2.1. Reconstructed Dynamic Displacement Algorithm
The objective of the reconstructed dynamic displacement algorithm is to translate measured accelerations into dynamic displacements, initially introduced by Lee et al. [
23]. The main concept is to minimize the difference between two accelerations: one is from the records, and the other one is transferred from the assumed displacements.
Figure 1 illustrates the procedure of the algorithm. As shown in
Figure 1, noise-contaminated acceleration data are the input, and the reconstructed displacement is the output. To be specific, at each time of reconstruction, a window length (
Nw) of acceleration are chosen, and only a single displacement is reconstructed. The window slides from left to right with the same increment of the acceleration data. Taking window
i = 1 as an example, as shown in
Figure 1, acceleration data inside window
i = 1 are the input for the reconstruction algorithm, and the output is a single displacement termed as
di=1. Such a process is repeatedly performed until window
i = j. The length of the window (
Nw) is determined by an optimization process; details are explained later.
Lee et al. [
22] adopted the central difference method to transfer the assumed displacement to acceleration, in which the reconstructed displacement does not contain noise. Instead of using the central difference method, the Newmark-beta method is utilized in this study to explore the possibility of enhancing the accuracy in displacement reconstruction. Equations (1) and (2) show the Newmark-beta equations, where
,
, and
are the displacement, velocity, and acceleration, respectively.
refers to the time interval,
and
are the Newmark constants. In this study,
and
are taken as 1/2 and 1/6, respectively. Such assumption represents a linear acceleration process:
Equation (1) could be rewritten in a matrix form, as indicated in Equation (3). As shown in Equation (3), matrix
a is a set of acceleration data, and in the proposed algorithm, matrix
a is the measured acceleration while
Ld and
L2 are the constant. The dimension of the matrix
a is adjusted to be the same length as each window (from
i = 1 to
Nw). Thus, the size of matrix
Ld and matrix
L2 are
Nw × (
Nw + 1) and
Nw ×
Nw, respectively. Matrix
v in Equation (3) can be considered as the reconstructed velocity. However, it is understood that the exact solution for
v is not possible.
where:
To find the reconstructed velocity, Equation (3) could be transformed into a minimization problem as described in Equation (4). Setting the first derivation with respect to
v equals to zero, leading to a new equation, as indicated in Equation (5).
Similarly, Equation (3) can be translated to a matrix form, as shown in Equation (6), in which the size of
L3 is
Nw ×
Nw.
where:
where
L3 is a constant and
u is defined as the reconstructed displacements, details of obtaining
u are provided below. Again, Equation (6) can be considered as a minimization problem as well, in which the objective is formulated, as shown in Equation (7). Equation (7) is differentiated with respect to
u to have Equation (2). Using
v in Equation (5) and combining it with Equation (8), one can have Equation (9), which is the formula of reconstructed displacement with measured acceleration (
a) data as input.
The Thikonov regularization is often used to tackle the ill condition of the matrix inverse problem encountered from the reconstruction process. In this study, instead of using Thikonov regularization, Moore–Penrose pseudo inverse is utilized to handle the inverse of matrix Ld. Please note that although Equation (9) yields a set of displacements, only the middle point will be taken as the reconstructed displacement, then the window will continue to slide to obtain another reconstructed displacement. For a given window, the constructed displacement at the middle is the most accurate one among others. To ensure accuracy, a window-sliding technique is utilized wherein each window, only the outcome at the mid-point data is used. That is, for each window operation, a single point is obtained as the reconstructed displacement.
Besides acceleration data, another parameter needed for Equation (9) is the window size (
Nw).
Nw is affected by the frequency and interval time of acceleration records. To determine the size of
Nw, this study utilizes a metaheuristic optimization technique to determine the appropriate
Nw value for each frequency and interval. A single degree of freedom (SDOF) structure with various frequencies from 1–10 Hz with an increment of 1 Hz is generated, in which the accelerations are recorded in different time intervals such as 0.005 s, 0.01 s, and 0.02 s. As a result, 30 different cases are generated, and each SDOF structure is excited by the El Centro earthquake, as shown in
Figure 2.
