## 2 Research Methodology
### Combined Faults
The measured data were used to construct a set of points representing the rotor orbit graph. However, the graph can be sparse and weakly structured, making it difficult to identify patterns or draw conclusions. To transform this information into a more structured representation, the ISOMAP algorithm was applied to the dataset, reducing its dimensionality while preserving interdependence. The authors clustered the data using the k-nearest neighbors' algorithm as the first step. Next, the Dijkstra algorithm was employed to determine the shortest path between all points. These processed data were then used to construct an Euclidean space model, to which multidimensional scaling (MDS) was applied. As a result, a map of points with a more distinct structure was obtained (Figure 28), where different shapes correspond to various types of rotor defects. Figure 27: Estimated PSD of the strain: (**a**) Kalman filter; (**b**) modal expansion algorithm; (**c**) joint input-state estimation algorithm. Adopted from [31]. However, the authors did not evaluate their method under combined defect conditions. Furthermore, a significant challenge for industrial applications could be even low-intensity noise, affecting the method's accuracy. Additionally, the method's reliability depends on the choice of initial clustering parameters, which should be determined empirically for each case. The study by [86] explores a diagnosis method for detecting various fault types in rotor systems using a radial basis function neural network (RBFNN). The signal is divided into 50% overlap frames, each processed by a synchronized Fourier transform (SFT). Special coefficient matrices are computed from the resulting spectra, which are used for training the neural network and analysis. The above matrices are then transformed into feature vectors with information about the system state. Learning is carried out at several rotational speeds, utilizing experience from previously known faulty systems. The result is matrices that relate the system's state vector to a feature vector, leading to a diagonalized representation. Classification of the vectors leads to identifying the type of faults being identified. The method was experimentally verified on a test rig using acoustic signals, vibration measurements, and motor current fluctuations as input signals. The authors obtained a fault detection rate of 97% or higher, demonstrating the effectiveness of the proposed approach in detecting multiple defects in the rotor system. Finally, classic methods with some improvements are still suitable for solving the problem of combining fault identification. The study by [30] is not directly connected to the problem but considers efficient improvements for the classical approaches, which can be used in combined fault identification. The study determines damage from the operating deflection shape ratio (ODSR) between the two supports by a vehicle on the bridge. The part of the equation for damping includes a term for the movement of the object along an assumed beam:  
\[\mu\frac{\partial^{2}u(x,t)}{\partial t^{2}}+\kappa\frac{\partial u(x,t)}{ \partial t}+\frac{\partial^{2}}{\partial x^{2}}\bigg{[}EI(x)\frac{\partial^{ 2}u(x,t)}{\partial x^{2}}\bigg{]}=f(t) \tag{40}\]  
where \(f(t)\) is the specific external force; \(u(x,t)\) is the function of displacement that depends on the longitudinal coordinate \(x\) and time \(t\); \(E\) is [PERSON]'s modulus of elasticity; \(I(x)\) is the cross-sectional moment of inertia; \(\mu\) is the density of mass distribution; \(\kappa\) is the distributed stiffness. The force magnitude in the model is described using the Dirac delta function, i.e., it represents the force magnitude at a point. Deflection is observed at the points of the bridge supports and is described as the ratio of the deflection of the current support to  
Figure 28: Two-dimensional features of ISOMAP based on multi-time domain. Adopted from [85]. that of the next.