Damage detection strategies based on propagation of guided elastic waves

The coordinates ξ_i and η_j of a two-dimensional spectral finite element in the element normalized coordinate system ξη can be expressed as the roots of a polynomial equation of the following form [27, 28]:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle \mathop{(1-\xi )}\nolimits ^{2}{P}_{n}^{^{\prime}}(\xi )=0,\tqs \xi \in \langle -1,+1\rangle \\ \displaystyle &\displaystyle \mathop{(1-\eta )}\nolimits ^{2}{P}_{n}^{^{\prime}}(\eta )=0,\tqs \eta \in \langle -1,+1\rangle \end{array} &&\displaystyle \end{eqnarray} \tag{ 1 }$$

where P_n(x) is the nth-order Legendre polynomial and the symbol ' denotes its first derivative in respect of x. In the case of the spectral shell finite element used by the authors the nodal coordinates of the element for n = 5 can be specified as:

$$\begin{eqnarray} \displaystyle \fl {\xi }_{i},{\eta }_{j}\in \left\{-1,-\sqrt{\frac{1}{3}+\frac{2}{3\sqrt{7}}},-\sqrt{\frac{1}{3}-\frac{2}{3\sqrt{7}}},\right.&&\displaystyle \\ \displaystyle \left.\sqrt{\frac{1}{3}-\frac{2}{3\sqrt{7}}},\sqrt{\frac{1}{3}+\frac{2}{3\sqrt{7}}},1\right\}&&\displaystyle \end{eqnarray} \tag{ 2 }$$

where i,j = 1,2,...,n + 1.

On the specified nodes a set of shape functions can be built in the local coordinate system ξη. A Lagrange interpolation function f(ξ,η) supported on the nodes ξ_i and η_j can be defined in the following manner:

$$\begin{eqnarray} &&f(\xi ,\eta )=\sum _{i=1}^{n+1}\sum _{j=1}^{n+1}{N}_{i}(\xi ){N}_{j}(\eta ){f}_{i j} \end{eqnarray} \tag{ 3 }$$

where N_i(ξ) and N_j(η) are one-dimensional shape functions of the element, while f_ij represent the nodal values of the function f(ξ,η). It should be emphasized that one-dimensional approximation functions N_i(ξ) are orthogonal in a discrete sense:

$$\begin{eqnarray} &&\int \nolimits _{-1}^{+1}{N}_{i}(\xi ){N}_{j}(\xi )\hspace{0.167em} \mathrm{d}\xi =\sum _{k=1}^{n+1}{w}_{k}{N}_{i}({\xi }_{k}){N}_{j}({\xi }_{k})={w}_{i}{\delta }_{i j} \end{eqnarray} \tag{ 4 }$$

where w_i is the well-known Gauss–Lobatto–Legendre (GLL) weight and δ_ij is the Kronecker delta [29, 30].

Assuming small strains the displacement field within the mid-plane of the spectral shell finite element, in the local coordinate system x'y'z', can be expressed in the form presented in [27] as:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle u(x ^{\prime},y ^{\prime},z ^{\prime})={u}_{0}(x ^{\prime},y ^{\prime})+z ^{\prime} \Phi (x ^{\prime},y ^{\prime})\\ \displaystyle &\displaystyle v(x ^{\prime},y ^{\prime},z ^{\prime})={v}_{0}(x ^{\prime},y ^{\prime})+z ^{\prime} \Psi (x ^{\prime},y ^{\prime})\\ \displaystyle &\displaystyle w(x ^{\prime},y ^{\prime},z ^{\prime})={w}_{0}(x ^{\prime},y ^{\prime})+z ^{\prime} \Omega (x ^{\prime},y ^{\prime}) \end{array} &&\displaystyle \end{eqnarray} \tag{ 5 }$$

