The inner product vector as an output-only cross-correlation-based feature to structural damage assessment

2.1. Single input case

A generic dynamic $N$-DOF system can be represented through the equations of motion for a linear time-invariant model as follows:

where $\left[M\right]$, $\left[C\right]$, $\left[K\right]\in {\mathbb{R}}^{N\times N}$ are respectively the mass, the damping and the stiffness matrices. The term $\left\{f\right(t\left)\right\}\in {\mathbb{R}}^{N\times 1}$ represents the vector of the forcing functions applied on the lumped masses and the arrays $\left\{\ddot{x}\right(t\left)\right\}$, $\left\{\dot{x}\right(t\left)\right\}$, $\left\{x\right(t\left)\right\}\in {\mathbb{R}}^{N\times 1}$ are the acceleration, the velocity and the displacement vectors, respectively. Using linear modal analysis, the displacement vector $\left\{x\right(t\left)\right\}$ can be written as a linear combination of mode shape vectors $\left\{{\mathrm{\Phi }}_r\right\}\in {\mathbb{R}}^{N\times 1}$ for $r=\mathrm{1,2},\dots ,N$, gathered in the modal matrix $\left[\mathrm{\Phi }\right]=\left[\right\{{\mathrm{\Phi }}_1\},\{{\mathrm{\Phi }}_2\},...,\{{\mathrm{\Phi }}_N\left\}\right]$, multiplied by the modal participation coefficients $q_r\left(t\right)$ ($r=\mathrm{1,2},\dots ,N$). Assuming zero initial conditions, the solution for $q_r\left(t\right)$ is provided by the Duhamel Integral and so the solution of the displacement vector $\left\{x\right(t\left)\right\}$ can be written as:

where $g_r\left(t\right)$ is the unit pulse response function related to the $r$th mode.

To simplify the derivation of the Inner Product Vector, let us first consider the case of a single input force applied at $k$th location. In the $\left\{x\right(t\left)\right\}$ vector, the component of the displacement at the $i$th location induced by an arbitrary force at the $k$th location is represented by:

By taking twice the derivative with respect to time of Eq. (3), the acceleration response at the $i$th location induced by an arbitrary force at the $k$th location can be expressed as:

where $\ddot{g}{\mathrm{\text{'}}}_r(t-\tau )=\left[{\ddot{g}}_r(t-\tau )+\frac{2}{m_r}\delta (t-\tau )\right]$, with $\delta (t-\tau )$ indicating the Dirac Delta function at time $t=\tau$ and ${\ddot{g}}_r\left(t\right)$ as:

In Eq. (5), the coefficients ${\xi }_r$, ${\omega }_{nr}$, ${\omega }_{dr}$ and $m_r$ are respectively the damping ratio, the undamped natural frequency, the damped natural frequency and the modal mass associated with the $r$th mode.

It is noteworthy that, up to this point, no assumption has been made about the nature of the excitation $f_k\left(t\right)$.

2.2. Multiple input case

For the case of multiple input forces applied at various locations, the acceleration at the $i$th location can be expressed from Eq. (4) using linear superposition as:

and the corresponding contribution from the $r\text{'}$th mode as:

Assuming that the forces at the various locations are uncorrelated with each other, it is possible to obtain an expression of the cross-correlation between acceleration time histories at the $i$th and $j$th locations following the same procedure as presented in the single force case:

If the input force is represented by white noise excitation, then, similarly to the single force case, the cross-correlation at $T=0$ between the $r\text{'}$th component of the acceleration time histories at the $i$th and $j$th locations can be written as:

which can be decomposed as the sum of $N$ cross-correlation elements from Eq. (8) having the common multiplier ${\mathrm{\Phi }}_{i,r^{\text{'}}}{\mathrm{\Phi }}_{j,r^{\text{'}}}$:

where ${\psi \text{'}}_{r\text{'}}={\sum }_{k=1}^N\mathrm{‍}\left({\alpha }_k{\mathrm{\Phi }}_{k,r\text{'}}^2{\int }_0^{\infty }\mathrm{‍}{\ddot{g}}_{r\text{'}}^{\text{'}2}\left(\lambda \right)d\lambda \right)$ indicates a sum of either zero or positive terms. It can then be concluded that the vector $\left\{R_{i,j,r^{\text{'}}}\right(0\left)\right\}$, containing the cross-correlations of the $r\text{'}$ contributions of all the acceleration time histories with that at the reference $j$th location, can be rewritten as:

