## 2 Research Methodology
### Misalignment  
Misalignment [66] is another of the most frequent faults of rotor systems because the rotor and drive axes deviate from ideal alignment. The principal reasons for misalignment include errors during assembly, thermal distortions, non-uniform bearing wear, coupling distortions, or manufacturing tolerances. Misalignment generates periodic loads on the bearings and shaft, additional bending moments, and a heightened degree of harmonic vibrations [73]. This can lead to structural element wear at an accelerated rate, a reduction in system efficiency, and even premature failure. The general concept of the approach in the study [27] is to create a novel approach to assess the unbalance and misalignment of flexible rotor systems based on vibration analysis of a single-stop cycle (run-down test). The approach will measure amplitude-frequency characteristics on bearing supports to compute unbalanced forces and moments simultaneously due to the misalignment of shaft connections. The influence of misalignment on the change in the bearing load distribution and the shape of the rotor deflection is analyzed based on the following dynamic rotor-bearing model:  
\[\widetilde{Z}_{F}r_{F,b}=Z_{B}\Big{(}P^{-1}Z_{B}-I\Big{)}r_{F,b}-Z_{B}P^{-1}Z_{ R,hi}Z_{R,ii}^{-1}f_{u}, \tag{22}\]  
where \(Z\) is the dynamic stiffness matrix; the subscripts \(i\) and \(b\) refer to the internal and bearing (connection) degrees of freedom, respectively; the subscripts \(F\), \(R\), and \(B\) refer to the foundation, the rotor, and the bearings; \(r\) is the response(s); \(f_{u}=\omega^{2}\cdot T_{e}+T_{m}\cdot e_{m}\) are the force vector(s) assumed to be applied only at the rotor's internal degrees of freedom; \(T_{e}\) is a selection matrix indicating the location of the balance planes; \(T_{m}\) is a transformation matrix indicating the location of the couplings; \(e_{m}\) is a vector of inner loads. Experimental testing was performed on a rotor stand, whose results confirmed the high accuracy of calculating the parameters of unbalance and misalignment with an error of no more than 5% and 0.2 mm, respectively. The formed technique allows for the diagnosing and balancing time of rotary machines without their dismounting or repeating measurements, which promises industrial applications. The principal outcome of the research is an experimental verification of the proposed method on a rotor stand, during which vibration was analyzed during a single-stop cycle. The results revealed that the approach guarantees more than 95% accuracy for the unbalance assessment and allows misalignment determination with an error of less than 0.2 mm and 0.7\({}^{\circ}\) (Figure 12). Figure 12: Measured and estimated responses at bearing: (**a**) horizontal direction and (**b**) vertical direction. Adopted from [27]. This confirms the approach's effectiveness for rapid diagnosis without disassembly or repetitive rotor balancing. The approach, nonetheless, has limitations that may limit its application in complex rotor systems. First, the method does not consider the thermal deformations of rotors and bearings, which can cause an additional change in misalignment when operating at high speeds. Second, it is sensitive to the accuracy of the rotor dynamic model, particularly stiffness calculations of the bearings and the connections; hence, it is prone to mistakes when applied to other machines. The paper by [48] examines a technique for determining axis misalignment and imbalance in rotors supported by magnetic bearings using a new trial displacement method. The rotor model is defined as a finite element model:  
\[M\ddot{\eta}^{mis}+(C-\omega G)\dot{\eta}^{mis}+K\eta^{mis}=f_{umb}+f_{UNB}^{ mis}, \tag{23}\]  
where \(\eta^{mis}\) is the global rotor displacement vector; \(M\), \(K\), \(C\), \(G\) are the global mass, stiffness, damping, and gyroscopic matrices, respectively.