## 3 Numerical Simulations and Discussion
### Accuracy of SDA and TODA  
#### 3.2.1 Results  
(1) Results of the first kind of simulation  
Figures 5 and 6 were obtained by using the computational example h1.4 based on SDA and TODA, respectively. They represent the maximum identification errors of each rotor unbalance. In the simulation, #10 point was used as one of the required measuring points and the other measuring points were #3, #47, #11, #21, #31, and #41 points at the location of the two bearings and the four discs. The form \(\alpha\)%\(\angle\)/\(\beta^{\circ}\), in which \(\alpha\)% is the identification error for the rotor unbalance amplitude and \(\beta^{\circ}\) represents the identification error for the rotor unbalance angle, is used to express the identification error for rotor unbalance. According to Figure 5, for SDA, the maximum relative errors of the amplitude of rotor unbalance for #1-#4 discs are almost equal to zero and are \(1.70545\times 10^{-8}\)%, \(1.4489\times 10^{-6}\)%, \(3.22562\times 10^{-5}\)%, and \(2.8886\times 10^{-5}\)%, respectively. For the rotor unbalance angle, the maximum absolute errors, which are \(5.505.9\times 10^{-9}\)%, \(5.99272\times 10^{-7}\)%, \(5.83537\times 10^{-5}\)%, and \(1.93098\times 10^{-5}\)%, respectively, also equal zero. According to Figure 6, for TODA, the maximum relative errors of the rotor unbalance amplitude of each disc are \(1.14223\times 10^{-6}\)%, \(2.66709\times 10^{-5}\)%, \(0.00102\)%, and \(1.6262\times 10^{-4}\)%, respectively. The maximum absolute errors of the rotor unbalance angle are \(4.68235\times 10^{-7}\)%, \(1.59148\times 10^{-5}\)%, \(4.41457\times 10^{-4}\)%, and \(5.47386\times 10^{-5}\)%, respectively. Although they are bigger than the maximum errors obtained by SDA, they can be also considered equal to zero. Similar results can be found for the first kind of calculation simulation example for g1.1, h1.1, h4.4, and g1.4, according to Table A1 and Figures A1-A4 in Appendix A. However, according to Figures 7 and 8, which show the maximum identification errors of each rotor unbalance for g4.4, for TODA, the first two errors are very small, but the maximum identification errors of rotor unbalance for #3 and #4 discs are too big, according to Figure 8. The first two errors are \(7.52405\times 10^{-5}\)%\(\angle\)/\(\angle\)\(1.81997\times 10^{-4}\)%, \(0.17108\)%\(\angle\)/\(0.03405\)deg, while for the errors for #3 and #4 discs, they are \(123.71543\)%\(\angle\)/\(21.8563\)deg and \(1281.45771\)%\(\angle\)/\(21.69524\)deg, respectively. Whereas, for SDA, the maximum identification errors of rotor unbalance for #1-#4 discs are \(4.70411\times 10^{-8}\)%\(\angle\)/\(1.49579\times 10^{-7}\)%, \(5.00642\times 10^{-4}\)%\(\angle\)/\(1.41912\times 10^{-4}\)%, \(0.28694\)%\(\angle\)/\(0.12378\)deg, and \(0.16497\)%\(\angle\)/\(1.57779\)deg, respectively, according to Figure 8.  
Figure 5: The maximum identification error for each rotor unbalance in h1.4 based on SDA using #10 point as one of the required measuring points: (a) relative error for the rotor unbalance amplitude; (b) absolute error for the rotor unbalance angle. Figure 8: The maximum identification error for each rotor unbalance in g4.4 based on TODA using #20 point as one of the required measuring points: (**a**) relative error for the rotor unbalance amplitude; (**b**) absolute error for the rotor unbalance angle. Figure 6: The maximum identification error for each rotor unbalance in h1.4 based on TODA using #10 point as one of the required measuring points: (**a**) relative error for the rotor unbalance amplitude; (**b**) absolute error for the rotor unbalance angle. Figures 10 and 11, which represent the maximum identification errors of each rotor unbalance for h1.4 are obtained based on SDA and TODA, respectively. In this simulation, #10 point was used as one of the required measuring points and the other measuring points were #3, #47, #11, #21, #31, and #41 at the locations of the two bearings and the four discs.