## 3 Numerical Simulations and Discussion
### Affect of Sensor Resolution
#### 3.4.1 Results
3. Sensor resolution of 0.1 um With the sensor resolution being reduced to 0.1 um, the minimum starting frequency of the identification was 4 Hz (SDA does not work at a low frequency). According to Figure 23, the results are as follows:  
For #1 disc, the peak values of the amplitude of identified rotor unbalance are 695.26%, 19.45%, 0.65%, and 0.62% at frequencies of 4 Hz, 268 Hz, 646 Hz, and 1266 Hz, respectively. The peak values of the absolute error for the angle are 140.04\({}^{\circ}\), 10.86\({}^{\circ}\), 0.55\({}^{\circ}\)and 0.45\({}^{\circ}\)at 4 Hz, 268 Hz, 646 Hz and 1262 Hz, respectively. For #2 disc, the peak values of the amplitude of identified rotor unbalance are 597.53%, 22.7%, 5.32% and 1.63% at frequencies of 4 Hz, 270 Hz, 644 Hz, and 1268 Hz, respectively. The peak values of the absolute error for the angle are 94.44\({}^{\circ}\), 5.72\({}^{\circ}\)1.9\({}^{\circ}\), and 0.71\({}^{\circ}\)at 4 Hz, 266 Hz, 644 Hz, and 1268 Hz, respectively. For #3 disc, the peak values of the amplitude of identified rotor unbalance are 1722.04%, 262.49%, 39.32%, 9.24%, and 11.76% at frequencies of 8 Hz, 268 Hz, 644 Hz, 814 Hz, and 1264 Hz, respectively. The peak values of the absolute error for the angle are 120.31\({}^{\circ}\), 20.6\({}^{\circ}\), 8.75\({}^{\circ}\), 3.49\({}^{\circ}\), and 11.3\({}^{\circ}\)at 16 Hz, 270 Hz, 644 Hz, 814 Hz, and 1268 Hz, respectively. For #4 disc, the peak values of the amplitude of identified rotor unbalance are 853.26%, 4.43%, 13.87%, and 3.3% at frequency 6 Hz, 268 Hz, 810 Hz, and 1266 Hz, respectively. The peak values of the absolute error for the angle are 165.1\({}^{\circ}\),10.04\({}^{\circ}\), 0.79\({}^{\circ}\), 4.34\({}^{\circ}\), and 1.5\({}^{\circ}\)at 10 Hz, 270 Hz, 644 Hz, 810 Hz, and 1264 Hz, respectively. 4. Sensor resolution of 1 um As the sensor resolution was reduced to 1 um, the starting frequency of the identification was 856 Hz (SDA does not work at a low frequency). According to Figure 24, for #1 disc, the peak value of the amplitude of identified rotor unbalance is 0.63% at a frequency of 1266 Hz. The peak value of the absolute error for the angle is 0.45\({}^{\circ}\)at 1262 Hz. For #2 disc, the peak value of the amplitude of identified rotor unbalance is 1.58% at a frequency of 1268 Hz. The peak value of the absolute error for the angle is 0.75\({}^{\circ}\)at 1268 Hz. For #3 disc, the peak value of the amplitude of identified rotor unbalance is 11.73% at a frequency of 1264 Hz. The peak value of the absolute error for the angle is 11.56\({}^{\circ}\)at 1268 Hz. For #4 disc, the peak value of the amplitude of identified rotor unbalance is 3.47% at a frequency of 1268 Hz. The peak value of the absolute error for the angle is 1.49\({}^{\circ}\)at 1264 Hz. 5. According to the above results, the peak values that were outside the assumed allowable range--the amplitudes bigger than 20% and the angles bigger than 10\({}^{\circ}\)--are all counted in Tables 7-10, with the sensor resolutions set at 0.1 nm, 1 nm, 0.1 um, and 1 um, respectively. From Tables 7-10, the frequencies at which these peak values appear are at or around the critical frequencies listed in Table 11 and the low frequencies such as 1 Hz, 4 Hz, and so on. This indicates that the rotor unbalance identification error is big when the rotor works at a low speed or near the critical speeds when using SDA. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Disc** & **Unbalance** & **Value** & **Value** & **Value** & **Value** \\ \hline
**\#1** & Amplitude & (10.84\%, 1 Hz) & - & - & - \\  & Angle & - & - & - & - \\  & Amplitude & - & - & (42.54\%, 645 Hz) & - \\  & Angle & - & - & - & - \\  & Amplitude & (26.02\%, 1 Hz) & (257.24\%, 269 Hz) & - & (20.64\%, 1265 Hz) \\  & Angle & (19.49\%, 1 Hz) & - & (11.72\%, 645 Hz) & - \\  & Amplitude & - & - & - & - \\  & Angle & - & (16.259\%, 269 Hz) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistical results of peak values outside the assumed allowable range when the resolution is 0.1 nm.