2014 - Damage Identification-Modal Curvature

8/19/2019 2014 - Damage Identification-Modal Curvature

1/11

Rapid Communications

Damage identification for beams in noisy conditions based onTeager energy operator-wavelet transform modal curvature

Maosen Cao a,b, Wei Xu a, Wieslaw Ostachowiczb,c,n, Zhongqing Su d

a Department of Engineering Mechanics, Hohai University, Nanjing 210098, Chinab Institute of Fluid Flow Machinery, Polish Academy of Science, 80-952 Gdansk, Polandc Faculty of Automotive and Construction Machinery, Warsaw University of Technology, Narbutta 84, 02-524 Warsaw, Polandd Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 

a r t i c l e i n f o

 Article history:

Received 15 April 2013

Received in revised form

28 October 2013

Accepted 1 November 2013

Handling editor: M.P. CartmellAvailable online 4 December 2013

a b s t r a c t

Modal curvatures have been widely used in the detection of structural damage. Attractive

features of modal curvature include great sensitivity to damage and instant determina-

tion of damage location. However, an intrinsic deficiency in a modal curvature is its

susceptibility to the measurement noise present in the displacement mode shape that

produces the modal curvature, likely obscuring the features of damage. To address this

deficiency, the Teager energy operator together with wavelet transform is tactically

utilized to treat modal curvature, producing a new modal curvature, termed the Teager

energy operator-wavelet transform modal curvature. This new modal curvature features

distinct capabilities of suppressing noise, canceling global trends, and intensifying the

singular feature caused by damage for a measured mode shape involving noise. These

features maximize the sensitivity to damage and accuracy of damage localization. The

proposed modal curvature is demonstrated in several analytical cases of cracked pinned–

pinned, clamped–free and clamped–clamped beams, with emphasis on characterizing

damage in noisy conditions, and it is further validated by an experimental program using

a scanning laser vibrometer to acquire mode shapes of a cracked aluminum beam. The

Teager energy operator-wavelet transform modal curvature essentially overcomes the

deficiency of conventional modal curvature, providing a new dynamic feature well suited

for damage characterization in noisy environments. (The Matlab code for implementing

Teager energy operator-wavelet transform modal curvature can be provided by the

corresponding author on request.)

&  2013 Elsevier Ltd. All rights reserved.

1. Introduction

Structural damage detection has been a research focus of increasing interest in mechanical, civil, aerospace, and military

fields during the last few decades  [1–9]. Damage detection relying on modal curvatures has been extensively discussed in

the literature [10–13]. Modal curvature is the curvature of a displacement mode shape for a structure. In the case of a beam,

Contents lists available at  ScienceDirect

journal homepage:   www.elsevier.com/locate/jsvi

 Journal of Sound and Vibration

0022-460X/$- see front matter  &  2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jsv.2013.11.003

n Corresponding author at: Institute of Fluid Flow Machinery, Polish Academy of Science, 80-952 Gdansk, Poland. Tel.: þ48 58 6995 258;fax: þ48 58 3416 144.

E-mail address:  [email protected] (W. Ostachowicz).

 Journal of Sound and Vibration 333 (2014) 1543–1553

http://www.sciencedirect.com/science/journal/0022460Xhttp://www.elsevier.com/locate/jsvihttp://dx.doi.org/10.1016/j.jsv.2013.11.003mailto:[email protected]://dx.doi.org/10.1016/j.jsv.2013.11.003http://dx.doi.org/10.1016/j.jsv.2013.11.003http://dx.doi.org/10.1016/j.jsv.2013.11.003http://dx.doi.org/10.1016/j.jsv.2013.11.003mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.jsv.2013.11.003&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jsv.2013.11.003&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jsv.2013.11.003&domain=pdfhttp://dx.doi.org/10.1016/j.jsv.2013.11.003http://dx.doi.org/10.1016/j.jsv.2013.11.003http://dx.doi.org/10.1016/j.jsv.2013.11.003http://www.elsevier.com/locate/jsvihttp://www.sciencedirect.com/science/journal/0022460X

8/19/2019 2014 - Damage Identification-Modal Curvature

2/11

its modal curvature can be interpreted as [10]

  M EI ð xÞ ¼ W ″ð xÞ

 W ð xhÞ2W ðhÞþW ð xþhÞh

2  (1)

where M  is the bending moment,  EI ð xÞ  is the bending stiffness,  W ð xÞ   is the displacement mode shape for the beam, andW ″ð xÞ is the modal curvature (the two primes denote second-order differentiation of  W ð xÞ). W ″ð xÞ is approximately obtainedby the second-order central difference of  W 

ð x

Þwith sampling interval h. Damage, commonly represented by an alteration in

EI ( x), can cause a change in  W ″ð xÞ, which in turn manifests the status of the damage. Thus the use of modal curvature istheoretically feasible for characterizing damage in beams.