The relation obtained is analogous to the one provided by Eq. (10). Thus, we can proceed to the normalization of the cross correlation vector $\left\{R_{j,r^{\text{'}}}\right(0\left)\right\}$ yielding $\left\{{\widehat{R}}_{j,r^{\text{'}}}\right(0\left)\right\}$ (Eq. (12)) and finally provide the damage index vector $\left\{D_{j,r^{\text{'}}}\right\}$ defined in Eq. (14).

2.3. Damage detection through a local damage index vector

Until now, the definition of a valid damage index vector has been the object of many studies conducted by researchers like Wang et al. [7], Trendafilova and Manoach [8], Kim and Stubbs [9]. In this paper, the damage index vector is defined according to Wang et al. [7] even though, as previously mentioned, Wang’s experiment relies on specific hypothesis about the input which has been strategically designed. According to Eq. (13), the damage index vector can be written as:

where $\left\{D_{j,r^{\text{'}}}\mathrm{}\right\}\in {\mathbb{R}}^{N\times 1}$ for a full sensors setup, when the acceleration response time history is monitored at every DOF of the system. A more general formulation of Eq. (21) can be expressed as:

where the vectors $\left\{D_{IPV,j,r^{\text{'}}}\right\}$, $\left\{{\widehat{R}}_{IPV,j,r^{\text{'}}}^d\right(0\left)\right\}$ and $\left\{{\widehat{R}}_{IPV,j,r^{\text{'}}}^u\right(0\left)\right\}$ are subsets of the vectors $\left\{D_{j,r^{\text{'}}}\right\}$, $\left\{{\widehat{R}}_{j,r^{\text{'}}}^d\right(0\left)\right\}$ and $\left\{{\widehat{R}}_{j,r^{\text{'}}}^u\right(0\left)\right\}$ respectively and have dimension $Q\le N$, given a sensors setup that monitors the structure at only $Q$ locations. An additional consideration about the dimensionality $Q$ of the vectors in Eq. (22) has to be pointed out. Despite the fact that it can be represented by a properly discretized model, any real (continuous) dynamic system has an infinite number of degrees of freedom. For such a reason, the restriction $Q\le N$ doesn’t affect in practice the number of sensors we can use to obtain the damage index vector.

A graphical representation of the damage index vector $\left\{D_{IPV,j,r^{\text{'}}}\right\}$ is provided by plotting its elements over their respective monitored locations (or DOFs). For sake of clarity, let’s consider the particular case of an 8-DOF system whose acceleration response time histories have been collected in both a damaged and an undamaged state at every DOF so to compute the corresponding damage index vector. In this example, the damaged state represents the condition of the system with a reduction of stiffness at a given location. Because of the localized damage, a local abrupt change in the damage index vector is expected.

Depending on the structural boundary conditions (structural constrains) and material properties, the damage index vector can present different types of jump discontinuity and so a local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ can be defined based on the type of jump discontinuity presented in the plot of the damage index vector. Three possible cases are here reported, considering the way that the structural damage affects the mechanical characteristics of the structure:

1. If all the DOFs but the closest to the damage location are more sensitive to the structural constrains than to the occurrence of damage, the damage index is approximatively null at those locations away from the damaged one. Fig. 1(a) shows a possible configuration plot of the damage index vector for the case in which, by considering a local damage between DOFs 3 and 4, only the 4th element of $\left\{D_{IPV,j,r^{\text{'}}}\right\}$ shows an appreciable variation (in this specific case, it is arbitrarily set equal to 1 to provide a clear graphical representation). The plot of the damage index vector is similar to one representing an impulse change. For such a specific case, the local damage index $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ is defined as the damage index vector itself so that $\left\{L_{IPV,j,r^{\text{'}}}\right\}=\left\{D_{IPV,j,r^{\text{'}}}\right\}$.

