## 2 Materials and Methods
### Data Acquisition and Preprocessing  
The acquired data are plotted, and a curve-fitting technique is employed to estimate the numerical data model. After observing the data, different mathematical models (polynomial, cubic spline, sinusoidal, composite sinusoidal, exponential, and their combinations, etc.) were employed, and the corresponding fitting errors were observed and analyzed. The results suggest that composite sinusoidal waves have the lowest fitting error, and the corresponding mathematical model is displayed in Equation (3) as follows. The general data model is represented as:  
\[f(x)=a_{1}*\sin(b_{1}*x+c_{1})+a_{2}*\sin(b_{2}*x+c_{2}), \tag{3}\]  
where function \(f\) mathematically indicates the crack response, input parameter \(x\) of function \(f\) is time, parameters _a1_ and _a2_ are the scaling parameters of sinusoidal functions to model the data, _b1_ and _b2_ are the frequencies of these functions, and _c1_ and _c2_ are the phases of the observed data. The proposed model of the system incorporates 6 free variables (_a1_, _b1_, _c1_, _a2_, _b2_, and _c2_), whose values describe the response based on the crack locations. To estimate these parameters in each scenario, the proposed model is reformulated in the form of a cost function \(J\), which describes the difference of actual values, _a(x)_, and calculated values by the mathematical model of _f(x)_:  
\[J=\sum_{n=1}^{N}(a(x)-a_{1}*\sin(b_{1}*x+c_{1})+a_{2}*\sin(b_{2}*x+c_{2}))^{2}. \tag{4}\]  
The free variables in the cost function are optimized to minimize the cost function; ideally, _J_\(\approx\) 0. This optimization is achieved using any regressive numerical algorithm that minimizes the residual error. For this purpose, an improved version of the simplex algorithm, known as the Nelder-Mead simplex method (NMSM), is used. The operating cycle of the NMSM is based on three steps: ordering, centroid, and transformation [41]. \[x_{j}=x_{0}+pe_{j}+\sum_{ \begin{array}{c}k=1\\ k\
eq j\end{array} }^{n}qe_{k};j=1,2,\ldots..,n, \tag{5}\]  
\[p=\frac{a}{n\sqrt{2}}\Big{(}\sqrt{n+1}+n-1\Big{)}\&\ q=\frac{a}{n\sqrt{2}} \Big{(}\sqrt{n+1}-1\Big{)} \tag{6}\]  
where _xj_ (_j_ = 1, 2,.., _n_) are the vertices, _x0_ is the initial guess, \(n\) is the number of free parameters (6 in this study), and _ej_ is the unit vector in the direction of the _j_th vertex. The following step is performed to sort the function in ascending order at all vertices:  
\[J(x_{l})<J(x_{s})<J(x_{h}). \tag{7}\]  
where _xl_ is the vertex with the minimum value, _xh_ is the vertex with the maximum value, and _xs_ is the vertex with the second highest value of the cost function. Now, the vertex with the highest value is discarded by defining the centroid:  
\[x=\frac{1}{n}\sum_{ \begin{array}{c}i=0\\ i\
eq h\end{array} }^{n}x_{i}, \tag{8}\]  
where \(x\) is the centroid. The upper bound vertex of the cost function is discarded by reflection, expansion, contraction, and shrinkage. These steps can be mathematically expressed as follows:  
\[\text{Reflection}:\ x_{r}=x+\delta_{1}(x_{h}-x) \tag{9}\]\[\text{Expansion}:\ x_{e}=x+\delta_{2}(x_{r}-x) \tag{10}\] \[\text{Contraction}:\ x_{c}=x+\delta_{3}(x_{e}-x)\] (11) \[\text{Shrinkage}:\ x_{s}=x+\delta_{4}(x_{c}-x) \tag{12}\]  
where \(\delta\)1, \(\delta\)2, \(\delta\)3, and \(\delta\)4 are the reflection, expansion, contraction, and shrinkage coefficients, respectively. Classification  
Support vector machines (SVMs) are powerful machine learning algorithms well-known for their superior classification performance. They define such a hyperplane which separates two classes with the largest possible margin. Learning is based on features that best-distinguish the classes during the training phase, and afterwards, SVMs predict the newly presented sample as a member of either of the two classes [42]. The SVM principle is shown in Figure 6.  
An SVM was designed in MATLAB(r) and simulation responses were provided to the model.