A Lamb-wave based SHM for multi-damage localizations in large composite plates by using piezoelectric transducer array

An input signal modulated by a Hanning window is used to excite the actuator. A Lamb wave emitted from the actuator is generated and received by sensor. The received signal contains the electromagnetic interference (EMI), FAW and successive wave packets, as shown in figure 1(a). In figure 1(a), ${S_b}$ and $S$ denote the signals before and after the damage exists in the specimen, respectively. The time duration ${T_{\text{H}}}$ of the Hanning window is generally identical to that of the FAW, which is formed by the wave packet of fastest Lamb wave mode. The starting time of the FAW can be estimated by calculating the group velocity of the fastest Lamb wave mode. The velocity can be used to determine the distance between the actuator and sensor. The distance must be large enough to avoid the overlap of EMI and FAW. Thus, it is possible to search the actual starting time of FAW between $t = {T_0}$ and $t = {T_1}$. The time ${T_0}$ is chosen such that ${T_0} > {T_{\text{H}}}$ in order to exclude EMI in the FAW identification. The time ${T_1}$ such that the time interval from ${T_0}$ to ${T_1}$ comprises FAW. In this time interval, the maximum baseline signal is denoted by ${S_{{\text{max}}}}$. Note that ${S_{{\text{max}}}}$ occurs in the FAW, not in the EMI. This value is used to normalize the signals shown in figure 1. Starting at $t = {T_0}$, the jth time stamping is denoted by ${t_j} = {T_0} + \left( {j - 1} \right){{\Delta }}t$, where ${{\Delta }}t$ is the time duration between two adjacent time stampings with a given sampling rate. With the notation ${S_{b,k}}$ being the baseline signal at time ${t_k}$, consider the absolute value of moving average ${\bar S_b}$ of the baseline signal. The value of the moving average at time ${t_j}$ is denoted by ${\bar S_{b,j}}$, which is defined by

$$\begin{align}{\bar S_{b,j}} = \left| {\frac{1}{{{N_{1/4}}}}\mathop \sum \limits_{k = j}^{j + {N_{1/4}}} \frac{{{S_{b,k}}}}{{{S_{{\text{max}}}}}}} \right|\end{align} \tag{ 1 }$$

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where ${N_{1/4}}$ denotes the number of data points in one-quarter period. The relation of ${N_{1/4}}$, excitation frequency $f$ and the sampling rate ${f_s}$ is given by

$$\begin{align}{N_{1/4}} = \frac{{{f_s}}}{{4f}}.\end{align} \tag{ 2 }$$

The moving average of the signal is calculated by averaging the signal data from ${t_j}$ to ${t_j} + T/4$, where $T/4$ denotes the one-quarter period of the sinusoidal excitation signal with frequency $f$. During the one-quarter period, there are ${N_{1/4}}\left( { = {f_s}/4f} \right)$ data points to be averaged. The moving average ${\bar S_b}$ is plotted with the time, which is also shown in figure 1(b). In the beginning of the ${\bar S_b}$ curve, the moving average is less than a threshold value. As the FAW comes, the ${\bar S_b}$ curve increases and exceeds the threshold value. By setting this threshold value, the algorithm can identify the starting time of the FAW. In this study, the threshold value is set to be 0.3, i.e. the starting time ${t_s}$ of FAW is to find ${t_j}$ with the lowest $j$ such that ${\bar S_{b,j}} > 0.3$. The final time ${t_f}$ of FAW can be estimated by ${t_f} = {t_s} + {T_{\text{H}}}$. A typical example to search starting time and final time of FAW is shown in figure 1(b).

The baseline signals before and after impact testing are denoted by ${S_b}$ and $S$, respectively. Theoretically, a difference between ${S_b}$ and $S$ within FAW can be used indicate the existence of a damage. The difference can be evaluated by a damage index DI which is defined by

$$\begin{align}{\text{DI}} = \frac{{{{\mathop \smallint \nolimits}}_{{t_s}}^{{t_f}}{{\left( {S\left( t \right) - {S_b}\left( t \right)} \right)}^2}{\text{d}}t}}{{{{\mathop \smallint \nolimits}}_{{t_s}}^{{t_f}}{S^2}\left( t \right){\text{d}}t}}.\end{align} \tag{ 3 }$$

In a subset shown in figure 2, the transducers are arrayed in two rows with horizontal spacing $w$ and vertical spacing $h$. There are ${N_a}$ transducers that are assigned as actuators in the first row, while there are ${N_s}$ transducers that are assigned as sensors in the second row. In this study, the number of actuators and sensors is ${N_a} = {N_s} = 4$. A lamb wave path is a straight line from an actuator to a sensor. A path group $P_a^{\left( i \right)}$ pivoted at the ith actuator in the subset is formed by four paths emitted from the ith actuator and received by the four sensors, as shown in red lines in figure 2. The jth paths belong to $P_a^{\left( i \right)}$ are denoted by $p_a^{\left( {i,j} \right)}$. The propagation angle of path $p_a^{\left( {i,j} \right)}$ is denoted by $\theta _a^{\left( {i,j} \right)}$.

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The path group can be also defined by pivoting at the jth sensor, as the blue dashed line shown in figure 2. A path group $P_s^{\left( j \right)}$ pivoted by the jth sensor in the subset is formed by paths from the four actuators that are received by the jth sensor. The ith paths belong to $P_s^{\left( j \right)}$ are denoted by $p_s^{\left( {j,i} \right)}$. The propagation angle of path $p_s^{\left( {j,i} \right)}$ is denoted by $\theta _s^{\left( {j,i} \right)}$.

It should be noted that $p_a^{\left( {i,j} \right)}$ and $p_s^{\left( {j,i} \right)}$ are actually the same path. They both refer the path from ith actuator to jth sensor. Thus, this path from ith actuator to jth sensor can be simplydenoted by ${p^{\left( {i,j} \right)}}$. The propagation angle for path ${p^{\left( {i,j} \right)}}$ is denoted by ${\theta ^{\left( {i,j} \right)}}$. It is obvious that

$$\begin{align}{\theta ^{\left( {i,j} \right)}} = \theta _a^{\left( {i,j} \right)} = \theta _s^{\left( {j,i} \right)}.\end{align} \tag{ 4 }$$

The damage index for path ${p^{\left( {i,j} \right)}}$ is denoted by ${\text{D}}{{\text{I}}^{\left( {i,j} \right)}}$.

Figure 3 shows the damage indices versus propagation angles for a path group. If the damage is within the subset, the distribution of the damage indices in figure 3(a) should be similar to a Gaussian distribution. The fitting function $\phi _a^{\left( i \right)}\left( \theta \right)$ of the data DI values for paths in path group $P_a^{\left( i \right)}$ is given by

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$$\begin{align}\phi _a^{\left( i \right)}\left( \theta \right) = m_a^{\left( i \right)}{{\Phi }}_a^{\left( i \right)}\left( \theta \right)\end{align} \tag{ 5a }$$

Similarly, the fitting function $\phi _s^{\left( j \right)}\left( \theta \right)$ of the data DI values for paths in path group $P_s^{\left( j \right)}$ is given by

$$\begin{align}\phi _s^{\left( j \right)}\left( \theta \right) = m_s^{\left( j \right)}{{\Phi }}_s^{\left( j \right)}\left( \theta \right)\end{align} \tag{ 5b }$$

where $m_a^{\left( i \right)}$, $m_s^{\left( j \right)}$ are constants and ${{\Phi }}_a^{\left( i \right)}\left( \theta \right)$ ${{\Phi }}_s^{\left( j \right)}\left( \theta \right)$ are the probability density functions of the Gaussian distribution:

$$\begin{align}{{\Phi }}_a^{\left( i \right)}\left( \theta \right) & = \frac{1}{{\sigma _a^{\left( i \right)}\sqrt {2\pi } }}{e^{ - \frac{1}{2}{{\left( {\frac{{\theta - \bar \theta _a^{\left( i \right)}}}{{\sigma _a^{\left( i \right)}}}} \right)}^2}}}, \notag\\ {{ \Phi }}_s^{\left( j \right)}\left( \theta \right) & = \frac{1}{{\sigma _s^{\left( j \right)}\sqrt {2\pi } }}{e^{ - \frac{1}{2}{{\left( {\frac{{\theta - \bar \theta _s^{\left( j \right)}}}{{\sigma _s^{\left( j \right)}}}} \right)}^2}}}.\end{align} \tag{ 6 }$$

