## 2 Research Methodology
### Modeling Methods Improvements
The standard analysis methods of rotor systems are ordinary second-order differential equations, where damping is used as a linear function of velocity and resistance force. However, in most practical situations, particularly in viscoelastic material systems, hydrodynamic bearings, and joint friction, damping is fractional, i.e., dependent on the historical evolution of the state of the system. In this paper, the authors introduce fractional derivatives to model the damping forces so that the rotor system's energy dissipation and accumulation can be modeled more precisely. The fractional derivative of order \(0<\alpha<1\) is employed in the motion equations to account for the non-inertial damping effects caused by anomalous vibration energy dissipation and material or lubricant layer hysteresis:  
\[\frac{d^{2}\xi(t)}{dt^{2}}+\beta_{0}\omega^{1-\alpha}\frac{d^{\alpha}\xi(t)}{ dt^{\alpha}}+\Big{(}\omega_{0}^{2}-i\theta_{0}\Big{)}\xi(t)=\omega^{2}e^{i( \omega t+\varphi)}, \tag{44}\]  
where \(\xi(t)=\xi_{0}(\omega)e^{i\omega t}\) is the displacement of the mass center; \(\xi_{0}\) is the complex amplitude of the dynamic response; \(\beta_{0}\) is the damping factor. After applying [PERSON]'s approach, the amplitude-frequency response is denoted as follows [35]:  
\[A(\
u)=\frac{\
u^{2}}{\sqrt{\big{[}1-\
u^{2}+\psi\
u cos\big{(}\frac{\pi\alpha }{2}\big{)}\big{]}^{2}+(\psi\
u)^{2}\big{[}sin\big{(}\frac{\pi\alpha}{2}\big{)} -k\big{]}^{2}}}; \tag{45}\]  
where the dimensionless values are denoted as \(\
u=\frac{\omega}{\omega_{0}}\), \(\psi=\frac{\beta_{0}}{\omega_{0}}\), and \(k=\frac{\theta_{0}}{\omega_{0}^{2}}\). The primary objective of the research is to investigate the effect of fractional-order damping on the rotor dynamics and analyze the system's stability at various rotational velocities. The authors are attempting to determine the critical parameters that ensure system stability and forecast modes of instability because of bifurcation and parametric resonances. The authors applied the Mittag-Leffler function to derive general amplitude-frequency characteristics of the rotor and to construct bifurcation diagrams, which display stability boundaries as functions of the fractional-order parameter. Numerical computations were performed using the Grunwald-Letnikov fractional derivative. Therefore, this research provides a universal method of modeling rotor dynamics, which can be applied to predicting vibration reliability in turbomachinery, compressor systems, and power generation units. Using fractional-order derivatives improves the accuracy of damping effects estimation and generalizes classical rotor dynamics by adding historical effects to the energy dissipation processes (Figure 29). Figure 29: Amplitude–frequency response. Adopted from [35]. The paper by [88] is dedicated to the reduced-order model (ROM) as an essential part of computational structural mechanics, as it significantly reduces computational costs in analyzing dynamically complex systems. The ROM is applied to many related applications in structural optimization, structural reanalysis, eigenfrequency analysis, dynamic properties, stability assessments, and structural parameter control. One of the serious problems that reduction techniques face is their inability to accurately capture the high-frequency dynamics, which is essential for diagnosing damage, structural acoustics, wave propagation studies, and ultrasonic sensor modeling. The complete and reduced system responses are presented in Figure 30.  
The drawbacks came from the high computational costs and the fact that high-frequency components were not accurately reproduced, thus imposing limitations on the extensive application scope along the frequency range.