## 3 Numerical Simulations and Discussion  
### Methodology of the Numerical Simulations  
The numerical simulations were conducted to validate the proposed methods by comparing the identified rotor unbalances with the set value of rotor unbalances. As shown in Figures 3 and 4, six computational examples, which represent single-span and single-disc rotors (g1.1, h1.1), single-span and four-disc rotors (g1.4, h1.4), and four-span and four-disc rotors (g4.4, h4.4) are used in the simulation. The rotors g1.1, g1.4, and g4.4 are supported by rolling bearings, and h1.1, h1.4, and h4.4 are sustained by oil journal bearings. The parameters of the rotors are summarized in Tables 1-6. The positions of each bearing and disc on the shaft are shown in Figures 3 and 4. There are 121 nodes in the rotors in Figure 3 by dividing the shaft into \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline r\_shaft & Radius of the shaft \\ p\_shaft & Density of the shaft \\ E\_shaft & Elastic modulus of the shaft \\ L\_shaft & Length of the shaft \\ r\_disc & Radius of the disc \\ p\_disc & Density of the disc \\ E\_disc & Elastic modulus of the disc \\ L\_disc & Width of the disc \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Meanings of symbols in the computational example of the rotor. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline r\_shaft of rotor g1.1, & & & & & \\ g1.4 and g4.4 & & & & & \\ r\_shaft of rotor h1.1, & & & & & \\ h1.4 and h4.4 & & & & & \\ p\_shaft & 7800 kgm\({}^{-3}\) & L\_shaft of the rotor h1.1 & & & \\ E\_shaft & \(2.1\times 10^{11}\) Pa & L\_shaft of the rotor h1.4 & & & \\ L\_shaft of the rotor g1.1 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the rotor shafts in the rotors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m_{1u}\) & 0.05 kg & \(m_{2u}\) & 0.15 kg & \(m_{3u}\) & 0.01 kg \\ \(e_{1}\) & \(30\times 10^{-3}\) m & \(e_{2}\) & \(10\times 10^{-3}\) m & \(e_{3}\) & \(20\times 10^{-3}\) m \\ \(\alpha_{1}\) & \(225^{\circ}\) & \(\alpha_{2}\) & \(120^{\circ}\) & \(\alpha_{3}\) & \(-120^{\circ}\) \\ \(m_{4u}\) & 0.10 kg & r\_disc of \#1 disc & \(60\times 10^{-3}\) m & p\_disc of of \#1-\#4 disc & 7800 kgm\({}^{-3}\) \\ \(e_{4}\) & \(15\times 10^{-3}\) m & r\_disc of \#2-\#4 & \(50\times 10^{-3}\) m & E\_disc of of \#1-\#4 disc & \(2.1\times 10^{11}\) Pa \\ \(\alpha_{4}\) & \(-170^{\circ}\) & L\_disc of of \#1-\#4 disc & \(10\times 10^{-3}\) m & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the eccentric discs in the rotors g1.1, g1.4, and g4. highest used at present. Finally, identification exercises were performed considering four kinds of typical sensor resolution (0.1 nm, 1 nm, 0.1 um, and 1 um). By limiting the number of digits after the decimal point in the unbalance responses, the resolutions of unbalance response measurement systems are considered. In the above simulations, the calculation frequency was from 1 to 2001 Hz and the interval was 2 Hz. The relative error for the identified rotor unbalance amplitude to the set rotor unbalance amplitude and the absolute error for the identified rotor unbalance angle to the set rotor unbalance angle were used for the analysis of the simulation results. Moreover, the statistical results of the allowable frequency points (AFPs), at which the identification error for the rotor unbalance amplitude was less than 20% and the rotor unbalance angle was less than 10\({}^{\circ}\), were obtained. In addition, as shown in Figure 2, SDA includes two algorithms which are represented by Equations (8), (10), (16), and (17), respectively. The difference between them is that one requires unbalance responses in the \(y\) direction as input and the other uses unbalance responses in the \(x\) direction. They can be considered the same algorithm.