In equation (6), $\bar \theta _a^{\left( i \right)}$ and $\sigma _a^{\left( i \right)}$ are mean angle and standard deviation for path group $P_a^{\left( i \right)}$, respectively, while $\bar \theta _s^{\left( j \right)}$ and $\sigma _s^{\left( j \right)}$ are mean angle and standard deviation for path group $P_s^{\left( j \right)}$, respectively. These statistical parameters can be calculated by

$$\begin{align}\bar \theta _a^{\left( i \right)} = \mathop \sum \limits_{j = 1}^{{N_s}} w_a^{\left( i \right)}{\theta ^{\left( {i,j} \right)}},{\text{ }}\sigma _a^{\left( i \right)} = \sqrt {\mathop \sum \limits_{j = 1}^{{N_s}} w_a^{\left( i \right)}{{\left( {{\theta ^{\left( {i,j} \right)}} - \bar \theta _a^{\left( i \right)}} \right)}^2}} \end{align} \tag{ 7a }$$

$$\begin{align}\bar \theta _s^{\left( j \right)} = \mathop \sum \limits_{i = 1}^{{N_a}} w_s^{\left( j \right)}{\theta ^{\left( {i,j} \right)}},{\text{ }}\sigma _s^{\left( j \right)} = \sqrt {\mathop \sum \limits_{i = 1}^{{N_a}} w_s^{\left( j \right)}{{\left( {{\theta ^{\left( {i,j} \right)}} - \bar \theta _s^{\left( j \right)}} \right)}^2}} \end{align} \tag{ 7b }$$

where $w_a^{\left( i \right)}$ and $w_s^{\left( j \right)}$ are the weighting according to ${\text{D}}{{\text{I}}^{\left( {i,j} \right)}}$:

$$\begin{align}w_a^{\left( i \right)} = \frac{{{\text{D}}{{\text{I}}^{\left( {i,j} \right)}}}}{{{{\mathop \sum \nolimits}}_{j = 1}^{{N_s}}{\text{D}}{{\text{I}}^{\left( {i,j} \right)}}}},{ }w_s^{\left( j \right)} = \frac{{{\text{D}}{{\text{I}}^{\left( {i,j} \right)}}}}{{{{\mathop \sum \nolimits}}_{i = 1}^{{N_a}}{\text{D}}{{\text{I}}^{\left( {i,j} \right)}}}}.\end{align} \tag{ 8 }$$

By least square method, the constants $m_a^{\left( i \right)}$ and $m_s^{\left( j \right)}$ appeared in equation (8) can be determined by

$$\begin{align}m_a^{\left( i \right)} & = \frac{{\sum\limits_{j = 1}^{{N_s}}{\text{D}}{{\text{I}}^{\left( {i,j} \right)}}\left( {{{\Phi }}_a^{\left( i \right)}{|_{\theta = {\theta ^{\left( {i,j} \right)}}}}} \right)}}{{\sum\limits_{j = 1}^{{N_s}}{{\left( {{{\Phi }}_a^{\left( i \right)}{|_{\theta = {\theta ^{\left( {i,j} \right)}}}}} \right)}^2}}},\notag\\{ }m_s^{\left( j \right)} & = \frac{{\sum\limits_{i = 1}^{{N_a}}{\text{D}}{{\text{I}}^{\left( {i,j} \right)}}\left( {{{\Phi }}_s^{\left( j \right)}{|_{\theta = {\theta ^{\left( {i,j} \right)}}}}} \right)}}{{\sum\limits_{i = 1}^{{N_a}}{{\left( {{{\Phi }}_s^{\left( j \right)}{|_{\theta = {\theta ^{\left( {i,j} \right)}}}}} \right)}^2}}}.\end{align} \tag{ 9 }$$

By using equations (5) and (6), the peak value $\hat \phi _a^{\left( i \right)}$ of the fitting function for path group $P_a^{\left( i \right)}$ is

$$\begin{align}\hat \phi _a^{\left( i \right)} = \phi _a^{\left( i \right)}{|_{\theta = \bar \theta _a^{\left( i \right)}}} = \frac{{m_a^{\left( i \right)}}}{{\sigma _a^{\left( i \right)}\sqrt {2\pi } }}.\end{align} \tag{ 10a }$$

Similarly, the peak value $\hat \phi _s^{\left( j \right)}$ of the fitting function for path group $P_s^{\left( j \right)}$ is

$$\begin{align}\hat \phi _s^{\left( j \right)} = \phi _s^{\left( j \right)}{|_{\theta = \bar \theta _s^{\left( j \right)}}} = \frac{{m_s^{\left( j \right)}}}{{\sigma _s^{\left( j \right)}\sqrt {2\pi } }}.\end{align} \tag{ 10b }$$

According to the transducer layout, there are eight path groups in one subset, four of which are pivoted by actuators and the rest four of which are by sensors. For the path group $P_a^{\left( i \right)}$, the damage is assumed to be located on the straight line $L_a^{\left( i \right)}$ passing through the ith actuator with a slope corresponding to the mean angle $\bar \theta _a^{\left( i \right)}$. Similarly, another straight line $L_s^{\left( j \right)}$ passing through the jth sensor can also be determined for the path group $P_s^{\left( j \right)}$. The straight lines are called the mean angle lines. For locations of ith actuator and jth sensor denoting by $( {x_a^{\left( i \right)},y_a^{\left( i \right)}} )$ and $( {x_s^{\left( j \right)},y_s^{\left( j \right)}} )$, respectively, the mean angle lines can be written by

$$\begin{align}L_a^{\left( i \right)}:{\text{ }}y & = y_a^{\left( i \right)} + \left( {x_a^{\left( i \right)} - x} \right)\tan \bar \theta _a^{\left( i \right)}, \notag\\ & \quad {\text{ for path group pivoted at }}i{\text{th actuator}}\end{align} \tag{ 11a }$$

$$\begin{align}L_s^{\left( j \right)}:{ }y & = y_s^{\left( j \right)} + \left( {x_s^{\left( j \right)} - x} \right)\tan \bar \theta _s^{\left( j \right)},\notag\\ & \quad {\text{ for path group pivoted at }}j{\text{th sensor}} .\end{align} \tag{ 11b }$$

It should be noted that not every path group can be used for damage localization. Some path groups are excluded from the damage localization calculations. The criterion to exclude a path group will be discussed later. Introduce two parameters $f_a^{\,\left( i \right)}$ and $f_s^{\,\left( j \right)}$ defined by

$$\begin{align}f_a^{\,\left( i \right)} = \left\{ \begin{array}{*{20}{l}} {0,{\text{ if path group }}P_a^{\left( i \right)}{\text{ is excluded}}} \\ {1,{\text{ if path group }}P_a^{\left( i \right)}{\text{ is preserved}}} \end{array} \right.\end{align} \tag{ 12a }$$

$$\begin{align}f_s^{\,\left( j \right)} = \left\{ {\begin{array}{*{20}{l}} {0,{\text{ if path group }}P_s^{\left( j \right)}{\text{ is excluded}}} \\ {1,{\text{ if path group }}P_s^{\left( j \right)}{\text{ is preserved}}} \end{array}.} \right.\end{align} \tag{ 12b }$$

Theoretically, by excluding path groups based on the above two exclusion rules, all the preserved mean angle lines pass through the damage location, i.e. these lines are concurrent at this location. However, due to measurement errors or other uncertain reasons, this concurrent does not occur. Suppose the damage location is estimated at a point $\left( {\bar x,\bar y} \right)$. In order to determine this point, consider the distance $d_a^{\left( i \right)}$ between $\left( {\bar x,\bar y} \right)$ and the mean angle line for path group $P_a^{\left( i \right)}$. According to equation (11), $d_a^{\left( i \right)}$ can be written by

$$\begin{align}d_a^{\left( i \right)} = \frac{{\left| {y_a^{\left( i \right)} + \left( {x_a^{\left( i \right)} - \bar x} \right)\tan \bar \theta _a^{\left( i \right)} - \bar y} \right|}}{{\sqrt {{{\tan }^2}\bar \theta _a^{\left( i \right)} + 1} }}.\end{align} \tag{ 13a }$$

Similarly, the distance $d_s^{\left( j \right)}$ for path group $P_s^{\left( j \right)}$ is

$$\begin{align}d_s^{\left( j \right)} = \frac{{\left| {y_s^{\left( j \right)} + \left( {x_s^{\left( j \right)} - \bar x} \right)\tan \bar \theta _s^{\left( j \right)} - \bar y} \right|}}{{\sqrt {{{\tan }^2}\bar \theta _s^{\left( j \right)} + 1} }}.\end{align} \tag{ 13b }$$

Define a function ${R^2}$ by

$$\begin{align}{R^2} = \sum\limits_{i = 1}^{{N_a}} \left( {f_a^{\,\left( i \right)}{{\left( {d_a^{\left( i \right)}} \right)}^2}} \right) + \sum\limits_{j = 1}^{{N_s}} \left( {f_s^{\,\left( j \right)}{{\left( {d_s^{\left( j \right)}} \right)}^2}} \right).\end{align} \tag{ 14 }$$

The geometric meaning of ${R^2}$ is the summation for the distance square from the point $\left( {\bar x,\bar y} \right)$ to the preserved mean angle lines. Then the solution for $\left( {\bar x,\bar y} \right)$ can be determined by solving the following simultaneous equations:

$$\begin{align}\frac{{\partial {R^2}}}{{\partial \bar x}} = 0,{ }\frac{{\partial {R^2}}}{{\partial \bar y}} = 0.\end{align} \tag{ 15 }$$

If all the paths are excluded, ${R^2}$ is zero and no damage is detected in this subset.