Modal curvatures allow not only the detection of damage in a structure but also determination of the location of damage

with a high degree of accuracy [14–15]. Pandey [10] originally proposed the dynamic feature of modal curvature for use in

damage detection in beams: the abrupt change of modal curvatures in the vicinity of a crack indicates the presence and

location of the damage. In damage detection applications, modal curvature is computed by the second-order central

difference of the displacement mode shape  [10–14]. Despite its popularity in characterizing damage, the modal curvature

has a noticeable drawback: it is susceptible to any slight noise existing in a displacement mode shape   [15–17]. This

susceptibility can be attributed to the fact that the second-order central difference used to generate a modal curvature

considerably amplifies any slight noise in the mode shape  [18]. The amplified noise easily masks features of damage,

possibly frustrating damage detection. Several researchers   [14–17]   have made efforts to study strategies to deal with

this deficiency of modal curvature. The resulting methods can be roughly categorized into two types: (1) the effect of 

measurement noise on modal curvature is diminished by taking the optimal sampling interval of a mode shape to generate

the modal curvatures; and (2) the immunity of modal curvature to noise is improved by sophisticated signal processingtechniques. Unfortunately, these methods either require a complex procedure to determine the optimal sampling interval

or exhibit strong dependence on the selected signal processing method. An accurate and reliable strategy capable of 

overcoming this drawback of modal curvature is yet to be presented. Unlike previous studies, this study focuses on

developing a new modal curvature based on the use of a wavelet transform (WT) [19,20] incorporating the Teager energy

operator (TEO) [21,22]. The new modal curvature, termed TEO-WT modal curvature, features noise suppression, cancellation

of global trends, and intensification of the local singular feature for the signal under inspection. These features are eminently

suitable for characterizing damage in noisy conditions. The capabilities of the TEO-WT modal curvature are investigated in

analytical cases of cracked beams. The applicability of the approach is validated by an experimental program using a

scanning laser vibrometer (SLV) [23,24] to acquire mode shapes of an aluminum beam with a crack.

2. TEO-WT modal curvature

 2.1. TEO

The discrete version of the TEO [21] was proposed by Kaiser  [22] with the aim of representing the transient energy of a

signal. Let  xn  be a sequence of sampling points of a discretized cosine signal:

 xn ¼ A   cos ð ΩnþϕÞ;   (2)

where n  is the sampling number,  ϕ  is the initial phase and  Ω  is the digital frequency specified by  Ω¼ 2π  f = f s,with f  being theanalog frequency and  f s   is the sampling frequency. The signal values at three successive points are

 xn1 ¼ A   cos ð Ωðn1ÞþϕÞ;   xn ¼ A   cos ð ΩnþϕÞ;   xnþ1 ¼ A   cos ð Ωðnþ1ÞþϕÞ:   (3)

According to the trigonometric identities, we can obtain

 x2n xn1 xnþ1 ¼ A2 sin 2ð ΩÞ:   (4)With the relation in Eq. (4), an algorithm to approximately calculate the point-wise energy  E n of a sole-component signal is

given as

E n ¼ A2 Ω2  A2 sin 2ð ΩÞ ¼ x2n xn1 xnþ1:   (5)

Considering Eqs. (4) and (5), the TEO for a discrete sequence  f ½n  is defined by

Ψ ð f ½nÞ ¼ f 2½n f ½n1 f ½nþ1:   (6)

where  Ψ  denotes the TEO that calculates the approximate transient energy of  f ½n.Currently, the TEO is widely used in speech engineering as an effective nonlinear operator to treat the local singularity of 

a speech signal [22,25,26], where it behaves like a supplement to the linear Fourier transform that is suitable for the analysis

of global characteristics [19]. As with local singularity characterization for a speech signal, the TEO in this study is utilized tocharacterize the abnormality of a mode shape that is caused by structural local damage.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –15531544

8/19/2019 2014 - Damage Identification-Modal Curvature

3/11

 2.2. WT modal curvature

For a measured displacement mode shape  W ð xÞ, the associated modal curvature  W ″ð xÞ  is obtained by the second-ordercentral difference (Eq.   (1)), which can considerably amplify the noise existing in   W ð xÞ, easily obscuring the features of damage [16–17]. To smooth noise, the classical Gaussian function  g ð xÞ is used to convolute W ″ð xÞ, i.e., W ″ g , with   beingthe notation of convolution. Owing to the differential property of convolution  [19], the following relation exists:

