei oU ei

2 :

ð4Þ

This is defined as the sum of the squares of the differences between the numerical Doi ðk; sÞ and experimental Doei values of the variations of frequencies between the undamaged and the damaged states, normalized with respect U to the frequencies of the undamaged arch, oU i and oei .

ð5Þ

The solution to the inverse problem is then given by the e over s. If this function exhibits one global minimum of GðsÞ minimum, the solution to the inverse problem exists and is unique. In this section, the first natural frequencies evaluated by solving the direct problem are assumed as pseudo-experimental data to test the reliability of the optimization procedure considered. This is then applied to the experimental data. The analysis is performed for the case of a notch located at s ¼ 0:3, with a depth corresponding to a 50% reduction of the height, to which k  20 corresponds. Figure 6 shows e the objective function GðsÞ calculated by using the first two, three and five frequencies. As shown in the analysis, based on the observation of the curves ki ðsj Þ, at least three frequencies are needed in order to achieve a unique solution. In fact, when only two frequencies are used, a flat area

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250 b)

a) 200 2

150 k

5

100

3

50 3

1 4 0.05

0.1

0.15

0.2

0.25 s

0.3

0.35

0.4

1

2 0.45

0.5

0

0

0.05

0.1

0.15

0.2

0.25 s

0.3

0.35

0.4

0.45

0.5

Figure 5. Curves ki (sj) for the first five modes of the arch (a) and three modes of a beam (b); s = 0.30, k = 20.

G(s)

0.0015

a) imax = 2

0.0010

b) imax = 3

c) imax = 5

0.002

0.0010 0.001

0.0005 0.0005

0

0 0

0.1

0.2

0.3

0.4

0.5

0

0.1

s

0.2

0.3

0.4

0.5

0

s

0.1

0.2

0.3

0.4

0.5

s

Figure 6. Objective functions using different frequencies in the summation: (a) i = 1,2; (b) i = 1,2,3; (c) i = 1,2,3,4,5.

in the neighborhood of the real solution appears in the diagram. This flat part of the curve is a consequence of the double close intersections of the curves k1 ðsj Þ and k2 ðsj Þ in the 0:30 range, as shown in Figure 5a. When three frequencies are used (Figure 6b), the objective function attains one absolute minimum in the exact abscissa where damage is located. In the case with five frequencies (Figure 6c), two relative minima occur, together with a global minimum in the right position.

5. Experimental tests and damage identification The experimental investigation was performed at the Laboratory of the Department of Structural and Geotechnical Engineering, Sapienza University of Rome, on a prototype double-hinged parabolic arch, whose geometry is depicted in Figure 1. The structure in Figure 7 is excited by an instrumented hammer at channels 2 and 3 and its response measured by seven uniaxial piezoelectric accelerometers. These have a bandwidth ranging from 1 to 7000 Hz, a dynamic range of 500 g and a sensitivity of

Figure 7. Experimental setup.

Table 1. Mechanical characteristics of the notches.

D1 D2 D3 D4

hU  hD (mm)

d

k

1 2 3 4

0.125 0.250 0.375 0.500

570 205 91 40

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Journal of Vibration and Control 17(4)

610

Table 2. Experimental natural frequencies of the undamaged (U) and damaged (D) arch (Hz) and their variations.