The maximum $\hat \phi _a^{\left( i \right)}$ and $\hat \phi _s^{\left( j \right)}$ values for preserved path groups in a subset can be used as an indicator to evaluate the damage size. This maximum value ${\hat \phi _{{\text{max}}}}$ can be calculated by

$$\begin{align}{\hat \phi _{{\text{max}}}} = {\text{max}}\left( {f_a^{\,\left( i \right)}\hat \phi _a^{\left( i \right)},f_s^{\,\left( j \right)}\hat \phi _s^{\left( j \right)}} \right).\end{align} \tag{ 16 }$$

For the case that no path group is preserved, ${\hat \phi _{{\text{max}}}} = 0$. The dimensionless distance between the detected location $\left( {\bar x,\bar y} \right)$ and the nearest damage location $\left( {{x_d},{y_d}} \right)$ is denoted by $\bar D$, which is defined as

$$\begin{align}\bar D = \frac{{\sqrt {{{\left( {\bar x - {x_d}} \right)}^2} + {{\left( {\bar y - {y_d}} \right)}^2}} }}{w}\end{align} \tag{ 17 }$$

where $w$ denotes the horizontal distance between two adjacent transducers, as indicated in figure 2.

In this study, two exclusion rules for path groups are proposed. The details for the rules are described as follows.

2.4.1. Exclusion rule 1.

2.4.2. Exclusion rule 2.

It is assumed that the damage index of a path is greater than a certain level if the path passes through the damage. It is expected that the damage indices of the paths in a path group are small if no damage exists. For this case, the path group should be excluded. Thus, the exclusion rule 2 states that a path group is excluded from the damage localization calculations if the damage indices of this path group is less than a specific value. By introducing a threshold damage index ${\text{D}}{{\text{I}}_{{\text{th}}}}$, a path group is excluded from damage localization calculations if

$$\begin{equation*}\hat \phi _a^{\left( i \right)} < {\text{D}}{{\text{I}}_{{\text{th}}}}{\text{ for path group }}P_a^{\left( i \right)}\end{equation*}$$

$$\begin{align}\hat \phi _s^{\left( j \right)} < {\text{D}}{{\text{I}}_{{\text{th}}}}{\text{ for path group }}P_s^{\left( j \right)}.\end{align} \tag{ 18 }$$

In another case that a damage occurs within a subset, the peak value $\hat \phi _a^{\left( i \right)}$ (or $\hat \phi _s^{\left( j \right)}$) exceeds ${\text{D}}{{\text{I}}_{{\text{th}}}}$. However, this path group does not directly interest with the damage. For such case, the damage indices of this path group should be lower than those of other path groups in the subset that are directly affected by the damage. Thus, another reference value ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ is introduced and the exclusion rule 2 is modified by

$$\begin{equation*}\hat \phi _a^{\left( i \right)} < {\text{max}}\left( {{\text{D}}{{\text{I}}_{{\text{ref}}}},{\text{D}}{{\text{I}}_{{\text{th}}}}} \right){\text{ for path group }}P_a^{\left( i \right)}\end{equation*}$$

$$\begin{align}\hat \phi _s^{\left( j \right)} < {\text{max}}\left( {{\text{D}}{{\text{I}}_{{\text{ref}}}},{\text{D}}{{\text{I}}_{{\text{th}}}}} \right){\text{ for path group }}P_s^{\left( j \right)}.\end{align} \tag{ 19 }$$

Equation (18) can be considered as a special case of equation (19) by letting ${\text{D}}{{\text{I}}_{{\text{ref}}}} = 0$. If ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ and ${\text{D}}{{\text{I}}_{{\text{th}}}}$ are both set to be zero, i.e. ${\text{D}}{{\text{I}}_{{\text{ref}}}} = {\text{D}}{{\text{I}}_{{\text{th}}}} = 0$, it means that the exclusion rule 2 is not applied. In the above procedure, the values of ${\text{D}}{{\text{I}}_{{\text{th}}}}$ and ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ are the key parameters that can affect the damage localization results. The threshold damage index ${\text{D}}{{\text{I}}_{{\text{th}}}}$ can be estimated by observing the largest $\hat \phi _a^{\left( i \right)}$ and $\hat \phi _s^{\left( j \right)}$ for paths in a subset without damage within it. The determination of ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ will be discussed latter.

As mentioned earlier, the introduction of ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ is used to exclude the path groups that do not directly interact with the damage. The paths within a subset can be categorized into two types based on whether they hit the damage or not. ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ can be defined as the minimum damage index from the paths that hit the damage. The following procedure provides a simple way to estimate ${\text{D}}{{\text{I}}_{{\text{ref}}}}$.

The reference damage index ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ is calculated subset by subset. The transducers in a subset are arrayed such that the numbers of actuators and sensors are ${N_a}$ and ${N_s}$, respectively. Therefore, the total number of paths is ${N_p} = {N_a}{N_s}$, and there are totally ${N_p}$ damage indices in a subset. In this study, ${N_a} = {N_s} = 4$ and ${N_p} = 16$. These damage indices can be ranked according to their values for all paths. Suppose that there exists a damage within a subset. By taking all the damage indices in a subset, they can be sorted in descending order and plotted figure as shown in figure 4. In the descending plot, ${\text{D}}{{\text{I}}_n}$ denotes the nth damage index, which means that ${\text{D}}{{\text{I}}_1}$ is the largest damage index, ${\text{D}}{{\text{I}}_2}$ is the second largest, etc. It is seen that some of the damage indices are significantly higher than the others, indicating that these paths are likely to hit the damage. It is also seen that the ${\text{D}}{{\text{I}}_n}$ curve can be roughly divided by two segments with different slopes, and the point where the slope changes can be assigned to determine ${\text{D}}{{\text{I}}_{{\text{ref}}}}$. However, it is not smooth for each of the two segments and the slope change location is not easy to be identified. In order to smooth the curve, introduce ${\text{DI}}_n^{\left( {{\text{ave}}} \right)}$ which is the average value of the damage indices from ${\text{D}}{{\text{I}}_n}$ to ${\text{D}}{{\text{I}}_{{N_p}}}$:

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$$\begin{align}{\text{DI}}_n^{\left( {{\text{ave}}} \right)} = \frac{1}{{16 - n + 1}}\sum\limits_{m = n}^{{N_p}} {\text{D}}{{\text{I}}_m} \end{align} \tag{ 20 }$$

Figure 4 also shows the ${\text{DI}}_n^{\left( {{\text{ave}}} \right)}$ plot, which shows two segments with different slopes. An apex can be identified at the point with slope change. In the ${\text{DI}}_n^{\left( {{\text{ave}}} \right)}$ plot, the coordinate of the nth point can be written as $( {n,{\text{DI}}_n^{\left( {{\text{ave}}} \right)}} )$. The first point $( {1,{\text{DI}}_1^{\left( {{\text{ave}}} \right)}} )$ and last point $( {{N_p},{\text{DI}}_{{N_p}}^{\left( {{\text{ave}}} \right)}} )$ make a line that makes an angle of $\alpha$ with the horizontal line. This angle can be calculated by

$$\begin{align}\alpha = {\tan ^{ - 1}}\left( {\frac{{{\text{DI}}_1^{\left( {{\text{ave}}} \right)} - {\text{DI}}_{{N_p}}^{\left( {{\text{ave}}} \right)}}}{{{N_p} - 1}}} \right).\end{align} \tag{ 21 }$$

Define the vector ${[ {\begin{array}{*{20}{c}} n&{{\text{DI}}_n^{\left( {{\text{ave}}} \right)}} \end{array}}]^{\text{t}}}$ that started from the origin point $\left( {0,{\text{ }}0} \right)$ to the point $( {n,{\text{DI}}_n^{\left( {{\text{ave}}} \right)}} )$, where the superscript t denotes the transpose. The rotation of this vector by angle $\alpha$ counterclockwise about the origin point forms the rotated vector ${[ {\begin{array}{*{20}{c}} {{n^{\left( {{\text{rot}}} \right)}}}&{{\text{DI}}_n^{\left( {{\text{rot}}} \right)}} \end{array}} ]^{\text{t}}}$, which can be calculated according to

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} {{n^{\left( {{\text{rot}}} \right)}}} \\ {{\text{DI}}_n^{\left( {{\text{rot}}} \right)}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{ - \sin \alpha } \\ {\sin \alpha }&{\cos \alpha } \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} n \\ {{\text{DI}}_n^{\left( {{\text{ave}}} \right)}} \end{array}} \right\}.\end{align} \tag{ 22 }$$

In general, the maximum damage index is approximately equal to 1 and ${\text{DI}}_{{N_p}}^{\left( {{\text{ave}}} \right)} \ll 1$. With ${N_p} = 16$, we have ${N_p} \gg {\text{DI}}_{{N_p}}^{\left( {{\text{ave}}} \right)}$. It gives that $\alpha$ is a small angle and

$$\begin{align}{n^{\left( {{\text{rot}}} \right)}} & \approx n\end{align} \tag{ 23a }$$

$$\begin{align}{\text{DI}}_n^{\left( {{\text{rot}}} \right)} & = n\sin \alpha + {\text{DI}}_n^{\left( {{\text{ave}}} \right)}\cos \alpha .\end{align} \tag{ 23b }$$

The ${\text{DI}}_n^{\left( {{\text{rot}}} \right)}$ plot is shown in figure 4. With the rotation, it is seen that ${\text{DI}}_1^{\left( {{\text{rot}}} \right)} = {\text{DI}}_{{N_p}}^{\left( {{\text{rot}}} \right)}$. It is also seen that the index $n = {n_{{\text{ref}}}}$ corresponding to the slope change in ${\text{DI}}_n^{\left( {{\text{ave}}} \right)}$ plot can be obtained by seeking the minimum of the ${\text{DI}}_n^{\left( {{\text{rot}}} \right)}$ plot. Then ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ is assigned to be the damage index ${\text{D}}{{\text{I}}_{{n_{{\text{ref}}}}}}$ corresponding to this minimum.