2. It might happen that, for specific boundary conditions and material properties, the presence of a local damage may induce a step change as jump discontinuity in the trend of the plotted elements of the damage index vector. An example is shown in Fig. 1(b) where the two trends are represented by arbitrarily setting the elements of the vector to zeros and ones. Again, the damage has been introduced between the DOFs 3 and 4. In this case, the local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ is provided by the first order difference of the damage index vector, $\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}}\right\}$. The elements of $\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}}\right\}$ related to the position halfway between two adjacent monitored positions $i$ and $i+1$ are given by:

where the index $(i+0.5)$ of the element $D_{IPV,\left(i+0.5\right),j,r^{\text{'}}}^{\text{'}}$ indicates the location halfway between the $i$th and $i+1$th elements of the vector $\left\{D_{IPV,j,r^{\text{'}}}\right\}$ indicated by $D_{IPV,\left(i\right),j,r^{\text{'}}}$ and $D_{IPV,(i+1),j,r^{\text{'}}}$. In this particular case $i=\mathrm{1,2},\dots ,Q-1$ so that $\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}}\right\}\in {\mathbb{R}}^{Q-1\times 1}$. In this case, the local damage index vector is defined as $\left\{L_{IPV,j,r^{\text{'}}}\right\}=\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}}\right\}$.

3. Finally, for some boundary conditions and material properties, it is possible that all the elements of the damage index vector are sensitive to the local damage, and their plot looks like a weak impulse. This is the case depicted by Fig. 1(c) in which a damage between the DOFs 4 and 5 is represented. For this particular case to locate the damaged area, the local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ can be defined as the second order difference of the damage index vector, $\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}\mathrm{\text{'}}}\right\}$. The elements of the vector $\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}\mathrm{\text{'}}}\right\}$ are computed as follows:

for $i=\mathrm{0,1},\dots ,Q-1\text{,}$ setting, for convenience, $D_{IPV,\left(0\right),j,r^{\text{'}}}=D_{IPV,(Q+1),j,r^{\text{'}}}=0$ so that $\left\{D_{IPV,j,r^{\text{'}}}^{\mathrm{\text{'}}\mathrm{\text{'}}}\right\}\in {\mathbb{R}}^{Q\times 1}$.

Fig. 1a) Impulse change, b) step change, c) weak change

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In summary, Fig. 1 represents the three abrupt changes in the damage index vector, each of which is related to particular structural cases. For each of them it is possible to derive a local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ linked to the change of the damage index vector: this operation is usually performed manually, but classification algorithms that rely on cross-correlation or classifiers can be designed in order to automate such process.

The final goal of this damage assessment algorithm is to detect abrupt changes in the elements of the local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ to assess the presence of locations of potential damage. In order to quantify the entity of these abrupt changes, the introduction of a threshold value for the elements of the local damage index vector is necessary, as shown in the next section.

2.4. Damage threshold for the local damage index vector

The most suitable approach for the definition of a threshold for local damage index vector is the one proposed by Wang et al. [6, 20] based on the statistics (mean and the standard deviation) of the elements of such a vector. The upper and lower threshold boundaries, $t_u$ and $t_l$, are defined as:

where ${\mu }_D$ and ${\sigma }_D$ are respectively the mean and the standard deviation of the local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$. The term ${\beta }_c$ is a constant value set to define a confidence interval and is commonly assumed to be $\text{1}\le {\beta }_c\le \text{1.8}$. For instance, in case of normal distribution of the values of the elements of the local damage index vector, the choice of ${\beta }_c=\text{1.2}$, ${\beta }_c=\text{1.5}$ and ${\beta }_c=\text{1.8}$ leads to a confidence interval respectively of 76.99 %, 86.64 % and 92.81 %. When the threshold is overcome by some values of the elements of the local damage index vector, the structure is claimed to be damaged and the location of the local damage is detected. A brief example is reported in Fig. 2.

