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Delamination detection in composite laminate plates using 2D wavelet analysis of modal frequency surface

Chen Yang1 and S Olutunde Oyadiji2 School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, M13 9PL, UK1 [email protected] [email protected]

Corresponding author Dr S Olutunde Oyadiji School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, M13 9PL, UKTel: 00 44 161 275 4348 Fax: 00 44 161 275 3844E-mail address: [email protected]

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Delamination detection in composite laminate plates using 2D wavelet analysis of modal frequency surface

Chen Yang and S Olutunde Oyadiji

School of Mechanical, Aerospace and Civil Engineering,The University of Manchester, M13 9PL, UK

ABSTRACT

A novel damage detection approach, using modal frequency surface (MFS), is presented. By attaching a point mass at different locations, the MFS is generated. Delamination causes a discontinuity in the MFS due to reduction of local stiffness. Finite element method is employed to simulate the modal frequency data. The frequency deviations show a quasi-exponential decrease as the delamination depth increases. MFS wavelet coefficient is computed and its magnitude under different random noise levels is discussed. It is shown that the MFS wavelet coefficient can characterise the location and shape of both near and far surface delaminations in laminated composite plates.

Keywords: delaminations detection; composite laminate; modal frequency surface; continuous wavelet transform; derivatives of Gaussian wavelet

1. Introduction

The use of composite materials, such as fibre reinforced polymers, in modern engineering structures is increasing due to the opportunities they present for weight reduction. In addition to their relatively high specific strength and stiffness, the variety of fibre and matrix combination enables the optimisation in material design and fabrication as well as the other physical and chemical properties. However, the structural integrity of composite structures is easily affected by certain types of damage, such as interlaminate failure [1 - 5]. Therefore, damage detection and localisation for composite structures at earlier stage in various engineering fields is a practical important aspect of structural integrity assessment. There are various non-destructive testing (NDT) methods used particularly in the aerospace sector for identification of delaminations in composite structures. These include high performance methods such as radiography, thermography, shearography and ultrasound techniques. For example, in previous work [6], the authors compared the performance of thermography and ultrasonics for identification of flaws in aerostructures. While they both give very good identification, they are very expensive for general

Corresponding author Tel: 00 44 161 275 4348 Fax: 00 44 161 275 3844E-mail address: [email protected]

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mailto:[email protected]

engineering applications. This cost disadvantage is true for the other high performance techniques listed. There are other less expensive approaches one of these being the structural dynamics-based approaches.

Structural dynamics-based damage detection methods can be categorised into vibration-based and wave propagation-based approaches. The wave propagation methods employ high-frequency waves which are reflected and diffracted by material and/or geometric discontinuities (e.g. inserts, contaminations, cracks and delaminations) to detect structural damages. Several researchers have investigated the application of high-frequency waves in damage detection in laminated aerostructures and some recent work can be found in [7 - 10]. The vibration-based approaches (e.g. modal frequency or mode shape) use the stiffness reduction effect in the low frequency structural responses to detect damage in structures. Due to the fact that low frequency responses are easier to measure for large-size structures (e.g. vehicle body panels and bridges) with a small number of conventional sensors, the use of vibration characteristics in structural damage detection has been studied by several researchers [11 - 34]. The wave propagation method uses a number of high frequency sensors/actuators to identify damage. However, high-frequency waves generally decay fast in large-scale structures due to material and geometric dispersion. Similarly, the vibration-based approach, using a scanning laser vibrometer, which enables fairly accurate damage identification at low frequencies, is also quite expensive. For common engineering fields, the cost-effectiveness of the method is important in application. The method presented in this paper is a vibration-based method which traverses a point mass over the surface of a structure to create a frequency surface. The point mass enhances the dynamic response of the structure such that any delamination in the structure can be identified by processing the frequency surface by wavelet transform. It should be noted that this method does not belong to the traditional frequency shift methods, which use only single modal frequency shift values. The dynamic enhancement due to the point mass and its interaction with the subsurface delamination has been experimentally validated in [20]. The advantage of the method compared to wave propagation and mode shape methods is that it is relatively inexpensive to implement, since it only requires the knowledge of modal frequencies of the structures and this is usually achieved with few sensors.

The literatures have reported the study of the relationship between structural integrity degradation and modal parameters. Many of these reported studies are on surface cracks. In fact, there are more publications on cracks than on delamination which is the focus of this paper. On a superficial level, it may seem that there is no difference between a crack and a delamination. Generally, the effect of a surface crack on vibration response increases as the depth of the crack increases. On the contrary, the deeper the location of a delamination in the through-the-thickness direction, the less effect it has on the vibration response. Nevertheless, the study of cracks of large size civil structures, which span many decades, may serve as a useful reference for the study on delaminations which is more recent with the development of composite structures.

There are many delamination detection approaches that are based on the concept of frequency shifting caused by the structural stiffness reduction [11 – 15]. Nevertheless, damages with different severity in two different locations may produce identical frequency shifting. Zhong and

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Oyadiji [16, 17] proposed a modified base-line free frequency-based method of using additional mass loading to detect and localise cracks in simply-supported beams. In their studies, a point mass is attached on the beam in order to enhance the local dynamic effect of the damage. The modal frequencies of the simply-supported beams with the point mass attached at different locations are derived and plotted against the spatial locations of the point mass to produce a modal frequency curve. The effects of crack depth, crack location, magnitude of auxiliary mass and spatial probing interval are also discussed. Based on the same approach, Zhang et al. [18, 19] numerically and experimentally studied damage identification of a damaged plate and cylinder with free-free conditions. They proposed a baseline-free damage index based on the 2D gap smoothing method (GSM). However, it is known that the numerical differentiation usually magnifies unwanted numerical noise. This becomes important when the detected systems are subjected to random environmental noise in real applications.

