# 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
### Single Parameter Uncertainty  
When there is a single uncertain parameter, the convex model evolves into an interval model. By the Chebyshev Convex Method (CCM), the uncertainty caused by the uncertain coefficient can be acquired. Several uncertain parameters are considered below, and the response boundary of the critical speeds calculated by CCM are compared with those by CMCS. In order to compare the influence of different single parameters of the rotor system, parameters are defined in same uncertain degree. According to the actual needs of the project, the degree of uncertainty is set to 10%. Firstly, the damping is viewed as an uncertain parameter, and the degree of uncertainty is 10%. Using CCM, the uncertain dynamic response affected by uncertain parameters can be obtained, and the results are shown in Figure 7.  
Compared with CMCS, the response boundary in the critical speed range calculated by CCM is similar, and the deterministic transient response curve of the system is symmetrically shifted up and down. The resonance peak position does not change and falls within the critical speed range. Generally, it is difficult to define the accurate value of the stiffness of a support. In this case, the stiffness of bearing 2 is taken as a convex variable to cover its variability and the uncertain degree of support stiffness \(K_{2}\) is 10%. The response range of the center of disk 2 is shown in Figure 8.  
As can been seen from Figure 8, we can find that there is a peak shift in second-order resonance peaks under the influence of the support stiffness uncertainty at no such phenomenon in the first-order. This means the different sensitivities of the system in different speed ranges to the first two critical speeds. Meanwhile, the upper and lower boundaries of the deflection amplitude increase at the peak of the second-order critical speed, which suggests that the inherent properties of the high-speed flexible rotor system are sensitive to the support stiffness. In addition, due to the simplification of the model and the calculation method in the theoretical analysis, the theoretical modeling cannot fully simulate the actual structure of the original high-speed flexible rotor system, resulting in small errors and overestimation of response. Figure 7: Deflection of the dual-disk rotor with damping uncertainty. In engineering practice, the rotor system may be affected by manufacturing error, material degradation, and wear. The actual mass often deviates from the theoretical value after long-term work. Therefore, the uncertainty of mass should be considered in the analysis. To consider the uncertainty of mass, the uncertain degree of mass of disk 2 is taken to be 10%. The ellipsoidal convex model is used to solve the analysis interval response, and the results are shown in Figure 9.  
The uncertainty of rotor disk mass leads to the deviation of system response in the whole speed range as show in Figure 9. More specifically, compared with the deterministic peak, the first peak of the uncertainty response shifts to the left and the second peak shifts to the right. That means the double rotors are very sensitive to the disk mass, which can be used as a key factor in the design and maintenance of such engineering systems. The results of the CCM also show that the upper and lower boundary gauges completely encapsulate the deterministic response values. Figures 7-9 show that the accuracy of CCM results is related to the sensitivity of critical speed to parameters. In the whole response range, the interval response results calculated by CCM are stable. Compared with the deterministic response results, the actual operating conditions of the rotor system can be well reflected.