U D1 D2 D3 D4

oe1

jDoe1 j % oUe1

oe2

jDoe2 j % oUe2

oe3

jDoe3 j % oUe3

oe4

jDoe4 j % oUe4

oe5

jDoe5 j % oUe5

50.40 50.34 50.18 49.91 49.23

0.12 0.44 0.97 2.32

118.94 118.92 118.92 118.91 118.91

0.02 0.02 0.03 0.03

224.80 224.63 224.62 223.96 222.05

0.08 0.08 0.37 1.22

346.33 346.08 345.02 343.20 339.48

0.07 0.38 0.90 1.98

508.55 508.28 507.89 507.05 505.18

0.05 0.13 0.29 0.66

10 mV/g. They were connected to a data acquisition system with 16-bit A/D converter, and anti-aliasing filters. Three accelerometers with a mass of 29  103 kg and four with a mass of 12  103 kg were used. They were located without altering the structural symmetry. Each test was repeated 10 times. An asymmetric notch 1 mm in width is made at the normalized abscissa s ¼ 0:71. The effects in terms of frequency changes of a notch at the abscissa s ¼ 0:71 are the same as those at s ¼ 0:29 because of structural symmetry. This is a drawback of the frequency approach, which could be overcome by adding some information on the mode shapes or on antiresonant data, as shown by Dilena and Morassi (2009). Four damage levels are taken into account, corresponding to different depths of the notch. In Table 1, the characteristics of the notches are reported, together with the values of the nondimensional stiffness according to equation 3. The frequency response functions (FRFs) obtained from the experimental acceleration time-histories are processed and the modal parameters extracted by means of a multimode method, using a nonlinear least-squares approach (Ewins, 2000). Table 2 reports the first five experimental natural frequencies of the structure in both its undamaged and damaged states and related variations. The frequencies are determined with great accuracy. The coefficients of variation cv , defined as the ratio of standard deviation to the mean, is of the order of 1/100 Hz for the different test repetitions. The modes presenting the highest cv are those that have a node in the vicinity of the measurement or excitation point. A slight increment of cv is observed when the damage increases. In this case, the frequencies determined with greater accuracy are 1–2–4–5 when the force is applied at Ch2 and 1–2–3–5 when the force is applied at Ch3. This is considered when determining the experimental frequencies, for which the most accurate evaluations are chosen between those of Ch2 or Ch3. Figure 8 shows two details of the FRFs in terms of accelerations of the undamaged and damaged structure in the vicinity of the peaks of the first (Figure 8a) and second (Figure 8b) modes. The decrease in the natural frequency is clear for the first mode, whereas the second natural frequency remains nearly unaltered (Table 2), since the notch is close to a node of the bending curvature of the second

Table 3. Finite-element modeling natural frequencies of the undamaged arch (Hz) and comparison with experimental values for different Young’s modulus.

oi ðEini Þ joUi  oUei j=oUei % oi ðEopt Þ joUi  oUei j=oUei %

1

2

3

4

5

49.58 1.63 49.95 0.89

120.86 1.61 121.77 2.38

220.56 1.89 222.21 1.15

345.48 0.25 348.07 0.50

500.12 1.66 503.87 0.92

mode. Figure 8 also shows another result regarding the damping coefficient, which remains constant or undergoes a slight reduction when the damage increases. This is evident from the peak amplitude and depends on how the damage is made, that is a sharp notch which does not increase the structural dissipation. As an initial step, the frequencies of the undamaged finite element model, in which the added masses due to the accelerometers are also taken into account, are compared with the experimental frequencies (Table 3). The frequencies, initially estimated and corresponding to a nominal value of Young’s modulus Eini ¼ 2  105 N/mm2 , are satisfactorily close to the experimental values, with a mean error of 1.41%. An updating of Young’s modulus, whose optimal value is Eopt ¼ 2:03  105 N/mm2 , provides a further reduction of the mean error, up to 1.17%. The procedure for damage identification described in the Section 4 is applied to determine the damage parameters k; sj in the finite-element model using the experimental data of Table 2. Figures 9a–d show the objective functions built with five frequencies for damage configurations D1, D2, D3 and D4, respectively. The global minimum of the objective function is unique, and well pronounced as in the pseudo-experimental case of Figure 6. Moreover, the two relative minima, already predicted by the pseudo-experimental data, can still be observed. Table 4 reports the values of spring stiffness, characteristics of the notches and damage positions, obtained at the global minimum of the objective function for the four damage configurations considered. It may be observed that the depth of the notch corresponding to the identified stiffness is close to the real one, with a mean error of 15.5%. The damage position is also identified with great accuracy, with a mean error of 1.5%. Only in the weakest damage scenario

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6000 U

1600

a)

b)

D1 D2 D3

1200

D4

|H12(f)|[g/N]

4000

800 2000 400

0

0 46

48

50

52

114

116

f [Hz]

118

120

122

124

f [Hz]

Figure 8. Details of the experimental frequency response functions (FRFs) in terms of accelerations for (a) the first and (b) the second natural frequencies.