W ″

  g 

¼W 

  g ″;   (7)

where g″ð xÞ   is the second-order derivative of the Gaussian function   g ð xÞ, just the well-known Mexican wavelet   [19,20]having two vanishing moments. Notate the Gaussian function involving translation parameter  u  and scale parameter   s,

 g u;sð xÞ ¼   1 ffiffisp   g ðð xuÞ=sÞ, from which the family of this wavelet can be derived as g ″u;sð xÞ ¼ s2

  d2

dx2  g u;sð xÞ ¼

  1 ffiffis

p   g ″  xus

:   (8)

where s  can scale the mother wavelet  g ″ð xÞ  s  times and  u  can enable the wavelet  g u;sð xÞ to translate along the x axis.Replacing g  with g u;s   and g ″ with  g 

″

u;s, Eq. (7)  can be converted into

W ″   g u;s ¼W    g ″u;s:   (9)

W    g ″u;s  is a quantity of multiscale property that measures the similarity between the wavelet  g ″u;sð xÞ and mode shape W ð xÞin the position x

¼u  and at the scale  s.

For convenience of expression,  W    g ″u;s  is represented by  W    g ″sðuÞ. Considering g u;sð xÞ ¼ g u;sð xÞ and g ″u;sð xÞ ¼ g ″u;sð xÞarising from the symmetry of the Gaussian function,  W    g ″u;s  can be expanded as

W    g ″sðuÞ ¼  1 ffiffi

sp 

Z  11

W ð xÞ g ″  xus

 dx

¼ s3=2W     d2

dx2 g 

  x

s

" #ðuÞ

¼ s3=2   d2

dx2  W    g    x

s

h iðuÞ

¼ s2   d2

dx2  W    g sðuÞ

  (10)

For brevity, the multiplying factor s2 in the output of Eq. (10) is provisionally omitted. Thus,  W    g ″sðuÞ in the left-hand-sideterm of Eq.  (10) comprises two consecutive actions that appear in the ultimate right-hand-side term: (i) the convolution

of  W ð xÞ with the s-scaled u-translated Gaussian function g u;sð xÞ:  W    g sðuÞ; and (ii) the second-order differentiation of theconvolution   W    g sðuÞ:   d

2

dx2  W    g sðuÞ

, i.e., the modal curvature of   W    g sðuÞ. To this end, Eq.   (10)   implies that theconvolution of   W ð xÞ   with the Mexican wavelet   g ″sð xÞ,   W    g ″sðuÞ, produces a new modal curvature of   W    g sðuÞ,d2

dx2  W    g sðuÞ

. Considering the factors of  wavelet  and   curvature   in Eq.  (10),  W ns ðuÞ ¼W    g ″sðuÞ  is defined as a WT modalcurvature.

The WT modal curvature exhibits the intrinsic multiscale property of wavelets  [19,27]. That multiscale property entails

decay of the noise-related wavelet coefficients in W ns ðuÞ with an increase in the scale  s  of the wavelet g ″sð xÞ, which is greatlyadvantageous for characterizing damage by tolerating noise. On the other hand, blunting of the wavelet coefficients for

a singular feature induced by damage occurs concomitantly, as a result of greater global fluctuation of wavelet coefficients

due to the wider effective support of the analyzing wavelets. This action of blunting of damage features is of course

disadvantageous for damage depiction.

 2.3. TEO-WT modal curvature

To take advantage of the merits of the WT modal curvature in suppressing noise, while counteracting its disadvantage in

portraying the feature of damage, the TEO described in Eq.   (6)   is adopted to enhance the WT modal curvature for

heightened characterization of damage. Let W ns ½n be the discrete form of the WT modal curvature  W ns ðuÞ. Implementation of the TEO on  W ns ½n  is expressed as

Ψ ðW ns ½n Þ¼ðW ns ½nÞ2W ns ½n1W ns ½nþ1:   (11)In Eq. (11),  Ψ ðW ns ½nÞ  is defined as the TEO-WT modal curvature. This new curvature displays some favorable features fordamage detection: (1) a multiscale mechanism inherited from WT appropriate for suppressing noise  [19,20]; and (2) strong

ability to intensify local singular features while removing the global trend of the signal under analysis  [21,22]. That being the

case, the TEO-WT modal curvature can maximize sensitivity to damage and accuracy of damage localization in a noisycondition.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –1553   1545

8/19/2019 2014 - Damage Identification-Modal Curvature

4/11

3. Analytical investigation

The proposed modal curvature is comprehensively investigated using a series of analytical cases of cracked pinned–

pinned (P–P), clamped–free (C–F) and clamped–clamped (C–C) Euler–Bernoulli beams, with particular emphasis on

characterizing damage in noisy conditions.

