## 3 Numerical Results and Discussion
### Without Sprung Eccentricity  
With the method described in Section 2, the finite element model of the rotor system is established with 189 beam elements to simulate the shaft and 5 mass elements to simulate the blades of each disk. Then, the steady-state unbalance response of the rotor system \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Compressor I** & **Compressor II** & **Compressor III** & **Compressor IV** & **Turbine** \\ \hline Mass (kg) & 0.17 & 0.36 & 0.28 & 1.269 & 2.12 \\ Polar moment of inertia & 4.35 & 1.15 & 0.79 & 9.03 & 10.00 \\ \hline \end{tabular}
\end{table} 
Table 3: Inertia properties of blades for each disk. Figure 3: Mode shapes of the first two critical speeds: (**a**) mode shape for critical speed 1588 rad/s; (**b**) mode shape for critical speed 3276 rad/s. Figure 2: Campbell diagram of the rotor system. is obtained when the rotor speed is in the range of 1000-4500 rad/s and the step size is 10 rad/s. The first step uses zero initial conditions. The remaining steps use the results of the previous step as the initial conditions. Figures 4-7 present the bifurcation diagram, the waterfall diagram and the frequency response of the rotor system without static sprung eccentricity under different unbalanced configurations. Here, \(e_{i},i=1,2,3,4,5\) represent the eccentric distance of compressors I, II, III, IV and the turbine disk respectively. Thus, the unbalanced force on the rotor would be the mass of the disk \(\times\)\(e_{i}\)\(\times\) square of the rotor speed. Also, the unbalance is applied to only one disk each time. For each disk, 10 different unbalanced configurations were considered with \(e_{i}=0.1,0.2,0.3\cdots 0.9,1.0\) mm. However, it was found that the results were quite similar if \(e_{1}\), \(e_{2}\) and \(e_{3}\) had adopted the same value, which is the same for \(e_{4}\) and \(e_{5}\). Therefore, only the results of \(e_{2}\) and \(e_{4}\) are presented and analyzed in this work. The following conclusions can be reached from Figures 4-7:  
1. From Figure 4, the bifurcation diagrams of support I, it can be seen that although the unbalanced configurations are different, the rotor executes periodic motion with period one in 1000-4500 rad/s. 2. Figure 4d shows an obvious break point at 3250 rad/s while the dots in Figure 4a-c form an almost continuous curve. The break point implies the occurrence of a jump phenomenon caused by the nonlinear interaction between the rotor and the SFD. By examining Figures 5d and 6f, the waterfall diagram and the frequency response curve corresponding to Figure 4d, it is confirmed that the jump phenomenon does occur at 3250 rad/s for \(e_{4}=1.0\) mm. 3. Increasing \(e_{i}\) would lead to a larger response, although the relationship between \(e_{i}\) and the response is nonlinear, as shown in Figure 7. Also, by comparing Figure 5a,b, it can be found that as \(e_{i}\) increases from 0.1 to 1 mm, the response of support I increases by approximately five times. The same phenomenon can also be observed in Figure 5c,d or Figure 6a,b. 4. Due to the nonlinearity of the SFD, the rotational speed at which the peak response occurs can be different from the critical speeds identified from the Campbell diagram, as shown in Figures 2 and 6. Two critical speeds, 1587 rad/s and 3274 rad/s, can be identified from the Campbell diagram shown in Figure 2. However, in Figure 6a, it can be seen that the peak response of the rotor system occurs at 1610-1680 rad/s, which is very close to 1587 rad/s. Also, Figure 6a shows that at approximately 1600 rad/s, the vibration amplitude of the turbine disk is the largest while that of support I is the smallest, which means that the operating deflection shape of the rotor system at about 1600 rad/s coincides with the mode shape shown in Figure 3a. This is also the situation for 3000-3500 rad/s, as shown in Figures 3b and 6a. 5. The peak response speed range is related to the amount of unbalance, as shown in Figures 6 and 7. Figure 7 shows the horizontal response of support I under different unbalanced configurations.