Impact localization on a composite tail rotor blade using an inverse filtering approach

Abstract

This article presents an in situ structural health monitoring imaging system for the localization of impacts on a composite complex structure such as a tail rotor blade. Unlike conventional plate-like panels, this composite structure presents a strong anisotropy and inhomogeneous elastic nature due to the presence of both glass fibre and carbon fibre, a geometrically complex shape due to the curvature of the blade’s airfoil section and variations in the mechanical behaviour due to local changes in the thickness. The proposed imaging technique is based on the inverse filtering or reciprocal time reversal approach applied to the waveforms originated from a point of the structure of unknown location (impact source) and a number of signals stored in a database containing the experimental Green’s function of the medium. Unlike other ultrasonic impact localization methods, the present technique allows achieving the optimal focalization of the impact point in the spatial and time domain, by taking advantage of multiple linear scattering and a small number of receiver sensors.

Keywords

Structural health monitoring active composites morphing piezoelectric embedded intelligence

Introduction

Developments in carbon fibre–reinforced plastic (CFRP) and glass fibre–reinforced plastic (GFRP) materials have allowed a radical advancement in lightweight aerospace applications such as rotor blade design. Indeed, laminated composite structures can be tailored to display desired properties in specific directions and areas, such as high stiffness and strength. However, these components are sensitive to low-velocity impact damages that can considerably degrade the structural integrity and, if not detected, they can result in loss of control and catastrophic failures of the vehicle. Hence, impact localization has become an important tool for structural health monitoring (SHM) systems based on ultrasonic guided waves (GW). A number of algorithm-based methods have been developed in the past for the localization of the acoustic emission (AE) source in isotropic and anisotropic plate-like structures (Ciampa and Meo, 2010a, 2010b; De Marchi et al., 2011; Meo et al., 2005). Most of these techniques rely on time of arrival (TOA) identification and the group velocity determination of the coherent part of the wave field (ballistic wave) reaching a network of transducers bonded on the structures. Usually, advanced signal processing techniques such as peak detection (Seydel and Chang, 2001), cross-correlation (Kosel et al., 2003), the Hilbert (Mustapha et al., 2011) and the wavelet transformation (Gaul et al., 2001) can be employed to extract physical parameters from measured data and then to use them to detect the impact location. However, the dispersive nature of GW and the presence of multiple scattering, mode conversion and reflection from the boundaries (diffuse wave field) can lead to waveforms recorded from the sensors dissimilar from the original elastic source. In other words, the effects of wave propagation in geometrically complex structures can degrade the quality of the TOA estimation, causing poor localization. Alternative approaches to algorithm-based methods are the pattern recognition algorithms such as the neural network (NN) approach (Sung et al., 2000). Such a technique is an intelligent data fusion system that works on the idea of training intensively a computer network with known input and output until the network converges. However, NN as any interpolation algorithm is able to identify the impact source with remarkable accuracy only within a training range. Hence, despite this method being suitable for complex structures, it requires large data sets and cannot provide an accurate solution. To overcome these limitations, time reversal (TR) process has been used as a tool for the imaging of the impact source in solid media (Ciampa and Meo, 2012; Ing et al., 2005). Indeed, in a TR experiment, due to the time invariance and spatial reciprocity of linear wave equation, an input signal can be focussed back on the original source if the output received by a set of transducers is time reversed and emitted onto the excitation point. The first improvement of TR process is to compensate the dispersive behaviour of GW (Park et al., 2009). Indeed, depending on the propagation frequency, dispersive GW have a number of wave packets that travel at higher and lower speeds towards the receiver sensor. After a TR process, the slower modes are emitted first, so that all the waveforms can converge at the original source point at the same time, thus compensating dispersion. However, the refocussing of TR holds only in the case of lossless media. Indeed, from the study of the elastodynamic wave equation, TR invariance is due to the presence of the even-order (second-order) time partial derivative operator. This condition cannot be fulfilled in dissipative media as the wave equation presents a time partial derivative operator of the first order (Tanter et al., 1998). That is, material dissipations can generate amplitude distortions of the propagating wave front and TR behaviour becomes difficult to be predicted. Nevertheless, TR can be easily converted to reciprocal TR or inverse filtering (IF) by inverting the impulsive response of the structure (Fourier transform of Green’s function). This simple but effective operation allows recovering the optimal focussing, even in dissipative media (Ciampa and Meo, 2011).