where the displacement components u₀(x',y'), v₀(x',y') and w₀(x',y') can be interpreted as the averages of the displacements measured at the upper and lower surfaces of the element. At the same time the displacement components Φ(x',y'), Ψ(x',y') and Ω(x',y') can be thought of as about the differences between those upper and lower surface displacements related to the element thickness h. Relations between the upper and lower surface displacements and the displacement components in the current formulation of the spectral shell finite element can be expressed as:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {u}_{0}(x ^{\prime},y ^{\prime})=\frac{u(x ^{\prime},y ^{\prime},\frac{h}{2})+u(x ^{\prime},y ^{\prime},-\frac{h}{2})}{2}\\ \displaystyle &\displaystyle \Phi (x ^{\prime},y ^{\prime})=\frac{u(x ^{\prime},y ^{\prime},\frac{h}{2})-u(x ^{\prime},y ^{\prime},-\frac{h}{2})}{h}\\ \displaystyle &\displaystyle {v}_{0}(x ^{\prime},y ^{\prime})=\frac{v(x ^{\prime},y ^{\prime},\frac{h}{2})+v(x ^{\prime},y ^{\prime},-\frac{h}{2})}{2},\\ \displaystyle &\displaystyle \Psi (x ^{\prime},y ^{\prime})=\frac{v(x ^{\prime},y ^{\prime},\frac{h}{2})-v(x ^{\prime},y ^{\prime},-\frac{h}{2})}{h}\\ \displaystyle &\displaystyle {w}_{0}(x ^{\prime},y ^{\prime})=\frac{w(x ^{\prime},y ^{\prime},\frac{h}{2})+w(x ^{\prime},y ^{\prime},-\frac{h}{2})}{2},\\ \displaystyle &\displaystyle \Omega (x ^{\prime},y ^{\prime})=\frac{w(x ^{\prime},y ^{\prime},\frac{h}{2})-w(x ^{\prime},y ^{\prime},-\frac{h}{2})}{h}. \end{array} &&\displaystyle \end{eqnarray} \tag{ 6 }$$

It should be noted at this point that the number of independent displacement components in (5) is directly linked with the number of wave modes that can be considered by the use of the spectral shell finite element, as shown below.

Following the notation used in [29] and under the assumption of small strains and by taking into account the form of the displacement field expressed by (5) the strain field within the element in the local coordinate system x'y'z', according to the first-order shear deformation theory, can be calculated based on the following relations [29, 30]:

$$\begin{eqnarray} &&\displaystyle \begin{array}{@{}l@{}} \displaystyle \fl {\epsilon }_{x ^{\prime}}=\frac{\partial {u}_{0}(x ^{\prime},y ^{\prime})}{\partial x ^{\prime}}+z ^{\prime}\frac{\partial \Phi (x ^{\prime},y ^{\prime})}{\partial x ^{\prime}}\\ \displaystyle \fl {\epsilon }_{y ^{\prime}}=\frac{\partial {v}_{0}(x ^{\prime},y ^{\prime})}{\partial y ^{\prime}}+z ^{\prime}\frac{\partial \Psi (x ^{\prime},y ^{\prime})}{\partial y ^{\prime}}\tqs {\epsilon }_{z ^{\prime}}=\Omega (x ^{\prime},y ^{\prime})\\ \displaystyle \fl {\gamma }_{x ^{\prime} y ^{\prime}}=\frac{\partial {u}_{0}(x ^{\prime},y ^{\prime})}{\partial y ^{\prime}}+\frac{\partial {v}_{0}(x ^{\prime},y ^{\prime})}{\partial x ^{\prime}}\\ \displaystyle +~z ^{\prime}\left(\frac{\partial \Phi (x ^{\prime},y ^{\prime})}{\partial y ^{\prime}}+\frac{\partial \Psi (x ^{\prime},y ^{\prime})}{\partial x ^{\prime}}\right)\\ \displaystyle \fl {\gamma }_{y ^{\prime} z ^{\prime}}=\frac{\partial {w}_{0}(x ^{\prime},y ^{\prime})}{\partial y ^{\prime}}+z ^{\prime}\frac{\partial \Omega (x ^{\prime},y ^{\prime})}{\partial y ^{\prime}}+\Psi (x ^{\prime},y ^{\prime})\\ \displaystyle \fl {\gamma }_{z ^{\prime} x ^{\prime}}=\frac{\partial {w}_{0}(x ^{\prime},y ^{\prime})}{\partial x ^{\prime}}+z ^{\prime}\frac{\partial \Omega (x ^{\prime},y ^{\prime})}{\partial x ^{\prime}}+\Phi (x ^{\prime},y ^{\prime}) \end{array}&&\displaystyle \end{eqnarray} \tag{ 7 }$$

leading to the definition of the characteristic inertia (diagonal) and stiffness matrices of the spectral element in the local coordinate system x'y'z', as shown in [27].