Figure 3 shows the ground accelerations of El Centro. Since the structure and excited vibration are given, the exact displacement and acceleration of the structure can be calculated using the numerical integration method. The calculated structural accelerations are then added with noise and transformed to displacements using Equation (9). The added noise is an arbitrary random wave with frequencies between 1 to 200 Hz. In addition, the maximum amplitude of the added noise is twice the absolute mean of structural accelerations.
Metaheuristic optimization is performed to find the most suitable
Nw value for each case. As illustrated in
Figure 2, the objective of the optimization is to minimize root-mean-square error (RMSE) between exact and reconstructed displacements. Symbiotic organism search (SOS) [
23] is used in this study. Although metaheuristic optimization probably needs more computational time compared to that of the conventional gradient-based optimization, metaheuristic optimization could avoid to trap at a local minimum point.
Table 2 summarizes the metaheuristic optimization used in this study, including the values of parameters adopted.
Figure 4 shows the optimization results of using the SOS algorithm. Based on the power regression lines, the formula for the
Nw value could be formulated as shown in Equation (10), where
f is the frequency of acceleration and Δ
t is interval time.
Please note that the value of
f in Equation (10) only has a moderate influence on the reconstructed displacement. Therefore, an approximate value of the fundamental frequency is often enough. To be specific, it should be acceptable to use a value of
f with a 10% error. Using the generated 30 cases, the proposed problem is compared to the previous work by H.S. Lee et al.
Table 3 shows the required
Nw values for the proposed and the literature methods, respectively. It is seen that the proposed method requires less
Nw that lessens the computational cost, and reduces the effect of time lag. In addition to the cost,
Table 4 shows that the proposed method also delivers displacements with higher accuracy. It is seen that the performance of the current algorithm is able to reduce the required window size and enhance the accuracy of constructed displacement compared to literature. The current algorithm incorporates the Newmark-Beta as the main component to construct the displacement and utilizes the Moore–Penrose pseudo inverse to tackle the non-symmetrical inverse problem. While the window size is optimized using symbiotic optimization search (SOS). However, it is difficult to distinguish the contribution of each component to the increase in performance.
2.3. Proposed Level 1 and Level 2 Damage Detection
This section explains the proposed method that intends to detect damage occurrence and location. The proposed method utilizes the displacement reconstruction algorithm explained in an earlier section. In the bridge monitoring task, accelerometers often spread along a bridge span, as illustrated in
Figure 7. Assuming that
i number of sensor points (SP) spread along a bridge, as shown in
Figure 7 and each sensor records acceleration data. If there is damage occurred, the collected accelerations naturally include two sets of records. Namely, the original and observed sets. These two sets of accelerations are then transformed into reconstructed displacements (
RDisp) using Equation (10). The root-mean-square (RMS) of
RDisp for each sensor point (
RMSDi) is calculated as formulated in Equation (11).
where
n is the quantity of data.
Each original and observed sets consist of
i number of
RMSD, and for either set and is normalized with its own maximum value if shown by a diagram with respect to the sensor position will create a deflection shape as shown in
Figure 7. Level 1 damage indicator is the shape difference between the original and observed
RMSDs, as shown in
Figure 8. Once there is a significant difference between these two
RMSDs, the bridge will be diagnosed as a damaged one. The larger the damage indicator, the more severely damaged the bridge. A threshold could be suggested based on available data. As shown in
Figure 8, the difference between two
RMSDs at the
SP4 position is zero since two sets of data are normalized with respect to the same midpoint.
For Level 2 damage indicator, the
dRMSD, which is the difference between adjacent
RMSDs as formulated in Equation (12), is calculated (
). Same as Level 1 damage indicator, original and observed
dRMSDs are computed as illustrated in
Figure 9. Subsequently, the difference between original and observed
dRMSDs are plotted with respect to the sensor point, as shown in
Figure 9. Each segment of the bridge is represented by a single value of damage indicator (i.e.,
dRMSD) and If one of the values is particularly prominent, indicating that a particular segment is in a damaged status. The more sensors, the more accurate in locating the damaged position.