Engineering Fracture Mechanics 74 (2007) 2992–3001

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Numerical and experimental verification of a techniquefor locating a fatigue crack on beams vibrating under

Gaussian excitation

S. Benfratello a, P. Cacciola b, N. Impollonia c, A. Masnata d, G. Muscolino b,*

a Dipartimento di Ingegneria Strutturale e Geotecnica, Universita di Palermo, Palermo 90128, Italyb Dipartimento di Ingegneria Civile, Universita di Messina, Messina 98166, Italy

c Dipartimento ASTRA, Universita di Catania, Siracusa 96100, Italyd Dipartimento di Tecnologia Meccanica, Produzione e Ingegneria Gestionale, Universita di Palermo, Palermo 90128, Italy

Received 28 November 2005; received in revised form 16 May 2006; accepted 1 June 2006Available online 9 November 2006

Abstract

The stationary vibrations of a beam excited by Gaussian noise are strongly affected by the presence of a fatigue crack.Indeed, as soon as the crack arises the system response becomes non-linear due to crack breathing and a non-Gaussianbehaviour is encountered. The paper presents both numerical and experimental investigations in order to assess the capa-bility of the non-Gaussianity measures to detect crack presence and position. Monte Carlo method is applied to evaluate intime domain the higher order statistics of a cantilever beam modelled by finite elements. The skewness coefficient of therotational degrees of freedom appears the most suitable quantity for identification purpose being very sensitive to thenon-linear behaviour of the cracked beam.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Fatigue crack; Non-linear system; Crack detection; Skewness; Non-Gaussian PDF

1. Introduction

Fatigue cracks are often present in structural components which are subjected to repeated loads and maycause structural failure if they are not promptly detected. A suitable model for identification of fatigue cracksmust take into account that the crack will open and close in time, depending on the rotation and vibrationamplitudes, causing a non-linear behaviour. As a consequence, the ‘‘linear spring hinge model’’, used in theliterature to describe the dynamic response of notched beam or beam with cracks always kept open, is noteffective for detecting fatigue cracks. This model is based on the concept that an open crack preserves the lin-ear behavior and introduces only a local stiffness reduction which alters the frequency response function, the

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doi:10.1016/j.engfracmech.2006.06.023

* Corresponding author. Tel.: +39 090 3977159; fax: +39 090 3977480.E-mail address: muscolin@ingegneria.unime.it (G. Muscolino).

S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001 2993

natural frequencies and the mode shapes. Such vibration data can be directly exploited to assess the presenceof a notch or an open crack but are less appropriate damage indicator for the case of fatigue cracks due to thedifficulty to relate their variation to non-linear effects produced by crack breathing [1,2]. Chondros et al. [3]evidenced that using an open crack model assumption to interpret vibration measurements for a fatiguebreathing crack will lead to incorrect conclusions, in particular one would presume that the crack severityis smaller than what it really is.

The accurate prediction of the dynamical response of a beam with a fatigue crack is not an easy task. Manymodels have been proposed in the literature employing various approaches with different assumptions anddegree of complexity. Shen and Chu [4] formulated a bilinear equation of motion for each vibration modeby using a Galerkin procedure. They evaluated time history and spectrum information of the beam under aconcentrated force and observed a clear non-linear behavior for each vibration mode. Pugno et al. [5] pro-posed a method based on the ‘‘harmonic balance’’ approach to evaluate the dynamic response of a beam withmultiple breathing cracks to an applied sinusoidal force. The procedure allows a remarkable time saving overdirect numerical integration and is able to reveal the super-harmonic components produced by the non-linearvibrations. The presence of super-harmonic components in the Fourier spectrum has been also proposed byChu and Shen [6] as an indicator of the discontinuity in stiffness resulting from a breathing crack. A morecomplex model was presented by Abraham and Brandon [7] who observed that the stiffness of a structure con-taining a real fatigue crack may change continuously with time as the load oscillates because of roughnessinterface.

