## 5 Non-Linear Dynamics of the Jeffcott System before and after Control  
The response curves of the considered Jeffcott system before and after control have been compared within this section. According to Figures 6, 13 and 15, the optimal control parameters have been selected such that \(\delta_{1}=0.83\) and \(\delta_{2}=0.02\). In Figure 20, the \(\sigma\)--response curve of the considered Jeffcott system has been compared before and after control. The figure demonstrates that the coupling of the proposed control strategy has forced the considered Jeffcott system to respond as a linear dynamical system with a single periodic attractor, where all non-linear bifurcation characteristics have been eliminated after control. In addition, the figure confirms that the high oscillation amplitudes of the uncontrolled Jeffcott system before control have been suppressed after control along the \(\sigma\)--axis. Numerical simulation of the Jeffcott system lateral oscillations (i.e., \(u(t)\) and \(v(t)\)) before and after control has been illustrated in Figure 21 via solving Equations (20) and (21) at zero initial condition according to Figure 20 when \(\sigma=0\). Figure 21 has been obtained via plotting the instantaneous lateral displacements \(u(t)\) and \(v(t)\) versus \(t\) by letting \(\beta_{j}=0\)\((j=1,2,\ldots,10)\) along the time interval \(0\leq t<600\), and at instant \(t=600\), the controller is turned on with the control gains \(\delta_{1}=0.83\) and \(\delta_{2}=0.02\) on the time interval \(600\leq t\leq 800\). By examining Figure 21, one can confirm the high efficiency of the proposed controller in mitigating the high lateral vibration of the considered Jeffcott system at a very small transient time.  
In Figure 22, the \(E\)--response curve of the Jeffcott system has been compared before and after control at \(\sigma=0\). It is clear from the figure that the strong lateral vibration amplitudes before control have been suppressed after control even at the strong shaft eccentricities. In addition, the non-linear relationship between the steady-state vibration amplitudes (\(a_{1}\), \(a_{2}\)) and the eccentricity (\(E\)) before control has become a straight line with a very small slope after control, as depicted in the figure. Figure 23 shows the temporal oscillations of the considered Jeffcott system before and after control according to Figure 22 when \(E=0.075\). The figure demonstrates the capability of the control algorithm in mitigating the system vibrations at a very small transient time even at large shaft eccentricity.  
Figure 20: \(\sigma\)—response curve of the studied Jeffcott rotor system before control and after optimal control (i.e., \(\delta_{1}=0.83\) and \(\delta_{2}=0.02\)): (a) oscillation amplitude \(a_{1}\) versus \(\sigma\) when \(E=0.03\), and (b) oscillation amplitude \(a_{2}\) versus \(\sigma\) when \(E=0.03\).  
Figure 21: Time-response and the corresponding frequency-spectrum of the Jeffcott-rotor system before and after control according to Figure 20 at \(\sigma=0.0\), \(E=0.03\), \(\lambda=0.05\), \(\delta_{1}=0.83\), \(\delta_{2}=0.02\) when turning the controller on at time \(t=600\): **(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.  
Figure 22: \(E\)—response curve of the studied Jeffcott rotor system before control and after optimal control (i.e., \(\delta_{1}=0.83\) and \(\delta_{2}=0.02\)): **(a)** oscillation amplitude \(a_{1}\) versus \(E\) when \(\sigma=0\), and **(b)** oscillation amplitude \(a_{2}\) versus \(E\) when \(\sigma=0\).