In the past few years, several signal processing algorithms for structural damage detection using mode shapes have been reported in the literature. Wang and Deng [21] reported the use of wavelet coefficient in a steel plate with induced crack under tensile loading. Rucka and Wilde [22] proposed the two-dimensional discrete wavelet approach in damage detection for plate-like structures. The location of the crack tip was found by a variation of the Haar wavelet coefficients. Fan and Qiao [23] showed that the 2D wavelet approach provide better damage sensitivity and noise-robustness compared to other damage detection algorithms such as GSM. The use of stationary wavelet transform in crack detection has been investigated in [24]. The combination of Ritz method and wavelet analysis has been proposed by Gallego et al. [25], where a new damage index based on the residual value between the wavelet results and the result generated by Ritz method has been studied numerically. Recently, the application of polynomial annihilation edge detection method for damage detection in cracked beams and plates has been reported by Surace et al. [26, 27]. Katunin [28, 29] proposed the 2D spline wavelet in damage detection of plate-like structures and applied the method in the honeycomb-core sandwich structures. The author also compared the mode shape approach with the conventional NDT methods for aerostructures.

Other vibration-based delamination detection approaches have also been reported in the literature by many authors. Shang et al. [30] provided a model-based method in damage detection of composite structures using modal testing. Aymerich and Staszewski [31] and Klepka et al. [32] reported the use of nonlinear vibro-acoustics responses in impact damage detection in laminate composites through displacement measurement. The use of a time-domain vibration characteristic in delamination detection by a multivariate statistical procedure is also reported by Garcia et.al [33]. Ooijevaar et al. [34] employed modal curvature for delamination detection in composite skin-stiffener structures using laser displacement measurement. Recently, Rama et al. [35] reported a numerical study of a novel delamination detection method based on frequency response functions and principal component analysis. Delamination detection based on vibration responses using sparse sensing piezoelectric sensors was experimentally investigated by [36]. A novel experimental delamination detection method using deflectometry and the virtual fields method has been demonstrated in [37].

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Generally, vibration-based methods using the mode shapes to deduce damage information require high accuracy measurement of displacement or strain responses. The literature has reported the use of the scanning laser vibrometer in the displacement measurement for aerostructures. However, due to the high cost of such advance instrumentation, their use is limited to advance industries. On the other hand, the frequency-based approaches can be easily implemented using few low cost sensors such as strain gauges and accelerometers as well as plastic fibre optic sensors [38]. It has been also reported in the literature that the combination of the wavelet coefficients of the structural mode shapes improves damage localisation for large-size structures [21 – 24, 27, 28]. As the wavelet transform can be used as an enhancement tool in damage detection, it is expected that the delamination with weaker frequency deviation can be detected through using the wavelet coefficients of frequency surfaces.

In this paper, a delamination detection method for plate-like structures using modal frequency surface and its wavelet coefficients is proposed. The main contributions of the paper are twofold. Firstly, the modal frequency curve approach used in crack detection of beam-like structures is extended to modal frequency surface (MFS) for delamination detection in two-dimensional composites laminated plates. It is found that the introduced local inertia greatly enhances the dynamic response at the delaminated section which produces significant frequency deviation on the MFS and is useful for detection and localisation of the delamination. The second innovation is the development of a damage detection approach using wavelet edge detection algorithm of the MFS to identify and locate delamination in composite laminate plate. By combining MFS with 2D CWT (two-dimensional continuous wavelet transform) gives an enhanced approach that can identify both near and far surface delaminations under noisy conditions. Glass fibre composite is selected for the study due to its relatively low cost for applications to large-size structures. Finite element method is employed to generate the modal frequency surface data. The obtained MFS is analysed using wavelet-based edge detection algorithm to highlight the damage characteristics from the modal data. Near and far surface delaminations at selected through-the-thickness positions and defects with different sizes are included in the analysis. The effect of random frequency noise of different magnitudes is studied by adding a random signal to the original noise-free MFS results. The performance of the proposed method is discussed according to the case studies.

2. Background

2.1 Two-dimensional continuous wavelet transform (2D CWT)

The wavelet transform can be used to analyse the nonstationary power signal with several different frequency components [39, 40]. It is a more powerful tool than traditional Windowed Fourier Transform (WTF). In general, the process of wavelet analysis of a two-dimensional signal f (x) can be defined as the inner product process given by

⟨ψb ,θ , a|f ⟩=∬ψ (x )b ,θ , a f (x)d x (1)

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ψ (x )b ,θ , a=1aψ (R−θ x−b

a ) and R−θ=[cosθ −sinθsinθ cosθ ] (2)

x=(x , y ) are the spatial variables which contained the information of locations, b=(bx , by ) are vectors of real components which represent the translation operation, R−θ is the rotation matrix operator in two-dimension and a is the scale factor which determines the central frequency of the wavelet function in Eq. (1). Hence, by choosing the different translation and the dilation factors, an isotropic family of wavelet functions can be generated from the mother wavelet function ψ (x ) for multiple scale analysis at different spatial location. In addition, by multiplying the mother wavelet by the rotation operator, an anisotropic family of wavelet function can be obtained.

