## 3 Verification of Mode Properties of a Dual-Rotor
### Modal Test and Analysis  
Figure 7 shows the frequency response function measured at a particular point. The frequency response function is an acceleration function corresponding to the ratio of the force and acceleration response, defined as:  
\[H_{i}(\omega)=A_{j}(\omega)/F_{i}(\omega) \tag{1}\]  
where the subscript \(i\) represents the impact point, and \(j\) represents the response point. In the test results, the natural frequency can be determined as the peak point at which the response occurs significantly. Four natural frequencies are located within 3500 Hz, with the first-order natural frequency being 15 Hz (0.9 krpm) and the second-order natural frequency being 160 Hz (9.6 krpm). The first- and second-order natural frequencies and the third-order natural frequencies are significantly separated, with values of 1835 Hz (110 krpm).  
Next, the natural frequency of the dual rotor was predicted through a three-dimensional finite element (FE) analysis in the same free-free condition as in the test. The analysis used Modal Analysis from ANSYS Workbench 16 [20], which is a three-dimensional FE analysis software. The key dimensions and material information of the motor rotor used in the analysis are described in previous reports [9; 21].  
Figure 6: Overall configuration of natural mode characteristic test environment.  
Figure 7: Measured frequency response function of a dual-rotor.  
Table 3 shows the first- to fourth-order natural frequencies derived from the test and analysis, and shows the relative error. The results of the test and analysis have an error of 5-7%, indicating that the tests and analyses were reliable. Furthermore, the analysis at all orders has a larger value than that of the test, and its error is relatively constant. This is judged to be a bias error that occurs when the material properties are different from the actual ones in analysis modeling. Therefore, if the accuracy of the analysis model is calibrated through material property adjustment, it can be inferred that model can be calibrated up to an error of approximately 1%.  
Figure 8 shows the mode shape at the first- to fourth-order natural frequencies derived from the test and analysis. The test and analysis results tended to be consistent across the board. In particular, in relatively low-order modes, such as first- and second-order modes, the motor and the test rotor behave rigidly, and the coupling with relatively weak stiffness deforms into a U or S shape. This is a typical dual-rotor bending mode. In addition, higher-order modes, such as third- and fourth-order modes, result in bending modes of the rotor itself, in which the test rotor and motor rotor deform into a U shape.  
Figure 8: Measured and predicted mode shapes of a dual rotor for first through fourth order.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Experiment (Hz)** & **Analysis (Hz)** & **Relative Error (\%)** \\ \hline
1 st & 15 & 17 & 6 \\
2 nd & 160 & 172 & 7 \\
3 rd & 1835 & 1931 & 5 \\
4 th & 3187 & 3377 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison between experimental and analysis results for natural frequencies of a dual rotor.  
Based on the results of the test, in the rotordynamic test, the maximum rotational speed is limited to before the third natural frequency. This is because rotors normally supported by gas bearings cannot support stable bearings in the rotor bending mode due to the low stiffness of the bearings. Thus, with API 612 criteria, the separation margin (_SM_) can be calculated as follows [22]:  
\[SM=\left(126-\frac{6}{AF-3}\right)-100 \tag{2}\]  
where _AF_ is an amplification factor of rotor vibration. In this work, the _AF_ is assumed to be infinitely large for a conservative approach. The _SM_ is then calculated to be 26%, and, when it is applied to the third natural frequency, the maximum rotational speed of the dual-rotor rotodynamic test rig is approximately 92 krpm.