Fig. 2a) Damage index vector {DIPV,j,r'} classified as step change and b) local damage index vector {LIPV,j,r'}

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Let’s focus, for instance, on an 8-DOF system whose damaged state is characterized by a localized damage between the DOFs 6 and 7. By looking at Fig. 2 (a) the damage index vector $\left\{D_{IPV,j,r^{\text{'}}}\right\}$ for $j=\text{1}$ and $r\text{'}=\text{1}$ can be easily classified in the step change class (Fig. 1(b)) while the corresponding local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$, is plotted in Fig. 2(b). Also, in Fig. 2(b), the thresholds are shown for ${\beta }_c=\text{1.0}$, ${\beta }_c=\text{1.2}$ and ${\beta }_c=\text{1.5}$. It is evident that the plot of the local damage index vector indicates that damage has occurred between the DOFs 6 and 7.

4.1. Numerical simulation: 8-DOF lumped model

The model of the structure is an 8-DOF shear-type model matching the one employed by [11] and is shown in Fig. 5(a). In its baseline undamaged conditions, the system is characterized by springs of stiffness $k_i=\text{25000}$ N/m and masses $m_i=\text{1}$ kg for $i=\mathrm{1,2},\dots ,8$. The frame is characterized by modal damping with a damping factor of ${\xi }_i=$ 1 % for each of the 8 vibration modes. The force excitation is applied horizontally on the 8-DOF model via zero-order-hold (ZOH) with a time sampling of 0.01 seconds: it is a zero-mean Gaussian signal with standard deviation $\sigma =\text{1}$ N and is applied at the top of the model (Fig. 5(a)). The input/output time histories are 100 seconds long leading to 10,000 time steps for each signal. In order to simulate a local damage, the 8-DOF system has been weakened by decreasing the stiffness of the spring between the 4th and 5th floors by 20 %.

Once the acceleration response time histories have been simulated, a spectral analysis can be carried out in order to select a suitable mode to be investigated. In this first example, in order to show the effectiveness of the proposed approach, let us assume that there is no measurement noise in the response signals (it will be included at a later stage).

Fig. 5a) 8-DOF shear type system, b) spectral analysis for the 1st floor: power spectral density

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In this case, all the modes appear to be well excited and separated, thus we can select one of them and filter out the contributions of the other modes. It is noteworthy to point out that generally, for civil engineering applications, e.g. buildings, it is common practice to explore lower frequencies rather than the higher ones because they have better resolution and they are more easily excited. Therefore, the proposed algorithm is applied to data obtained by considering only the contribution of the first mode. Based on the spectral analysis, the lower and upper cut-off frequencies of the band pass filter have been set to 4.3 Hz and 5 Hz respectively (see Fig. 5 (b)).

In the proposed methodology, the computation of the IPVs (Eq. (12)) and, consequently, of the damage index $\left\{D_{j,r^{\text{'}}}\right\}$ (Eq. (14)) requires the selection of an arbitrary $j$th reference point. The components of the damage index vector $\left\{D_{IPV,j,1}\right\}$ for any choice of the reference point $j=\mathrm{1,2},\dots ,8$ are reported in Fig. 6.

Fig. 6Damage index vector for j=1,2,…, 8 as defined in Eq. (14), r'=1

Looking at Fig. 6, it can be concluded that, no matter what $j$th reference position is selected, the damage index vector always shows a jump discontinuity between the lumped elements bounding the damaged spring. Such kind of discontinuity let the damage index vector $\left\{D_{IPV,j,1}\right\}$ be classified into the step change class (Fig. 1(b)). Thus, the elements of the local damage index vector $\left\{L_{IPV,j,r^{\text{'}}}\right\}$ can be obtained using Eq. (23). The local damage index vector for $j=$ 1 is plotted over the monitored DOFs in Fig. 7(b) with the threshold set for three different values of ${\beta }_c$ (1.0, 1.2 and 1.5). Clearly, from the analysis of the results, it appears that the proposed algorithm is successful in detecting the damage between the 4th and 5th floors for any of the three values of ${\beta }_c$. Fig. 7(a) shows an analogous result for a damage simulated by 10 % drop in stiffness between the DOFs 4 and 5.