The wavelet analysis for a certain signal is done by multiplying the signal by a wavelet function of a certain scale along different spatial locations. The inner product process is repeated for the analysed signal and all the wavelets are generated using these procedures. From the spatial domain, the above processes can be regarded as a special case of cross-correlation process which returns the similarity of the two local waveforms evaluated by the normalised wavelet coefficients. For example, the wavelet coefficients (real and imaginary parts) reach the maximum value if there is a high degree of local similarity of two functions and vice versa. If the local profile of the signal (e.g. modal frequency surface) changes continuously, the corresponding wavelet coefficient changes smoothly as well. When the local signal profile contains a contribution of damage-dependent disturbance, this feature will lead to an abnormal change of the wavelet coefficients that can be used for damage detection purpose. Applying Fourier transform to Eq. (1), the above process can be expressed through convolution theorem as,

F [∬ψ (x )b , θ ,a f ( x )d x ]=F [ψ ]×F [ f ] (3)

where F [ f ] stands for the Fourier transform of the analysed signal and F [ψ ] represents the complex conjugate of the Fourier transform of the wavelet. In the frequency domain, correlation can be explained by the overlap of the spectrum between the analysed signal and the selected wavelet. Therefore, the frequency spectrum of the selected wavelet determines the performance of the edge detection in modal frequency surface.

2.2 Derivatives of Gaussian (Dog) wavelet

The selection of mother wavelet is generally dependent on the type of problem, as discussed in the literature [22 – 24, 28, 39]. In order to balance the spatial localisation and the spectrum localisation, the family of Gaussian wavelets are generally adopted due to its invariance form during Fourier transform. In spite of the general conditions of the wavelet function (e.g. zero-mean, square integrable), the selection of the wavelets should also match the specific feature of the damage. In detail, the aim of damage detection is to extract the local deviation that is present in the response of a damaged structure. From vibration theory, the increase in flexibility of a structure at a delaminated zone causes an increase of modal displacement. This effect is further magnified by introducing a local mass loading to increase the deviation of modal frequencies.

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The traditional signal enhancement achieved by using the derivatives of modal frequency surface (curve) is already shown to be sensitive to localised defect but is also sensitive to numerical noise. Therefore, a wavelet which contains the differential operator is ideal to perform the task. One of the candidates is the derivative of Gaussian wavelet, which can be obtained directly through differentiation of a Gaussian function given by,

G ( x )= 1σ √2 π

exp(−x2

2σ2 ) (4)

When the Gaussian function is differentiated in 2D, it gives the Derivative of Gaussian (DoG) as

ψ DoG ( x )=∇2G ( x )= r2−σ2

σ4 exp(−x2

2σ2 ) and r2=x2+ y2(5)

Substituting Eq. (5) into Eq. (3) and using the property of convolution integral, the Laplace operator is separated from the wavelet and applied to the analysed signal to give

F [ψ ]× F [ f ( x ) ]=F [∇2G ( x ) ]×F [ f (x ) ]=F [G ( x ) ]×F [∇2 f ( x ) ] (6)

From Eq. (6), the process of wavelet transform using derivative of Gaussian wavelet can be interpreted as the multiplication of the Gaussian wavelet by the derivative of the response signal in the frequency domain. This is equivalent to application of the differential operator to the original response signal followed by post-processing using a Gaussian filter. In a numerical processing software (e.g. MATLAB), direct high order numerical differentiation is usually avoided due to the round-off errors and the time cost. In the numerical processing, the derivative of Gaussian in Eq. (5) is replaced by the difference of Gaussian. Starting from Eq. (5), the second derivative of Gaussian can be approximated by the difference between the two Gaussian functions of different variances. Therefore, Eq. (5) can be expressed by an approximate form given by,

ψ DoG ( x )=G1 ( x )−G2 ( x )= 1σ1 √2π

exp(−x2

2σ12 )− 1

σ2 √2 πexp (−x

2

2σ 22 ) (7)

where the ratio of the two variances σ 1and σ 2 of the Gaussian functions equal to 1:1.6 [41]. Fig. 1 shows the surface plot of the derivative of Gaussian and the difference of Gaussian calculated using this ratio. The Fig. shows that the difference of Gaussian given by Eq. (7) is a very good approximation of the derivative of Gaussian given by Eq. (5).

According to Eq. (8), the DoG wavelet transform can be regarded as the difference of two individual wavelet transforms on two Gaussian functions, where one is shaper and the other is flatter, as shown in Fig. 2 (a). In general, the convolution between the signal and Gaussian function with wider variance is not sensitive to the local detail, as the spectrum of wider variance Gaussian is dominated by lower frequency components. The narrower variance Gaussian function usually has wider spectrum and can sense the higher spatial frequency components in the signal,

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as shown in Fig. 2(b). As a result, the difference between these two Gaussian wavelets shows the residual of the analysed signal in certain frequency ranges, as shown in Fig. 2(d). The corresponding difference of Gaussian in the time or spatial domain is shown in Fig. 2(c). As the high frequency noise components are filtered, the use of DoG wavelet shows attractive noise robustness than direct numerical differentiations.

(a): Derivative of Gaussian (b): Difference of Gaussian

Figure 1 – Comparisons of derivative and difference of Gaussian wavelet

(a): Gaussian functions (time) (b): Gaussian functions (spectrum)

(c): Difference of Gaussian (time) (d): Difference of Gaussian (spectrum)

Figure 2 – Spatial and frequency domain of difference of Gaussian wavelet

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Combining Eq. (1) with Eq. (6) and using Eq. (3), the inner product process can be expressed as the difference of the wavelet transforms, given by,

F [∬ψ (x )b , θ ,a f ( x )d x ]¿ F [∬G1 ( x ) f ( x )d x ]−F [∬G2 ( x ) f ( x )d x ]=F [G1 ]× F [ f ]−F [G2 ]×F [ f ]

(8)

In the present study, a complex wavelet in the family of Gaussian derivatives, the power of DoG wavelet (dogpow), is employed for wavelet analysis. The explicit form of the dogpow wavelet in the frequency domain can be expressed as [39],

ψdogpow (ωx ,ω y )=[exp(−α2(ωx2+ωy

2

2 ))−exp (−( ωx2+ω y

2

2 ))]p (9)

where ωx and ω y refer to the spatial frequency from Fourier transform. The parameter α is the shape factor that controls the ratio of the two variances of the Gaussian functions and the parameter p is the power factor of the DoG wavelet. The complex wavelet enables damage detection through both real and imaginary coefficients of the wavelet transform. Generally, the real coefficients are sensitive to features such as peaks in the signal and the imaginary coefficients have better performance in detecting the oscillations in the signal. The magnitude coefficients are sensitive to the zero-crossing lines which are important for delamination detection. Therefore, the imaginary and magnitude coefficients can be regarded as a useful complement of the real coefficients. A more detailed review of the features of the real and complex wavelets can be found in [42].