It can be clearly seen from equation (7) that in the current formulation of the spectral shell finite element the additional displacement component Ω(x',y') denotes, in fact, the _z' strain component vital for the element performance as well as its capability of modelling simultaneously the symmetric and antisymmetric dynamic behaviour associated with the propagation of elastic waves.

The effectiveness and correctness of a particular shell theory in modelling various dynamic or wave-propagation-related problems by the SFEM can be easily evaluated and assessed by analysing the dispersion curves associated with these theories. Moreover the dispersion curves give a great insight into the performance of spectral finite elements under various conditions and allow one to estimate the applicability of the elements by comparison of numerical results with known analytical results.

Dispersion curves can be calculated in two straightforward steps. Firstly it is necessary to determine equations of motion associated with the theory under investigation. This can be easily achieved by the use of Hamilton's principle. Next, based on the given displacement field the virtual work W related to deformation and motion of a spectral finite element is expressed in terms of its strain energy U and kinetic energy T as well as the work of some external forces F. Subsequent application of Hamilton's principle leads to a set of equations of motion that are derived for each component of the displacement field.

Secondly, propagation of harmonic waves is assumed, which helps to transform the equations of motion from a set of partial differential equations, defined in the time domain for each displacement component, into a set of linear homogeneous equations defined in the frequency domain for the amplitudes of each displacement component. This system can be solved only then when its determinant vanishes, which gives rise to the characteristic polynomial equation. The roots of this equation define dispersion relations between particular modes of harmonic waves that are allowed to propagate by the theory investigated, the wavenumber k and the angular frequency ω as well as the dispersion curves for the phase c_p and group c_g velocities, as shown in figure 1, where c_l and c_t denote the velocities of longitudinal (irrotational, voluminal, dilatational or primary) and torsional (rotational, equi-voluminal, shear or secondary) waves propagating in three-dimensional unbounded isotropic media, respectively [31, 35]. The values c_l = 6.3 km s⁻¹ and c_t = 3.2 km s⁻¹ correspond to aluminium alloy 2024-T3. The computational procedure described above and applied by the authors is based on the procedure presented in [31], which is also described in detail in [34].

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In the present case the spectral shell finite element under consideration was assessed in terms of its accuracy expressed by the relative error δ of the group-to-phase velocity ratio c_g/c_p, obtained based on the displacement field given by (5), and the same velocity ratio calculated analytically based on the Lamb solutions [1].

The relative error δ is defined as follows:

$$\begin{eqnarray} \displaystyle \delta (f h)=\frac{{f}_{\mathrm{T}}-{f}_{\mathrm{L}}}{{f}_{\mathrm{L}}}100\% ,\tqs {f}_{i}={\left(\frac{{c}_{\mathrm{g}}}{{c}_{\mathrm{p}}}\right)}_{i},\tqs i=\mathrm{T},\mathrm{L}&&\displaystyle \\ \displaystyle &&\displaystyle \end{eqnarray} \tag{ 8 }$$

where T denotes the group-to-phase velocity ratio c_g/c_p obtained for the given shell theory, while L denotes the same ratio obtained for the Lamb solutions [1]. Furthermore, in order to estimate a broad frequency accuracy of the theory an average error Δ can be defined in the following manner:

$$\begin{eqnarray} &&\Delta (f h)=\frac{1}{f h}\int \nolimits _{0}^{f h}\vert \delta (s)\vert \hspace{0.167em} \mathrm{d}s. \end{eqnarray} \tag{ 9 }$$

The obtained results are presented in figure 2 for the fundamental symmetric S₀, antisymmetric A₀ and shear symmetric SH₀ wave propagation modes. It should be noted that the relative error δ is a function of not only the frequency f but also the thickness of the element h. In the case of the symmetric wave mode S₀ the error is positive and increases gradually with an increase in the frequency parameter fh. It is equal to 1.47% for 1 MHz mm, while its value reaches 34.1% for 2 MHz mm. In the case of the antisymmetric wave mode A₀ the error stays small and negative in the whole range of the frequency parameter fh. It is equal to −1.56% for 1 MHz mm and has a local minimum equal to −1.6% for 1.35 MHz mm. It is interesting to note that within the whole range of the frequency parameter fh the value of the relative error δ for the fundamental shear symmetric wave mode SH₀ is equal to zero. The corresponding values of the average error Δ calculated up to 1 MHz mm and 2 MHz mm are equal to 0.39% and 5.41% for the symmetric wave mode S₀ and 1.01% and 1.29% for the antisymmetric wave mode A₀.

