## 2 Research Methodology
### Residual Rotor Bending
The procedure presented proved reliable for identifying two faults that co-occurred and had common symptoms (Figure 16). Nevertheless, an essential limitation of the proposed method exists. Since the technique is concerned with a model with a limitation on degrees of freedom, the estimation accuracy can be unsuitable for systems with highly noisy vibration signals or with significant nonlinear effects. Various signal processing methods were employed in the study to improve the identification accuracy of the most common rotor fault--residual bending or bow [55]. The rotor model is defined as a finite element model in a complex form:  
\[M\frac{d^{2}r}{dt^{2}}+C\frac{dr}{dt}+Kr=M\alpha\omega^{2}e^{i\omega t}+Kr_{0}e ^{i(\omega t+\alpha_{0})}, \tag{29}\]  
where \(r_{0}\) is the residual shaft bow; \(\alpha_{0}\) is the location of a bow with respect to the eccentricity; \(M\), \(C\), \(K\) are the mass, damping, and stiffness matrices, respectively. In the first step, wavelet filtering (Db5 wavelet) was applied to signal preprocessing, effectively removing the noise components and highlighting the vibration signal's crucial features corresponding to the residual shaft bow. Dimensionality reduction is followed by using the ISOMAP technique to map the high-dimensional data onto a low-dimensional space to simplify classifying the signal features. For a simpler identification of the effect of the residual shaft bow over the other possible rotor defects, such as unbalance or bearing support misalignment, the principal component analysis (PCA) technique was used. This approach enabled the identification of the primary sources of variation in the signal and the unbundling of the components immediately responsible for the residual bow presence (Figure 17). Figure 16: Frequency response functions of the defective rotors: the orange dots are the experimental results; the blue lines are the numerical results. Adopted from [28]. The final process in validating the results achieved was the enhancement of the rotor motion equations, considering the impact of eccentricity and potential measurement errors. For this purpose, a modified motion equation incorporated supplementary corrective coefficients to compensate for sensor installation inaccuracy. With wavelet analysis, dimensionality reduction algorithms, and statistical analysis methods, it was possible to accurately describe the residual shaft bow and its influence on rotor dynamic behavior. The primary objective of the study [53] was to assess the potential of the finite element model updating in machine tool design:  
\[[M]\ddot{x}+[K]x=[F], \tag{30}\]  
where \([M]\) and \([K]\) are the inertia and stiffness matrices, respectively; \([F]\) is the column vector of cutting forces and moments; \(x\) is the column vector of displacements. The present article presents the use of the iterative eigenvalue sensitivity (IES) method to finite element model updating on the lathe boring bar model [53]:  
\[P_{i+1}=P_{i}+[S_{i}]^{-1}[R_{m}-R_{i}], \tag{31}\]  
where \(P_{i}\) is the vector of the current model parameters (i.e., [PERSON]'s modulus, stiffness); \(P_{i+1}\) is the vector of the updated model parameters; \(R_{m}\) is the vector of the measured responses; \(R_{i}\) is the vector of the numerically calculated responses; \(S_{i}\) is the sensitivity matrix. The boring bar model was represented as an Euler-Bernoulli beam. Experimental modal analysis was then conducted on the boring bar mounted on a lathe machine. To limit the degrees of freedom (DOF) of the FEM model to match that of the experimental model, the system equivalent reduction expansion process (SEREP) was applied. Both the experimental model and reduced-order FEM model were correlated in a correlation study using the natural frequency difference (NFD) correlation coefficient and modal assurance criterion (MAC). The FEM model was then revised using an iterative inverse eigenvalue sensitivity method with elemental stiffness as the updating parameter.