The objective of this work is to report, both theoretically and experimentally, an in situ SHM system able to identify the occurrence and location of low-velocity impacts in a ‘real’ aerospace component. In particular, this research study extended a recent work published by the same authors on the localization of the impact source in a reverberant plate-like structure (Ciampa and Meo, 2011), to a complex construction such as a tail rotor blade of an helicopter. Indeed, unlike conventional plate-like panels, this composite structure presents a strong anisotropy and inhomogeneous elastic nature due to the presence of a double material (CFRP and GFRP), a geometrically complex shape due to the curvature of the blade airfoil section and rivets and variations in the mechanical behaviour due to local changes in the thickness. The proposed imaging technique is based on the IF approach applied to the waveform originated from a point of the structure of unknown location (impact source) and a number of signals stored in a database containing the experimental Green’s function of the medium. Unlike other ultrasonic impact localization methods, the present method allows achieving the optimal focalization of the impact point in the spatial and time domain, by taking advantage of multiple linear scattering and a small number of receiver sensors.

The layout of this article is as follows: in section ‘Theoretical analysis of reciprocal TR’, the imaging method for the localization of the impact source is theoretically presented. In section ‘Influence of diffuse wave field on the imaging process’, the benefits of a reverberant fully diffuse wave field on the imaging process using IF are illustrated. Section ‘Experimental set-up’ reports the experimental set-up, while section ‘Impact localization results’ illustrates the imaging results for a number of impact points. Then, the conclusions of the article are presented.

Theoretical analysis of reciprocal TR

If the TR invariance and spatial reciprocity of the elastodynamic wave equation are satisfied, either a TR or an IF imaging process can be used to focus ultrasonic waves in diffuse wave fields and anisotropic media. The last technique, however, allows compensating some detrimental effects such as the limited transducer bandwidth and the material absorptions, in order to recover the optimal focussing at the impact source. In addition, according to Huygens’s principle in diffraction theory (Landau and Lifshitz, 1960), the reconstruction of the wave function in a generic volume at any time can be obtained by the knowledge of its sources located on a two-dimensional (2D) surface. The reciprocal TR experiment is usually split into two steps. In the first step, a number of signals representing a library of impulse responses from M points (‘excitation points’) along the plane of the structure are recorded by one receiver transducer and stored (Figure 1).

The second step consists of the recognition of the optimal refocussing procedure at the source location. The basic idea is to correlate the impulsive transfer function associated with each excitation point with that of the new impact of unknown location. In this manner, the information on the AE source location is accomplished by the imaging process as the maximum of the correlation at the focus point (referred as ‘virtual IF experiment’).

Elastodynamic wave equation with Dirac point-like source

Reciprocal TR theoretical description for elastic wave propagation involves exploiting properties of temporal and frequency convolution, Borel’s theorem and Green’s function theory. As a starting point, assuming the wave field ψ(r, t), with $r = x_{1} \hat{i} + x_{2} \hat{j} + x_{3} \hat{k}$ in Lagrangian coordinates x_j, as a generic component of the displacement vector field in the case of a solid, either elastically isotropic or anisotropic medium, we can write (Achenbach, 1984)

c^{2} \nabla^{2} ψ (r, t) - \frac{\partial^{2} ψ (r, t)}{\partial t^{2}} = - e (r, t)

(1)

Equation (1) is an inhomogeneous, linear, partial differential equation (PDE) where c is a generic component of the velocity of propagation in the medium and e(r, t) is the excitation of a known impulsive point-like source given by

e (r, t) = δ (r, t)