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Based on the results shown in figure 2 it can be concluded that the practical range of application of the shell theory under consideration is limited by the cutoff frequency of the antisymmetric A₁ and shear antisymmetric SH₁ wave modes, as presented in figure 1. This is approximately equal to 1.6 MHz mm. Within this range of the frequency parameter fh numerical results are characterized by small errors and stay close to the analytical solutions. However above 1.6 MHz mm the errors increase significantly due to the inability of the current theory to reproduce the appropriately complex multimode behaviour observed.

For the square aluminium plate shown in figure 3 the IMV and RMS damage maps obtained and based on the results of numerical simulations are presented in figures 4 and 5. They were calculated for a force excitation of 1 N acting transversely at the top surface of the plate and at the plate centre. The maps were plotted for all three displacement components u, v and w and as damage a through hole of 10 mm diameter was assumed.

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It can be seen from the results presented in figures 4 and 5 that both the IMV and RMS damage maps can indicate well the location of the through-hole damage. This is true for all displacement components investigated. However, in the case of the RMS damage map the position of the damage is more clearly visible if only the sampling time interval Δt is long enough for the propagating waves to reach the damage. It is interesting to note that the use of the in-plane displacement components u and v results in good sensitivity in both cases. However, from a practical point of view measurements of the out-of-plane displacement components by LSV is much easier due to their higher values and signal-to-noise ratios. Because of that only the transverse displacement component w was taken into consideration in all the following results presented.

The results of numerical simulations presented in figure 6 show the influence of the weighting factor w_k used for generation of RMS damage maps on the effectiveness of damage detection and localization in the plate under investigation. In the case of unweighted RMS damage maps (w_k = 1) the position of the through-hole damage is visible for all displacement components u, v and w, but the patterns observed are strongly dominated by the excitation source located at the plate centre. In certain cases this effect can be dominating and too strong to allow one to pinpoint any damage. This undesired behaviour can be avoided when linearly weighted RMS damage maps are used (w_k = k), while square weighted RMS damage maps (w_k = k²) offer further improvement. However, it should be pointed out at this place that the use of weighted RMS damage maps requires appropriately chosen sampling time intervals Δt in order to minimize the influence of the boundary reflected waves that can mask any effects originating from the damage.

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It should be noticed that, due to the 2D nature of the plate, the IMV and RMS damage maps are virtually based on the behaviour of the fundamental antisymmetric A₀ wave propagation mode only that is dominating in the current case. For that reason it is visible from the RMS damage maps that the transverse displacement component w is more sensitive to the weighting process, while the in-plane displacement components u and v indicate smaller sensitivity to it.

Contrary to the previous case of the 2D plate the following two numerical examples are concerned with 3D shell structures. Their 3D geometry results in a much more complex nature of their responses and because of that in these two cases the RMS damage maps were based on the amplitude $A=\sqrt{{u}^{2}+{v}^{2}+{w}^{2}}$ of the total displacement rather than each particular displacement components u, v and w.

Next two numerical cases are concerned with a cracked fuselage section with two stiffeners, as presented in figure 7, and a cracked wing section skin, shown in figure 8. It was assumed that both these structures are damaged and the damage has the form of an open crack of 10 mm length located on the circumference of the inner opening (window hole) of the fuselage section or is located on the leading edge of the wing section skin. The crack was modelled using a technique developed by the authors in the past for various applications of the finite element method (FEM), an approach described in details in [40–42]. The stiffness loss due to the crack was calculated according to the laws of fracture mechanics.

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The results of numerical simulations corresponding to both analysed cases presented in figures 9 and 10 refer to the linearly weighted RMS damage maps (w_k = k) and their evolution in time. It can be clearly seen from the results of numerical simulations presented in figures 9 and 10 that the patterns observed originate from a complex interaction and conversion between the symmetric S₀ and SH₀ and the antisymmetric A₀ wave propagation modes in both cases. Despite their complexity it should be strongly emphasized that, as soon as propagating waves fully reach the crack position, which best corresponds to the time intervals t₂ − t₁ = 0.3 ms and t₂ − t₁ = 0.4, the crack location and orientation become very well visible.