Three specimens made of CFRP were prepared for the experiments in this study. Each specimen has identical area of 27 in by 21 in (685.8 mm by 533.4 mm). In applications to aircraft engineering, the thickness of the composite plates is generally non-uniform. The tapered thickness is a common seen design. It is well known that the Lamb wave propagation depends on the plate thickness. Thus, three specimens labeled A, B and C are considered, for which three thickness arrangements are presented: no taper (uniform thickness), T1 thickness taper and T2 thickness taper, respectively. The taper design is illustrated in figure 5. According to thickness, the T1 and T2 tapers can be divided into five and nine taper zones, respectively. For tapered specimen, the bottom surface is not flat due to the thickness variation in two adjacent taper zones. The sensors are adhered on the top surface.

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The reference stacking sequences is symbolized by ${R_0}$, for which the detail is listed in table 1. The ${R_0}$ laminate has 24 layers, in which a single lamina has a thickness of 0.005 in (0.127 mm). Table 2 lists the details of the stacking sequences of the three specimens. In table 2, ${R_{ - n}}$ denotes a stacking sequence by deleting the last n laminas of ${R_0}$. For specimen A, there is no taper design and the stacking sequence is ${R_{ - 4}}$ ($n = 4$). It means that the specimen A is stacked by 20 layers. For tapered specimen, the stacking sequence is denoted according to taper zones. For example, the T1 taper is utilized in specimen B and the stacking sequences for taper zones T1–1 to T1–5 are denoted by R_-8/R_-6/R_-4/R_-2/R₀. All the specimens were fabricated by Aerospace Industrial Development Corporation (Taichung, Taiwan).

In the experiment, each specimen was placed on the drop-weight impact testing machine for creating impact damages, as shown in figure 6. The ASTM D7136 test standard was followed. The composite plate was placed on clamping fixture and impacted by an impactor with a mass of 5988 g from a pre-set height. The impact energy can be determined by impactor mass and the height. Each specimen was conducted two experiments. For the first experiment, the impact testing was conducted at three selected locations. In the second experiment, a second impact testing was conducted again at the same locations in each specimen. Two additional locations in each specimen were selected for impact testing in the second experiment. Figure 7 shows the specimens after impact testing. Table 3 lists the details of the impact testing experiments. By using ultrasonic NDI (Olympus OmniScan SX), a damage area was marked for each damage spot. This area can be approximated by an ellipse with horizontal semi-axis $c$ and vertical semi-axis $a$. The damage size for each damage in table 3 is denoted by $2c \times 2a$.

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Figure 8 shows the transducer array in one specimen. In each CFRP plate, 24 Smart Layer transducers (Acellent Technologies, Sunnyvale, California) numbered from 1 to 24 are mounted on the top surface. The transducers are arrayed in three rows and eight columns. The distances between two adjacent rows and between two adjacent columns are 63.5 mm (2.5 in) and 152.4 mm (6 in), respectively. The damage locations listed in table 3 are also marked in figure 8 for reference.

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For generating Lamb waves, the actuator is excited by a Hamming window signal. The input signal ${S_{{\text{input}}}}$ can be formulated by

$$\begin{align}{S_{{\text{input}}}} = {V_0}H\left( t \right)\sin \left( {2\pi ft} \right)\end{align} \tag{ 24a }$$

where ${V_0}$ is the peak voltage of the signal and $H\left( t \right)$ is the Hanning window:

$$\begin{align}H\left( t \right) = \frac{1}{2}\left( {1 - \cos \left( {\frac{{2\pi ft}}{m}} \right)} \right)\end{align} \tag{ 24b }$$

where $m$ is an integer which defines the width of the Hanning window. In this study, $m = 5$ is used in the experiments. A system composed of Smart Layer transducers, hardware ScanGenie III and a laptop was utilized for signal excitation and acquisition, as shown in figure 9(a). The software SHM Composite (version 2.18.1, Acellent Technologies, Sunnyvale, California) installed in the laptop was utilized to control the hardware and transducers, as shown in figure 9(b). The sampling rate of the receiving signals is ${f_s} = 48$ MHz. Figure 10 shows the dispersion curves for various laminate with different layer sequences. The group velocities shown in figure 10 are calculated with the code GUIGUW (Graphical User Interface for Guided Ultrasonic Waves). The stiffness matrix ${\mathbf{C}}$ for a lamina with its principal coordinate is given by

$$\begin{align}{\mathbf{C}} = \left( {\begin{array}{*{20}{c}} {111.68}&{3.3594}&{3.3594}&0&0&0 \\ {3.3594}&{10.128}&{3.3096}&0&0&0 \\ {3.3594}&{3.3096}&{10.128}&0&0&0 \\ 0&0&0&{4.5}&0&0 \\ 0&0&0&0&{3.58}&0 \\ 0&0&0&0&0&{3.58} \end{array}} \right){\text{ GPa}}\end{align} \tag{ 25 }$$

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The density and thickness of one lamina is 1750 kg m⁻³ and 0.005 in (0.127 mm), respectively. By taking the path 1–9 and path 1–11 as examples, the taper zones for the three specimens include ${R_{ - 4}}$, ${R_{ - 6}}$ and ${R_{ - 8}}$. Note that the plate simulation with GUIGUW must be with uniform thickness. The wave propagates through various thickness along path 1–11 for specimens B and C. For simplification, various thickness conditions are considered for the dispersion curves solved with GUIGUW in figure 10. Also note that the wave propagation characteristics vary with the propagation direction due to the anisotropic behavior of the composite plate. According to the results shown in figure 10, it is seen that the group velocity of the S0 mode approximately keeps constant for frequency less than 300 kHz. For a frequency greater than 300 kHz, the group velocity of S0 mode may be no longer the fastest and the trends of the dispersion curves becomes complex. According to the group velocity shown in figure 10, the shortest time of flight (TOF) between any two transducers is approximately ranged from 30 $\mu {\text{s}}$ to 33 $\mu {\text{s}}$. The time span of the EMI is approximately equal to that for Hanning window, which is inversely proportional to the frequency $f$. For Hanning window with lower frequency, the time span of the EMI could be long and overlap the FAW, causing difficulties in the ${t_s}$ calculations. According to the experimental observations, the time span of EMI for $f = 250$ kHz is approximately 28 $\mu {\text{s}}$, which does not overlap the EMI. Thus, three frequencies, 250 kHz, 275 kHz and 300 kHz are utilized to generate the Lamb waves in this study.

The wave propagation in the composite plate was also simulated by finite element software ABAQUS (version 6.14). Due to the limitation of the computation power, partial area with 15 in by 15.5 in (381 mm by 393.7 mm) of the composite plate with transducers 1, 9 and 11 are established. The material properties of the composite are as the same as those used in the dispersion curves shown in figure 10. Unlike the group velocity calculations with GUIGUW, the wave propagation calculations in ABAQUS were conducted with transient analysis. In addition, the three-dimensional geometry in ABAQUS model includes the taper zones with variation of thickness. The C3D8R element in ABAQUS (8-node hexagonal element) is utilized to mesh the composite plate. The transducers are meshed by piezoelectric elements (C3D8E). The input voltage modulated according to equation (24) is applied on the piezoelectric actuator and the output voltage can be monitored as the Lamb waves pass through the sensors. According to a mesh sensitivity test, the element size in the model is approximately equal to 0.8 mm. Figure 11 shows the transient displacement response of the Lamb wave generated by transducer 1 with ABAQUS for specimen C. At the beginning of the wave propagation as shown in figure 11(a), it is seen that the group velocities along vertical and horizontal directions are not the same, indicating the anisotropy of the laminate plates. Figure 11(b) shows the first wave packet, the S0 mode, passing through the transducers 9 and 11 at $t = 46.2{\text{ }}\mu {\text{s}}$. Figure 11(c) shows the displacement contour at $t = 76.5{\text{ }}\mu {\text{s}}$, revealing that there exist multiple complicated wave packets, which comprise higher-order modes and S0 mode reflected from the boundaries. Thus, it is difficult to identify the modes of the signals received by sensors after $t = 70{\text{ }}\mu {\text{s}}$. Fortunately, these complicated wave packets can be ignored as the FAW only contains the S0 mode, which has the fastest group velocity.

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For damage localization, some transducers are grouped to belong to one subset. Note that the cover area of a subset cannot be too large. For example, if transducers 1–16 are grouped in one subset, then the wave propagation distance for path 1–9 differs a lot from that for path 1–16. Thus, the comparison of damage indices between these two paths becomes improper. In addition, the signal of a path with long distance may also be weak and become unavailable for damage localization. A subset with too small covering area is also unavailable because of the limited detection area. In this study, ten subsets are assigned, each of which is composed of 8 transducers, as listed in table 4. It should be also noted that the cover area of a subset overlaps that of the adjacent one. This arrangement can increase the detectability of the damage.