2.3 Effect of Random noise

The random numerical noise due to the FFT frequency resolution will generally lead to data oscillation from one point to another during the modal parameter estimation process. For a given structure, the modal frequency is generally more robust. Thus, the effect of background noise on frequency is relatively small [2]. In practise, the frequency resolution limits the level of minimum frequency difference that can be distinguished by the system and is another source of noise in practise. The resolution of frequency ∆ f is usually defined as,

∆ f=( f max− f min)

N(10)

where f max and f min denote the upper and lower frequency limits of interest, and N represents the number of spectral lines used in the FFT analysis. If the original data points are used (without numerical interpolation of the FRF), the random noise in the frequency estimation is generally within the magnitude of the resolution which is∆ f .

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For applications such as structural damage detection, the effect of small delamination on frequency shifting is usually subtle compared with global dynamic behaviour of the structure. The damage-induced deviation of the modal parameters (modal frequency and modal displacement) are generally small and require relatively sensitive level of modal parameter estimation in practise. However, the introduced local inertia increases the dynamic responses of the structure, which make the deviation of the frequency (up to 20%) much greater than traditional frequency shifting. Hence, the deviation of the frequency can be easily measured in practise. If the random noise from the frequency estimation is assumed to be normally distributed within the magnitude of the frequency resolution, an additional signal (noise term) can be applied to the noise-free frequency data to study the random noise effect under different frequency resolution levels. In the present study, noise associated with two FFT resolution levels of 0.125 Hz and 0.25 Hz are selected for noise analysis. Although, the current commercial modal testing instruments support fine frequency resolution than the above assumption, the cost for using fine resolution needs to be considered for practical applications. Therefore, the assumed resolution enables time-efficiency and acceptable accuracy of the analysis. In addition, several noise compensation methods are currently available for modal parameter estimation. The performance of noise reduction using different numerical compensation methods is not considered in this study.

Another important factor for vibration-based damage detection methods is to determine the suitable grid for experimental measurement. Literatures have reported the effect of the measurement noise under different spatial sampling distances for modal displacement shape measurement under laboratory conditions [43], where the very coarse grid results in a distortion of the original modal shape and the fine grid may cause numerical processing noise. Similar effects are also present in the measurement of modal frequency surface. The coarse grid may not highlight the local disturbance information of the damage and should be avoided in the analysis if there is no base-line information available. In addition to the measurement time required, the employment of very fine grid will introduce data oscillation due to resolution limits as well as increases the operational cost. Consequently, a 21 x 21 grid of point-mass loading is used to investigate the response of damaged composite plates in the present study.

3. Finite element modelling and data processing

In the case studies, 550 mm x 550 mm x 4 mm six-layer [0/+45/-45/-45/+45/0] glass/epoxy laminate plates with pre-defined delaminations are modelled, as shown in Figs. 3 (a) and (b). The finite element model contains 44 x 44 x 6 elements along the length, width and thickness directions. The dimensions of the individual element are 12.5mm in length and width directions and 0.67mm in the thickness direction. Three-dimensional (3D) solid elements with quadratic shape function interpolation are employed to discretise the models. These elements are designated as element type C3D20 in the commercial finite element package ABAQUS (v6.13) that is used. In the present study, four cases of delaminations of different sizes and located at different layers are investigated. The centre of the delaminated section is located at( x , y )=(340 ,310 ), as shown in Fig. 3 (a). Delaminations of lengths a = 50 and 100 mm are

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created between layers 3 and 4 (the mid-plane). This is done by specifying the delamination as two planes which are defined by exactly the same coordinates but are not tied (or glued) together. Similarly, near surface delaminations are created between layers 5 and 6 (near the top surface) in the simulation. In the present study, the proposed method is employed for lower modes of vibration. Therefore, nonlinear effects, such as interlaminate contacts which are exhibited at high frequencies, are not included in the FE model.

The material properties employed in finite element analysis are given in Table 1. The boundary conditions of the composites plates are set to be free-free. The auxiliary mass loading is modelled as a 1D point element where the constant mass magnitude is set to be 0.2 kg, which is about 10% of the mass of the composites laminate plates. For low frequency vibration, the mounting of the mass and structure is generally rigid. Thus a rigid connection between the point mass and the structure is assumed in the numerical model. For high frequency analysis, the dynamic effect of the mounting medium may need to be considered. It should be noted that the placement of the point mass in the discrete finite element nodes can be approximately achieved in practice. For example, in previous studies, point contact between the mass loading and structural surface was achieved experimentally via the head of an M4 screw which made a circular contact area of diameter of about 7 mm with the surface. The screw was connected to a calibrated mass and its head was fixed to the surface of the structure by using an adhesive [20]. An even smaller contact area of say a circle of 2 mm diameter can be achieved by bonding a thin spacer between the mass and the surface. The governing principle is to minimise the contact area as shown in Fig. 3(b). A similar procedure was adopted in [16 - 19].

