## 3 Results and Discussion  
The study analyses experimental data on natural frequency, structural amplitude, temperature, and crack position. According to [29], Spearman's correlation coefficient is alternative measure of correlation that can capture non-linear relationships and are suitable for ranking or ordinal data. Spearman's correlation coefficient (\(\rho\)) is calculated based on the ranks of the variables. Let us denote the ranks of X and Y as R(X) and R(Y), respectively. The Spearman correlation coefficient is given by:  
\[\rho=1-[(6\times\Sigma(D^{2}))/(n\times(n^{2}-1))]\ldots\ldots. \tag{1}\]  
where:  
\(\Sigma\) denotes the sum of the respective values. D represents the difference between ranks (R(X) \(-\) R(Y)). n is the number of data points. The value of \(\rho\) ranges between \(-1\) and \(1\), where:  
\(\rho=1\) indicates a perfect monotonic increasing relationship. \(\rho=-1\) indicates a perfect monotonic decreasing relationship. \(\rho=0\) indicates no monotonic relationship. Results show that ABS, concrete, and aluminum exhibit similar patterns in natural frequency, with frequency decreasing as crack depth increases (as shown in Figure 4) in a nonlinear manner. Additionally, the impact of temperature on natural frequency is more pronounced in aluminum than in concrete and ABS. This is due to the higher elastic modulus and greater consistency of isotropic sheet metal properties relative to additive layer manufactured ABS and concrete [1; 5]. As the crack depth and temperature increase in aluminum specimens, the amplitude also increases, consistent with the expected effect of reduced natural frequency resulting from a crack in the specimen. In contrast, ABS exhibits the opposite behavior: as crack depth increases, the natural frequency drops similarly to aluminum, but amplitude decreases instead of increasing, as shown in Figure 5. This discrepancy can be explained by the different temperature conditions during the experiments. The maximum temperature for aluminum specimens is 200 \({}^{\circ}\)C, well below the temperature threshold for any significant material transformations during the short test duration. In contrast, the maximum temperature for ABS samples is 70 \({}^{\circ}\)C, which is close to the glass transition temperature. Figure 4: The correlation between crack depth (mm) and natural frequency (Hz) for (**a**)—Aluminum, (**b**)—ABS, (**c**)—Concrete. Figure 6 provides a more comprehensive view of the frequency-amplitude relationship in the experimental data, revealing an inverse relationship between frequency and amplitude in aluminum and concrete, while ABS shows a positive relationship. No clear relationship is evident between crack depth and position in ABS and concrete data, but in aluminum specimens, increasing crack position results in increased amplitude and decreased frequency. To enable a proper comparison and account for the diverse material behavior discussed earlier, three identical but independent models were trained and validated with unseen data. The proposed models were evaluated for training and validation accuracy using a mean squared error (MSE) metric. MSE is a commonly used statistical metric for evaluating the performance of regression models. MSE measures the average squared difference between the predicted values and the actual values of the target variable. For each data point in the test set, the model generates a predicted value based on the input features. The squared difference between the predicted value and the corresponding actual value is calculated. These squared differences are then averaged across all data points in the test set. MSE provides an estimate of the average squared error between the predicted values and the true values. It gives higher weight to larger errors due to the squaring operation. The value of MSE is always non-negative, with lower values indicating better performance. A value of 0 indicates a perfect fit, where the predicted values match the actual values exactly.