Non-linear effects make the response of beams with a fatigue breathing crack more difficult to predict thanthat of a notched beam. However, on the other hand, their presence accentuate the difference of the damagedstate from the undamaged one. Then, if a vibration measure which is sensitive to presence of non-linear behav-iour is selected, the identification of a fatigue crack, which is more likely to appear on real structure, is moreeasy to achieve than that of an always open crack. Some researchers [8,9] have found that the stochasticresponse produced by a random excitation can be very sensitive to the non-linear behaviour of the system,that is to say to the presence of the fatigue crack in the structure.

In fact, if the beam is undamaged (linear state) and a Gaussian load is applied, the structural response willbe Gaussian too. On the contrary, as soon as non-linearity arises, due to the appearance of a crack, theresponse becomes non-Gaussian. Inspection of the integrity of the structural element may be performed byestimating the non-Gaussianity of the response, through the evaluation of higher order statistics. The non-Gaussianity may be revealed in several ways. For example the kurtosis or skewness coefficient for one or moredegrees of freedom may be adopted as a measure of the non-Gaussianity of the system. Moreover, time orfrequency domain analysis can be performed. If the crack depth is important (let say greater than half ofthe beam height) the non-linearity is evident and any of such measures is suitable. However in practical appli-cation the crack dimension is usually quite small and it is much more cumbersome to assess the presence of thecrack. This is due to the slight non-linearity of the system whose behaviour is very similar to the linear one (thebeam without any crack). In this case the choice of the measure to detect the non-Gaussianity is a crucial pointto give clear information on the beam condition.

A procedure based on the bispectral analysis for the detection of a crack in a straight beam, modelled as asingle degree of freedom system, has been employed in the frequency domain [8], using the response dataresulting from Gaussian white noise excitation. The bispectrum, or third-order polyspectrum, can be viewedas a decomposition of the skewness of a signal over frequency, therefore can be adopted to detect the non-Gaussianity in the response due to the non-linearity arising in the system. However, if the breach is smallone can achieve ambiguous interpretation of the results, as numerical applications reveal [8]. Moreover, theprocedure, when extended to multi-degree-of-freedom systems, seems unable to give any information onthe position of the crack when it is present.

The stochastic dynamic analysis of a cracked cantilever beam under white noise has also been addressed inthe time domain [9]. The beam was discretised by finite elements in which a so-called closing crack model, withfully open or fully closed crack, is used to describe the damaged element. Then, the response was evaluated byapplying a numerical procedure based on the philosophy of structural systems with dynamic modifications [10].

In this paper numerical and experimental verification of the procedure proposed in [9] are performed. Thestatistics of the stochastic response are evaluated by means of Monte Carlo simulation. The skewness and

2994 S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001

kurtosis coefficients are evaluated at nodal degrees of freedom. A remarkable variability of the coefficientsevaluated at the rotational degrees of freedom for different crack locations and depths is encountered, assess-ing the capability of this measure to detect both the presence and the position of the crack along the beam evenif the crack depth is small.

2. Preliminary concepts

The presence of a crack in the beam, according to the Saint Venant’s principles, causes a perturbation of thestress field in the neighbourhood of the breach. Such a perturbation is relevant especially when the crack isopen and determines a local reduction of the flexural rigidity. On the other hand, when the crack is closedthe beam acts, approximately, as a homogeneous beam with no crack. A natural choice is a finite element for-mulation of the problem. In this case, the properties of the cracked element, changing during motion, deter-mine the non-linearity of the whole system, being constant the parameter modelling all the other elements.According to several authors [5,11,12] the stiffness matrix is the structural property that is most affected fromthe breathing of the crack, as damping and mass matrices do not change appreciably during the opening andclosure of the crack.

In the following the finite element model for the cracked beam with an on-edge non-propagating crack, pro-posed by Qian et al. [12] is adopted and the formulation presented in [9] is briefly summarized. Undamagedelements of the beam are modelled by Euler type finite elements with two nodes and two degrees of freedom(transverse displacement and rotation) at each node. The cracked element is modelled as an undamaged ele-ment if the crack is closed whereas it exhibits a more flexible behaviour if the crack is open. A non-propagatingcrack will be considered so that the characteristics of the cracked element in the crack open phases do notchange during the motion. Furthermore, the crack has two perfectly flat surfaces so that can only exist inthe fully open or fully closed states. The problem is of piecewise interval nature in time domain, the closureand opening phases being governed by two different sets of linear equations.