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The first object of experimental measurements was a square, 1 mm thick, plate made of pure aluminium (99.5%). Figure 12 schematically illustrates the test plate with the location of the measurement area, the position of a PZT actuator and damage arrangement. The measurement area consisted of 331 vertical lines and 281 horizontal lines, which was approximately equal to a rectangle of 550 mm × 490 mm including 93 011 grid points. The intersection point of the rectangle's diagonals coincided with the plate centre and was the source of Lamb waves. The PZT actuator was fixed to the plate by electrical conductive adhesive, while the plate itself was working as the negative electrode, as shown in figure 12 in detail. Damage to the plate was simulated by a number of small additional masses: 1.6 g (3–5), 2.5 g (6) and 9.8 g (7), fixed to the top surface of the plate as well as by a narrow notch cut 25 mm × 1 mm × 0.5 mm (8). Certain additional information about the correlation between the transmitted and reflected signals due to various damage-related discontinuities, including the additional mass, can be found in [31–33].

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During measurements as the excitation signal a 35 kHz five-cycle sine pulse modulated by the Hann window was used. For each grid point measurements were repeated 20 times and the resulting signals were averaged to improve the signal-to-noise ratio. Between each measurement a time interval of 50 ms was kept in order to ensure fading out of previously generated elastic waves. The sampling frequency was set up to 512 kHz and 1024 time samples were registered during each measurement that covered 2 ms. Table 1 presents certain selected parameters of the PZT transducer used in experimental measurements [43].

It is typical for LSV measurements that, for a small number of points, a lower signal-to-noise ratio is achieved based on the fact that not enough reflected light returns back to a scanning head. All such points are characterized by much bigger RMS values. In order to reduce this effect median filtering in the space domain with 3 × 3 point window was applied. Examples of registered elastic wave propagation patterns in the aluminium plate under consideration are presented in the form of six arbitrary chosen time snapshots. Figure 13 illustrates the propagation of Lamb waves in the plate with one additional mass attached to its surface.

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Due to the fact that the introduced damage in the form of additional mass was a structural discontinuity, propagating elastic waves were reflected from the discontinuity and, as a consequence, damage became a source of new waves. This effect is clearly visible in figure 13 at time t = 292.97 µs and may be easily adopted for damage detection and localization purposes. However, in the case of multiple damage it can be difficult to detect and localize all defects, as is presented in figure 14. For more precise recognition of damage-related patterns the RMS function can be applied, as proposed theoretically in section 4.

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Because elastic waves are reflected from various damage-related discontinuities the distribution of associated scattered energy within the plate is also changed. These subtle changes can also be associated with changes in the RMS values and therefore can be used for damage detection and localization purposes. However, relatively large values of the RMS function usually dominate in the areas near the excitation points and, as a result, almost the whole range of scale values is associated with this small area. In order to overcome that problem the authors propose to use a logarithmic scale for the values of the RMS function calculated. Figures 15 and 16 present RMS and logarithmic RMS damage maps for the square aluminium plate under investigation calculated based on all 1024 time samples, i.e. covering the total time span of 2 ms.

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Following the results of numerical simulations presented in section 4 it can be said that in the case of unweighted RMS damage maps (w_k = 1) the patterns observed in RMS maps are strongly dominated by the excitation source. This has a masking effect on any damage-related features, especially those originating from damage distant from the excitation source.

In order to equalize and smooth the distribution of the RMS function in the whole area under investigation the subsequent time samples were multiplied by a weighting factor, as proposed in equation (13) in section 4. A comparison of such evaluated RMS damage maps for the first 256 time samples, i.e. covering the total time span of 0.5 ms, and for different values the weighting factors w_k are presented in figures 17 and 18.

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In general it can be said that the resolution of RMS damage maps can be easily and effectively increased by the use of different values of the weighting factors w_k, as presented earlier in section 4 in the case of the results of numerical simulations by the SFEM. The results obtained from experimental measurements by LSV confirm that better resolution corresponds to higher values of the weighting factor w_k, i.e. for linear w_k = k or square w_k = k² weighted RMS damage maps. However, attention must be paid in order to minimize the influence of the boundary reflected waves. This can be very well observed in the case of results presented in figure 17.

