## 2 Research Methodology
### Unbalance
Through bifurcation analysis, frequency response analyses, and phase-space mapping, they demonstrated the influence of uncertainties in system parameters on stability, periodic responses, and chaos (Figure 5). \[X(z)=\sum_{i=0}^{m}A_{i}z^{i};\;Y(z)=\sum_{i=0}^{m}B_{i}z^{i}, \tag{3}\]3 Bifurcation frequencies for the rotary system before and after the damping coefficient, tuning. Adopted from [37]. The results show that the fuzzy approach provides a more robust and adaptive framework for rotor prediction, particularly where classical models cannot capture complex nonlinear interactions. The rotor model does not usually include properties of the entire system. The study by [67] demonstrates how the support structure's properties can influence the oscillatory system's behavior. In the paper, the authors examine a two-degrees-of-freedom system, modeling a vibration-generating system (e.g., a machine tool) and a damp support on which the system is mounted. The study seeks to develop a methodology that optimizes the vibration characteristics of the system through adjustment of the damping of the support. Motion equations for the system are formulated based on [PERSON]'s method (11) of tuning mass dampers for vibration-critical structures (e.g., a machine tool). However, in the original [PERSON] model, damping of the primary structure (the support) is ignored:  
\[ \begin{gathered} m_{1}\ddot{x}_{1}+(c_{1}+c_{2})\dot{x}_{1}-c_{c} \dot{x}_{2}+(k_{1}+k_{2})x_{1}-k_{2}x_{2}=F_{1}(t);\\ m_{2}\ddot{x}_{2}+c_{2}\big{(}\dot{x}_{2}-\dot{x}_{1}\big{)}+k_{2}(x_{2}- x_{1})=F_{2}(t).\end{gathered}  \tag{11}\]  
The author derived a system of equation modeling, such as a two-mass model, and parametric optimization in MATLAB R2024b was utilized to determine the optimum damper parameters (Figure 6). However, the method has several limitations. The analysis was conducted for harmonic excitation, which is suitable for wind loading or machine-induced vibrations but is not very realistic for random or impulse excitations. The model also presupposes that the system is linear, with the damper elements and the structural attachments behaving linearly. However, in real life, nonlinear behavior will express itself as plastic deformation, frictional losses, and time changes in damping, none of which are included in the present model. Another study examines the influence of the overall system properties on rotor behavior [25] and aims to determine imbalance masses because of the nonlinear relationships between bearing stiffness, rotor axis deflection, and rotating speed using a neural network (NN). Initially, a regression equation for estimating imbalance was derived from the matrix equation of the vibration process:  
\[[W]=\omega^{2}\Big{(}[C]-\omega^{2}[M]\Big{)}^{-1}; \tag{12}\]  
\[\{D\}=\Big{(}[W]^{\mathsf{T}}[W]\Big{)}[W]^{\mathsf{T}}\{Y\}, \tag{13}\]  
Figure 6: Variations in damping and frequency with mass ratio. Adopted from [67]. where \(\{Y\}\) is the row vector of axial deflections; \([D]\) is the row vector of imbalances; \([C]\) is the stiffness matrix; \([M]\) is the matrix of inertia; \(\omega\) is the angular velocity. This approach does not account for the variations in stiffness with rotating speed and rotor deflection. To improve the model, the authors have proposed an equation for the elements of the stiffness matrix, which includes rotational speed and axis deviation effects through influence coefficients [25]:  
\[C_{ij}^{(k)}=C_{ij}^{(0)}+a_{ij}\omega_{k}^{2}+b_{ij}\Big{|}Y_{j}^{(k)}\Big{|} \tag{14}\]  
The issue is how to determine these coefficients for each stiffness matrix element. Since solving this equation is analytically impossible, the authors propose using a neural network to cope with nonlinearities. For training NN, actual deflections and imbalance values were converted to dimensionless relative quantities. The network was trained using measured imbalance values, and the deflection values corresponding to the given imbalance and speed conditions were calculated through simulations.