Rapid early damage detection using transmissibility with distance measure analysis under unknown excitation in long-term health monitoring

2.1. Transmissibility estimation

For a harmonic applied force at a given coordinate of a linear elastic system, the transmissibility between one point $i$ and $j$ is defined as:

where $X_i$ and $X_j$ are the complex amplitudes of the responses, $x_i\left(t\right)$ and $x_j\left(t\right)$, respectively, and $\omega$ is the natural frequency.

Transmissibility can be estimated by choosing another reference node, for instance $P$, i.e. transmissibility can be derived as:

where $T_{\left(i,j\right)}\left(\omega \right)$ is also called the power spectral density transmissibility (PSDT) [9]. In this study, the transmissibility is estimated using Eq. (2). The choice of $P$ in Eq. (2) may influence the results, and this has been addressed in reference [3]. One possible selection of $P$ is discussed in details in reference [2].

2.2. Relation between Mahalanobis distance and Hotelling T-square

Mahalanobis distance, as a measure of distance between a point and a distribution, was introduced by Mahalanobis [10] in 1930. Mahalanobis distance takes the correlations of the data sets into consideration, and gives a better classification than Euclid distance.

Mahalanobis distance (MD): For two independent data sets $U$, $V$, it is defined as:

Mahalanobis squared distance (MSD) is written as:

where ${\sum }_{}^{-1}$ means the inverse covariance of the baseline, herein it is $\stackrel{-}{V}$, which denotes the sample mean matrix, i.e. the centroid of data set $V$, and $U$ denotes the testing vector, ‘${\left(\right)}^T$’ indicates that the vector should be transposed. In this study, $U$ and $V$ mean transmissibility estimated under damaged condition and intact condition.

In order to overcome the deficiency of high computational cost, one may use Hotelling T-square, which can be considered as partially averaged MSD. This can significantly reduce the computation. Hotelling T-square was first proposed by Harold Hotelling (1931) [11].

Hotelling T square: For the two data sets described above, the Hotelling T-square is defined as:

where $\stackrel{-}{U}$ means the averaged data sets. Herein, note that the testing matrix is averaged for each segment. This will extremely reduce the computational time.

3.1. Damage sensitive feature

Modal Assurance Criterion (MAC) is used to estimate the change of structural dynamic properties by measuring the transmissibility based MSD and Hotelling T-square. The indicators are demonstrated as MSD assurance criterion (MSDC) and Hotelling T-square assurance criterion (TSAC):

where ${MSD}^d$ and ${HT}^d$represent MSD and Hotelling T-square for damaged structure, respectively; while ${MSD}^u$ and ${HT}^u$ denote the MSD and Hotelling T-square for intact structure, respectively. [${\omega }_L$,${\omega }_H$] means the frequency band chosen for analysis. The choice of the frequency band in Eqs. (6) and (7) depends on the engineer’s experience. It is worth mentioning that the noise contaminated part should be avoided in this choice. This has also been addressed in references [2, 3].

3.2. Damage detection procedure

The damage detection procedure will be divided into two stages, namely stage 1 (fast detection) and stage 2 (exact detection). Details description of both stages is given as follows:

Stage 1: Fast detection. In this stage, all the scenarios are averaged in three scenarios and the averaged scenarios are analyzed using Hotelling T Square. Herein, note that the intact pattern is only set in the beginning as baseline, and without any change or modification if the loading and boundary conditions are the same for all scenarios. Note that in this stage, as all the scenarios are averaged, the computational time will be much less than in case if all original data was analyzed. Furthermore, when the minor damage is detected, the damaged scenarios are not all originally measured scenarios. One needs to identify the exact scenario in the original measurements.

Stage 2: Exact detection. In this stage, the key idea is to identify which damage scenario is the exact one. Note that in this stage, the damage detection procedure is almost the same as Stage 1, with only one difference is that herein MSD replaces the T-square, while the baseline is the same as before.

In summary, this damage detection procedure intends to reduce the computational time, while it also keeps the exact damage detection capability. Note that to decide on the averaging length is a key issue depending on the engineering’s experience, and it may greatly affect the results. In order to show the rapidness of the proposed damage detection procedure, MSD based detection for all damage scenarios is used for comparison.

4.1. Model description

A two-sided simply supported beam is simulated and analyzed as shown in Fig. 1(a). The beam is discretized with 20 elements and 21 nodes, which are labeled as 1, 2,…, 21, from left to right. Fig. 1(b) shows a possible occasion of single-damage scenario. A double-damage scenario means two elements are damaged. A unit force is applied at node 9 with spectrum along frequency domain [0, 1000] (Hz). The material properties of the beam are as follows: density = 2700 Kg/m³, dimensions = 1000×50×6 mm, Poisson’s ratio = 0.3, damping ratio = 0.002 and elastic modulus = 70 GPa. Damage is modeled by stiffness loss, i.e. only stiffness reduction based damages are considered [12].