 3.1. Analytical models of cracked beams

The procedure for establishing analytical models of cracked beams is exemplified on the P–P beam shown in Fig. 1(a). The

beam is divided into two segments by the crack. Given that transverse vibration of an Euler –Bernoulli beam concerns only

the bending deformation of the beam, the crack is modeled as a linear rotational spring  [28–31] with its bending spring

constant determined by fracture mechanics principles  [32–33]:

K ¼ 1=c ;   c ¼ 6π hJ ða=hÞ=bEI ;   (12.1)where E  is Young's modulus,  I  is the area moment of inertia, K  is the bending spring constant, b is the width of the beam,  a

and h  are the depths of the crack and the beam, respectively,  ξc ¼ a=h is the crack depth ratio, and J ða=hÞ or J ðξc Þ is given by J ðξc Þ ¼ 1:86ðξc Þ23:95ðξc Þ3þ16:37ðξc Þ4þ37:22ðξc Þ5

þ76:81ðξc Þ6þ126:9ðξc Þ7

þ172:5ðξc Þ8

144ðξc Þ9

þ66:6ðξc Þ10

:   (12.2)

As described in [34], the data on which Eq. (12.1) is based was obtained experimentally by Brown and Srawley [35], and the

related formulas for the stress intensity factor of the crack depth can be found in several handbooks, e.g., Ref.  [36].

Spatial governing equations for transverse vibration of beam segments, jointly forming the basis for the global solution to

the vibration of the entire beam, can be described as

W ″″;iðζ Þ λ4W iðζ Þ ¼ 0;   i¼ 1; 2;   (14)where  ζ A ½0; 1, the normalized length of the beam,  W iðζ Þ  is the transverse deflection shape of the  ith beam segment, thefour primes denote the fourth order derivative of  W iðζ Þ and λ ¼ L

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 ρS =EI 4

p   with L,  ω,  ρ  and  S  being the length of the beam,

circular frequency, material density, and cross-section area, respectively.

The general solution of Eq.  (14) is expressed as [37]

W iðζ Þ ¼ Ai   cos ð λζ ÞþBi   sin ð λζ ÞþC i  coshð λζ ÞþDi sinhð λζ Þ;   i¼ 1; 2:   (15)The four boundary conditions at the ends of the P–P beam are specified by

W 1ð0Þ ¼ 0;   W ″;1ð0Þ ¼ 0;   W 2ð1Þ ¼ 0; W ″;2ð1Þ ¼ 0;   (16)and the four compatible conditions at the location of the crack are described as

W 1ðζ c Þ ¼W 2ðζ c Þ;   K ðW ′;1ðζ c ÞW ′;2ðζ c Þ Þ¼EIW ″;1ðζ c Þ;   W ″;1ðζ c Þ ¼W ″;2ðζ c Þ;   W ″′;1ðζ c Þ ¼W ″′;2ðζ c Þ;   (17)where ζ c  is the crack location ratio, specifying the distance from the left end of the beam.

Substituting Eq. (15) into Eqs. (16) and (17), a group of simultaneous equations with respect to the circular frequency can

be obtained as

DðωÞC¼ 0;   (18)where  C   is a column vector of   Ai,  Bi,   C i   and  Di,   i¼ 1; 2;   and  DðωÞ   is an 8 8 matrix. To find the nontrivial solution, thedeterminant of  D

ðω

Þ should be equal to zero:

detðDðωÞÞ ¼ 0:   (19)

Fig. 1.  Mode shapes of cracked beams. (a) P–P beam; (b) C–F beam; and (c) C–C beam.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –15531546

8/19/2019 2014 - Damage Identification-Modal Curvature

5/11

where det denotes the operator to calculate the determinant of  DðωÞ. According to Eq. (19), a series of natural frequencies  ω jcan be solved. Provided with ω j, we can obtain the coefficient vector  C from Eq. (18). Substituting ω j  and C into Eq. (15) yields

the corresponding mode shape  W  j.