On the other hand, in the case of logarithmic RMS damage maps the influence of the weighting factors w_k is much less prominent and relatively good resolution of the maps can be obtained, even of the unweighted case, i.e. when the weighting factor w_k = 1, as clearly seen from figure 18. As shown earlier in section 4 an appropriately chosen sampling interval Δt has great influence on the resulting RMS damage maps. Figure 19 illustrates this influence on the sensitivity of weighted logarithmic RMS damage maps.

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As the second object of experimental measurements a square, 3.2 mm thick, composite plate was investigated. The plate was composed out of four material layers, for which the reinforcement was glass fibres and the matrix was epoxy resin. The layers were arranged symmetrically as [0°/90°]_s. The position of a PZT actuator with the location of the measurement area together with damage arrangement is presented in figure 20. A similar measurement area was used as for the square aluminium plate previously investigated, which consisted of 331 vertical lines and 281 horizontal lines, forming a rectangle of 600 mm × 500 mm and including 93 011 points. The intersection point of the rectangle's diagonals coincided with the plate centre and was the source of Lamb waves. In a similar manner as before, damage to the composite plate under investigation was simulated by four small additional masses: 1.6 g (3) and 3.2 g (4–6), fixed to the top surface of the plate.

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During measurements as the excitation signal a 10 kHz four-cycle sine pulse modulated by the Hann window was used. As before for each grid point measurements were repeated 20 times and the resulting time signals were averaged to improve the signal-to-noise ratio. Between each measurement a time interval of 50 ms was kept in order to ensure fading out of previously generated elastic waves. The sampling frequency was set up to 256 kHz and 1024 time samples were registered during each measurement that covered 4 ms. Certain selected parameters of the PZT transducer used in experimental measurements [43] are presented in table 1.

The results of experimental measurements presented in figure 21 correspond to the case of the plate with multiple damage, where two additional masses 5 and 6 were attached to the plate surface. It can be clearly seen from the wave propagation patterns registered and shown in figure 21 that, due to plate material orthotropy, the elastic waves that propagate within the plate have different velocities in two perpendicular directions. Also damage-reflected waves are very well visible at time t = 585.94 and 781.25 µs. However, it should also be pointed out that a relatively small distance between masses 5 and 6, in comparison to the length of propagating waves, causes that their simultaneous detection and localization cannot be successfully performed based only on the wave propagation patterns registered by LSV.

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Figure 22 presents logarithmic RMS damage maps obtained for two different damage scenarios. Only two additional masses at the same time were considered in each damage scenario, i.e. masses 3 and 4 (case 1) or masses 5 and 6 (case 2). In both these cases the logarithmic RMS damage maps were calculated based on the first 256 samples, i.e. covering the total time span of 1 ms. As can be clearly seen from figure 22 the use of the RMS or logarithmic RMS damage maps is very effective, enabling both detection and localization of the additional masses. It should be pointed out that the use of the RMS damage map revealed clearly the location of additional mass 6 hidden behind mass 5 (case 2) on a direct wave propagation path from the PZT transducer.

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The third object of experimental measurements was a composite stabilizer of a PZL W-3A helicopter, shown in figure 23. In this case the measurement area consisted of 475 vertical lines and 291 horizontal lines, which was approximately equal to a rectangle of 845 mm × 510 mm and included 138 225 grid points. As before, the intersection point of the rectangle's diagonals was the source of Lamb waves and the location of a PZT actuator. For that purpose a Sonox CeramTec PZT element of 10 mm diameter was used and the actuator was fixed to the stabilizer by PCB Petro Wax, as presented in figure 24.

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During measurements as the excitation signal a five-cycle sine pulses of various values of carrier frequencies f_c modulated by the Hann window were used. Measurements for each point of the grid were repeated 20 times and the signals obtained in this way were averaged in order to improve the signal-to-noise ratio. Between each measurement of 1024 time samples a time interval of 20 ms was kept in order to ensure the fading out of previously generated elastic waves. The time span and the sampling frequency of the measurements was dependent on the carrier frequency f_c of the excitation signal and were equal to: 4 ms and 256 kHz for the carrier frequency f_c = 17.5 kHz, 2 ms and 512 kHz for the carrier frequency f_c = 35 kHz, 0.8 ms and 1280 kHz for the carrier frequency f_c = 100 kHz and 0.4 ms and 2.56 MHz for the carrier frequency f_c = 250 kHz. Damage to the stabilizer was simulated by a number of small additional masses: 0.2 g (3, 4), 0.5 (5–7) and 1.6 g (8, 9), fixed to the top surface of the stabilizer, as presented in figure 24.

