## 2 Research Methodology
### Cracks
The crack model consisted of an array of point elastic elements evenly distributed along the crack edges, connecting them and allowing for deformation. The experimental results revealed the presence of so-called anti-resonance zones, where the vibration amplitude decreases sharply. The authors attribute this phenomenon to the self-damping effects occurring at specific rotational frequencies. Additionally, they note that this effect can indicate a crack presence in the rotor system. [PERSON]'s method produced more accurate results than [PERSON]'s but required several times more computational time. Nonetheless, the accuracy of [PERSON]'s method was still deemed satisfactory, making it a viable alternative for cases where computational efficiency is a priority. A similar problem is considered in the study by [42]. The effort focused on developing a stiffness matrix of a cracked rotor and studying it to find instability regimes. The model under consideration is a Jeffcott rotor with a crack in the midspan of the shaft:  
\[ \begin{split} m\ddot{u}+c\dot{u}+k_{1}(t)u+k_{12}(t)v=m_{ed} \Omega^{2}sin(\Omega t+\beta);\\ m\ddot{v}+c\dot{v}+k_{21}(t)u+k_{2}(t)v=m_{ed}\Omega^{2}cos( \Omega t+\beta)-mg,\end{split}  \tag{20}\]  
where \(k_{1}(t)\), \(k_{2}(t)\) is the instantaneous time-varying stiffnesses; \(k_{12}(t)\), \(k_{21}(t)\) is the cross-coupling stiffness; \(m\) is the mass of the rigid disk; \(c\) is the external damping; \(m_{ed}\) is the mass unbalance; \(\beta\) is the angle between the mass unbalance and the weak direction of the crack; \(\Omega\) is the rotating speed; \(g\) is the gravitational acceleration. The calculation of the stiffness matrix is reduced to merely calculating the variation of the moment of inertia along the cracked section and the law of the variation of the eccentricity of the centroid with respect to the rotation axis [42]:  
\[ \begin{split}\overset{\sim}{I}_{X}(t)=I-f_{1}(t)I_{11};\\ \overset{\sim}{I}_{Y}(t)=I+f_{1}(t)I_{11}+f_{2}(t)I_{22},\end{split}  \tag{21}\]  
where \(f_{1}(t)\) and \(f_{2}(t)\) are the special functions that depend on the open and closed crack state angles, respectively. For this purpose, rotational angles of the shaft at which the crack opens and closes were developed for the shaft, thereby defining the boundaries of the variations in the moment of inertia and the eccentricity of the centroid of the cracked cross-section. Stability analysis was carried out using the Floquet theory. The results were used to construct bifurcation diagrams for the damped and undamped rotors. It was found that each macrozone of instability contains subzones with all three forms of instability. The subzones are bounded  
Figure 10: Stability charts of a cracked rotor. Adopted from [41]. by domains of neutral stability, which are related to eigenvalues on the unit circle in the complex plane. The authors further note that damping affects the width of the instability zone. The bifurcation diagrams for the damped rotor show that instability zones appear at larger values of the relative crack depth, while some instability zones in the undamped rotor diagrams disappear completely (Figure 11). In addition, to ensure that the results are correct, the crack should be in the middle section of the shaft. Otherwise, the method should be modified to consider the varied effects of the location of the crack. A comparative analysis of the methods for rotor crack detection is summarized in Table 3.