The procedure for solving mode shapes for the P–P beam is of generality aside from the particularity in the boundary

conditions. In what follows, this procedure is also used to solve mode shapes for a C–F beam (Fig. 1(b)) with the boundary

conditions of 

W 1

ð0Þ ¼

0;   W ′;1

ð0Þ ¼

0;   W ″;2

ð1Þ ¼

0;   W ″′;2

ð1Þ ¼

0;   (20)

and for a C–C beam (Fig. 1(c)) with the boundary conditions of 

W 1ð0Þ ¼ 0;   W ′;1ð0Þ ¼ 0;   W 2ð1Þ ¼ 0;   W ′;2ð1Þ ¼ 0:   (21)

 3.2. Damage identification

The performance of the TEO-WT modal curvature in identifying damage under noisy conditions is investigated for a wide

spectrum of crack scenarios in a progressive manner in terms of modal curvature, WT modal curvature, and TEO-WT modal

curvature. Each crack scenario is specified by beam type, crack location ratio  ζ c , crack depth ratio  ξc , and mode order j  used

for damage detection.  Table 1 lists the representative nine crack scenarios, where each mode shape has a signal-to-noise

ratio (SNR) of 80 dB as a result of adding Gaussian white noise to the mode shape yielded by the analytical procedure

described in Section 3.1.

 3.2.1. Method demonstration

Crack scenarios I, II and III in Table 1, as graphically displayed in Fig. 1, are employed to demonstrate the TEO-WT modal

curvature by comparing it with the WT modal curvature and the conventional modal curvature.

(1) Modal curvature

For mode shapes W 1, W 3  and  W 5  (Fig. 1) incorporating noise of 80 dB, related to crack scenarios I, II and III, respectively,

the associated modal curvatures W ″1, W ″

3  and  W ″

5  computed using the second-order central difference are presented in

Figs. 2–4(a), respectively. In each figure, the noise amplified by the second-order central difference severely impairs the

profile of the modal curvature, from which it is impossible to differentiate any features of damage. These scenarios

clearly illustrate the susceptibility of modal curvature to noise.

 Table 1

Crack scenarios used in damage identification.

Crack scenario Beam type Crack location ratio ðζ c Þ   Crack depth ratio ðξc Þ   Mode order ( j)

I P–P 0.65 0.2 1

II C–F 0.3 0.15 3

III C–C 0.45 0.25 5

IV P–P 0.3 0.2 1

V P–P 0.5 0.2 1

VI P–P 0.8 0.2 1

VII C–C 0.45 0.15 5

VIII C–C 0.45 0.2 5

IX C–C 0.45 0.3 5

Fig. 2.   Modal curvature (a), WT modal curvature (b), and TEO-WT modal curvature (c) for crack scenario I.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –1553   1547

8/19/2019 2014 - Damage Identification-Modal Curvature

6/11

Fig. 3.   Modal curvature (a), WT modal curvature (b), and TEO-WT modal curvature (c) for crack scenario II.

Fig. 4.   Modal curvature (a), WT modal curvature (b), and TEO-WT modal curvature (c) for crack scenario III.

Fig. 5.  TEO-WT modal curvatures for crack scenarios IV (a), V (b), and VI (c).

Fig. 6.  Planforms of TEO-WT modal curvatures for crack scenarios IV (a), V (b), and VI (c).

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –15531548

8/19/2019 2014 - Damage Identification-Modal Curvature

7/11

(2) WT modal curvature

According to Eq.  (9), mode shapes  W 1,  W 3   and  W 5   (Fig. 1) are processed by the Mexican wavelet, producing the WT

modal curvatures W n1;s,  W n

3;s  and W n

5;s, as shown in Figs. 2–4(b), respectively. In each WT modal curvature, the noise is

Fig. 7.  TEO-WT modal curvatures for crack scenarios VII (a), VIII (b), and IX (c).

c=0.3 (IX)

c=0.25 (III)

c=0.2 (VIII)

c=0.15 (VII)

c=0.3 (IX)

c=0.25 (III)

c=0.2 (VIII)

c=0.15 (VII)

c=0.3 (IX)

c=0.25 (III)

c=0.2 (VIII)

c=0.15 (VII)

Fig. 8.   Correlation between maximum height of singular peak with crack depth for crack scenarios VII, VIII, III, and IX at the scales s ¼ 10 (a),  s ¼ 15 (b),and  s ¼ 20 (c).

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –1553   1549

8/19/2019 2014 - Damage Identification-Modal Curvature

8/11

considerably suppressed, with an increase in the scale, conducive to damage identification, but the global trend of the

WT modal curvature is so dominant that the damage feature, signified by the small upsurge around the crack position, is

too insignificant to ascertain the crack.

