## 2 Research Methodology
### Cracks
To determine the system's stiffness, an energy-based fracture mechanics approach (SERR) was applied:  
\[g_{ij}=\frac{\partial^{2}U^{0}}{\partial P_{j}\partial P_{j}}+\frac{\partial^{ 2}U^{C}}{\partial P_{i}\partial P_{j}}, \tag{15}\]  
where \(U^{0}\) is the crack-free element strain energy; \(U^{C}\) is the crack-induced strain energy:  
\[U^{C}=\int_{A}G(A)dA, \tag{16}\]  
where \(G(A)\) is the strain energy release rate:  
\[G(A)=\frac{1}{E^{\prime}}\left[\left(\sum_{i=1}^{6}K_{i}^{I}\right)^{2}+\left( \sum_{i=1}^{6}K_{i}^{II}\right)^{2}+(1+v)\left(\sum_{i=1}^{6}K_{i}^{III} \right)^{2}\right], \tag{17}\]  
where \(E^{\prime}=E/\left(1-v^{2}\right)\); \(K_{i}^{I}\), \(K_{i}^{II}\), and \(K_{i}^{III}\) are the different types of stress intensity factors (SIFs). The primary assumption in the experiment was that the crack breathes solely at speeds lower than a given critical speed. When the neutral axis does not coincide with the crack, it opens and closes, and the cross-sectional stiffness varies. Otherwise, it is locked in a fixed location, depending on the angle between the opening and unbalanced force directions. The simulations fully justified the authors' assumptions. When the critical speed was attained, the location of the neutral axis abruptly shifted, locking the crack in a fixed location. The authors relate the sudden shift around the neutral axis to a change in dominant forces. When the unbalanced force begins to dominate the gravity force, the \"rocking\" motion of the crack ceases. However, because the rotor has uneven stiffness, the whirl starts to form, which degrades as the force of imbalance increases [71; 72]. Experimental testing validated that the simulation results are presented in Figure 9.  
The mathematical model, however, does not account for all nonlinear effects. For instance, the potential contact between the edges of the crack during rotation is not considered, which could introduce additional friction forces and alter the system's stiffness. This limitation is especially significant for highly cracked rotors or systems with changing eccentricity effects with operating conditions. The existence of a crack affects rotor stability, so defining unstable regimes caused by cracks is an essential problem for rotor systems diagnostics. The study by [41] compares the efficiency of Floquet and Bolotin methods in assessing the stability of a reduced-order model of a cracked rotor. The full-order equation of motion takes the following form:  
\[M\tilde{q}+(D_{d}+\omega D_{G})\dot{q}+K[q(t)]q=p_{u}+w, \tag{18}\]  
where \(M\), \(D_{d}\), \(D_{G}\), \(K\) are the mass, damping, gyroscopic, and stiffness matrices, respectively; \(p_{u}\), \(w\) are the vectors of unbalance and gravity forces; \(q\) is a vector of generalized coordinates of the mass centers of subsequent rigid finite elements; \(\omega\) is the rotor spin speed. The order of the model was reduced to two dominant modes using modal transformation techniques [41]:  
\[\widetilde{M}\ddot{p}+\widetilde{D}_{d}\dot{p}+\left[\widetilde{K}_{0}+\widetilde {K}_{c}\text{cos}(\eta t)\right]p=0, \tag{19}\]  
where the matrices subscripted with the \"Tilda\" sign are reduced from the corresponding full-order matrices. Stability analyses were then performed on second-order systems, obtained by combining models of individual modes (Figure 10). The results of these analyses were compiled into a comprehensive stability map. The stability map from the reduced-order model was found to fully correspond to the stability map calculated using the full-order rotor model. At the same time, the computational time required to generate the map, as well as its overall size, was significantly reduced. The approach was demonstrated on a mathematical model of a rotor with a breathing crack, simulated using the finite element method (FEM). The rotor was non-rotating, but the shaft stiffness was periodically varied to simulate parametric excitation.