# Dynamic Response of Dual-Disk Rotor System with Uncertainties Based on Chebyshev Convex Method
## 4 Numerical Analysis of the Rotor Dynamic Response Based on CCM
### Experimental for Uncertainty Response  
The dual-disk rotor system shown in Figure 11 is used for the uncertainty response experimental, and the physical parameters of the rotor are tabulated in Table 1. The rotor system consists of a straight shaft and two discs overhung on the right. The shaft is fixed on the bracket by two bearing supports, and the left end of the shaft is connected with the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Uncertainty and**} & \multicolumn{2}{c}{**Critical Speeds**} \\ \multicolumn{2}{c}{**Nominal Critical Speeds**} & & & \\ \hline \multicolumn{2}{c}{**Multi-Parameters**} & \multicolumn{2}{c}{**First-Order (rpm)**} & \multicolumn{2}{c}{**Second-Order (rpm)**} & \multicolumn{2}{c}{**Third-Order (rpm)**} \\ \multicolumn{2}{c}{**Uncertainty (\(E\), \(\rho\), \(K_{2}\))**} & \multicolumn{2}{c}{**[Lower, Upper]**} & \multicolumn{2}{c}{**[Lower, Upper]**} \\ \hline \multirow{6}{*}{Uncertainty} & 2838.67 & 6406.81 & 9985.07 \\  & 10\% & [2752.25, 2918.21] & [6188.02, 6624.86] & [9444.11, 10,593.81] \\  & 15\% & [2704.75, 2959.06] & [6071.80, 6739.12] & [9196.52, 10,909.02] \\ \cline{2-4}  & \multicolumn{4}{c}{Maximum error in interval} \\ \cline{2-4}  & 10\% & 3.04\% & 3.41\% & 6.10\% \\ \cline{2-4}  & 15\% & 4.72\% & 5.23\% & 9.25\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Calculation of critical speeds under different uncertainty. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Uncertainty and Nominal**} & \multicolumn{2}{c}{**Critical Speeds (rpm)**} \\ \multicolumn{2}{c}{**Critical Speeds**} & \multicolumn{2}{c}{**First-Order**} & \multicolumn{2}{c}{**Second-Order**} & \multicolumn{2}{c}{**Third-Order**} \\ \multicolumn{2}{c}{} & **[Lower, Upper]**} & \multicolumn{2}{c}{**[Lower, Upper]**} & \multicolumn{2}{c}{**[Lower, Upper]**} \\ \hline \multirow{6}{*}{Nominal value} & 2838.67 & 6406.81 & 9985.07 \\  & \(K_{2}\) & [2796.70] & [6255.72, & [9894.02, \\  & 2873.55] & 6544.52] & 10,079.23] \\  & \(E\) & [2762.54, & [6369.19, \\  & 2905.78] & 6437.51] & 10,246.43] \\  & \(\rho\) & [2800.36, & [6397.32, \\  & 2864.39] & 6444.52] & 10,035.64] \\ \cline{2-4}  & \multirow{2}{*}{Multiple} & [2752.25, \\  & 2918.21] & 6624.86] & 10,593.81] \\ \cline{2-4}  & & \multicolumn{4}{c}{Maximum error in interval} \\ \cline{2-4}  & \(K_{2}\) & 1.48\% & 2.36\% & 0.94\% \\  & \(E\) & 2.68\% & 0.59\% & 2.68\% \\  & \(\rho\) & 1.30\% & 0.58\% & 0.89\% \\ \cline{2-4}  & Multiple & 3.04\% & 3.41\% & 6.10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculation of critical speeds under different uncertainty parameters. drive motor by the coupling. The maximum speed of the motor is 10,000 r/min. The two displacement sensors (NO. 3 and 4 in Figure 11) are arranged horizontally on the right side of the bearing supports of the discs to measure the dynamic displacement of the rotor. And the rotor speed is measured by two displacement sensors (NO. 1 and 2 in Figure 11) which receive the displacement information on the coupling. The deflections of the discs are obtained by Hilbert transform under the condition of neglecting the anisotropy of rotor. In order to verify the effectiveness of CCM, the uncertainty experiment of mass parameter is carried out. The simulation of mass uncertainty is realized by adding screws to the disk. The screws shown in Figure 12 with the weight of 4.73 g are inserted into symmetrical holes on the disk 2 to simulate the change of mass (4 screws correspond to the mass uncertainty of 4% and 8 screws correspond to the mass uncertainty of 8%). Due to the existence of uncertainty, the dynamic response of the system can be any possible value in the response interval.