## 3 Numerical Results and Discussion
### Validation with Experiment Results  
The rotor model from Reference [22] is modeled to validate the method presented in Section 2. Detailed configurations of the rotor system can be found in Reference [22].  
The rotor model diagram is shown in Figure 16. This test rig is used to study the dynamic characteristics of a co-rotating and counter-rotating dual-rotor system with four or five supports. The additional supports in Figure 16 are removed because this paper focuses on a four-support counter-rotating dual-rotor system. The rotor system consists of four supports and four discs, of which each rotor has two discs. Each shaft is driven by a separate motor, so the rotation speed of the inner and outer rotors can be different with a speed ratio of \(-1.65\). The angular acceleration for the inner rotor during run-up and run-down is approximately 1.05 rad/s. The operational ranges for the inner and outer rotor are 0-232 rad/s and 0-382 rad/s, respectively. The excitation is only due to the residual unbalance of the rotors.  
Figure 17 is the counter-rotating coaxial system in this paper. The model parameters of the rotor system are listed in Tables 4-8 contains the geometric dimensions and information in each test. Table 5 contains the stiffness coefficients of the elastic supports. Tables 6 and 7 show the parameters of the intermediate bearing and SFDs. Table 8 shows the unbalanced configuration and inertia properties of each disk.  
Figure 16: Dual-rotor test rig.  
Figure 17: Structural diagram of the coaxial rotor system.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Node No.** & **Axial Location (m)** & **Bearing/Disk** & **Element No.** & **Outer Diameter (m)** & **Inner Diameter (m)** \\ \hline  
earing no.2 & 28 & 0.014 & 0.00 \\
29 & 1.21838 & & 29 & 0.014 & 0.00 \\
30 & 1.23038 & & \multicolumn{3}{c}{End of inner rotor} \\  
earing no.4 & 46 & 0.060 & 0.03 \\
48 & 1.07380 & & & & \\ \hline \end{tabular}
\end{table}
Table 4: Dimension and elements information of the rotor system.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Support I** & **Support II** & **Support III** & **Support IV** \\ \hline Stiffness (N/m) & \(1.45\times 10^{6}\) & \(2.21\times 10^{5}\) & \(9.29\times 10^{5}\) \\ \hline \end{tabular}
\end{table}
Table 5: Stiffness of elastic supports (squirrel cage) in the test rotor system.  
Figure 18 is borrowed from Reference [22] for comparison with the results obtained by the method in this work, which can be seen in Figure 7. By comparing Figures 18 and 19, it can be seen that the numerical results of this work show a great agreement with those in Reference [22], which has demonstrated the validity of the method presented in this paper.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Radius of Inner** & **Radius of Outer** & **No. of Rollers** & **Contact Stiffness** & **Radial** \\
**Ring (mm)** & **Ring (mm)** & **No. of Rollers** & **(N/m\({}^{3/2}\))** & **Clearance (\(\mu\)m)** \\ \hline
9.37 & 14.13 & 9 & 7.055 \(\times\)\(10^{9}\) & 6 \\ \hline \end{tabular}
\end{table}
Table 6: Parameters of the intermediate bearing in the test rotor system.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{2}{c}{**Inner Rotor**} & **Outer Rotor** \\  & **Disk 1** & **Disk 2** & **Disk 3** & **Disk 4** \\ \hline Unbalance (\(\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\)) & 2 & 4 & 1 & 2 \\ Mass (kg) & 2.3386 & 2.3386 & 3.2590 & 1.6303 \\ Polar moment of inertia (kg\(\cdot\)m\({}^{2}\)) & 0.00815 & 0.00815 & 0.01561 & 0.00661 \\ \hline \end{tabular}
\end{table}
Table 8: Unbalanced configuration and inertia properties of disks.  
Figure 18: _Cont._  
Figure 18: Experimental results from Reference [22] (Chapter 5, Figure 5.25): (**a**) orbit of the inner rotor disk \(-\omega=188\) rad/s; (**b**) orbit of the outer rotor disk \(-\omega=188\) rad/s; (**c**) orbit of the inner rotor disk \(-\omega=230\) rad/s; (**d**) orbit of the outer rotor disk \(-\omega\)= 230 rad/s.