(a): View in X-Y plane (b): View along Z direction (through-the-thickness)

Figure 3 - Geometrical parameters of a six-layer composite laminate plate contain pre-defined delamination

Table 1 – Measured material property of six-layer E-glass fibre plate used in finite element simulation

E11

(GPa)E22 (GPa) E33 (GPa) v12 v13 v23 G12

(GPa)G13

(GPa)G23

(GPa)25.1 10.2 10.2 0.37 0.37 0.41 3.6 3.6 3.6

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For each delamination case, the point-mass loading of 21 x 21 grid locations (with interspacing of about 25 mm along length and width directions) are modelled via the input file generation. The frequencies of the first two flexural modes of the laminate plates are extracted from the output results of the ABAQUS eigenvalue analysis. The extracted frequency data are further processed under the numerical software package MATLAB (R2014a). First, the frequency values for the point-mass at various locations are reconstructed to produce the modal frequency surface (MFS) with x-y coordinate associated with the point-mass at different locations. The noise effect is simulated using random numbers with levels of about 0.125 Hz and 0.25 Hz and added to the MFS. Before performing the wavelet transform, the MFS (with noise) is further interpolated using the 2D biharmonic function provided in the MATLAB programme. The wavelet coefficients of the MFS are calculated through the 2D CWT toolbox with the Dog wavelet.

4. Case studies and discussions

Numerical frequency data as well as the results using wavelet-based edge detection algorithm are presented in each subsection. Delaminations at near and far interface locations are studied separately for each selected size. A summary and discussion of the detection approach and influential factors are given in section 4.3 to highlight some key features of the analysis.

4.1 Delamination with size 100 x 100 mm

4.1.1 Case 1: near surface delamination

The MFS and corresponding MFS wavelet coefficients are presented in Fig. 4 for the first two flexural modes of vibration. In the region without delamination, smooth contours are observed in the MFS data for both modes. In the region with delamination, however, a significant frequency reduction is exhibited in the MFS of both modes, as shown in Fig. 4. This frequency reduction is due to the enhanced dynamic effect that the auxiliary mass applies to the plate at or near to the delaminated section where the local flexibility is increased. Due to the enhancement of local inertia, delamination (local) modes are produced at the delaminated region, as shown in Fig. 4 (c1) and (c2). Moreover, comparing the result obtained from the fundamental and second modes, the indicated defect size in the second mode is more accurate than that in the fundamental mode. This is because the degree of MFS deviation is larger in the higher mode than in the fundamental mode. In general, for the near surface delamination, the MFS can be used as a direct damage indicator to localise the damage.

Fig. 5 presents the contour plots of the obtained result using wavelet edge detection algorithm under noise-free condition. The figure comprises 6 figlets which are labelled as a1, b1 and c1 for mode1 and a2, b2 and c2 for mode 2. The labels a, b and c denote the real, imaginary and magnitude of the MFS wavelet coefficients. The 2D CWT calculates the similarity of local MFS profile with respect to the selected wavelet. As shown in Fig. 5, the real and imaginary wavelet coefficients return a quasi-homogeneous background in the defect-free region. The reason for this

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is because at lower modes of vibration the MFS has low wavenumber and any local region of the MFS has a very large radius of curvature. The peaks in the wavelet coefficients for the second bending mode create local minima and maxima in the wavelet coefficient plots, which are due to the differentiation of the original peaks in the MFS shown in Fig. 4 (b2).

In the region with delamination, the correlation between the wavelet and the MFS changes significantly in magnitude, this reflects the change of the wavelet coefficients. Therefore, this region becomes highlighted in the above quasi-homogeneous background of the damage-free region. For the fundamental modes, the deviations of both the real and imaginary coefficients are shown at the boundary between the intact and defect areas. A local minimum coefficient is reached in the real coefficient. This is because the real part is not sensitive to the sudden change (considered as higher order oscillation terms in MFS). The imaginary coefficient, however, shows maximum value at the edge locations because the imaginary part is more sensitive to the oscillations. When the wavelet shifts to the centre of the damage, where the MFS reaches a peak, the real coefficient reaches the maximum values while the imaginary coefficient attains the minimum value. From the edges to the centre of the defect, a smooth change from minimum to maximum is shown in the real coefficient and the opposite trend is displayed in the imaginary plot.

Similar results are also presented for the second mode, but this time the size of damage is more accurately indicated, as can be seen from Figs. 5 (a2) to (b2). In Fig. 5 (a2), the real plane shows the defect region as the edge region with the maximum wavelet coefficients (in red) surrounded by four edges with minimum wavelet coefficients (in blue). The interface between these two regions defines the exact size and location of the delamination damage. Conversely, in Fig. 5 (b2), the imaginary plane shows the defect region as the edge region with minimum wavelet coefficient (in blue) surrounded by the four edges with maximum wavelet coefficients (in red). Similarly, the interface of the two regions defines the exact size and location of the delamination. Besides the real and imaginary coefficients, the magnitude coefficient presents some unique features of MFS, as can be seen in Figs. 5 (c1) and (c2). In detail, the zero-crossing lines (in white) of the second derivative of the MFS are clearly presented in the magnitude plot. It can be seen that the delamination edges also form zero-crossing lines which indicates the shape of the delamination. These zero-crossing lines may be used to identify the pattern of the delamination.

Figs 6 and 7 present the effects of random noise of different levels on the wavelet coefficients. For the fundamental mode, the effects of noise in the real and imaginary plots are not significant even up to 0.25 Hz, as shown in Figs. 7 (a1) and (b1). A certain degree of noise-induced oscillations can be seen in the zero-crossing lines plotted in the magnitude coefficients, as shown in Figs 6 (c1) and 7(c1). From Figs. 6 (a2) to (c2) and Figs. 7 (a2) to (c2), there is no obvious change in the wavelet coefficients of the second modes under different noise levels. This is because the resolution-dependent noise is relatively small compared with the adjacent spatial change between neighbouring points in the MFS. Consequently, the second mode MFS wavelet coefficients are more robust under various noise conditions in terms of the prediction of the location and size of the defect.