The results of experimental measurements presented in figure 25 were obtained for the lowest value of the carrier frequency f_c = 17.5 kHz and were related to the case of the stabilizer with multiple damage, where all additional masses were attached to the upper surface of the stabilizer. However, contrary to the two cases of the aluminium and composite plates previously studied, the complex geometry of the stabilizer and its material properties resulted in wave propagation patterns carrying out very little visible information about any presence of damage. This was also observed for the remaining higher values of the carrier frequencies f_c considered. Despite this fact, calculation of RMS or logarithmic RMS damage maps based on the signals registered by LSV revealed the presence of damage, but the maps obtained were strongly dependant on the values of the carrier frequency of propagating elastic waves f_c, which, on the other hand, was very closely related to the damping properties of the composite material used for the stabilizer manufacturing and resulting energy distributions.

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As can be clearly seen from the logarithmic RMS damage maps presented in figure 26 simultaneous detection and localization of all additional masses could not be performed for one single carrier frequency f_c due to different penetration ranges of propagating elastic waves within the structure of the stabilizer (i.e. different inspection areas). For the lowest value of the carrier frequency f_c = 17.5 kHz the largest penetration range was obtained and, at the same time, additional masses 8 and 9 located relatively far from the excitation point could be successfully detected and localized as well as masses 3 and 6 located on the wave propagation path along the spur of the stabilizer. It is interesting to note that in this case the close range resolution was relatively low, and detection and localization of the remaining additional masses was difficult based on the RMS damage map obtained. When the value of the carrier frequency was increased to f_c = 35 kHz the energy distribution associated with propagating elastic waves within the structure of the stabilizer was reduced to the area closer to the excitation point. This resulted in an improved resolution of the map obtained and, as a consequence, masses 4, 5 and 7 could be easily detected and localized, whereas detection and localization of masses 8 and 9 became difficult. Any further increase in the value of the carrier frequency f_c resulted in a following decrease in the wave penetration range as well as in the signal-to-noise ratio. In the case of the carrier frequency f_c = 100 kHz only masses 4 and 5 as the closest to the excitation point could be detected and localized together with masses 3 and 6 located on the spar. For the highest value of the carrier frequency f_c = 250 kHz only mass 6 could be effectively detected and localized due to a very low signal-to-noise ratio.

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All results presented in the three previous sections were obtained based on very precise experimental measurements. Unfortunately the assurance of this high precision required 14 h long tests. Such time-consuming measurements are, however, unacceptable from any practical point of view and should be significantly reduced in order to make this technique attractive for more realistic applications. Thus the authors examined an approach in which the long experimental testing was significantly shortened, preferably without any loss in the quality of the final result. This is demonstrated in the case of the square aluminium plate previously analysed. For experimental measurements the total number of grid points was reduced from an initial 93 011 to 9000 and any signal averaging was skipped. In this way the time required for the measurements was reduced nearly 200 times to 5 min only. In order to verify the quality of the signals registered by LSV and their further applicability for damage detection purposes and generation of RMS damage maps linear and cubic spline interpolations in the space domain were tested. The results obtained are presented in figure 27.

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As can be seen from figure 27 linear interpolation in the space domain of the RMS damage map based on the raw signal had no significant improvement in the resulting RMS damage map. The obtained patterns in both raw and interpolated RMS damage maps are practically identical. A certain improvement in comparison with the linearly space-interpolated RMS damage map is obtained when cubic spline interpolation in the space domain is used. However, the best results were obtained in the case when the interpolation process was firstly performed on each time frame registered rather than on the resulting RMS damage map itself, which again was equivalent to cubic spline interpolation in the space domain. It should be emphasized that the resulting RMS damage map interpolated in this manner had practically the same resolution and quality as the original RMS damage map shown in figure 18 in section 4.