## 2 Materials and Methods
### Uncertainty  
In rotor fault detection, uncertainty refers to the inherent variability and unpredictability in the physical and operational conditions of the rotary systems. This variability can cause variations in the signals generated by faulty systems, making it difficult to identify and classify the faults accurately. The uncertainty concept in rotor fault detection aims to account for these variations and ensure that the data set used for modeling and simulation represents the diverse conditions that the rotary systems may experience in real-world scenarios. By taking uncertainty into account, the goal is to develop more robust and reliable models for detecting and classifying rotor faults.  
In the forming of the data set, the concept of uncertainty is employed. There are various sources of uncertainties in a system, such as model parameters, boundary conditions, external (operational) loads, faults, observations (measurements), and model uncertainty. All of them can be aleatory or epistemic in nature. Different techniques, such as the probabilistic distribution function (PDF), fuzzy (membership) function, and interval function, can be used to model uncertainty. Additionally, uncertainty can be quantified using either probabilistic or non-probabilistic methods. The first category includes methods such as Monte Carlo simulation, polynomial chaos expansion, random matrix theory, Bayesian inference, and Kriging surrogate, whereas the second category contains techniques, such as the fuzzy, evidence-based, imprecise, Taylor, and Chebyshev interval strategies [26]. The equation of motion of a rotary machine can be written as Equation (1):  
\[\mathrm{M}\mathrm{Y}^{\prime\prime}(\mathrm{t})+\mathrm{C}\mathrm{Y}^{\prime} (\mathrm{t})+\mathrm{K}\mathrm{Y}(\mathrm{t})=\mathrm{F}(\mathrm{t}) \tag{1}\]  
in which M, C, and K represent the mass, damping, and stiffness matrices, respectively. Furthermore, Y is the displacement vector, and F represents the force vector.  
Since as mentioned earlier, the effects of faults are simulated by variations of the systems' geometries in this work, uncertainties in the model parameters are considered to produce the sample tests. As a result, by changing the values of [PERSON]'s modulus and rotational speed, the modeling and simulation parameters of systems varied. To apply uncertainty in [PERSON]'s modulus, the probabilistic method with perturbation technique is used. If the initial value of [PERSON]'s modulus is \(\mathrm{E}_{\mathrm{j0}}\), the jth uncertain parameter can be calculated using the following equation [50].  
\[\mathrm{E}_{\mathrm{j}}=\mathrm{E}_{\mathrm{j0}}(1+\epsilon_{\mathrm{j}}) \tag{2}\]where \(\epsilon_{\rm j}\) is the jth uncertain parameter that should satisfy \(\mid\epsilon_{\rm j}\mid<1\). The data set is created by selecting five rotational speeds, 15, 50, 75, 100, and 150 rad/s; E\({}_{\rm j}\) is 207,000 MPa; uncertain parameters are then calculated for this parameter by applying Equation (2) until the 15 th order. The perturbation term is selected between 0.001 and 0.09 with a constant increment. As a result, for each of the unbalanced and bowed systems, 70 samples are achieved.  
Several key considerations support the selection of the five specified rotational speeds. Firstly, opting for values below the lower limit of 15 rad/s would result in a rotation that is excessively slow, making it difficult to clearly observe the noticeable effect of the shaft bow. Hence, values below this limit are deliberately avoided. Conversely, the upper limit of 150 rad/s has been carefully determined to effectively minimize the impact of imbalance, ensuring that it remains well below the first critical torsional speed and does not overshadow the bowing effects.