## 4 Sensitivity Investigations and Numerical Simulations
### System Dynamics with Control (\(\beta_{j}\
eq 0,j=1,2,\ldots,10\))  
According to the above discussions given in Section 4.1, the main objective of this article is to eliminate the non-linear behaviors of the considered Jeffcott-rotor (i.e., eliminate the hard-spring characteristic, jump-phenomenon, sensitivity to the initial conditions) via designing a suitable control system. Within this section, the non-linear interactions between the studied Jeffcott-rotor and the suggested control system (i.e., PD-controller and the 8-pole electro-magnetic actuator) have been explored. Figure 6 shows the controlled system oscillation amplitudes (\(a_{1},\ a_{2}\)) against \(\sigma\) at three values of eccentricity (i.e., \(E=0.01,\ 0.02,\)  
Figure 5: \(E\)—response curves of the uncontrolled Jeffcott rotor at various values of \(\sigma\): (**a**) oscillation amplitude in \(X\)—direction (\(a_{1}\)) versus the eccentricity (\(E\)), and (**b**) oscillation amplitude in \(Y\)—direction (\(a_{2}\)) the eccentricity (\(E\)). and 0.03) when the control parameters \(\delta_{1}=0.83\) and \(\delta_{2}=0.002\). Comparing Figure 5 (i.e., ncontrolled rotor) with Figure 6 (i.e., controlled rotor), one can note that the controlled rotor vibration amplitudes do not exceed the unity (i.e., \(a_{1}<1,\ a_{2}<1\)) regardless of the rotor angular speed (i.e., \(\Omega=1+\sigma,-0.2\leq\sigma\leq 0.2\)) and the shaft eccentricity (i.e., \(0\leq E\leq 0.03\)), which means that the impossibility of rub-impact occurrences between the Jeffcott-rotor and the magnetic actuator. However, the controlled Jeffcott system exhibits bifurcation characteristics that completely differ from the uncontrolled system. It is clear from Figure 6 that the controlled Jeffcott system has a bi-stable solution in the vicinity of \(\sigma=0\), which is not the case with the uncontrolled Jeffcott system. In addition, the controlled system may lose its periodic motion at the large eccentricity as shown in Figure 6e,f when \(E=0.03\) and \(-0.01143<\sigma<0.007408\). Moreover, Figure 6 demonstrates that the hard-spring behavior of the uncontrolled Jeffcott system that is reported in Figure 4 has been turned into a soft-spring behavior due to non-linear interaction between the rotor and the electro-magnetic actuator. Figure 6: _Cont._  
The system eccentricity response curve has been obtained as shown in Figure 7a,b when \(\sigma=0,\ \delta_{1}=0.83\), and \(\delta_{2}=0.002\) to explore the dynamical behaviors of the controlled Jeffcott system at the perfect resonance (i.e., \(\Omega=1+\sigma,\ \sigma=0\)) for a wide range of the eccentricity \(E\) (i.e., \(0<E\leq 0.075\)). It is clear from Figure 7a,b that the system responds with a mono-stable solution as long as \(0<E\leq 0.006\), but the rotor system may have a bi-stable periodic solution if the eccentricity magnitude has been increased such that \(0.006<E\leq 0.025\). In addition, the figures confirm that the controlled Jeffcott system may lose its periodic oscillation if the shaft eccentricity has been increased beyond 0.025. To investigate the nature of the unstable periodic solution that is reported in Figure 7a,b when \(E>0.025\), the Poincare-map for the controlled Jeffcott system has been constructed via solving Equations (20) and (21) numerically (using ODE45) utilizing \(\sigma\) as a bifurcation parameter via replacing \(\Omega=1+\sigma\) as shown in Figure 7c,d. It is clear from Figure 7c that the controlled Jeffcott system oscillates with a periodic motion as long as \(E\leq 0.025\). However, increasing the eccentricity beyond the critical value \(E=0.025\) destabilizes the stable periodic motion, where the rotor system responds with bounded aperiodic oscillations in the eccentricity span \(0.025<E\leq 0.075\).