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(a1): MFS of 1st mode (surf) (b1): MFS of 1st mode (contour) (c1): Displacement mode shape of 1st mode

(a2): MFS of 2nd mode (surf) (b2): MFS of 2nd mode (contour) (c2): Displacement mode shape of 2nd mode

Figure 4 – MFS and mode shapes of near surface delamination (100mm x 100mm) without additional noise

(a1): real coefficient of 1st mode (b1): imaginary coefficient of 1st mode

(c1): magnitude coefficient of 1st mode

(a2): real coefficient of 2nd mode (b2): imaginary coefficient of 2nd mode

(c2): magnitude coefficient of 2nd mode

Figure 5 – MFS wavelet coefficients of near surface delamination (100 mm x 100mm) without random noise

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Figure 6 – MFS wavelet coefficients of near surface delamination (100 mm x 100mm) with random noise at level of 0.125 Hz





Figure 7 – MFS wavelet coefficients of near surface delamination (100 mm x 100mm) with random noise at level of 0. 25 Hz

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4.1.2 Case 2: far surface delamination

The MFS and MFS wavelet coefficients for the far surface delamination are shown in Figs 8 - 11. It is seen in Figs 8 (a1) and (b1) that the MFS of the first mode is generally smooth and there is no sudden change observed at the highlighted defect area. The results of the second mode display a greater deviation at the defect area as can be seen in Fig. 8 (b2). However, the MFS itself in this case is not able to provide detail information of the delamination pattern. This can be addressed by applying wavelet analysis to the MFS. By comparing Fig. 4 with Fig. 8, one can see that the delamination at different through-the-thickness positions has different effect on the MFS. As shown in Fig. 8 (c1) and (c2), in this case, there is no delamination mode produced, therefore the deviation of modal frequency is smaller than in case 1. This observation indicates that the severity of the interface delamination on the dynamic response of the composite plate is influenced by its through-the-thickness position. Consequently, the near surface and the far surface delaminations may produce different values of modal frequency deviations between the intact and damaged composite plates.

Fig. 9 shows the contour plots of the 2D CWT of the noise-free MFS of the first two bending modes. The MFS wavelet coefficients of the fundamental mode, which is plotted in Figs. 9 (a1) to (c1), show an abnormal region in the real, imaginary and magnitude coefficient planes. However, the contrast of this feature, with respect to the background, is not obvious as in the case of the near surface delamination which is shown in Figs. 5 (a1) to (c1). The damage profile is more clearly presented in the second mode wavelet coefficient planes, as shown in Figs. 9 (a2) to (c2). From the results in the real and imaginary coefficient planes, the pattern as well as the spatial location of the defect is accurately predicted compared with the highlighted area shown in Fig. 8 (b2). Figs. 9 (a2) to (c2) also show a symmetric pattern of maximum points in the real and imaginary coefficient planes, which are due to the peaks of the MFS shown in Fig. 8 (b2). These results demonstrate the sensitivity of the MFS wavelet coefficient compared with MFS in terms of far surface delamination detection.

Comparing Fig. 9 with Fig. 5, it is seen that the defect edges of the far surface delamination are not as well defined as in the near surface delamination case. In detail, the trend of the variations of the wavelet coefficients is similar to that of the near surface delamination discussed above. However, the contrast ratio between the intact region and the edges of the defect in the far surface delamination case is relatively shallower than the near surface delamination case. As a result, the severity of the delamination shown by the wavelet coefficients generally corresponds to the difference indicated by the original MFS data. When the frequency deviation of the far surface delamination is less distinct, the noise level is expected to be an important factor for performance of the damage detection. For the fundamental mode, there is a visible change of the wavelet coefficients in the defect location under noise-free condition, as shown in Figs. 9 (a1) to (c1). However, this feature is shown to be noise sensitive. Figs. 10 (a1) to (c1) and Figs. 11(a1) to (c1) show that the delamination is no longer visible or distinguishable. Thus, it is not feasible to identify and localise the defect using the fundamental mode when the noise level is greater than 0.125 Hz.

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(a1): MFS of first mode (surf) (b1): MFS of first mode (contour) (c1): Displacement mode shape of 1st mode

(a2): MFS of second mode (surf) (b2): MFS of second mode (contour) (c2): Displacement mode shape of 2nd mode

Figure 8 – MFS and mode shapes of far surface delamination (100mm x 100mm) without additional noise





Figure 9 – MFS wavelet coefficients of far surface delamination (100mm x 100mm) without random noise

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Figure 10 – MFS wavelet coefficients of far surface delamination (100mm x 100mm) with random noise at level of 0.125 Hz






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For the second mode, both the location and the size of the defect are also clearly displayed in the wavelet coefficient plots of the noise-free case shown in Figs. 9(a2) to (c2). However, in the presence of noise of about 0.125 Hz, there is a reduction in the sharpness of the defect edges defined by the wavelet coefficients due to the decrement of the contrast between the intact and damaged regions as can be seen in Figs. 10 (a2) to (c2). As the noise level increases to 0.25 Hz, there is further reduction in the contrast as can be seen from the zero-crossing lines in the magnitude plot shown in Fig. 11 (c2). Nevertheless, the results show that the second mode still provides a clear identification and localisation of the delamination even in the presence of noise of up to 0.25 Hz. As a consequence, if the noise level is not known, delamination detection using only MFS of the fundamental mode may not be sufficient. If the noise level is well controlled (e.g. by using fine resolution), the defect can be located using the fundamental mode MFS wavelet coefficient for far surface delamination. The use of the MFS data of the second bending mode is generally robust in terms of damage identification and localisation. The pattern of the defect can be detected under a noise level of 0.125 Hz. However, a certain degree of defect pattern distortion occurs under higher noise conditions. For example, Fig. 11 (c2) shows that the left bottom corner of the defect is not accurately indicated.