For the transducer layout shown in figure 8, it should be confirmed that the distance between two transducers of a path is long enough that the starting time of the FAW is longer than the time duration of the EMI. Figure 12 shows the signals for paths 1–9 and 1–11. The excitation frequency is $f = 250$ kHz. For specimen A, no taper design in the layer sequences. For specimens B and C, the taper definitions are shown in figure 5 and table 2. For paths 1–9 and 1–11, the wave propagation paths pass through different thickness region for the three specimens. The FEM results for specimens A, B and C show that the thickness variations do not significantly affect the group velocity of Lamb waves. Figure 12 also shows the measured baseline signals for the two paths of the specimens A, B and C. It is seen that the measured group velocity of the Lamb wave propagating along different paths agree well with the FEM results. For specimen A, the number of layers is 20. For specimen B, the path 1–9 lies on the taper zone T1–2, which has 18 layers. For specimen C, the path 1–9 lies near the border of the taper zones T2–2 and T2–3, which have 18 layers and 20 layers, respectively. The measured results in figure 12 show that no significant difference exists for the starting time of the FAW of path 1–9 in the three specimens. Due to the longer distance, the TOF of the wave for path 1–11 is larger than that for path 1–9. For specimens B and C, the path 1–11 across different taper zones, and similar results for path 1–11 are observed. It should be noted that the group velocity theoretically varies with different plate thickness. Due to the insignificant thickness variation in the tapered plate, the results in figure 12 show that the group velocity is not significantly affected by the thickness variations in the taper designs. The time duration of EMI for $f = 250$ kHz is ${T_{\text{H}}} = 24{ }\mu {\text{s}}$. The starting time of FAW of path 1–9 for $f = 250$ kHz is about 40 $\mu {\text{s}}$. As mentioned in the discussions on figure 11, the FAW contains the S0 mode. For the succeeded wave packets after FAW, they comprise the higher-order modes or the S0 mode reflected from the boundary, and it is difficult to identify the wave modes. For figure 12 shows the wave signals for an excitation frequency of 250 kHz. For the other two driving frequencies, 275 kHz and 300 kHz, the similar results can be obtained.

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Based on the FEM results and measured data, it is found that the starting time ${t_s}$ of FAW is longer than the time duration ${T_{\text{H}}}$ of Hanning window. For the current transducer layout, the FAW is not interfered by the EMI.

Figure 13 shows the comparisons of baseline signal ${S_b}$ and signal $S$ with damage for paths 6–14 and 11–19. Note that the curves in figure 13 are original, non-normalized signals. The signals are extracted with $f = 300$ kHz. Note that these signals are original and not normalized. These two paths have vertical path angle, i.e. $\theta _a^{\left( {6,14} \right)} = \theta _a^{\left( {11,19} \right)} = 90^\circ$. However, the thickness where the paths located are vary with different specimen. For specimen A, the plate has 20 layers with uniform thickness. For specimen B, the path 6–14 is located in taper zone T1–4 (22 layers), while path 11–19 is in taper zone T1–3 (18 layers). For specimen C, the paths 6–14 and 11–19 are located in taper zones T2–7 (20 layers) and T2–5 (24 layers). Recall that the thickness of each layer is 0.005 in (0.127 mm). The two paths for the three specimens are located in the taper zones ranged from 18 layers to 24 layers. It is seen that the starting time of the FAW (S0 mode) has no significant variations for the two paths in the three specimens. Again, the signals in figure 13 show that the layer sequences for different plates have no significant effect on the group velocity of S0 mode. Recall the damage locations shown in figure 13, there is no damage on the path 6–14 for the three specimens, indicating that the signals with damage are identical to the baseline signal. For path 11–19, there is a damage on this path for specimen A and the signal $S$ has a difference by comparing ${S_b}$, as shown in figure 13(a). For path 11–19 in other two specimens, no damage exists on the path and signal $S$ has no significant difference from ${S_b}$, as shown in figures 13(b) and (c).

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Figure 14 shows the comparisons of baseline signal ${S_b}$ and signal $S$ with damage for paths 6–16 and 11–21. The signals are extracted with $f = 300$ kHz. These two paths have the same path angle, i.e. $\theta _a^{\left( {6,16} \right)} = \theta _a^{\left( {11,21} \right)} = 50.2^\circ$. For specimens B and C, the two paths pass through different tapered zones. Similar to the results shown in figure 13, the signals in figure 14 show that the group velocity of S0 mode is nearly independent of the thickness variations in the tapered composite plate. It is also observed that the signal $S$ has significant change from baseline signal ${S_b}$ if there exists a damage on the path.

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The dispersion characteristics for Lamb waves propagating through the damage are quite complicated. However, the arrival times of the S0 mode for the baseline signal are equal to that for the signal with damage, as shown in figures 13 and 14. It suggests that the group velocities from the dispersion curves as shown in figure 10 are also valid for waves propagating through damage.

By analyzing the signals from all paths, the damage indices for these paths can be calculated. Figure 15 shows the damage indices for paths in the experiments. By comparing with the damage location shown in figure 8, it is observed large damage indices of the paths that pass through the damages. For example, damages exist on the path 6–16 for the three specimens, and the damage indices for this path in figure 15 are greater than 0.1. It is also observed that the damage indices for path 6–16 after the second impact are greater than those in the first impact.

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In figure 8, it is observed that no damage exists within subset 3 for the three specimens. The ${\hat \phi _{{\text{max}}}}$ values are shown in figure 16. In order to determine ${\text{D}}{{\text{I}}_{{\text{th}}}}$, the values of ${\hat \phi _{{\text{max}}}}$ in figure 16 are calculated from the path groups thar are preserved by applying the exclusion rule 1 only, i.e. the exclusion rule 2 is not applied in the calculations of ${\hat \phi _{{\text{max}}}}$. It is seen that the ${\hat \phi _{{\text{max}}}}$ values in subset 3 for the first impact experiment are typically lower than 0.005. After the second impact, the ${\hat \phi _{{\text{max}}}}$ values for specimen A and specimen B are greater than 0.01 because two damages A-4 and B-5 are located near subset 3. In specimen C, no damage is located near subset 3 after the second impact. Thus, the ${\hat \phi _{{\text{max}}}}$ values in subset 3 of specimen C for the second impact are typically unchanged by comparison those for the first impact. According to the ${\hat \phi _{{\text{max}}}}$ values in subset 3 after first impact for the three specimens, the ${\hat \phi _{{\text{max}}}}$ values less than 0.005 can be considered that no damage exists within a subset. Thus, the threshold value of damage index ${\text{D}}{{\text{I}}_{{\text{th}}}}$ is set as 0.005. For comparisons, two additional ${\text{D}}{{\text{I}}_{{\text{th}}}}$ values, ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$ and ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$ are also considered in this study.

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Tables 5 and 6 list the reference damage index ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ and ${\hat \phi _{{\text{max}}}}$ for the subsets in the three specimens. Two sets of ${\hat \phi _{{\text{max}}}}$ values are presented here for each case. The first set of ${\hat \phi _{{\text{max}}}}$ values determined by applying exclusion rule 1 are presented. By applying equation (19) (the exclusion rule 2) on the first set of ${\hat \phi _{{\text{max}}}}$ values and letting ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$, the second set of ${\hat \phi _{{\text{max}}}}$ values are also presented. For the ${\hat \phi _{{\text{max}}}}$ values are zero for some subsets, it means that all the path groups have been excluded by applying the exclusion rules. For a subset that a damage exists within it, the ${\hat \phi _{{\text{max}}}}$ values are higher. A typical example is the subset 5 for the three specimens. Large ${\hat \phi _{{\text{max}}}}$ values in subset 5 for all cases are observed. For such cases, no further path groups can be excluded by applying the exclusion rule 2.

Table 5.  ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ and ${\hat \phi _{{\text{max}}}}$ at first impact.

