## 4 Sensitivity Investigations and Numerical Simulations
### System Dynamics with Control (\(\beta_{j}\
Comparing Figure 5 with Figures 7 and 11, one can notice from Figure 5 that the uncontrolled Jeffcott system is stable along the eccentricity span (i.e., as long as \(0<E\leq 0.075\)) regardless of the shaft angular speed. In addition, the system can perform symmetric lateral oscillation in both \(X\) and \(Y\) directions. On the other hand, Figures 7 and 11 demonstrate that the integration of the radial PD-control algorithm and the 8-pole actuator has completely changed the bifurcation behavior of the studied system, where the controlled Jeffcott system may perform quasi-periodic and chaotic motion beside the periodic motion. The effect of the normalized cubic non-linear stiffness coefficient (\(\lambda\)) on oscillation amplitudes of the controlled Jeffcott system has been illustrated in Figure 12, where Figure 12a,b show the system response curves at \(\lambda=0.3\), while Figure 12c,d illustrate the system response curves when \(\lambda=-0.3\). It is clear from Figure 12a,b that the response curves have been bent to the right at the positive values of \(\lambda\) (i.e., \(\lambda=0.3\)), which leads to the hard-spring characteristic. On the other hand, Figure 12c,d demonstrate that the response curves have been bent to the left at the negative values of \(\lambda\) (i.e., \(\lambda=-0.3\)), which leads to the soft-spring characteristic. Figure 11: \(E\)—response curve of the controlled Jeffcott rotor (i.e., \(\delta_{1}=0.83\) and \(\delta_{2}=0.002\)) and the corresponding bifurcation diagrams at \(\sigma=-0.05\): (**a**,**b**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(E\) when \(\sigma=0\), and (**c**,**d**) Poincaré-map of the instantaneous oscillations \(u(t)\) and \(v(t)\) versus \(E\) when \(\sigma=0\). Figure 6:0 _Cont._  
The effect of the proportional gain \(\delta_{1}\) on the oscillatory motion of the Jeffcott system has been illustrated in Figure 13, where Figure 13a,b show the system oscillation amplitudes versus \(\sigma\) at \(\delta_{1}=0.85\), while Figure 13c,d illustrate the vibration amplitudes at \(\delta_{1}=0.8\). It is clear from the figure that the increase of \(\delta_{1}\) from 0.83 to \(\delta_{1}=0.85\) (See Figure 6c,d) has shifted the system response curves to the right as shown in Figure 13a,b, where the quasi-periodic motion that is reported in Figure 6c,d at \(\sigma=0\), became a mono-stable periodic solution. However, Figure 13a,b confirm that the Jeffcott system may respond with an aperiodic motion (i.e., for example, at \(\Omega=1+\sigma,\ \sigma=0.1\)). On the other hand, Figure 13c,d demonstrate the shift of the response curve to the left when \(\delta_{1}\) has been decreased from 0.83 (See Figure 6c,d) to 0.8. Accordingly, the parameter \(\delta_{1}\) can be used as a control key to shifting the system response curves either to the right (i.e., via increasing \(\delta_{1}\)) or to the left (i.e., via decreasing \(\delta_{1}\)) in order to avoid the complex oscillatory behaviors of the controlled Jeffcott system at the resonance condition (i.e., when \(\Omega\equiv 1\)). However, Figure 13 in general shows that the Jeffcott system response curve at \(\delta_{1}=0.8\) is better than the response curves when \(\delta_{1}=0.83\) and \(\delta_{1}=0.85\). Figure 6:1 _Cont._  
To validate the accuracy of the Figure 13, the system equations of motion (i.e., Equations (20) and (21)) have been solved numerically using ODE45 according to Figure 13 (i.e., when \(E=0.03,\ \delta_{2}=0.002\)) at \(\sigma=0.1\) and \(\delta_{1}=0.85\), \(0.8\) as shown in Figure 6:2 The figure has been obtained via simulating Equations (20) and (21) by letting \(\delta_{1}=0.85\) along the time interval \(0\leq t\leq 700\), where at instant \(t=700\), the control parameter \(\delta_{1}\) has been decreased to \(0.8\) on the interval \(700<t\leq 1000\).