(3) TEO-WT modal curvature

As per Eq.  (10), the WT modal curvatures  W n1;s,  W n

3;s   and  W n

5;s  are further treated by the TEO, producing the TEO-WT

modal curvatures Ψ ðW n1;sÞ, Ψ ðW n3;sÞ, and Ψ ðW n5;sÞ, as shown in Figs. 2–4(c), respectively. In each TEO-WT modal curvature,the global trend of the modal curvature is largely eliminated and a unique singular peak stands out dramatically, clearly

indicating the presence and the location of the crack.

These results of crack identification for P–P, C–F and C–C beams demonstrate that the capability of the WT modal

curvature surpasses that of conventional modal curvature, and the capability of the TEO-WT modal curvature further

exceeds that of the WT modal curvature in characterizing damage in noisy conditions.

 3.2.2. Method verification

After the conceptual method demonstration using the first three crack scenarios, crack scenarios IV –IX are used to

illustrate the capabilities of the TEO-WT modal curvature to locate and quantify cracks.

(1) Crack location

Use of the TEO-WT modal curvature to locate damage is investigated for crack scenarios IV, V, and VI ( Table 1) of SNR 

80 dB, pertinent to a crack with a stepwise shift in location. For each crack scenario, the TEO-WT modal curvature

(Fig. 5) obtained from the fundamental mode shape is dominated by an abrupt peak around the crack location,

signifying the presence of the crack; moreover, the singular peak behaves like a unique strip of higher intensity in

the planform (Fig. 6) of each TEO-WT modal curvature, and the finer end of the strip pinpoints the crack location.

The identified crack locations 0.29, 0.51, and 0.82 for crack scenarios IV, V, and VI are in good agreement with the actual

crack locations 0.3, 0.5, and 0.8 (Table 1), respectively, demonstrating that the TEO-WT modal curvature can locate a

crack under noisy conditions with great accuracy.

Fig. 9.   Experimental setup. (a) Electromagnetic shaker; (b) Beam component containing the crack.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –15531550

8/19/2019 2014 - Damage Identification-Modal Curvature

9/11

(2) Crack quantification

Use of the TEO-WT modal curvature to quantify damage is examined with crack scenarios VII ðξc ¼ 0:15Þ, VIII ðξc ¼ 0:2Þ,III ðξc ¼ 0:25Þ, and IX ðξc ¼ 0:3Þ (Table 1) in the order of evenly increased depth for a crack. For crack scenarios VII, VIII, IX,the TEO-WT modal curvatures (Fig. 7(a)–(c), respectively) arising from the fifth mode shape are dominated by a singular

peak caused by the crack. Clearly, the magnitude of the singular peak increases with the increase in crack depth. To

clarify the augmentation, Fig. 8 presents three groups of  ζ   aligned slices of singular peaks for crack scenarios VII, VIII,III, and IX, at the scales 10, 15, and 20, respectively. In each group of slices, the maximum height of the singular peak

distinctly correlates with the depth of the crack. In Figs. 7 and 8, the dependence of the maximum height of the singularpeak on the severity of damage indicates that the TEO-WT modal curvature can serve to quantify a crack in beams in

noisy conditions.

4. Experimental validation

4.1. Experimental setup

A cantilever beam (aluminum 6061) of length 543 mm, width 30 mm and height 8 mm is considered. A through-width

transverse crack, 1.2 mm long (along the beam span) and 2 mm deep is located 293 mm distant from the fixed end.

The mode shapes of the beam are obtained using an electromechanical shaker (B&K 4890) as an actuator and a SLV (Polytec

PSV-400) as a sensor. The experimental setup is shown in Fig. 9.

4.2. Results

Identification of damage using the TEO-WT modal curvature is exemplified by the fourth and the seventh mode shapes.

Fig. 10(a) presents the fourth mode shape at the natural frequency (655 Hz), for which the modal curvature, WT modal

curvature, and TEO-WT modal curvature are shown in  Fig. 10(b)–(d), respectively. In Fig. 10(b), several higher-magnitude

singular peaks appear in the noisy modal curvature, which could induce false alerts of the occurrence of damage; in  Fig. 10

(c), a small upsurge around 300 mm from the fixed end can be slightly discerned in the WT modal curvature but it is too

insignificant to indicate unequivocally the occurrence of damage; in  Fig. 10(d), a marked singular peak rises sharply in the

Fig. 10.  Experimental identification of a crack in a cantilever beam. (a) the fourth mode shape; (b) modal curvature; (c) WT modal curvature; (d) TEO-WTmodal curvature.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –1553   1551

8/19/2019 2014 - Damage Identification-Modal Curvature

10/11

TEO-WT modal curvature, pinpointing the crack at 295 mm from the fixed end of the beam. This identification result is in

good agreement with the actual crack at 293 mm from the fixed end. A better result of the crack identification can be foundin the seventh mode shape, as shown in  Fig. 11, where an enhanced singular peak in the TEO-WT modal curvature much

more clearly indicates the crack.