In order to show the performance of the rapid dual-step damage detection procedure, 300 scenarios (#1 to #300) are generated by Latin Hypercube sampling (LHS). Scenarios #1 to #50 under intact state are introduced using random maximum 2 % Gaussian white noise, while scenario #51 to #100 are introduced using random maximum 5 % Gaussian white noise, which are used to model operational conditions. And #101 to #200, #201 to #300 are single- and double-damage scenarios, respectively. Note that noise is not considered for these damaged scenarios.

Fig. 1A schematic diagram of the simply supported beam

a) Intact state

b) Single-damage scenario

4.2. Results and discussions

In order to validate and illustrate the applicability of the proposed approach, the aforementioned damage detection procedure is conducted and the obtained results are shown in the next subsections.

4.2.1. Damage detection by MSD

As for comparison, MSD based damage detection is firstly derived. Fig. 2 shows the MSDC calculated for the 300 scenarios. Herein, note that in Figs. 2 to 4, ‘Zone I’ means MSDC for intact scenario with 2 % Gaussian white noise; and ‘Zone II’ means MSDC for intact scenario with 5 % Gaussian white noise; ‘Zone III’ means MSDC for single-damage scenarios, and ‘Zone IV’ means MSDC for double-damage scenarios.

Fig. 2MSDC for all the damage scenarios and noisy cases

Fig. 3TSAC for all folded scenarios and MSDC for unfolded scenarios

From Fig. 2, one can find the following: (i) It is easy to identify the damages in most single and double-damage scenarios. However, for single-damage scenarios, two damage scenarios (denoted by ‘A’ and ‘B’) failed to be detected; (ii) For damage scenarios ‘A’ (#139) and ‘B’ (#178), the damage severities are 1.09 % in element 19 and 0.4 % in element 12, respectively. For less damage severities, it is challenging for MSDC to detect the damage; (iii) As the MSDC is based on MAC, if a monotonically changing range of MSD is chosen, MSDC might be used for quantify the damage in a relative manner. It should be noted that the severity of damage is defined by a stiffness reduction, i.e. 1 % damage = 1 % stiffness loss.

Fig. 4TSAC for all folded scenarios and MSDC for unfolded scenarios

5.1. Structure description

In order to validate the proposed rapid damage detection procedure using real experimental data, a four-story and two-bay by two-bay steel frame scale model structure shown in Fig. 5 (American Society of Civil Engineers (ASCE) benchmark problem) [13] is used. Details about the experimental set-up can be found in [14-16]. Different damage levels are modeled using adding additional mass, changing braces or loosening bolts at the beam-column connections. Three different types of excitation are modeled [17]. The various damage levels are indicated in Table 2 [16]. One of the reasons why we considered this example for validation of our damage detection technique, is that this benchmark has been widely used in the literature and previous researches have proved its well performance. In this study, the experimental data obtained using a hammer in Phase II is used.

Fig. 5Model of the benchmark structure [16]

5.2. Results and discussion

Figs. 6 and 7 show MSDC and TSAC calculated for all the scenarios considered in Table 2, respectively. From the Fig. 6, one can find that: (i) All damage scenarios can be detected via MSDC. However, it is challenging to determine the damage quantitatively from the MSDC as various damage patterns are presented. (ii) Comparing damage scenario ‘#6’ and ‘#9’, theoretically speaking, they should be the same, however, some differences occur. This might be caused by the operational variability or systematic error. (iii) If all the damages caused by braces modification, i.e. damage scenario ‘#2’, ‘#3’, ‘#4’, ‘#5’, ‘#6’ and ‘#9’, it will be difficult to draw out a general conclusion to quantify damage. Hence, further investigation should be conducted to give a better understanding. (iv) In Fig. 7, each three damage scenarios are averaged, so that the nine damage scenarios are combined as three new damage scenarios. From Fig. 7, it is clear that damage is successfully detected as damage occurs in the first damage scenario (#1), while TSAC decreased in the second damage scenario (#2). Then, by unfolding the damage scenario, one can find that damage occurs in the second scenario (#2 in MSDC in Fig. 7).

Table 3 shows the computational time of MSD based damage detection procedure and rapid damage detection procedure. It can be observed that rapid damage detection procedure consumes less time than MSD based damage detection procedure. However, due to the small number of damage scenarios, the rapid damage detection procedure did not save much time than MSD based damage detection procedure.

Fig. 6MSDC for all damage scenarios

Fig. 7TSAC for all damage scenarios