	A 250 kHz			A 275 kHz			A 300 kHz
		${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$
Subset	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2
1	0.002237	0.004892	0	0.002558	0.007833	0.007833	0.005298	0.014592	0.014592
2	0.003237	0.004768	0	0.003539	0.010662	0.010662	0.005899	0.020309	0.020309
3	0.003272	0.004438	0	0.002336	0.003549	0	0.006587	0.006875	0.006875
4	0.007258	0.112085	0.112085	0.004919	0.138972	0.138972	0.013038	0.175116	0.175116
5	0.007258	0.088022	0.088022	0.068046	0.114532	0.114532	0.076643	0.14311	0.14311
6	0.004052	0.015721	0.015721	0.007268	0.044187	0.044187	0.012349	0.063501	0.063501
7	0.004014	0.015391	0.015391	0.007118	0.041041	0.041041	0.012349	0.059177	0.059177
8	0.002892	0	0	0.007828	0	0	0.007053	0.006482	0
9	0.001841	0.002423	0	0.000732	0.00296	0	0.002197	0.005899	0.005899
10	0.000736	0.00274	0	0.000732	0.003306	0	0.002197	0.005494	0.005494
	B 250 kHz			B 275 kHz			B 300 kHz
		${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$
Subset	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2
1	0.000371	0.00132	0	0.001005	0.001295	0	0.000974	0.002039	0
2	0.000403	0.001479	0	0.000709	0.001751	0	0.000974	0.001967	0
3	0.000276	0.00036	0	0.000477	0.000448	0	0.000508	0.001229	0
4	0.00912	0	0	0.006574	0.000618	0	0.005945	0.001009	0
5	0.032622	0.128403	0.032622	0.032964	0.141268	0.141268	0.033568	0.228174	0.228174
6	0.011476	0.069699	0.011476	0.011613	0.078845	0.078845	0.008179	0.088286	0.088286
7	0.001945	0.001823	0	0.003243	0.003512	0	0.005377	0.004988	0
8	0.000789	0.001781	0	0.002673	0.003727	0	0.004968	0.006111	0.006111
9	0.001114	0.000931	0	0.003243	0.005872	0.005872	0.006263	0.011106	0.011106
10	0.000771	0.002581	0	0.002958	0.005397	0.005397	0.005984	0.01041	0.01041
	C 250 kHz			C 275 kHz			C 300 kHz
		${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$
Subset	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2
1	0.000505	0.001242	0	0.000993	0.001199	0	0.000823	0.001701	0
2	0.000803	0.000793	0	0.001075	0.002237	0	0.001517	0.002703	0
3	0.000971	0.001191	0	0.001109	0.002039	0	0.001517	0.0027	0
4	0.030832	0	0	0.043142	0	0	0.045035	0	0
5	0.071987	0.430603	0.071987	0.088001	0.533777	0.533777	0.090257	0.590221	0.590221
6	0.001322	0.001384	0	0.002014	0.001907	0	0.003635	0.003699	0
7	0.003876	0.033914	0.003876	0.01029	0.047066	0.047066	0.016414	0.10309	0.10309
8	0.006738	0.029297	0.006738	0.01029	0.043364	0.043364	0.024138	0.12118	0.12118
9	0.005255	0.023021	0.005255	0.008865	0.028498	0.028498	0.009286	0.015392	0.015392
10	0.001327	0.003174	0	0.004193	0.007685	0.007685	0.006221	0.008338	0.008338

Table 6.  ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ and ${\hat \phi _{{\text{max}}}}$ at second impact.

	A 250 kHz			A 275 kHz			A 300 kHz
		${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$
Subset	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2
1	0.003503	0	0	0.005534	0.00305	0	0.007188	0.00482	0
2	0.003503	0.013928	0.013928	0.002868	0.002974	0	0.003194	0.006149	0.006149
3	0.005189	0.015201	0.015201	0.002222	0.011456	0.011456	0.005744	0.015052	0.015052
4	0.014061	0.015953	0.015953	0.0118	0.013646	0.013646	0.01464	0.017541	0.017541
5	0.074664	0.675766	0.675766	0.064139	0.741718	0.741718	0.063119	0.747812	0.747812
6	0.038815	0.380774	0.380774	0.044088	0.486001	0.486001	0.044205	0.562601	0.562601
7	0.038693	0.380111	0.380111	0.029563	0.516052	0.516052	0.026241	0.60401	0.60401
8	0.038693	0.465831	0.465831	0.029563	0.507919	0.507919	0.026241	0.555014	0.555014
9	0.029942	0.616899	0.616899	0.020298	0.644286	0.644286	0.026241	0.700275	0.700275
10	0.010239	0.641028	0.641028	0.017434	0.672212	0.672212	0.02373	0.749241	0.749241
	B 250 kHz			B 275 kHz			B 300 kHz
		${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$
Subset	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2
1	0.00416	0	0	0.003613	0	0	0.00289	0.008651	0.008651
2	0.003164	0.003121	0	0.002158	0.008646	0.008646	0.002364	0.011631	0.011631
3	0.002542	0.003082	0	0.002158	0.00879	0.00879	0.001883	0.010951	0.010951
4	0.021272	0	0	0.013282	0	0	0.011551	0	0
5	0.107497	0.449442	0.449442	0.106646	0.499151	0.499151	0.110235	0.569612	0.569612
6	0.111617	0.612702	0.612702	0.120673	0.683718	0.683718	0.076422	0.611747	0.611747
7	0.027451	0.147941	0.147941	0.042361	0.12822	0.12822	0.042905	0.121631	0.121631
8	0.027451	0.025581	0	0.042361	0.036622	0	0.052929	0.046805	0
9	0.013795	0.033626	0.033626	0.024567	0.034819	0.034819	0.012553	0.046012	0.046012
10	0.012014	0.015485	0.015485	0.024567	0.023241	0	0.044383	0.044586	0.044586
	C 250 kHz			C 275 kHz			C 300 kHz
		${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$			${\hat \phi _{{\text{max}}}}$
Subset	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2	${\text{D}}{{\text{I}}_{{\text{ref}}}}$	Exclusion rule 1	Exclusion rule 1 + 2
1	0.002228	0.002781	0	0.001444	0.001774	0	0.00132	0.002271	0
2	0.001737	0.002352	0	0.000752	0.001714	0	0.001371	0.002284	0
3	0.001001	0.00173	0	0.001703	0.001969	0	0.001482	0.002299	0
4	0.051323	0	0	0.090943	0	0	0.08858	0	0
5	0.087734	1.041138	1.041138	0.090943	1.17228	1.17228	0.08858	1.175354	1.175354
6	0.008287	0	0	0.009084	0.002581	0	0.010456	0.003732	0
7	0.01583	0.144415	0.144415	0.015484	0.188227	0.188227	0.016415	0.264003	0.264003
8	0.065414	0.132029	0.132029	0.07341	0.182225	0.182225	0.073041	0.268669	0.268669
9	0.049657	0.214551	0.214551	0.071631	0.222933	0.222933	0.07708	0.250494	0.250494
10	0.034132	0.297522	0.297522	0.037361	0.327166	0.327166	0.049269	0.363936	0.363936

Figures 17–19 show the damage localization results for specimens A, B and C, respectively. Three different threshold damage indices (${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$, 0.005 and 0.01) are considered in the calculations. In each specimen, both the actual damage locations and the damage localization locations are marked for comparison. Each specimen was undergone two impact tests. The damage localization results obtained for the first impact and second impact are presented. In general, if a damage is within the transducer layout, it can be detected by one or more subsets. For example, consider damage B-1 shown in figure 18. It can be detected by the paths in subset 6. For the other damages that are within the transducer layout, most of them are identified by the present method. Conversely, damages A-3, A-5, B-3, C-3 and C5 are not within the transducer layout and are difficult to be identified.

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Tables 7–15 list the damage localization results for each subset of specimens under various driving frequencies and threshold damage indices. If two or more mean angle lines are preserved in a subset, a damage location prediction $\left( {\bar x,\bar y} \right)$ is obtained. Then the damage identified by the calculation is assumed to be the nearest actual damage to $\left( {\bar x,\bar y} \right)$. Also, the dimensionless distance $\bar D$ between the predicted location and actual location of the damage is also given in tables 7–15. For example, consider the subset 5 in specimen A, the damage localization results of which are listed in table 7. The damage localization location with 250 kHz after the 2nd impact is (504.16, 337.12) in mm. This location remains the same as the threshold damage index ranged from 0 to 0.01. The nearest actual damage is A-2, which is located at (501.65, 337.60) in mm. The distance between the predicted damage location actual damage location is 2.56 mm, resulting a dimensionless distance $\bar D = 0.040$. For subsets that $\bar D > 1$, the damage localization results are inaccurate and are marked by 'star'. For the case of ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$, some path groups are preserved due to the low threshold value. These path groups have low ${\hat \phi _{{\text{max}}}}$ values, indicating that no significant damage signals detected. It results in more subsets are marked by 'star' for ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$. By increasing the threshold value, say ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$ or ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$, the number of the subsets marked by 'star' are significantly reduced. It is concluded that by assigning proper value of ${\text{D}}{{\text{I}}_{{\text{th}}}}$, the damage localization results are more accurate.

Table 7. Damage localization results for specimen A at 250 kHz.

	1st impact								2nd impact
	${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset	Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$	Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1	A-3	0.933
2	A-3 a	2.122
3	A-1 a	2.797
4	A-2	0.401		A-2	0.401		A-2	0.401
5	A-2	0.412		A-2	0.412		A-2	0.412	A-2	0.040		A-2	0.040		A-2	0.040
6	A-1	0.626		A-1	0.626		A-1 a	1.122	A-1	0.076		A-1	0.076		A-1	0.076
7	A-1	0.890		A-1	0.890		A-1	0.983	A-1	0.024		A-1	0.024		A-1	0.024
8									A-4	0.965		A-4	0.965		A-4	0.965
9	A-1 a	3.188							A-4	0.261		A-4	0.261		A-4	0.261
10	A-2 a	1.382

^a indicates $\bar D > 1$.

Table 8. Damage localization results for specimen A 275 kHz.

	1st impact								2nd impact
	${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset	Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$	Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1	A-3 a	1.514		A-3	0.997
2
3	A-1 a	2.154							A-4 a	1.166		A-4	0.556
4
5									A-2	0.116		A-2	0.116		A-2	0.116
6	A-1	0.673		A-1	0.673		A-1	0.183	A-1	0.066		A-1	0.066		A-1	0.066
7	A-1	0.260		A-1	0.260		A-1	0.260	A-1	0.046		A-1	0.046		A-1	0.046
8									A-4	0.493		A-4	0.493		A-4	0.493
9									A-4	0.239		A-4	0.239		A-4	0.239
10	A-2 a	2.579

^a indicates $\bar D > 1$.

