## 4 Sensitivity Investigations and Numerical Simulations
### System Dynamics with Control (\(\beta_{j}\
To validate the accuracy of \(\delta_{2}\)-response curve that is given in Figure 16, the system temporal Equations (20) and (21) have been solved using ODE45 according to Figure 16 (i.e., when \(E=0.03\), \(\sigma=0\), \(\delta_{1}=0.83\)) at \(\delta_{2}=0.001\), \(0.005\), and \(0.04\) as shown in Figures 17 and 18. Figure 17 is obtained via plotting \(u(t)\) and \(v(t)\) versus the normalized  
Figure 6:7 Time-response and the corresponding frequency-spectrum of the controlled Jeffcott-rotor according to Figure 13 when \(\sigma=0.1\), \(E=0.03\), \(\lambda=0.05\), \(\delta_{2}=0.002\) when decreasing online the proportional gain from \(\delta_{1}=0.85\) to \(\delta_{1}=0.8\) at time \(t=700\): **(a,b)** the normalized temporal oscillations \(u(t)\) and the corresponding frequency spectrum in \(X\) direction, **(c,d)** the normalized temporal oscillations \(v(t)\) and the corresponding frequency spectrum in \(Y\) direction. time \(t\) at zero initial conditions by letting \(\delta_{2}=0.001\) along the time interval \(0\leq t\leq 1000\), where at \(t=1000\), the parameter \(\delta_{2}\) has been increased to \(0.005\) on the time interval \(1000<t\leq 1200\), then the parameter \(\delta_{2}\) has been increased again to \(0.04\) on the time interval \(1200<t\leq 1400\). Figure 18 is a repetition of Figure 17 but at the non-zero initial conditions \(u(0)=\dot{u}(0)=0,=v(0)=\dot{v}(0)=0.5\). It is clear from Figures 17 and 18 that the controlled Jeffcott system can perform quasi-periodic oscillation (i.e., unstable periodic solution) at \(\delta_{2}=0.001\) on the time interval \(500\leq t<1000\), but the quasi-periodic oscillation has been turned into a stable periodic solution on interval \(1000\leq t<1200\) when \(\delta_{2}\) is increased to \(0.005\). Moreover, the figures show that the increase of \(\delta_{2}\) to \(0.04\) has suppressed the non-linear oscillations and eliminated the sensitivity of the Jeffcott system to the initial conditions. Based on Figure 6, the increase of the eccentricity \(E\) can destabilize the stable periodic motion of the controlled system. In addition, Figure 15 demonstrates that the increase in the control parameter \(\delta_{2}\), increases the damping coefficient, which ultimately stabilizes the unstable motion of the controlled Jeffcott system. Accordingly, the stability chart of the studied system in \(E\delta_{2}\)--plane has been established as shown in Figure 19 at two different values of the rotor angular speed (i.e., when \(\Omega=1+\sigma,\ \sigma=0,-0.05\)). The figure demonstrates that the increase of the control parameter \(\delta_{2}\) (i.e., \(\delta_{2}>0.006\)) can stabilize the  
Figure 6:8 \(\sigma\)—response curve of the controlled Jeffcott rotor (i.e., \(E=0.03,\ \lambda=0.05\), and \(\delta_{1}=0.83\)) at three different value of the derivative gain \(\delta_{2}\): (**a**,**b**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(\sigma\) when \(\delta_{2}=0.005\), and (**c**,**d**) oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) versus \(\sigma\) when \(\delta_{2}=0.01,\ 0.02\). unstable motion regardless of both the eccentricity magnitude (i.e., \(0<E\leq 0.1\)) and the shaft angular speed. Figure 6:9 \(\delta_{2}\)—response curve of the controlled Jeffcott rotor (i.e., \(E=0.03\), \(\lambda=0.05\), and \(\delta_{1}=0.83\)): (**a**) oscillation amplitude \(a_{1}\) versus \(\delta_{2}\) when \(\sigma=0\), and (**b**) oscillation amplitude \(a_{2}\) versus \(\delta_{2}\) when \(\sigma=0\).