3E−005

4E−006 3E−006 G(s)

2E−005 2E−006 1E−005 1E−006

a)

b)

0

0

0.00016

0.0012

0.00012 G(s)

0.0008 8E−005 0.0004 4E−005

c)

d) 0

0 0.5

0.6

0.7

0.8

0.9

1

0.5

0.6

0.7

0.8

0.9

1

s

s

Figure 9. Objective functions for damage configurations (a) D1, (b) D2, (c) D3 and (d) D4.

are notable errors present; in all other cases satisfactory results are obtained with a mean error of 13.5% on the equivalent spring stiffness, 5.9% on the notch depth and, much smaller, 0.8% on the location. The error strongly decreases with increasing damage, where the systematic errors are less important.

However, as clearly emerges in Figure 6, only three frequencies are strictly needed to identify the damage parameters: more data do not necessarily yield better results. In the present case, by comparing the variations of natural frequencies represented in Figure 3 for a notch at s ¼ 0:29 to the experimental variations in Table 2, the experimental

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Journal of Vibration and Control 17(4)

612 Table 4. Identified stiffness, corresponding notch depths, damage locations and errors. k 1161 249 98 36

G(s)

D1 D2 D3 D4

Error (%)

hU  hD (mm)

Error (%)

103.7 21.7 8.2 10.7

0.6 1.8 2.9 4.1

44.3 11.1 3.3 3.3

2E-006

2.5E-005

1.6E-006

2E-005

1.2E-006

1.5E-005

8E-007

1E-005

4E-007

s

Error (%)

0.74 0.72 0.71 0.70

3.6 0.8 0.0 1.4

5E-006

a)

b) 0

0.00012

0.0006

8E-005

0.0004

4E-005

0.0002

G(s)

0

c)

d)

0 0.5

0 0.6

0.7

0.8

0.9

1

0.5

0.6

s

0.7

0.8

0.9

1

s

Figure 10. Objective function with frequencies 1–2–5 for damage configurations (a) D1, (b) D2, (c) D3 and (d) D4.

Table 5. Identified stiffness, corresponding notch depths, damage locations and related errors when only frequencies 1–2–5 are considered in the objective function.

D1 D2 D3 D4

k

Error (%)

hU  hD [mm]

Error (%)

s

Error (%)

527 205 89 36

7.5 0.2 1.4 10.1

1.1 2.0 3.0 4.1

6.2 0.1 0.6 3.1

0.65 0.71 0.72 0.71

8.5 0.0 0.8 0.0

variation of the third frequency appears as an outlier. In fact, its experimental variation should be greater than it is and approximately equal to 2/3 of Do1 =o1 . This is confirmed by the fact that, when this frequency is included in the objective function, the identified position tends to be displaced toward the right hinge constraint, which provides a smaller ratio ðDo3 =o3 Þ=ðDo1 =o1 Þ. Furthermore, the inclusion of the fourth frequency produces a flattening of the objective function and yields inaccurate results.