These numerical simulations based on an output-only approach can be compared with those obtained following the approach by Wang et al. [6] which rely on input-output information. By considering this noise free experiment, a band pass filter applied to the acceleration response time histories leads to the same results obtained by filtering the excitation source. It is worthy to recall that, in this specific case, the frequency contribution provided by modes other than the first one ($r\text{'}\ne \text{1}$) have been considered negligible in the filtered frequency band. Of course, this condition implies that the natural frequencies related to those neglected modes should be far enough from the one of the analyzed mode or, at least, the contribution of those modes to the total response should not be significant in the vicinity of such a frequency.

Fig. 7Local damage index vector, a) 10 % damage, b) 20 % damage, r'=1

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Fig. 8Damage index vector for j=1,2,…, 8 ad defined in Eq. (14), r'=2

As the spectral analysis suggests, also the mode related to the second peak appearing in Fig. 5(b) ($r\text{'}=\text{2}$) can be isolated and used in the estimation of the IPVs taking advantage of the flexibility of the proposed methodology. For this purpose, the lower and upper cut-off frequencies of the Finite Impulse Response (FIR) filter have been set equal to 13 Hz and 14.5 Hz respectively. Again, the damage index vector $\left\{D_{IPV,j,2}\right\}$ is computed for any reference point $j=\mathrm{1,2},\dots ,8$ and such vectors are presented in Fig. 8. A jump discontinuity, similar to the one already shown in Fig. 6, appears between the DOFs 4 and 5, revealing the presence of damage. However, an interesting observation can be made by looking at the plots in Fig. 8: it appears that the damage index vectors computed for $j=1,\dots ,5$ are ’mirror’ images of those obtained for $j=6,\dots ,8$. The reason why two different types of damage index vectors appear is due to the fact that, recalling Eq. (14), the sign of the damage index vector depends on the sign of the component of the mode in question at the $j$th reference location ($sign\left({\stackrel{-}{\mathrm{\Phi }}}_{j,r^{\text{'}}}^u\right)$). Thus, moving from the reference point $j=\text{5}$ to the reference point $j=\text{6}$, the damage index changes its sign because there is a sign change between the 5th and 6th components of the second mode. Since this is a peculiarity of the mode analyzed (in this case the second), it can be concluded that the damage index vectors also provide information about the selected mode shape.

Analogously to the previous analysis ($r\text{'}=\text{1}$), also in this case the damage index vector $\left\{D_{IPV,j,2}\right\}$ is claimed to belong to the class defined as a step change and the local damage index vector $\left\{L_{IPV,\mathrm{1,2}}\right\}$ (for $j=\text{1}$) is shown in Fig. 9 for a drop in stiffness of the 10 % (a) and for a drop in stiffness of the 20 % (b).