(2)

where $δ (r, t)$ represents the Dirac distribution in space and time domains. Equation (2) is a causal function and the temporal excitation only exists on a finite time interval [0, T], where T is the temporal duration of the excitation function e(r, t). The time-derivative operator of equation (1) appears only at the second order, hence if ψ(r, t) is a solution, then ψ(r, −t) is also a solution. It should also be noticed that the presence of the Laplacian differential operator guarantees the invariance of equation (1) under the transformation $r \to - r$ , that is, it guarantees a spatial reciprocity condition. According to Green’s theorem (Duffy, 2001), to solve equation (1) it is useful to define a Green space–time function G that depends on the variables $(r, t; r_{0}, t_{0})$ . In other words, $G (r, t; r_{0}, t_{0})$ is the field produced in r at time t by an impulsive force located in r₀ and excited at time t₀. Green’s function allows the calculation of the wave field $ψ$ at a given position and time (r, t) without explicitly solving the differential equation (1). From a mathematical point of view, Green’s function has to satisfy the following conditions (Barton, 1989):

$G (r, t; r_{0}, t_{0})$ must be a solution to the following linear inhomogeneous differential equation

\begin{matrix} c^{2} \nabla^{2} G (r, t; r_{0}, t_{0}) - \frac{\partial^{2} G (r, t; r_{0}, t_{0})}{\partial t^{2}} \\ = - δ (r - r_{0}) δ (t - t_{0}) \end{matrix}

(3)

where $δ (r - r_{0})$ and $δ (t - t_{0})$ are Dirac delta functions in space and time, respectively. For $t < t_{0}$ , causality requires that $G (r, t; r_{0}, t_{0}) \equiv 0$ .

$G (r, t; r_{0}, t_{0})$ is a reciprocal function in the sense of the reciprocity theorem

G (r, t; r_{0}, t_{0}) \equiv G (r_{0}, t_{0}; r, - t) \equiv G (r_{0}, t; r, t_{0})

(4)

$G (r, t; r_{0}, t_{0})$ must satisfy the homogeneous boundary conditions on S (Dirichelet, Neumann or mixed)

G (r, t; r_{0}, t_{0}) = 0 r \in S

(5.1)

\frac{\partial G (r, t; r_{0}, t_{0})}{\partial n} = 0 r \in S

(5.2)

where S is a closed surface of a solid $Ω$ , $\partial G (r, t; r_{0}, t_{0}) / \partial n$ is Green’s function normal derivative and n is the vector field normal to the closed surface in each of its points, directed outwards.

$G (r, t; r_{0}, t_{0})$ must satisfy the Cauchy initial conditions

G (r, t; r_{0}, t_{0}) = 0 t = t_{0}

(6.1)

\frac{\partial G (r, t; r_{0}, t_{0})}{\partial t} = 0 t = t_{0}

(6.2)

Hence, considering equation (3) and conditions (5.1)–(6.2), the solution to the elastodynamic wave equation (1) can be expressed in terms of Green’s function, boundary conditions and initial condition as follows

c^{2} {\nabla_{0}}^{2} ψ (r_{0}, t_{0}) - \frac{\partial^{2} ψ (r_{0}, t_{0})}{\partial {t_{0}}^{2}} = - e (r_{0}, t_{0})

(7.1)

c^{2} \nabla^{2} G (r, t; r_{0}, t_{0}) - \frac{\partial^{2} G (r, t; r_{0}, t_{0})}{\partial {t_{0}}^{2}} = - δ (r - r_{0}) δ (t - t_{0})

(7.2)

Multiplying equation (7.1) by $G (r, t; r_{0}, t_{0})$ and equation (7.2) by $ψ (r_{0}, t_{0})$ , performing the difference and then integrating over the solid $Ω_{0}$ and over the time t₀ from 0 to $t'$ , with $t' >> t$ in order to avoid the integration exactly at the peak of the delta function, we obtain

\begin{array}{l} \int_{0}^{t^{'}} d t_{0} \underset{Ω}{∭} {c^{2} [{\nabla_{0}}^{2} ψ (r_{0}, t_{0}) G (r, t; r_{0}, t_{0}) - {\nabla_{0}}^{2} G (r, t; r_{0}, t_{0}) ψ (r_{0}, t_{0}) \\ + \frac{\partial^{2} G (r, t; r_{0}, t_{0})}{\partial {t_{0}}^{2}} ψ (r_{0}, t_{0}) - \frac{\partial^{2} ψ (r_{0}, t_{0})}{\partial {t_{0}}^{2}} G (r, t; r_{0}, t_{0})]} d Ω_{0} \\ = \int_{0}^{t^{'}} d t_{0} \underset{Ω}{∭} [δ (r - r_{0}) δ (t - t_{0}) ψ (r_{0}, t_{0}) - f (r_{0}, t_{0}) G (r, t; r_{0}, t_{0})] d Ω_{0} \\ = ψ (r, t) - \int_{0}^{t^{'}} d t_{0} \underset{Ω}{∭} [e (r_{0}, t_{0}) G (r, t; r_{0}, t_{0})] d V_{0} \end{array}