4.2 Delamination with size 50 x 50 mm

4.2.1 Case 3: near surface delamination

The MFS results of the first two modes are shown in Fig. 12. Figs 12 (a1) and (a2) show that the MFS have a local contour distortion around the defect region. As shown in Fig. 12 (c1) and (c2), delamination modes are produced for the two modes when the mass loading is located at the delaminated section. As the frequency deviation is greater for the higher modes, the second mode MFS shows stronger evidence than the first mode and enables the defect localisation directly through the MFS result as shown in Figs 12 (a2) and (b2). Compared to the results for the larger delaminations presented in the case 1, it can be found that the deviations of modal frequency in both modes are reduced as the size of the delamination is reduced. However, the delamination modes are retained even though the size of the delamination is less than 1% of the area of the whole composite plates.

Fig. 13 shows the wavelet coefficient of the noise-free case. In Figs. 13 (a1) to (c2), the defect section is shown as the red section surrounded by blue edges in the real coefficient and the blue section surrounded by the red edges in the imaginary coefficient, for both the fundamental and the second modes. The predicted sizes of the defect from the coefficient planes of both modes are similar and are close to the actual dimensions. Figs. 14 (a1) to (c1) and Figs. 15 (a1) to (c1) show that when random noise is present, there is a reduction of the contrast between the defect and intact regions with increasing random noise levels. This reduction in contrast is more severe for the fundamental mode. Nevertheless, the defect is detectable even under relatively strong noise condition.

(a1): MFS of first mode (surf) (b1): MFS of first mode (contour) (c1): Displacement mode

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shape of 1st mode


Figure 12 – MFS and mode shapes of near surface delamination (50mm x 50mm) without additional noise





Figure 13 – MFS wavelet coefficients of near surface delamination (50mm x 50mm) without random noise

(a1): real coefficient of 1st mode (b1): imaginary coefficient of 1st (c1): magnitude coefficient of 1st

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mode mode



Figure 14 – MFS wavelet coefficients of near surface delamination (50mm x 50mm) with random noise at level of 0.125 Hz





Figure 15 – MFS wavelet coefficients of near surface delamination (50mm x 50mm) with random noise at level of 0.25 Hz

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4.2.2 Case 4: far surface delamination

The results of the small size far surface delamination are presented in Figs 16 to 19. There are no direct information on the delamination in the result of the fundamental mode MFS, as in Fig. 16 (b1). Fig. 16 (b2) shows a small abnormal distortion at the defect locations for the second mode. However, the distortion does not provide detail information on the defect in terms of the size and the location. Additionally, there are no delamination modes produced in this case, as shown in Fig. 16 (c1) and (c2). This is due to the fact that as the depth of the delamination increases, the detectability of the delamination decreases as can be deduced by comparing the MFS and displacement mode shapes to those of case 3. Moreover, comparing the present case to case 2, it is seen that the reduction of the size of delamination (from 4% to 1% of the whole plate area) causes smaller influence on the vibration responses.

The results of wavelet coefficients of MFS under noise-free condition are shown in Fig. 17. The wavelet analysis of the second mode MFS indicates the approximate size and location of the delamination as shown in Figs. 17 (a2) to (c2). In detail, the magnitude of the wavelet coefficients manifests a sudden change from the intact part to the delaminated part. These sudden changes in coefficients refer to the edges of the local frequency deviations caused by interactions between a delamination and the point mass loading. Moreover, this magnitude change preserves the original shape of the delaminated section, which (as discussed previously) is an unique pattern of the delamination compared to random noise effect. However, the fundamental mode MFS wavelet coefficients do not provide strong indication of the defect in terms of the size and location. This indicates the risk of using only the fundamental mode in delamination detection. The wavelet coefficients for the noisy MFS are shown in Figs. 18 and 19. As expected, the increasing of the random noise level reduces the visibility of the delamination pattern. The random noise effect is usually dependent on the measurement parameters as well as the environmental factors. The numerical analysis may provide a guide to the suitable parameters in the physical measurement.

Generally, if the local frequency deviation is larger than the noise magnitude, the delamination will manifest a local pattern in the wavelet coefficient plots, while the damage pattern may not be achievable if the noise level is greater than the deviation caused by the delamination. According to the four case studies above, the detectability of delamination using frequency surface approach has shown its dependency on the delamination size, the through-the-thickness (depth) position as well as the magnitude of the noise. Both the depth and the size tend to have a relatively significant influence on the detectability of the delamination from vibration characteristics. Among the two, it is also noted that the depth of the delamination tend to be more influential compared to the sizes/area of the delamination. This will be discussed in more detail in section 4.3.

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(a1): MFS of first mode (surf) (b1): MFS of first mode (contour) (c1): Displacement mode shape of 1st mode


Figure 16 – MFS and mode shapes of far surface delamination (50mm x 50mm) without additional noise





Figure 17 – MFS wavelet coefficients of far surface delamination (50mm x 50mm) without random noise

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(a1): real coefficient of first mode (b1): imaginary coefficient of first mode

(c1): magnitude coefficient of first mode









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4.3 Discussions and summary

Table 2 shows the frequency deviations for different interface cases. The deviations are obtained by subtracting the delamination data set from intact data set. In detail, Table 2 shows that the maximum frequency deviations are about 28 and 59 Hz for the near surface delaminations with sizes of about 50 mm and 100 mm, respectively. These deviations are easily detectable for noise magnitudes of up to 0.25 Hz. However, when delaminations of the same sizes occur at the mid-plane position, the maximum frequency deviations are only about 1.2 and 6.7 Hz for noise magnitudes of up to 0.25 Hz. Nevertheless, the delamination is still detectable even in the mid-plane position in the presence of the spectral noise. This is because the traversing of the point-mass to different locations of the plate, which is the basis of the modal frequency surface method, enhances the dynamics of the plate at those locations. Therefore, the frequency deviations are much larger than in the conventional approach that does not make use of an additional point mass. Fig. 20 shows the frequency deviations due to the delaminations of different sizes and at different interface (depth) locations with the point mass located on the top surface (see Fig. 3). Due to the symmetry of the lay-up, the frequency responses are the same when the point mass is located on the bottom surface.