Fig. 9Local damage index vector, a) 10 % damage, b) 20 % damage, r'=2

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To show the effect of the sign of the component of the selected mode at the reference location on the damage index vector, the sign of the first two mode shapes $sign\left({\stackrel{-}{\mathrm{\Phi }}}_{j,r^{\text{'}}}^u\right)$ ($r\text{'}=\text{1}$ and $r\text{'}=\text{2}$) for reference locations $j=\mathrm{1,2},\dots ,8$ is shown in Fig. 10 together with the corresponding mode shapes. The bars, providing an estimate of $sign\left({\stackrel{-}{\mathrm{\Phi }}}_{j,1}^u\right)$ given by changes in the sign of the damage index vector $\left\{D_{IPV,j,1}\right\}$, have been arbitrarily set equal to +0.5 when $sign\left({\stackrel{-}{\mathrm{\Phi }}}_{j,1}^u\right)$ is positive and equal to -0.5 when it is negative. According to Fig. 6, for $r\text{'}=\text{1}$ the damage index vector $\left\{D_{IPV,j,1}\right\}$ never changes its sign over different reference locations since the components of the first mode have all the same sign (Fig. 10(a)). On the contrary, looking at Fig. 8, the damage index vector $\left\{D_{IPV,j,2}\right\}$ obtained for $r\text{'}=\text{2}$ changes its sign moving from the reference location $j=\text{5}$ to $j=\text{6}$, in agreement with the second mode shape (for both damaged and undamaged conditions).

Fig. 10a) First mode shape estimation, b) second mode shape estimation

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Fig. 11a) Double local damage: damage index vector, b) local damage index vector (for step change)

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It is important to mention that, even in the case of multiple damage locations, the proposed methodology is successful in locating the damaged areas. Fig. 11 shows the damage index vector $\left\{D_{IPV,\mathrm{1,1}}\right\}$ and the corresponding local damage index vector $\left\{L_{IPV,\mathrm{1,1}}\right\}$ for a double damage occurrence between the DOFs 3 and 4 and the DOFs 6 and 7.

Finally, it should be remarked that, as long as the single mode’s contribution to the acceleration responses can be isolated, the normalization in Eq. (11) removes any effect of the input. Consequently, any change in the position and/or magnitude of the input from the undamaged to the damaged configuration doesn’t affect the final solution. This is one of the advantages of the proposed methodology because tests are generally performed under different excitation configurations and so removing the requirement of identical testing conditions from the undamaged and damaged tests free engineers from unnecessary constrains.

4.2. Fully excited system: effects of measurement noise

To look at the impact of external disturbances on the accuracy of the results, let’s consider the same 8-DOF shear-type system subjected to an external excitation at every DOF. Each input force is represented by a zero-mean Gaussian white noise signal, with standard deviation of 1 N, and it is uncorrelated with the others. The disturbance representing measurement noise has been modelled as an additional zero-mean white noise signal, having a root mean square (RMS) equal to a certain percentage of the RMS of the output, and added to the output signals. Because of the stochastic nature of the external white noise excitation, a statistical approach based on Monte Carlo simulations has been used to highlight the effect of noise disturbances on the damage index vector.

Fig. 12a) Excitation setup, b) waterfall plot

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The baseline configuration of the structure and the excitation setup are shown in Fig. 12(a), whereas the waterfall plot of the magnitude of the spectrum of the acceleration time histories, at each DOF, is presented in Fig. 12(b). Also, in this case, the first mode ($r\text{'}=1$) seems to be the perfect candidate for the structural damage assessment through the proposed methodology. Through Monte Carlo simulations, 50 realizations of the cross-correlation vectors $\left\{{\widehat{R}}_{IPV,j,1}^d\right(0\left)\right\}$ and $\left\{{\widehat{R}}_{IPV,j,1}^u\right(0\left)\right\}$ have been generated in order to obtain the damage index vector $\left\{D_{IPV,j,1}\right\}$. Damage has been simulated by introducing a 20 % stiffness reduction between the DOFs 4 and 5. Local damage index vectors $\left\{L_{IPV,j,1}\right\}$ for a noise with RMS of 1 %, 5 %, 10 % and 20 % are reported in Fig. 13. These plots show that, as long as the assumptions behind of the theory of the IPV are fully respected, the damage index shows a remarkable robustness to white noise disturbances and remains a good indicator of damaged areas.

