88 G. Karyofyllas et al. The optimization starts with an initial set of parameters in the vector θ, which represent the parameters for the drivetrain’s numerical model. The CMA-ES algorithm iteratively updates the parameters by calculating the numerical model’s response and minimizing the cost function J(θ), which is based on normalized least squares error. The process continues until the parameter values are refined, aligning the numerical and experimental frequency responses, and ensuring a more accurate representation of the drivetrain’s dynamics. Next, the optimal MBD model is utilized to generate labeled data for training the proposed supervised machine learning framework. This numerical model produces a comprehensive dataset that encompasses various health states in the rotor dynamics model under investigation, providing detailed information about both the health status and specific instances of damage occurring at discrete locations within the structure. To account for the inherent uncertainty in real-world systems, key hyperparameters θ are sampled from a Gaussian distribution. In practical experiments, uncertainty is always present, resulting in discrepancies between experimental responses and those predicted by the model. By sampling the model’s hyperparameters, these issues are addressed, mitigating discrepancies observed between numerical and experimental data across different trials. To facilitate the generation of Nlabeled datasets, an automated algorithm, as shown in Figure 1, has been developed. This algorithm leverages the numerical model’s response to compute Fast Fourier Transforms (FFTs) from the acceleration data in the X and Y directions, which serve as the primary input for the machine learning framework. At the end of the process, the algorithm determines the model’s health status and returns both the FFTs and the health label Y. This label, represented as a single integer, indicates the simulated health state, designating whether the corresponding MBD model instance is healthy or exhibiting specific damage forms. Essentially, Y acts as a categorical indicator linked to the configuration of the solved MBD model, distinguishing between healthy and compromised states. The simulated FFT response data is then organized into a complete labeled dataset, expressed as: Train set = {(FFT1,Y1), (FFT2,Y2)· · · (FFTn,Yn)}, withFFTn = [FFT X n ,FFT Y n] (5) Fig. 1 Numerical data generation framework. The simulated dataset for supervised damage detection and identification in the drivetrain system is processed using a Support Vector Machine (SVM) with a Radial Basis Function (RBF) kernel [20]. SVM is a robust classification algorithm that works by transforming the input data into a higher-dimensional space, allowing it to handle complex decision boundaries. The classification problem is expressed as: y(x)=wTφ(x)+b (6) where φ(x) is the feature-space transformation function, andb is the bias term. The RBF kernel provides flexibility through hyperparameters and is defined as: K(x, y)= e−γ∥x−y∥ 2 (7) Where K(x, y) represents the kernel function applied to input vectors x and y, and γ is a hyperparameter controlling the decision boundary’s smoothness. The training dataset T = {xi ∈ Rn, i = 1, . . . , N} and labels from two classes yi ∈ {−1, 1} are used to optimize the model. During training, the SVM seeks to maximize the margin between classes by solving the optimization problem: Maximize : W(a)= NX i=1 ai − 1 2 NX i=1 NX j=1 (ai · aj · yi · yj · K(xi,xj)) (8)
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