This is due to the fact that the variations of the different frequencies do not have the same sensitivity to the damage parameters along the s-axis. In particular, the fourth frequency has a low sensitivity to the position parameter in the neighborhood of s ¼ 0:29, as can be inferred from the shape of the curve Do4 =o4 in Figure 3. Therefore, the analysis is repeated using only frequencies 1–2–5. The objective functions are reported in Figure 10. They show two close minima; however, the global minimum provides

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Table 6. Finite-element modeling and experimental frequency variations with respect to the undamaged model. D1

1 2 3 4 5

D2

D3

jDoi j % oUi

jDoei j % oUei

jDoi j % oUi

jDoei j % oUei

jDoi j % oUi

jDoei j % oUei

jDoi j % oUi

jDoei j % oUei

0.12 0.01 0.19 0.07 0.05

0.12 0.02 0.08 0.07 0.05

0.42 0.03 0.19 0.48 0.14

0.44 0.02 0.08 0.38 0.13

0.96 0.07 0.43 1.07 0.31

0.97 0.03 0.37 0.90 0.29

2.30 0.17 1.03 2.45 0.70

2.32 0.03 1.22 1.98 0.66

updated model is able to correctly reproduce these differences would also be of interest. These variations, for the analysis with three frequencies, are reported in Table 6, which shows that the updated damaged model accurately captures the variations for the frequencies included in the summation. The other frequencies present greater errors, suggesting the presence of modeling and experimental errors.

1E−006

Log(G(s))

D4

1E−007

1E−008

1E−009 0.5

0.6

0.7

s

0.8

0.9

1

Figure 11. Comparison between objective functions obtained from the model with (bold) and without reduction of axial stiffness.

very good results. Table 5 reports the identified stiffness, corresponding notch depths and damage positions at the true global minimum of the new objective function. The errors on the damage parameters largely decrease with respect to Table 4. For cases D2–D4, the equivalent spring stiffness is determined with a mean error of 3.9% and the corresponding notch depth with a mean error of 1.3%. The damage position is identified with a mean error of 0.3%. Here also, the largest errors regard the weakest damage D1, for which the error on the stiffness greatly decreases with respect to the case with five frequencies, but the error on the location slightly increases. In this case, the second close minimum provides the right position, but greater errors on the stiffness and notch depth. When the frequency variations are small, the accuracy of the model becomes a key factor. In fact, in this case, the addition of an axial spring modeling the axial stiffness reduction helps in identifying the right notch location, as shown in Figure 11. The procedure of damage identification can be carried out without adding new parameters if the axial and flexural stiffness are expressed as a function of the reduction of the section height due to damage. The results are essentially the same for the other damage configurations and are not reported for the sake of brevity. Since the identification process (equation 4) is based on the variation of frequencies due to damage, whether the

6. Conclusions A double-hinged parabolic arch with concentrated damage of a notch type has been studied here. The identification of damage parameters, i.e. location and intensity, has been conducted using a procedure where a finite-element model updating is performed taking into account the natural frequencies of the undamaged and damaged structure. The identification technique is based on the minimization of an objective function which measures the differences between numerical and experimental variations of frequencies due to damage. Different damage configurations are considered, either making use of pseudo- or real experimental data. The study of the uniqueness of the inverse problem has shown that, differently to simply supported straight beams, at least three natural frequencies are needed to characterize the two damage parameters k and s for the arch considered. Comparative studies on the shape of the objective function taking different combinations of natural frequencies have been performed. Although the problem is fairly illconditioned, satisfactory results have been obtained, apart from the case of very weak damage. However, the study shows that discrepancies between the model and the structure have an influence on the parameter estimates, and therefore both the quality of the experimental data and an accurate modeling are fundamental if reliable results are to be obtained. Acknowledgment This research was partially financed by MIUR (PRIN, FY 2005-06).

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Journal of Vibration and Control 17(4)

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Appendix Here LD is determined by requiring that the deformability of a beam with a distributed stiffness decay described by the function f ðxÞ equals the deformability of a beam with a step variation of stiffness with length LD , that is ð ðLLD Þ=2 ðL 1 1 dx ¼ U f ðxÞEI EI 0 0 ð6Þ ðL ð ðLþLD Þ=2 1 1 dx þ dx: þ D U ðLLD Þ=2 EI ðLþLD Þ=2 EI If the exponential function presented by Christides and Barr (1984) is assumed for f ðxÞ, the value LD ¼ h=2 is obtained.

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