Similar results can be observed for most of the other mode shapes, demonstrating that the TEO-WT modal curvature can

be used to identify a crack in an actual aluminum beam.

5. Conclusions

In structural health monitoring, modal curvature is a dynamic feature commonly used for detection of structural damage.

A challenge to modal curvature, however, is its susceptibility to noise, which severely impairs its applicability to identify the

damage in noisy conditions. To tackle this challenge, this study develops a TEO-WT modal curvature that exhibits two

prominent features for characterizing damage: high sensitivity to damage and strong immunity to noise. These features are

comprehensively demonstrated by comparisons with conventional modal curvatures on a wide spectrum of numerical casesof cracked beams. The practicability of the TEO-WT modal curvature is experimentally validated using an aluminum beam

with a slight crack. It should be noted that, aside from technological aspects, the formulation of the WT modal curvature is a

theoretical highlight of this study, which creates a new regime for observing curvature features of a mode shape in a

multiscale space. This regime endows the TEO-WT modal curvature with the ability to pinpoint the essential feature of 

damage by eliminating noise interference.

 Acknowledgments

M. Cao and W. Ostachowicz are grateful for a Marie Curie Industry Academia Partnership and Pathways Grant (Grant no.

251309 STA-DY-WI-CO) within the 7th European Community Framework Programme). W. Xu acknowledges the partial

support provided by a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201050)and a National Natural Science Foundation of China (Grant no. 11172091).

Fig.11.  Experimental identification of a crack in a cantilever beam. (a) the seventh mode shape; (b) modal curvature; (c) WT modal curvature; (d) TEO-WT

modal curvature.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –15531552

8/19/2019 2014 - Damage Identification-Modal Curvature

11/11

References

[1]   F.P. Kopsaftopoulos, S.D. Fassois, Vibration based health monitoring for a lightweight truss structure: experimental assessment of several statisticaltime series methods,   Mechanical Systems and Signal Processing  24 (7) (2010) 1977–1997.

[2]  M. Kulisiewicz, R. Iwankiewicz, S. Piesiak, An identification technique for non-linear dynamical systems under stochastic excitations,  Journal of Soundand Vibration  200 (1) (1997) 31–40.

[3]  S.S. Wang, Q.W. Ren, P.Z. Qiao, Structural damage detection using local damage factor,   Journal of Vibration and Control 12 (9) (2006) 955–973.[4]  F. Semperlotti, K.W. Wang, E.C. Smith, Localization of a breathing crack using nonlinear subharmonic response signals,   Applied Physics Letter  95 (25)

(2009). (254101-1–254101-3).

[5]  L. Cheng, G.Y. Tian, Surface crack detection for carbon fiber reinforced plastic (CFRP) materials using pulsed eddy current thermography,   IEEE Sensors Journal 11 (12) (2011) 3261–3268.

[6]  N. Viet Ha, J.-C. Golinval, Damage localization in linear-form structures based on sensitivity investigation for principal component analysis,  Journal of Sound and Vibration  329 (21) (2010) 4550–4566.

[7]  W. Ren, G. De Roeck, Structural damage identification using modal data. I: simulation verification,  Journal of Structural Engineering  128 (2002) 87–95.[8]  G.Y. Xu, W.D. Zhu, B.H. Emory, Experimental and numerical investigation of structural damage detection using changes in natural frequencies,  ASME 

 Journal of Vibration and Acoustics  129 (2007) 686–700.[9]   Y.H. An, J.P. Ou, Experimental and numerical studies on model updating method of damage severity identification utilizing four cost functions,

Structural Control and Health Monitoring  20 (1) (2013) 107–120.[10]  A.K. Pandey, M. Biswas, M.M. Samman, Damage detection from changes in curvature mode shapes,  Journal of Sound and Vibration  145 (1991) 321–332.[11]  M.M. Abdel Mahab, G. De Roeck, Damage detection in bridges using modal curvatures: application to a real crack scenario,   Journal of Sound and