(8)

The volume integral over the two terms in left side of equation (8) involving Laplaceans turns into a surface integral employing Green’s second identity as follows

\begin{array}{l} \underset{Ω}{∭} [\frac{\partial G (r, t; r_{0}, t_{0})}{\partial t_{0}} ψ (r_{0}, t_{0}) - \frac{\partial ψ (r_{0}, t_{0})}{\partial t_{0}} G (r, t; r_{0}, t_{0})] d Ω_{0} \\ + \int_{0}^{t^{'}} d t_{0} \iint_{S} c^{2} [\nabla_{0} ψ (r_{0}, t_{0}) G (r, t; r_{0}, t_{0}) - \nabla_{0} G (r, t; r_{0}, t_{0}) ψ (r_{0}, t_{0})] \cdot n d S_{0} \end{array}

(9)

Hence, equation (9) can be rewritten as

ψ (r, t) = g (r, t) + h (r, t) + s (r, t)

(10)

where

\begin{array}{l} g (r, t) = \int_{0}^{t^{'}} d t_{0} \underset{Ω}{∭} [e (r_{0}, t_{0}) G (r, t; r_{0}, t_{0})] d Ω_{0} \\ h (r, t) = c^{2} \int_{0}^{t^{'}} d t_{0} \iint_{S} {[G (r, t; r_{0}, t_{0}) \nabla_{0} ψ (r_{0}, t_{0}) - \nabla_{0} G (r, t; r_{0}, t_{0}) ψ (r_{0}, t_{0})] \cdot n (r_{0})} d S_{0} \\ s (r, t) = \underset{Ω}{∭} [\frac{\partial G (r, t; r_{0}, t_{0})}{\partial t_{0}} ψ (r_{0}, t_{0}) - \frac{\partial ψ (r_{0}, t_{0})}{\partial t_{0}} G (r, t; r_{0}, t_{0})] d Ω_{0} \end{array}

Equation (10) gives the complete solution of the inhomogeneous problem (1) including the initial conditions. The above terms, $g (r, t)$ and $h (r, t)$ , represent the effect of the source and the boundary conditions, respectively. The last term $s (r, t)$ is related only to the initial conditions (Duffy, 2001). Assuming a free unbounded space, supposing that we are able to measure at any point of a closed surface S the wave field $ψ (r, t)$ and its normal derivatives (set to zero at infinity), the general solution of the elastodynamic wave equation (1) takes the form

\begin{matrix} ψ (r, t) & = \int_{0}^{t'} d t_{0} \underset{Ω}{\int \int \int} [e (r_{0}, t_{0}) G (r, t; r_{0}, t_{0})] d Ω_{0} \\ = \underset{Ω}{\int \int \int} [G (r, t; r_{0}) \otimes_{t} e (r_{0}, t)] d Ω_{0} \end{matrix}

(11)

where the symbol $\otimes_{t}$ represents a convolution over time (note that t = t₀ = 0 for causality reasons). Equation (11) represents a linear combination of the acoustic field and its normal derivatives measured by the receiver sensor located in r at time t by an impulsive force located in r₀. Substituting equation (2) in equation (11), the above equation represents a particular case of the Borel’s theorem, in which the excitation function is the Dirac delta function. Therefore, only in this particular case, the wave field acquired by the sensor is equal to the structural impulsive response.

