## 2 Research Methodology
### Combined Faults
As the procedure is based on normalized (dimensionless) quantities, the deflection shape in a particular vibration mode for the bridge is not a function of the passage parameters and can be used as a meaningful indicator of the structure's condition. The measurement is kept relative to the position of the supports instead of time. The authors suggest that the object crosses the bridge at a constant velocity to transform the data into the time domain. To obtain the response's instantaneous phase and amplitude value, the authors used the Hilbert transform to obtain the required complex number representation. Numerical computations showed that the dimensionless shape deviation is indeed damage-sensitive: in the presence of flaws in some regions of the bridge, the value of the ODSR for the corresponding mode deviates from the reference state [30]:  
\[ODSR_{n}(t)=\frac{\sin\bigl{(}\frac{n\pi\mathit{v}t}{l}\bigr{)}}{sin\Bigl{[} \frac{n\pi(\mathit{v}t-d)}{l}\Bigr{]}}, \tag{41}\]  
where \(n\) is the mode number; \(\mathit{v}\) is velocity; \(l\) is the full length of the beam; \(t\) is time; \(d\) is the length of a current beam segment. The suggested damage indicator using the ODSR quantifies the damage level by comparing the ODSR in damaged and intact states. Even though the work is not directly related to rotor dynamics, the system modeling techniques can be applied to model instantaneous values of parameters for imbalance identification. Table 8 presents a comparative analysis of the methods for detecting combined faults. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Method** & **Advantages** & **Disadvantages** & **Domain \({}^{*}\)** \\ \hline Joint input-state estimation algorithm (JISEA) [31] & Provides simultaneous estimation of both system states and unknown excitatory forces. & High sensitivity to noise and errors in input data. & N/A \\ \hline Modal expansion algorithm (MEA) [31] & Does not require knowledge of natural frequencies or damping ratios. & Not suitable for low-frequency (quasi-static) strain estimation. & FD \\ \hline ISOMAP-based orbit identification [85] & Captures nonlinear features from high-dimensional orbit data. & It requires careful parameter tuning and may be sensitive to sampling density. & N/A \\ \hline Digital filter method [85] & It allows for the effective reduction of noise and extracts the rotor orbit’s basic shape. & It is not effective for nonlinear or complex signals. & TD/FD \\ \hline RBFNN [86] & Achieves good accuracy on balanced and imbalanced datasets across three sensor modalities (acoustic, vibration, current). & It requires careful choice of window size and overlap, as well as careful ANN architecture design planning. & N/A \\ \hline Synchronized Fourier transform (SFT) [86] & Allows reconstruction. & Requires proper tuning of squeezing parameters. & TFD \\ \hline Operating deflection shape (ODS) method [30] & It does not require knowledge of input forces, as it is based only on vibration measurements. & Determining local changes can be difficult due to the scaling of defective forms. & TD/FD \\ \hline Artificial neural network (ANN) & High accuracy in estimating or identifying [59,86]. & Depends on reliable FEM input or experimental data [59]. & \\ \hline \multirow{3}{*}{
\begin{tabular}{} \end{tabular} } & Suitable for nonlinear and implicit process modeling [59]. & Sensitive to class imbalance ratios [86]. & N/A \\ \cline{1-1}  & Works with limited sensor setups [86]. & Performance depends heavily on careful feature engineering and parameter tuning [59,86]. & \\ \hline \hline \end{tabular} \({}^{*}\)Signal processing domain (if applicable). TD—time domain; FD—frequency domain; TFD—time-frequency domain.  
\end{table} 
Table 8: Comparative analysis of methods for rotor combined faults detection.