The stiffness matrix of the undamaged element with rectangular cross section is here assumed to be thatgiven by Bernoulli–Euler theory with Hermite shape functions:

ku ¼

Ebh3

l3Ebh3

2l2 � Ebh3

l3Ebh3

2l2

Ebh3

3l � Ebh3

2l2Ebh3

6l

Ebh3

l3 � Ebh3

2l2

sym Ebh3

3l

266666664

377777775

ð1Þ

where E is the Young’s modulus, l is the length of the finite element, b and h are the width and the height of thebeam cross section.

If an edge crack with depth a is present in the element, the stiffness matrix of that damaged element will bestill given by Eq. (1) if the crack is in the close phase. By applying fracture mechanics methods, it is possible todetermine the modification introduced in the stiffness matrix when the crack is in the open phase. To this aimone can evaluate the additional strain energy related to an element with an open crack so to determine first itsflexibility matrix and then its stiffness matrix accordingly to the principal of virtual work [12]. The stiffnessmatrix for a cracked element during the open phase, kc, is a quite involved expression in terms of the beamparameters and crack depth. The following convenient relationship has been determined in [9] by simple ana-lytical manipulation:

kc ¼ a1

ku11a2 ku12

a2 ku13a2 ku14

a2

ku22a3 ku23

a2 ku24a4

ku33a2 ku34

a2

sym ku44a3

2666664

3777775

ð2Þ

S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001 2995

In Eq. (2) kuij are the elements of the matrix in Eq. (1) whereas the coefficients ai (i = 1,2,3,4) can be easilydetermined once the ratio between the crack depth and the height of the element s = a/h is fixed [9]. For exam-ple, assuming s = 0.4 one gets

a1 ¼1

0:556r4 þ 0:303r3 þ 1:832r þ 1

a2 ¼ ð1:832r þ 1Þa3 ¼ ð0:0758r3 þ 1:374r þ 1Þa4 ¼ ð�0:152r3 þ 2:748r þ 1Þ

ð3Þ

being r = h/l the ratio between the height and the length of the cracked element. The matrix form given by Eq.(2) along with the coefficients a1, a2, a3, and a4 (Eq. (3)) allow consistent exemplification in the evaluation ofthe stiffness property of the cracked element. The value of the coefficients ai (i = 1, . . . , 4) for different ratios s

are reported in [9].

3. Crack detection by measure of response to Gaussian excitation

The dynamic response of the beam in the time intervals the crack is closed may be regarded as that of abeam without crack, because the crack interfaces are completely in contact with each other. Under the actionof the excitation force, crack opening and closure will alternate as a function of time.

The equations of motion of a cracked beam discretised by Ne finite elements and subjected to an externalexcitation vector f(t) can be written as

M€uðtÞ þ C _uðtÞ þ ðKu � cDKÞuðtÞ ¼ fðtÞuð0Þ ¼ u0; _uð0Þ ¼ _u0

ð4Þ

in which M is the mass matrix, C is the damping matrix, Ku is the stiffness matrix of the undamaged beam, u(t)is the displacement vector of the nodal points of order N · 1. A dot over a variable denotes differentiation withrespect to time. It has been assumed that the change between fully open and fully closed takes place instan-taneously giving rise to a bilinear-type stiffness non-linearity. The change in the global stiffness matrix due tothe crack is

DK ¼ Ku � Kc ð5Þ

where Kc is the stiffness matrix of the damaged beam and

c ¼1 when the crack is open

0 when the crack is closed

�: ð6Þ

The global matrix Ku is evaluated assembling Ne matrices ku whereas the global matrix Kc assembling Ne � 1matrices ku and the matrix kc.

To determine the state of the crack it is sufficient to evaluate the slopes /i and /i+1 of the response defor-mation at the nodes i and i + 1, closest to the crack. In particular, it is assumed the condition of closing if/i+1 P /i, whereas the crack is open if /i+1 < /i.