Vibration 226 (2) (1999) 217–235.[12]  T.M. Whalen, The behavior of higher order mode shape derivatives in damaged, beam-like structures, Journal of Sound and Vibration  309 (3–5) (2008)

426–464.[13]   K. Roy, S. Ray-Chaudhuri, Fundamental mode shape and its derivatives in structural damage localization, Journal of Sound and Vibration  332 (21) (2013)

5584–5593.[14]   M.S. Cao, P.Z. Qiao, Novel Laplacian scheme and multisolution modal curvatures for structural damage identification,   Mechanical Systems and Signal

Processing  29 (2009) 638–

659.[15]  E. Sazonov, P. Klinkhachorn, Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes,   Journal of Sound and

Vibration 285 (2005) 783–801.[16]   B.H. Kim, T. Park, G.Z. Voyiadjis, Damage estimation on beam-like structures using the multi-resolution analysis,  International Journal of Solids and

Structures 43 (2006) 4238–4257.[17]  M. Chandrashekhar, R. Ganguli, Damage assessment of structures with uncertainty by using mode shape curvatures and fuzzy logic,   Journal of Sound

and Vibration  326 (2009) 939–957.[18] R. Chartrand, Numerical differentiation of noisy, nonsmooth data,   ISRN Applied Mathematics;   http://dx.doi.org/10.5402/2011/164564. (Article ID

164564).[19]   S. Mallat,   A Wavelet Tour of Signal Processing , third edition, Academic Press, San Diego, 2008.[20]  Y.Q. Bao, H. Li, Y.H. An, J.P. Ou, Dempster–Shafer evidence theory approach to structural damage detection,   Structural Health Monitoring  11 (2012)

13–26.[21]  H.M. Teager, S.M. Teager,  A Phenomenological Model for Vowel production in the Vocal Tract , College-Hill Press, San Diego, 1983.[22] J.F. Kaiser, On a simple algorithm to calculate the energy of a signal,   IEEE Proceeding, ICASSP-90, 1990, pp. 381–384.[23]   A.B. Stanbridge, M. Martarelli, D.J. Ewins, Measuring area vibration mode shapes with a continuous-scan LDV,  Measurement  35 (2004) 181–189.[24]  H. Weisbecker, B. Cazzolato, S. Wildy, S. Marburg, J. Codrington, A. Kotousov, Surface strain measurements using a 3D scanning laser vibrometer,

Experimental Mechanics  52 (2012) 805–815.

[25] E. Kvedalen, Signal Processing Using the Teager Energy Operator and Other Operators, Candidatus Scientiarum Thesis, University of Oslo, 2003.[26]  A.C. Bovik, P. Maragos, T.F. Quatieri, AM–FM energy detection and separation in noise using multiband energy operators,   IEEE Transactions on Signal

Processing  41 (1993) 3245–3265.[27]  S. Mallat, W.L. Hwang, Singularity detection and processing with wavelets,   IEEE Transaction on Information Theory  40 (1992) 617–643.[28]  W.M. Ostachowicz, M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam,  Journal of Sound and Vibration  150 (2)

(1991) 191–201.[29]   M. Behzad, A. Ebrahimi, A. Meghdari, A new continuous model for flexural vibration analysis of a cracked beam,  Polish Maritime Research  15 (2008)

32–39.[30]   M.S. Cao, L. Ye, L.M. Zhou, Z.Q. Su, Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams,

Mechanical Systems and Signal Processing  25 (2011) 630–643.[31]  S.A. Paipetis, A.D. Dimarogonas, Identification of crack location and magnitude in cantilever beam from the vibration modes,  Journal of Sound and

Vibration 138 (1990) 381–388.[32]   H. Liebowitz, H. Vanderveldt, D.W. Harris, Carrying capacity of notched column,   International Journal of Solids and Structures  3 (1967) 489–500.[33]  H. Liebowitz, W.D. Claus, Failure of notched columns,  Engineering Fracture Mechanics  1 (1968) 379–383.[34]   A.D. Dimarogonas, Vibration of cracked structures: a state of the art review,  Engineering Fracture Mechanics  55 (1996) 831–857.[35]  F.W. Brown, J.E. Srawley, Plane strain crack toughness testing of high strength metallic materials,  ASTM STP  410 (1966) 12.[36]  H. Tada, P.C. Paris, G.R. Irwin,  The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, Pennsylvania, USA, 1985.

[37]   S.S. Rao,   Vibration of Continuous Systems, John Wiley &  Sons, Inc., New Jersey, 2007.

M. Cao et al. / Journal of Sound and Vibration 333 (2014) 1543 –1553   1553