Table 9. Damage localization results for specimen A 300 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		A-3 a	1.417		A-3	1.417		A-3	1.384
2
3											A-4 a	1.416		A-4 a	1.416		A-4	0.744
4
5		A-2 a	1.572		A-2 a	1.572		A-2 a	1.572		A-2	0.122		A-2	0.122		A-2	0.122
6		A-1	0.194		A-1	0.194		A-1	0.194		A-1	0.062		A-1	0.062		A-1	0.062
7		A-1	0.166		A-1	0.166		A-1	0.166		A-1	0.041		A-1	0.041		A-1	0.041
8											A-4	0.498		A-4	0.498		A-4	0.498
9		A-1 a	2.532		A-1 a	2.523					A-4	0.215		A-4	0.215		A-4	0.215
10		A-2 a	2.150		A-2 a	4.668

^aindicates $\bar D > 1$.

Table 10. Damage localization results for specimen B 250 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		B-1 a	1.649
2		B-3 a	1.761
3											B-5 a	1.425
4
5		B-2	0.091		B-2	0.091		B-2	0.228		B-2	0.109		B-2	0.109		B-2	0.109
6		B-1	0.147		B-1	0.147		B-1	0.368		B-1	0.145		B-1	0.145		B-1	0.145
7											B-5	0.122		B-5	0.122		B-5	0.122
8		B-1 a	3.403
9
10		B-2	0.454								B-4	0.434		B-4	0.434		B-4	0.434

^a indicates $\bar D > 1$.

Table 11. Damage localization results for specimen B 275 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		B-1 a	1.812
2											B-5	0.527		B-5	0.527
3											B-5 a	1.233		B-5 a	1.233
4
5		B-2	0.077		B-2	0.077		B-2	0.193		B-2	0.084		B-2	0.084		B-2	0.084
6		B-1	0.108		B-1	0.108		B-1	0.270		B-1	0.219		B-1	0.219		B-1	0.219
7		B-1 a	3.117								B-5	0.169		B-5	0.169		B-5	0.169
8		B-1 a	3.303
9											B-4 a	1.637		B-4 a	1.637		B-4 a	1.637
10		B-2	0.924

^a indicates $\bar D > 1$.

Table 12. Damage localization results for specimen B 300 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		B-3 a	1.747
2											B-5	0.530		B-5	0.530
3		B-2 a	2.985								B-5	0.731		B-5 a	1.117
4
5		B-2	0.121		B-2	0.121		B-2	0.302		B-2	0.086		B-2	0.086		B-2	0.086
6		B-1	0.134		B-1	0.134		B-1	0.334		B-1	0.256		B-1	0.256		B-1	0.256
7											B-5	0.191		B-5	0.191		B-5	0.191
8		B-1 a	3.329		B-1 a	3.391
9											B-4 a	1.192		B-4 a	1.192		B-4 a	1.192
10		B-2	0.758		B-2	0.758

^a indicates $\bar D > 1$.

Table 13. Damage localization results for specimen C 250 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		C-3 a	1.851								C-1 a	1.710
2											C-1 a	2.266
3											C-1 a	2.611
4
5		C-2	0.325		C-2	0.325		C-2	0.813		C-2	0.098		C-2	0.098		C-2	0.098
6
7		C-1	0.333		C-1	0.333		C-1	0.832		C-1	0.281		C-1	0.281		C-1	0.281
8		C-1	0.038		C-1	0.038		C-1	0.094		C-1	0.401		C-1	0.401		C-1	0.401
9											C-4	0.805		C-4	0.805		C-4	0.805
10		C-2	0.873								C-4	0.113		C-4	0.113		C-4	0.113

^a indicates $\bar D > 1$.

Table 14. Damage localization results for specimen C 275 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		C-3	0.864								C-3	0.781
2		C-3 a	1.803								C-3 a	2.098
3		C-1 a	2.349
4
5		C-2	0.256		C-2	0.256		C-2	0.640		C-2	0.134		C-2	0.134		C-2	0.134
6
7		C-1	0.356		C-1	0.356		C-1	0.890		C-1	0.305		C-1	0.305		C-1	0.305
8		C-1	0.235		C-1	0.235		C-1	0.589		C-1	0.482		C-1	0.482		C-1	0.482
9		C-1	0.799		C-1	0.799		C-1	1.996		C-4	0.787		C-4	0.787		C-4	0.787
10		C-2	0.910		C-2	0.910					C-4	0.120		C-4	0.120		C-4	0.120

^a indicates $\bar D > 1$.

Table 15. Damage localization results for specimen C 300 kHz.

		1st impact									2nd impact
		${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$			${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.01$
Subset		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$		Damage identified	$\bar D$
1		C-1 a	2.837								C-3 a	2.356
2											C-3 a	2.539
3		C-1 a	1.986								C-1 a	2.338
4
5		C-2	0.071		C-2	0.071		C-2	0.177		C-2	0.120		C-2	0.120		C-2	0.120
6
7		C-1	0.405		C-1	0.405		C-1	1.011		C-1	0.352		C-1	0.352		C-1	0.352
8		C-1	0.707		C-1	0.707		C-1	1.768		C-1	0.501		C-1	0.501		C-1	0.501
9											C-4	0.790		C-4	0.790		C-4	0.790
10		C-2	0.563		C-2	0.563					C-4	0.143		C-4	0.143		C-4	0.143

^a indicates $\bar D > 1$.

There are two possible cases that result in no damage detected in a subset, for which the fields for damage localization and $\bar D$ in tables 7–15 are left as blanks. The first case is that no path group or only one path group is preserved in that subset. The second is that two path groups are preserved, but the two corresponding mean angle lines are nearly parallel to each other such that the location $\left( {\bar x,\bar y} \right)$ is far away from the subset.

For the damages identified by the subsets, the relationship between damage sizes and ${\hat \phi _{{\text{max}}}}$ values are shown in figure 20. The ${\hat \phi _{{\text{max}}}}$ values are calculated by using ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$. Note that the inaccurate damage localization that are marked by 'star' as listed in tables 7–15 are not included in figure 20. A trend can be confirmed that the ${\hat \phi _{{\text{max}}}}$ value increases with the increase of damage size and can be regressed by a straight line with a slope of $0.001852{\text{ m}}{{\text{m}}^{ - 1}}$.

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Before more detail discussions for the damage localization results, it is worthy to mention that the damages appeared in the specimens can be categorized by four types: no damage within a subset, a damage within a subset, a damage near border of a subset and two damages within a subset. In the following, some selected examples categorized in the four types are discussed in detail.

4.4.1. Results for no damage within a subset.

Figure 21(a) shows the DI variations with path angles and the calculation results of damage localizations for subset 3 of specimen C. The damage indices are obtained from signals after first impact with driving frequency 275 kHz. In subset 3, the four transducers 3, 4, 5 and 6 are assigned as actuators while the four transducers 11, 12, 13, 14 are assigned as sensors. It is seen that the path groups $p_a^{\left( 3 \right)}$ $p_a^{\left( 4 \right)}$, $p_a^{\left( 6 \right)}$, $p_s^{\left( {11} \right)}$, $p_s^{\left( {13} \right)}$, and $p_s^{\left( {14} \right)}$ are preserved after utilizing exclusion rule 1. In table 5, the ${\hat \phi _{{\text{max}}}}$ values for the six path groups is 0.002039 and the reference damage index ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ is 0.001109 for this subset. Thus, all the six path groups are preserved for damage localization calculations by applying exclusion rule 2 with ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$. As shown in figure 21(b), the red sold line and blue dashed line indicate the mean angle lines $L_a^{\left( i \right)}$ and $L_s^{\left( j \right)}$, respectively. The damage localization in mm is $\left( {\bar x,\bar y} \right) = \left( {343.7,343.4} \right)$, which is marked in figure 21(b). The actual damages are also marked in figure 21(b) for comparison. The nearest actual damage to the predict damage is C-1. The distance between them is 149.1 mm, which is equivalent to a dimensionless distance of $\bar D = 2.349$.

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The small ${\hat \phi _{{\text{max}}}}$ value after first impact for driving frequency 275 kHz in subset 3 of specimen C agrees with the fact that no damage exists in this subset. However, there is a damage detected in figure 21(b). If the threshold damage index ${\text{D}}{{\text{I}}_{{\text{th}}}}$ is adjusted to 0.005, all the six path groups are excluded and no damage localization is reported within subset 3.

It should be noted that two of the mean angle lines $L_a^{\left( 4 \right)}$ and $L_a^{\left( 6 \right)}$ in figure 21(b) toward to damage C-1. It means that some of the preserved mean angle lines may be useful for predicting a damage in the exterior of the subset. Figure 22 shows an example for this. The damage indices and predicted damage shown in figure 22 are for subset 1 of specimen A after first impact with driving frequency 250 kHz. No damage exists within this subset. With ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$ and ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.002237$ as listed in table 5, four mean angle lines are preserved and the predicted damage location in mm is $\left( {\bar x,\bar y} \right) = \left( {241.2,473.7} \right)$. The predicted location is near the actual damage A-3 with a dimensionless distance of $\bar D = 0.933$. This distance is approximately equal to the horizontal spacing between two adjacent transducers. The accuracy of the damage localization results is not as good as the case that a damage exists within a subset. By applying the exclusion rule 1, the value of ${\hat \phi _{{\text{max}}}}$ in subset 1 of specimen A at the first impact is 0.004892 as listed in table 5. This value is lower than 0.005. Thus, by increasing ${\text{D}}{{\text{I}}_{{\text{th}}}}$ to 0.005, all the four mean angle lines are excluded by applying the exclusion rule 2 and the calculation results suggest that no damage exists within this subset.