A convenient numerical procedure able to solve the bilinear equation Eq. (4) is given in [13] and will beutilised in the numerical computation. It is well known from stochastic mechanics that a linear system drivenby a Gaussian stationary stochastic process undergoes steady state Gaussian stationary oscillations. Thismeans that only first and second order statistical information are sufficient to completely describe the station-ary stochastic response of the system. Moreover, such statistics, namely the mean vector and the covariancematrix of the degrees of freedom, can be easily determined and the exact solution is available in the case theforcing function is a white noise.

As soon as the system is non-linear the stochastic response under Gaussian input is not Gaussian anymore.Then, the presence of non-Gaussianity in the response may be exploited to assess the presence of damage inthe structure. This means that higher order measures such as skewness (third order), kurtosis coefficients

2996 S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001

(fourth order), higher order cumulants or statistical moments may provide details about the response that theconventional second order statistics cannot.

The major issue is the choice of the right parameters to be estimated so to detect the non-Gaussian behav-iour of the structural response. For the case of a beam with an edge crack, the opening and closure of the crackcauses the non-Gaussianity of the nodal displacement and rotation processes. However, if the crack depth issmall in comparison with the beam height then such non-Gaussianity may be overlooked. For these reasons itis important to select the proper higher order statistics at given measure points, in such a way these are muchsensitive to the crack presence. Skewness and kurtosis coefficients can be conveniently used in order to detectthe crack presence. Such coefficients are, respectively, given by:

c3 ¼k3

k3=22

; c4 ¼k4

k22

ð7Þ

where k2, k3 and k4 are the second, third and fourth cumulant of the generic degree of freedom #, namely,

k2 ¼ E½ð#� lÞ2� ¼ r2

k3 ¼ E½ð#� lÞ3�k4 ¼ E½ð#� lÞ4� � 3r4

ð8Þ

being E[Æ] the mean operator, l the mean value and r2 the variance. The skewness coefficient c3 quantifies theasymmetry of the probability density function present in the response due to non-symmetric non-linearity inthe stiffness matrix of the cracked element. The kurtosis coefficient c4 is a measure of whether the data arepeaked or flat relative to a Gaussian distribution. That is, data sets with high kurtosis tend to have a distinctpeak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flattop near the mean rather than a sharp peak. Under white noise Gaussian stationary excitation, the skewnessand kurtosis coefficients vanish if the system is linear, i.e. the beam is undamaged, whereas the coefficients donot vanish if the system is non-linear, i.e. a crack is present. As the crack depth increases, the coefficients gen-erally increase too due to more pronounced non-Gaussian behaviour.

In [9] it has been shown that the skewness coefficient c3 of the rotations appears the best parameter to beinvestigated in order to detect both the presence and the position of the crack. The evaluation of the shapeindex c3 requires the application of stochastic analysis so to get the cumulants needed. Monte Carlo simulationwill be adopted to this aim, and the numerical and experimental results pertaining a cantilever beam underwhite noise Gaussian excitation are reported in the next section.

4. Numerical and experimental analysis

In this section the random response of the cantilever beam with a fatigue crack depicted in Fig. 1 is inves-tigated. The finite element model and the pertinent mechanical and geometrical parameters are also reportedin Fig. 1. The modulus of elasticity and the mass density have been, respectively, obtained performing exper-imental tests on the undamaged beam by a pulse-echo ultrasonic test and by weighting the beam on a balancewith 1 g resolution. The ultrasonic test has been carried out by a narrow-band straight probe with 10 MHzcentral frequency connected to a Gilardoni RG20 instrument and by measuring the response signal in five dif-ferent locations by a Tektronix 540D digital oscilloscope. The recorded times of flight have been comparedwith those obtained by the same test on a calibrated block of aluminium. The results lead to the conclusionof an high non-homogeneity of the material with Young modulus varying in the range 63000–64500 MPa,with the mean value of 63 850 MPa adopted in the numerical application.

