## 3 Numerical Simulations and Discussion
### Affect of Sensor Resolution
#### 3.4.2 Discussion  
In the fourth kind of numerical simulation, the proposed adjustment point is applied and four kinds of typical sensor resolutions are considered for the validation of SDA. At or around the critical frequencies which are obtained by CRDAM and low frequencies, the identification error for rotor unbalance is bigger than at other frequencies. The resolution of sensors is applied by limiting the number of digits after the decimal point in the unbalance responses. At a low frequency, the unbalance responses are very small. The rounding digits have a considerable influence on small values and causes big input errors for unbalance responses. At or near the critical frequency, the unbalance responses vary greatly with frequencies. It is indicated that the rounding digits may lead to big differences among the measured errors of unbalance responses. The repeatability precision of the input responses is poor. Hence, big identification error occurs at low frequencies or at critical frequencies. Similarly, the peak values that are outside the assumed allowable range are at or around the critical frequencies and low frequencies. Far from the critical frequencies, the identification errors are small. The rotating machines' operational frequency (speed) is designed to be far from the critical frequencies to avoid resonance. This indicates that SDA can be applied to rotors not working under very low rotating speeds.  
The statistical results for peak values of the identification error for #1 disc of rotor g1.4 are listed in Table 12. According to Table 12, it is indicated as follows:  
1. At a low frequency, the peak values of identification errors become bigger as sensor resolution is reduced and rotor unbalance cannot be identified when the sensor resolution is 1 um.
2. At or around the first order critical frequency, the identification error, which is obtained using a sensor resolution of 0.1 nm, is little different from the identification error obtained using a sensor resolution of 1 nm. The identification error becomes much bigger when the sensor resolution is 0.1 um and the rotor unbalance cannot be identified when the sensor resolution is close to 1 um.
3. At or around the second order critical frequency, the identification error obtained when the sensor resolution is 0.1 nm is equal to the identification error obtained when the sensor resolution is 1 nm. However, when the sensor resolution is 0.1 um, the identification error becomes a little bigger. Moreover, when the sensor resolution comes close to 1 um, the rotor unbalance cannot be identified.
4. At or around the third order critical frequency, the identification error obtained when the sensor resolution is 0.1 nm is equal to the identification error obtained when the sensor resolution is 1 nm. However, when the sensor resolution comes close to 0.1 um and 1 um, the identification errors become a little bigger.  
The above indicates that sensors with a resolution of 1 um should be avoided and sensors with a resolution of 1 nm and 0.1 nm are recommended for practical application. Similar results can also be obtained from the identification errors for rotor unbalance for #2-#4 discs. These indicate that SDA requires high sensor resolution. The higher the sensor resolution is, the smaller the identification error. In engineering, the resolution of most vibration displacement sensors is 1 um and sensors with resolutions of 0.1 nm and 1 nm are very expensive. Hence, the cost of SDA can be very high when there are multiple discs and bearings in a rotor.  
\begin{table}
\begin{tabular}{c c c} \hline
**First Order** & **Second Order** & **Third Order** \\ \hline
269 Hz & 645 Hz & 1267 Hz \\ \hline \end{tabular}
\end{table}
Table 11: First three critical frequencies of g1.4 from 1 Hz to 2000 Hz.