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As a summary, if the maximum damage index in a subset is approximately equal to or less than a certain level, say 0.005, it can be concluded that no damage exists within the subset. With ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0$, it might result in a false damage location within the subset or a predicted damage in the exterior of the subset at low accuracy.

4.4.2. Results for a damage within a subset.

Figure 23 shows the DI variations with path angles and the calculation results of damage localization for subset 5 of specimen B. The damage indices are obtained from signals after 1st impact with driving frequency 275 kHz. In subset 5, the four transducers 5, 6, 7 and 8 are assigned as actuators while the four transducers 13, 14, 15, 16 are assigned as sensors. The value of ${\hat \phi _{{\text{max}}}}$ for this case is 0.141268, as listed in table 5. This value is much greater than the threshold value of 0.005. It indicates that there exists a damage within this subset. In figure 23, it is seen that four path groups are preserved to form four mean angle lines and a predicted damage location is calculated. In figure 23, it is seen that the nearest actual damage is B-2. The damage localization is quite accurate with a dimensionless distance of $\bar D = 0.077$, as seen in table 11.

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Figure 24 shows the results subset 5 of specimen B. The damage indices are obtained with driving frequency 275 kHz. The difference of this case from figure 23 is that the results in figure 24 are analyzed after the second impact. The damage size of B-2 is greater after 2nd impact. The value of ${\hat \phi _{{\text{max}}}}$ for this case is 0.499151, which is higher than that for the case of figure 23. By comparing the result for the first impact, the increasing ${\hat \phi _{{\text{max}}}}$ for second impact suggests that this parameter can reflect the damage level. The damage localization in figure 24 is also similar to that in figure 23, showing high accuracy in prediction of damage location. The dimensionless distance for this case is $\bar D = 0.084$, as seen In table 11.

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In one subset with more than one preserved mean angle lines, an intersection point is formed by taking any two of the lines. There may exist more than one intersection points. With the damage localization results shown in figures 23 and 24, it can be concluded that, by applying the two exclusion rules with ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$ within one subset, only one damage can be identified in this subset if the intersection points are distributed in a small region, and the damage can be localized within this region.

4.4.3. Results for a damage near boarder of a subset.

There are two cases for a damage existing near border of a subset. The first is that a damage exists near a transducer of a subset. Figure 25 shows the DI variations with path angles and the calculation results of damage localization for subset 9 of specimen A. The damage indices are obtained from signals after second impact with driving frequency 300 kHz. In figure 25(a), it is seen that the damage indices for path group $p_a^{\left( {13} \right)}$ are greater than those for path groups pivoted by the other three actuators since the damage A-4 is near actuator 13. The damage indices for paths 13–22 and 13–23 are the largest as these two paths directly hit the damage. However, distributions of the DI's for path groups pivoted by actuators do not fit the Gaussian distribution curves. For the four path groups pivoted by the sensors, the damage index variations fit the Gaussian distributions and are preserved by applying the two exclusion rules. The damage localization results in an accurate damage location prediction with a dimensionless distance of $\bar D = 0.215$.

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Figure 26(a) shows the DI variations with path angles for subset 7 of specimen B after the second impact with driving frequency 250 kHz. This is another demonstration for a damage existing near a transducer. For the eight path groups, five of them are preserved by applying the exclusion rule 1. For $p_a^{\left( {10} \right)}$, one of the five preserved path groups, the peak value of the fitting function is $\hat \phi _a^{\left( {10} \right)} = 0.01457$. The reference damage index for this subset is ${\text{D}}{{\text{I}}_{{\text{ref}}}} = 0.027451$. The damage index ranking plot that is used to calculate this ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ value has been shown in figure 4. By further applying the exclusion rule 2 with ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$ and ${\text{D}}{{\text{I}}_{{\text{ref}}}} = 0.027451$, it is seen that $p_a^{\left( {10} \right)}$ is excluded since $\hat \phi _a^{\left( {10} \right)} < {\text{D}}{{\text{I}}_{{\text{ref}}}}$. By using the four remaining path groups that are preserved, the damage localization results are shown in figure 26(b). It is seen that the predicted damage location is near actual damage B-5 with a dimensionless distance of $\bar D = 0.122$. The damage localization is quite accurate. If the exclusion rule 2 is not applied, the path group $p_a^{\left( {10} \right)}$ is preserved, and the damage localization result is also shown in figure 26(b). It is seen that the mean angle line for $p_a^{\left( {10} \right)}$ has a large error by comparing other four preserved mean angle lines. It results in a more inaccurate location prediction with $\bar D = 0.4972$. In the case study of figure 26, ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ dominates the exclusion results by applying the exclusion rule 2 since ${\text{D}}{{\text{I}}_{{\text{ref}}}} > {\text{D}}{{\text{I}}_{{\text{th}}}}$. It suggests that the damage localization can be more accurate if the value of ${\text{D}}{{\text{I}}_{{\text{ref}}}}$ that is determined by the procedure shown in figure 4 is utilized.

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The second case is that a damage does not exist near a transducer, but near the right or left border of the subset. Two examples will be demonstrated for this case: subset 1 in specimen A at second impact with 275 kHz and subset 4 in specimen B at first impact with 300 kHz. Figure 27 shows the DI variations with path angles for these two examples. For subset 1 in specimen A at second impact with 275 kHz, the damage A-5 is near the left border of the subset, as indicated in figure 8. In figure 27(a), it is seen that no path group is preserved by applying the two exclusion rules. It is obvious that the path 1–9 has the highest damage index of 0.049 97 as it is the nearest path to the damage. For the other path, the damage indices are much less than that for path 1. The second largest damage index in this subset is 0.009 854. The subset 1 is the most left one in the transducer layout, and the path 1–9 is the most left one in the subset 1. Thus, the information shown in figure 27(a) can conclude that a damage exists one the left side area of the subset. This conclusion coincides with the actual location of damage A-5. Similar results can be seen in figure 27(b), where the DI variations with path angles for subset 4 in specimen B at first impact with 300 kHz are presented. The path 7–15, the most left path in this subset, has the highest damage index of 0.1510, while the damage indices for the other paths are much smaller. The variations of the damage index can be accepted since damage B-2 is located one the right border of this subset. Again, no path group in this case can be preserved. The damage B-2 cannot be identified in subset 4, however, it can be identified by the next subset, which encloses this damage.

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By summary, the criterion to identify the case with only one damage near boarder of a subset is similar to the case with only one damage within the subset. The intersection points by applying the two exclusion rules with ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$ are located in a small region.

4.4.4. Results for two damages within a subset.

For second impact in specimen C, there are two damages in subset 9. Figure 28 shows the damage index variations and damage localization results for this subset. The driving frequency and threshold damage index are 250 kHz and 0.005, respectively. By looking at the damage index variations with the path angles as shown in figure 28(a), it is observed that the damage index for path hitting the damage is higher. For example, consider the path group $p_s^{\left( {22} \right)}$ pivoted at sensor 22. The four paths in this path group are 12–22, 13–22, 14–22 and 15–22. The second path of this path group (13–22) does not hit the damage. The path 12–22 passes near the damage C-1. Both the other two paths, paths 14–22 and 15–22, hit damage C-4. Thus, the distribution of damage index for path group $p_s^{\left( {22} \right)}$ reveals its minimum at path 13–22. The damage indices for the other three paths are relatively higher. The path-angle dependent DI variation for path group $p_s^{\left( {22} \right)}$ does not follow the Gaussian distribution and has been excluded in the damage localization calculations. For another path group $p_a^{\left( {12} \right)}$, the two damages affect the path-angle dependent DI variation in a similar manner. However, it follows the Gaussian distribution and is preserved after applying the exclusion rule 1. As a result, there are five path groups are preserved and the intersection points formed by the mean angle lines are distributed in a relatively large region. Based on the present algorithm, only one damage location can be reported. In this case, a predicted damage location is obtained at the middle of the two real damages. Nevertheless, the predicted damage location is more near to damage C-4 with $\bar D = 0.805$.

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The damages C-1 and C-4 can be separately predicted by subset 7 and subset 10, as shown in figures 29 and 30. The subset 7 encloses only one damage C-1 and has a predicted damage location with $\bar D = 0.281$. Similarly, the predicted damage location by subset 10 is near to damage C-4 with $\bar D = 0.113$. It is concluded that for the case that one subset encloses two damages, these locations of the two damages can be separately identified by the neighborhood subsets.

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With the observations from figures 28 to 30, a subset is identified that there are two damages within it if the intersection points are distributed in a relatively large region by applying the two exclusion rules with ${\text{D}}{{\text{I}}_{{\text{th}}}} = 0.005$. The predicted damage may be located between the two actual damages. The localizations for the two actual damages can be obtained from the results of the two adjacent subsets.