In the first step, impact loads are applied in points arbitrarily chosen on the plane of the structure (‘focussing plane’), in order to create a database of impulse responses from different locations r_m. In other words, the surface of the structure is converted into a discrete domain composed of M‘excitation points’, and the spatial integral of equation (11) is turned into a sum over the ‘focussing plane’ as follows

ψ (r, t) = \sum_{m = 1}^{M} G (r, t; r_{m}) \otimes_{t} e (r_{m}, t)

(12)

In the frequency domain, equation (12) is (the sum term is omitted for clarity reasons)

\hat{ψ} (r, ω) = \hat{G} (r, ω; r_{m}) \hat{e} (r_{m}, ω)

(13)

where the symbol ‘ $\hat{\cdot}$ ’ corresponds to a Fourier transformation operation and $\hat{G} (r, ω; r_{m}) = \int_{- \infty}^{+ \infty} G (r, t; r_{m}) e^{- j ω t} dt$ is the impulsive transfer function of the system from each of the m $(1 \leq m \leq M)$ excitation points. In particular, $\hat{G} (r, ω; r_{m})$ represents the amplitude of the ith mode, that is, the eigenvectors associated with ith eigenfrequency contained in the structural response due to impulsive excitation. On the other hand, according to the spatial reciprocity condition (equation (4)), $\hat{G} (r_{m}, ω; r)$ corresponds to the inverse propagation from the receiver transducer to the mth excitation point. It should be noted that in matrix notation, this condition corresponds to a transposition operation. Hence, the M impulse responses are recorded and stored in a memory.

Reciprocal TR focussing approach

A classical TR experiment can be obtained by time reversing the received signals during the first step, that is, throughout the transformation $t \Rightarrow T - t$ . From a mathematical point of view, this means that we are able to create secondary source on the surface of the structure, such that the boundary conditions on S are the TR component of the wave field $ψ (r, t)$ received during the first step (Cassereau and Fink, 1992). In a more practical case, the second step of a reciprocal TR process is ‘virtually’ performed in the computer. In particular, it consists of correlating the waveform emitted by a point of unknown location r_m0 with all the impulse responses stored in the memory. Hence, a new impact is applied in one of the points of the focussing plane and its impulse response is measured by the same receiver sensor used in the first step. Therefore, the new spectral distribution ${\hat{ψ}}_{new} (r, ω)$ measured by the receiver, when the impact $\hat{e} (r_{m 0}, ω)$ is generated, can be obtained as follows (Ciampa and Meo, 2011)

{\hat{ψ}}_{new} (r, ω) = \tilde{G} (r, ω; r_{m 0}) \hat{e} (r_{m 0}, ω)

(14)

where $\tilde{G} (r, ω; r_{m 0}) = {[\hat{G} (r, ω; r_{m 0})]}^{*} / {‖ \hat{G} (r, ω; r_{m 0}) ‖}^{2}$ is the inversion of the impulsive transfer function $\hat{G} (r, ω; r_{m 0})$ . The asterisk denotes complex conjugate operation (that is equivalent to a TR operation in the time domain) and $‖ \hat{G} (r, ω; r_{m 0}) ‖^{2}$ is the squared norm of the impulsive response, representing the square modal energy of the system. The effect of weighting the impulsive transfer function acquired in the second step is to create a ‘whitening process’, wherein the number of modes employed for the back propagation can be increased. Indeed, the modes with weak and strong amplitude are emitted back at higher and lower energies, respectively. In this manner, the material dissipation can be compensated and the contrast can be enhanced (Catheline et al., 2007). That is, the optimal focussing at the impact source is performed through a ‘virtual’ IF experiment, which relies on calculating the maxima correlation coefficients given by the following equation

{\hat{e}}_{IF} (r_{m}, ω) = \hat{G} (r_{m}, ω; r) \tilde{G} (r, ω; r_{m 0}) \hat{e} (r_{m 0}, ω)

(15)

where the operator $\hat{G} (r_{m}, ω; r) \tilde{G} (r, ω; r_{m 0})$ is called the IF operator. The maxima correlation coefficients are used to create a 2D map representing the structural surface, and the maximum of ${\hat{e}}_{IF} (r_{m 0}, ω)$ , that is, when r_m = r_m0, allows obtaining the image of the impact source at the focus point.

Influence of diffuse wave field on the imaging process

From a statistical point of view, fully diffuse wave fields are characterized by a superposition of different fields with uncorrelated amplitude and random direction of propagation and phase (Weaver, 1982). Furthermore, in complex structures such as the tail rotor blade tested, the measured signals are typically non-stationary and feature an exponentially decaying ‘coda’ which is dominated by multiple scattering (Figure 2).