Table 2 – Modal frequency deviations for delamination of different sizes and at different interfaces

Modal frequency deviation (Hz)Mode 1 Layer1&2 Layer2&3 Layer3&4 Layer4&5 Layer5&650 mm 3.56 0.34 0.11 0.05 0.04100 mm 11.04 2.11 0.58 0.23 0.10

Mode 2 Layer1&2 Layer2&3 Layer3&4 Layer4&5 Layer5&650 mm 28.25 3.63 1.24 0.58 0.25100 mm 59.57 20.63 6.77 2.55 0.88

According to Fig. 20, the greater the depth of the delamination from the mass-loading surface, the lower the deviation of modal frequency manifested. For delaminations that occur near the surface positions, a delamination (local) mode is generally produced which causes a significant reduction of modal frequencies at the delaminated section. As the through-the-thickness location of the delamination moves far from the working surface, the level of frequency deviation shows a quasi-exponential decrease. This is due to the fact that the breathing effect of the delamination at far surface positions are more constrained than near the surface. This leads to a relatively small flexibility with respect to the near surface cases. As a result, when using the mass-loading scanning approach, the delaminations near the working surface are easily identified due to their large modal frequency deviations. Although the frequency deviation reduces as the through-the-thickness location of the delamination moves towards the central plane from the working surface, the wavelet transform remains an effective enhancement tool to highlight the abnormal region in the composite laminate plates using the same set of available data. Generally, compared to the delaminations near the working surface, the far surface delamination produces less structural stiffness degradation and less effect on the structural integrity. Satisfactory damage detection also depends on the noise levels associated with the detection.

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Figure 20 – Modal frequency deviation curves for delamination at different sizes and depth

It should be noted that the presented method is not limited by the boundary conditions of the structure. This is because it uses the point mass to enhance the dynamic response of the delaminated section. The reason for the enhanced dynamics is due to the local stiffness reduction caused by the breathing effect of the delaminated section. This breathing behaviour of the interlaminar damage will occur in a composite plate of any boundary condition. The boundary conditions mainly influence the magnitude of the breathing effect if displacement mode shapes are used for the detection of delamination. However, the presented method uses the modal frequencies of the structure which will always be present irrespective of the boundary conditions which generally affect the amplitudes of the displacement mode shapes.

Through the case studies presented in section 4.1 and 4.2, the performance of the proposed damage indicator is demonstrated by the numerical tests. Several key features of the proposed method can be deduced from these case studies.

The level of delamination-induced frequency deviation in MFS shows a quasi-exponential decrease with depth of the delamination from the detection surface. As a result of delamination mode, the MFS is very sensitive to the near surface delaminations. By visually inspecting the MFS of the first and second flexural modes, the location as well as the approximate pattern of the near surface delamination can be identified.

When the delamination occurs at the far surface positions, the MFS is not as sensitive as in the near surface cases. Local distortion of the MFS contour is obvious for near surface delamination but is less obvious for delamination at far surface. The area of the above distortion is controlled by the size of the delamination.

The wavelet coefficients greatly magnify the damage pattern from the modal frequency surfaces. This enables the detection of delaminations at greater depth using the same vibration testing techniques. Numerical analysis shows that the MFS wavelet coefficients have the capability to identify the approximate pattern of the delaminations even under noisy conditions. This makes the wavelet-based approach a robust technique for delamination detection. In addition, the selected dogpow wavelet in 2D CWT shows a satisfactory

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performance in terms of localisation and sizing of the defects, which agrees with the previous studies [23].

For the low noise or noise-free condition (e.g. less than 0.1 Hz), the use of fundamental mode MFS wavelet coefficient is acceptable in defect identification and localisation. When the noise level is higher (e.g. around 0.125 Hz), a certain degree of noise-induced effect is observed. When the magnitude of the random noise is at a higher level (e.g. 0.25 Hz or greater), the fundamental mode may not show any useful information about the defects. Nevertheless, even in the presence of such relatively high levels of random noise, the second mode MFS wavelet coefficients still shows a robust performance in terms of the defect localisation and sizing.

5. Conclusions

A vibration-based damage detection method for delaminated plate-like structures using modal frequency surface (MFS) has been proposed in this study. The proposed method is a base-line free method that only requires information of modal frequencies of the damaged structures. The concept of the method has been investigated through selected numerical examples of composite plates with delaminations of various sizes and different through-the-thickness (depth) positions. The analysis indicates that the through-the-thickness position of the delamination has a more significant effect on the detectability of the delamination than the size of the delamination. A significant frequency deviation is observed in the MFS for the near surface delamination. However, this effect is weaker for the delaminations at far surface locations. Overall, as the through-the-thickness location of a delamination moves from the detection surface towards the central plane, the level of modal frequency deviation shows a quasi-exponential decrease. The numerical examples have shown that the use of the wavelet coefficient of MFS enhances the detectability and localisation of delaminations at near and far surfaces. The effect of numerical noise associated with the FFT resolution has been studied under the random noise assumption. The satisfactory performance of the delamination identification approach proposed under random noise for a combination of the first and second modes has been demonstrated.

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