Fig. 13Local damage index vector for varying RMS of noise: a) 1 %, b) 5 %, c) 10 %, d) 20 %

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4.3. Numerical simulation: 100 DOF model

This section extends the investigation about the applicability of the proposed IPV-based damage index vector to a more complex system, (e.g. a plate) represented by a two-dimensional frame (Fig. 14(a)). The structure is a 2-D square grid of 10×10 lumped masses of 1 kg each connected by spring elements placed horizontally, vertically and diagonally, each one having stiffness of 1000 N/m, 900 N/m and 800 N/m respectively. The modal damping has been set to $\xi =$ 1 % for all vibration modes. The structure is doubly-fixed at the top and at the bottom and a set of excitation forces acts perpendicular to the plane of structure. Using the assumption of zero-order-hold (ZOH) with a time sampling of 0.01 seconds, these forces are represented by zero-mean Gaussian signals (uncorrelated to each other) with standard deviation of $\sigma =$ 1 N and a length of 100 seconds.

Fig. 14a) 100 DOF mock up, b) spectrum

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Fig. 14(b) shows the spectral magnitude of the response of the system at DOF 50, i.e. 5th row from the bottom, 10th column from left of the model shown in Fig. 14(a). The spectral analysis suggests that the first mode contribution is the most suitable to be analyzed through the proposed IPV-based approach since it appears well isolated by the other structural modes (0.72 Hz).

Fig. 15a) Simulated damage spot, damage index vector, b) 2-D representation

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The damage is simulated through a drop in stiffness of 25 % for two of the diagonal springs, as shown in Fig. 15(a). After checking that the damage index vector for each $j$th reference point of the 100 DOFs is consistent with the others, Fig. 15 (b) shows the value of the computed damage index vector for a randomly picked reference point $j=$ 92, i.e. 10th row from the bottom, 2nd column from left, far from the damage location. It is clear that the damage location can be identified by just looking at the plot of the damage index vector $\left\{D_{IPV,\mathrm{92,1}}\right\}$. The threshold for the damage has been set accordingly to the mean and the standard deviation of the elements of the damage index vector so that, for a value of the standard deviation multiplier ${\beta }_c=$ 1.8, the ’Upper Bound’ is set at 0.0334 and the ’Lower Bound’ at –0.0356. The latter is reported in the colorbar of Fig. 15(b).

Increasing the level of measurement noise in the response output signals to 5 % RMS and 10 % RMS does not prevent the proposed IPV-based approach to find the damage location, as seen in Fig. 16. In this case, the corresponding ’Upper Bound’ and ’Lower Bound’ are respectively 0.0347 and –0.0375, for the case in Fig. 16(a), and 0.0362 and –0.0387, for the case in Fig. 16(b).

These numerical tests confirm the effectiveness of the proposed IPV-based methodology for damage identification and localization even in the case x more complicated structural models.

Fig. 16Damage index vector, 2-D representation: a) 5 % RMS, b) 10 % RMS

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5.1. 50 % stiffness reduction between floors 1-2 and 2-3

As previously done, the first step is the spectral analysis of the output response in order to evaluate the contribution of the various modes for both the damaged and undamaged configurations. The magnitude of the spectral response of the system in the undamaged and damaged configuration for damage case 1 is shown in Fig. 18 where the spectra of the structural accelerations at each DOF are superimposed. It appears that, after the occurrence of damage, a huge drop of the stiffness between floors 1-2 induces the first natural frequency to shift of almost 2 Hz. For such a reason, a band pass filter with cut-off frequencies at 29 and 33 Hz has been adopted in undamaged conditions whereas in damaged configuration the cut-off frequencies have been changed to 27 and 31 Hz.

Fig. 182-D representation of the Magnitude of the spectral response in a waterfall plot: a) undamaged structure, b) damaged structure, damage between floors 1-2

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Fig. 19 shows the plots of the calculated damage index vector as function of the DOFs for different $j$th reference points ($j=$ 1,…, 4). As seen in the numerical example, when moving from reference point $j=$ 2 to $j=$ 3, the values of the damage index change sign because of the sign change between the two corresponding modal components in Eq. (14). This indicates that the mode considered in the damage assessment analysis is not the first mode: this would have prevented the application of the original formulations of the IPV based approach (Wang et al. [6]). However, this obstacle has been overcome with the proposed methodology.