In particular, the presence of heterogeneities such as a composite double material (CFRP and GFRP), local variations in the thickness and the curvature of the rotor blade airfoil influence the acoustic properties of the structure, generating reverberations within the medium. In this section, we propose to analyse the IF experiment in terms of correlation of a diffuse field (CDF). This concept has been widely used in seismology to retrieve the structural impulse response at two sensor locations (Larose et al., 2006) and only recently it has been examined through a TR process (Wapenaar et al., 2005). In particular, in a reciprocal TR experiment, CDF is defined as the inverse Fourier transformation of the IF operator at r_m = r_m0

\begin{matrix} CDF & = \frac{1}{{‖ G (r, t; r_{m 0}) ‖}^{2}} G (r_{m 0}, t; r) \otimes_{t} G (r, - t; r_{m 0}) \\ = \frac{1}{{‖ G (r, t; r_{m 0}) ‖}^{2}} \int_{- \infty}^{+ \infty} G (r_{m 0}, t; r) G (r, t + τ; r_{m 0}) d τ \end{matrix}

(16)

where the right term of the above equation represents the cross correlation of Green’s function of the system. The above equation is maximum at the focussing time t = 0 and equal to the energy of the impulsive function $G (r, t; r_{m 0})$ . Although a fully diffuse wave field is negative for impact localization techniques based on the TOA estimation, in the IF process the presence of reverberations becomes extremely important to image the impact source. In other words, in real applications where the waveforms under analysis are very dispersive, the cross-correlation function may cease to work for techniques that rely on the TOA measurement of the ballistic wave. Conversely, in the IF process (equation (16)), as the entire signal is needed for the back propagation, a fully diffuse wave field improves the focussing at the original source. From a physical point of view, this concept can be explained in terms of conversion of the modes with complex wave number into modes with real wave number. Indeed, multiple scattering in a complex medium behaves like a ‘mirror’, allowing the non-propagating (evanescent) modes to be converted into propagating modes. In this manner, they can reach the far field where the receiver sensor is located, thus carrying additional information on the impact location. The result of such operation, known as kaleidoscopic effect, creates a virtual enlargement of the transducer angular aperture, enabling a radical reduction of the number of transducers (Fink, 2008). Therefore, the number of sensors employed is not related to the construction of a baseline to take into account the anisotropic features of the composite structures in each direction, as misinterpreted by Qui et al. (2011), but simply to create a database of waveforms needed for the virtual IF process. Indeed, the same methodology would be valid for a reverberant isotropic structure. That is, only one receiver transducer can be used to acquire the signals.

Experimental set-up

The experiments were carried out on a composite tail rotor blade (125 cm × 20 cm × 2 cm) of a helicopter (Figure 3). Although no quantitative information was given by the manufacturer regarding the blade, a number of assumptions were made on the structural make-up (Figure 3(a)).

In particular, the leading edge of the blade is made of GFRP for impact damage tolerance, while CFRP has been used for the rest of the blade to increase the structural strength and stiffness. The section of the blade is split into two parts by a CFRP spar web. The leading edge section is a cavity and the trailing edge section is filled with foam. In addition, the surface of the blade was divided into six distinct regions, as shown in Figure 3(b), based on the expected behaviour of the propagating GW. The dashed black lines mark the boundaries of the leading edge cavity space into which sensors could be placed in practical applications. The greatest complications were expected in region 3, the CFRP/GFRP boundary region, where the number of CFRP plies was reduced incrementally until GFRP made up the surface. It was not known what effect this would have on GW propagating through it. The passive sensors employed were four piezoelectric transducers (APC sensors) with diameter of 6.35 mm, thickness of 2.5 mm and central frequency of 100 kHz (broadband spectrum), mainly used for ultrasonic applications. The waveforms were recorded using an oscilloscope (Picoscope 4224) with a sampling rate of 99.5 kHz. The baseline was acquired at different temperatures (with a variation of approximately 15°C) and the time histories of the signals were normalized in amplitude and averaged in the first step to eliminate any random noise. From the time histories and the associated spectra (Figure 2), the main energy of the amplitude spectrum of the waves induced by an uncontrolled system (modal hammer) was confined below 12 kHz. All the impacts were carried out manually in order to avoid damaging the structure, and as a result, negligible energy in the spectrum at frequencies higher than 25 kHz was found. Moreover, the dispersion diagrams were not successful because of the unknown mechanical properties and layout of the composite blade. Hence, it was difficult to discriminate vibration modes with fundamental GW modes in this particular spectrum range. Moreover, as reported by Moulin et al. (2009), the duration time window T of the signals acquired must be higher than the Heisenberg time τ_w or break time, which is equal to the modal density of the structure. However, since it is impossible to estimate τ_w due to the lack of knowledge of the mechanical properties of the blade, according to the Nyquist theorem and the long reverberation present in the waveforms recorded, a 100-ms duration time window T was chosen (Figure 2(a)). Sensor positions are reported in Table 1.