The damping coefficient f = 0.015 has been obtained by means of the half-power bandwidth methodapplied to the response of accelerometer n#5 (see Fig. 2) to an impact hammer test. The amplitude of thetransfer function is reported in Fig. 3. Note that the frequencies of 118 Hz, 263 Hz and 352 Hz are due tothe connection of the beam to the shaking table.

The first two natural frequencies determined by the finite element model are x1 = 233.03 Hz andx2 = 1477.54 Hz in agreement with experimental results.

Acc. Dist.1 382 723 944 1045 245,56 255,5

1

234

5 6

Fig. 2. Sketch of the beam along with accelerometer (Acc.) positions The distance (mm) is from the restrain.

Fig. 3. Amplitude of the accelerance of accelerometer 5 to an hammer test.

L

c

c

ü(t)

L

h1

2

3

4

5

6

8

9

10

7

E 63.850 MPa

ρ 2.860 kg/m³ L 333,4 mmh 83,0 mmc 35,0 mm

Fig. 1. Cantilever beam (a) and finite element model (b).

S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001 2997

2998 S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001

4.1. Numerical results

It has been assumed that the beam is subjected to a ground acceleration modelled by a white noise process withpower spectral density S€ug ¼ 0:0031 m3=s2. The numerical step-by step procedure proposed in [13], adopting atime integration step Dt = 0.0001 s and duration 50 s, has been used for the numerical analysis. The randomresponse of the beam has been investigated for various crack depths and positions via a pertinent Monte Carloanalysis. Ergodicity has been exploited to get the stationary coefficients of the response. The skewness, c3, andkurtosis, c4, coefficients have been determined using Eqs. (7) and (8). The results pertinent to nodal displacementsand rotations are reported in Figs. 4 and 5. The figures show the sensitivity of the skewness coefficient to crackpresence. In particular the high sensitivity of the skewness coefficient of the rotational degrees of freedom to thedamage presence is evident in Fig. 4b where a remarkable jump appears between the values of c3 at the left andright side of the crack. The jump is present wherever the crack, shaded in grey in the figure, is positioned. Thecoefficient values for each beam segment bounded by the crack are almost constant, putting on evidence thegap. The jump is then due to crack existence, whereas its position identifies the position of the cracked element.

The values of the kurtosis coefficients for vertical displacements and rotations, respectively, are plotted inFigs. 5a and 5b. Such values differ from zero quite sensibly so they could give some information about theexistence of the crack. On the other hand they do not give any information about the crack position. Onlyfor a remarkable crack depth it can be seen a jump corresponding to the crack position. Note that, for a givensample number the evaluation of the kurtosis coefficient is intrinsically less accurate than that of the skewnesscoefficient, the former requiring higher order statistics with respect to the latter. So that skewness coefficientsgive more reliable results during an experimental test in order to detect both crack presence and position withlower computational effort.

4.2. Experimental results

In order to validate the proposed procedure experimental tests have been carried out for the case of a ver-tical cantilever aluminium beam subjected to an horizontal shaking white noise. The fatigue crack has been

1 10-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

a=0.5ha=0.3ha=0.1h

γ3

γ3

γ3

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

γ3

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

γ3

element

a=0.5ha=0.3ha=0.1h

-0.2

-0.1

0

0.1

0.2

a=0.5ha=0.3ha=0.1h

a=0.5ha=0.3ha=0.1h

a=0.5ha=0.3ha=0.1h

-0.2

-0.1

0

0.1

0.2

a=0.5ha=0.3ha=0.1h

2 3 4 5 6 7 8 9

1 10element

2 3 4 5 6 7 8 9 1 10element

2 3 4 5 6 7 8 9 1 10element

2 3 4 5 6 7 8 9

1 10-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

γ3

element2 3 4 5 6 7 8 9 1 10

element2 3 4 5 6 7 8 9

Fig. 4. Skewness coefficents for various crack depths and positions: (a) displacements; (b) rotations.