Fig. 19Damage index vector over different reference points

Usually the threshold for the damage localization relies on calculations of the mean and standard deviation of the elements of the local damage index vector: however, monitoring only 4 positions, the damage index vector contains 4 values only (i.e. Fig. 19) and its first derivative (representing the local damage index vector) only 3 (i.e. Fig. 20). Hence, finding outliers among just 3 points can be difficult when using only mean and standard deviation. However, in this case, having only 3 points does not seem to be a limiting factor: in fact, even with just $3$ points, the algorithm is capable to locate the damaged columns as shown in Fig. 20(a) for case 1 (damage between DOF 1 and DOF 2) and in Fig. 20(b) for case 2 (damage between DOF 2 and DOF 3).

Fig. 20Local damage index vector for a) case 1 and b) for case 2. Reference point j= 4

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5.1.1. 50 % stiffness reduction between floors 3-4

In this case, even though the analysis of this damage scenario is conducted along the same line as the previous ones, something different happens. As shown in Fig. 21, where the damage index vectors for $j=$ 1, 2, 3, 4 are plotted, an interesting behaviour is shown for the case of reference point $j=$ 3 with the corresponding damage index vector showing an inconsistent pattern with respect to the other vectors.

As discussed in [24], the imaginary part of the Frequency Response Function (FRF) provides information about the shape of a particular mode. In Fig. 22, the imaginary part of the response spectra in the frequency range in which the signal is filtered is plotted over the DOFs for the undamaged (a) and damaged (b) conditions. It can be noted that passing from the former (undamaged) to the latter (damaged), the imaginary part for the FRF for DOF 3 changes its sign (from negative -undamaged- to positive -damaged-) violating the theoretical assumption allowing to pass from Eq. (13) to Eq. (14). As a consequence, the reference point $j=$ 3 does not constitute a reliable reference point. All the other DOFs ($j=$ 1, 2, 4) are valid reference points, leading to the correct solution.

Using the damage index vectors in Fig. 22, it is possible to compute the corresponding local damage index vector $\left\{L_{IPV,j,1}\right\}$ and successfully locate the damaged area, Fig. 23(a).

It is interesting to observe that, for this particular study case, the presence of a rigid body mode does not allow to use the mode related to the first natural frequency of the system in the calculation of the IPVs. Since the first mode has a very low natural frequency, its contribution is mixed with that of the rigid body mode and a low pass filter (0-20 Hz) applied to the input excitation helps removing such a contribution in the original acceleration response time histories. This would impair the use of the original formulation of the IPV approach presented by Wang et al. [6]; instead, it does not represent a problem for the current formulation because of its ability to handle higher modes. In addition, the next mode, the one with the lowest natural frequency in the spectrum, presents a structural node at the third floor for the structure in damaged condition (damage between DOFs 3 and 4) and so the arbitrary selection of the reference point might lead to some numerical inaccuracies (Fig. 21, Reference $j=$ 3). It is then recommended, when using the proposed methodology, to compute the damage index vector for multiple reference points in order to verify their compatibility and correctly assess the presence of local damage.

Fig. 21Damage index vector over different reference points

Fig. 22FRF, imaginary part. a) Undamaged configuration, b) damaged configuration

a)

b)

Finally, as a further validation of the effectiveness of the proposed method in assessing the damage location, the case of a unit pulse excitation is analyzed. Considering the acceleration response time histories for case 3 (damage between floors 3-4), the unit pulse response can be obtained by considering the Markov parameters extracted through an input-output identification algorithm, e.g. Observer Kalman Identification [23, 25]. By using the system’s Markov parameters sequences as unit pulse responses into the proposed IPV-based methodology, it is possible to obtain the corresponding local damage index vector. A comparison between the local damage index vectors obtained using a Gaussian white noise excitation (a) and a unit pulse response (b) is shown in Fig. 23. The results are consistent and a successful damage localization is achieved.

Fig. 23Local damage index vectors for a) white noise and b) unit pulse excitation

a)

b)