Table 1.

Sensors coordinates.

	x-coordinate (cm)	y-coordinate (cm)
Sensor 1	36	4
Sensor 2	40	4
Sensor 3	54	10
Sensor 4	58	10

Impact localization results

According to section ‘Theoretical analysis of reciprocal TR’, the structural surface was divided in M = 17 × 6 excitation points, distributed along a grid of rectangular cells (Figure 1). The size of the single cell is not constant, but varies from smaller cells of dimensions 6 cm × 2 cm on the GFRP region, to greater cells of dimensions 6 cm × 3 cm on the CFRP section. Such a configuration was designed to improve the resolution at the leading edge of the blade, as that is the area mostly subjected to impacts. The refocussing wave fields at the impact source location are represented as a 2D map and the maxima of equation (15) are deduced from highest values of the correlation coefficients (Figure 4).

In Figure 4, the asterisk symbol corresponds to the position of sensor 1, the red circle to the real impact location and the dashed white line to the distance between the sensor and the impact point. However, it was found that for some excitation points, the incoherent measurement noise (e.g. sensor noise, electronic noise) could negatively influence the library of signals acquired in the first step, leading to ambiguities in the focussing at the source. This further effect was eliminated by acquiring simultaneously the wave fields in both steps with additional transducers, and then averaging the maxima of the correlation coefficients according to the following formula

{\hat{e}}_{IF} (r_{m}, ω) = \frac{1}{N} \sum_{n = 1}^{N} [max {\hat{e}}_{n} (r_{m}, ω)]

(17)

where N is the number of total sensors used in the IF process. In our tests, four sensors were used as they provided satisfactory results for the impact location in each excitation point (Figures 5 and 6).

Indeed, it can be seen from Figures 5 and 6 that small ambiguities (highlighted by a blue circle) in the imaging of the impact source using only one sensor were clearly resolved using the averaged values of the correlation coefficients from four receivers. Although the methodology is suitable for only one sensor, we found that four sensors were enough to avoid any ambiguities in the imaging of impact location in any point of the structure. Naturally, for larger structures, due to the material attenuation of the elastic wave propagating through it, this number should be carefully investigated. Hence, compared to other ultrasonic impact localization techniques based on TOA evaluation, this methodology needs a simple signal processing to locate the source, with a computational time less than 1 s using a standard PC (Intel Dual Core System @ 3.16 GHz and 4 GB of RAM). It also does not require any numerical routines as well as a priori knowledge of the mechanical properties and the GW dispersion behaviour. Further work is ongoing to test this imaging technique from the laboratory to a full-scale, operational activity.

Conclusion

This article proposes an in situ SHM system for the localization of the impact source in real aerospace structures with diffuse field conditions. This technique, based on reciprocal TR process, is applied to a number of waveforms recorded by a number of passive sensors containing the impulse response of the medium. Unlike conventional ultrasonic impact localization systems based on the TOA estimation, with the present method, the benefits of scattering, mode conversions and boundary reflections enable the focussing at the impact source. Since the imaging process is obtained through a virtual focussing procedure, this system does not require any iterative algorithms and a priori knowledge of the mechanical properties of the structure and the anisotropic behaviour. The robustness of this technique was experimentally demonstrated on a tail rotor blade undergone to low-velocity impacts. The imaging results showed that the location of the impact point could be achieved with a high level of accuracy in any point of the structure.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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