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

a=0.5ha=0.3ha=0.1h

γ4

γ4γ4

γ4γ4

γ4

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

a=0.5ha=0.3ha=0.1h

a=0.5ha=0.3ha=0.1h

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

a=0.5ha=0.3ha=0.1h

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

a=0.5ha=0.3ha=0.1h

a=0.5ha=0.3ha=0.1h

1 10element

2 3 4 5 6 7 8 9

1 10element

2 3 4 5 6 7 8 9 1 10element

2 3 4 5 6 7 8 9 1 10element

2 3 4 5 6 7 8 9

1 10element

2 3 4 5 6 7 8 9 1 10element

2 3 4 5 6 7 8 9

Fig. 5. Kurtosis coefficients for various crack depths and positions: (a) displacements; (b) rotations.

S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001 2999

generated by first producing a small notch at the height of 83 mm and therefore by subjecting the beam to athree point bending test with an Instrom resonance testing machine obtaining an apparent depth of the fatiguecrack of 13.5 mm.

The experimental set-up is composed by an Unholtz-Dickie vibration system (shaking table and amplifiermodel TA250), six Bruel &Kjaer model 4382S accelerometers with two Nexus amplifiers. The whole system isdriven by a National Instruments PCI-MIO 16XE10 board and Signal Express 1.1 software. Preliminarily, theUnholtz-Dickie vibration system has been appropriately characterized and the geometric characteristics of thebeam have been selected in such a way that the fundamental bending frequency lies in the flat zone of the sys-tem response. The experimental test has been performed considering a single specimen under four differentvalues of the white noise amplitude with sampling frequency set equal to 10 kHz. The specimen has beenexcited by a single time-history of duration 50 s and Dt = 0.0001 s. The first 0.1365 s have been cutoff. So thatjust the steady-state response has been considered exploiting ergodicity. Consequently, 498634 samples havebeen processed. The accelerometers signals have been double integrated in order to obtain the displacementsand therefore the rotations of the beam at the measuring points. The skewness coefficient of the rotations hasbeen determined by Monte Carlo analysis exploiting the ergodicity of the response. Remarkably, also theexperimental results (see Fig. 6) show a jump where the crack is positioned. This interesting result assessesthe capability of the proposed procedure to detect both the presence and position of the crack. Note thatthe different values of the experimental skewness coefficients with respect to the numerical ones are due tothe simplified closing-crack model assumed in this paper. Specifically, displacements and rotations have beendetermined neglecting the contribution of static forces due to the self weight that contrasts crack opening. Fur-thermore, crack propagation, partial opening or closing of the crack and impact between its surfaces have notbeen considered in the model.

At the end the beam has been cracked in order to evidence the real characteristics of the crack. In Fig. 7 thecross-section of the beam in correspondence of the crack is reported. Three different zones are evident: thenotch (4.5 mm) is located at the bottom of the figure; above the notch is located the fatigue crack (beach-marks), whose path is orthogonal to the edge of the beam; lastly, the zone in which the fracture is obtainedby clivage. The dashed line represents the mean depth of the crack (14.56 mm).

Fig. 7. Cross-section of the beam after rupture (the dashed line represents the mean depth).

-0.01

0

0.01

0.02

0.030 0.11 0.22 0.33

γ3

element

xr [m]

1 102 3 4 5 6 7 8 9

Fig. 6. Skewness coefficients of the rotations experimentally determined.

3000 S. Benfratello et al. / Engineering Fracture Mechanics 74 (2007) 2992–3001

5. Conclusions

In this paper a procedure able to determine the presence and the position of a fatigue crack in a cantileverbeam has been presented. The procedure exploits the idea that the response of linear undamaged beam sub-jected to a Gaussian random process is Gaussian too. On the other hand the presence of a fatigue crack intro-duces a non-linearity in the response behavior. As a consequence the response process will be non-Gaussian. Itfollows that the measure of non-Gaussianity gives information about the presence of the crack. In this paper ithas been shown that the most suitable parameter in order to detect the crack presence is the skewness.Remarkably, the skewness coefficients of the rotation manifest a jump where the fatigue crack is located.Numerical and experimental results confirm the accuracy and the applicability of the proposed procedurein order to detect crack presence and position.

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