## 4 Bifurcation Analysis and Control Performance
Based on the above discussion, it is clear from Figure 3 that the uncontrolled system may be subjected to rub-impact force between the rotor and pole housing if the eccentricity magnitude became greater than \(0.013\). Therefore, the main goal of this work is to mitigate the system lateral vibration close to zero even if the considered system has a large rotor eccentricity. Based on the introduced control strategy, Equations (21) and (22) represent the equations of motion of the controlled rotor system, while Equations (23)-(26) are the equations of motion of the proposed controller (i.e., LIPPF-controller), where the LIPPF-controller is a combination of both the conventional PPF-controller (i.e., Equations (23) and (24)) and the linear integral resonant controller (i.e., Equations (25) and (26)). Therefore, we can explore the performance of the PPF-controller only by mitigating the non-linear oscillation of the rotor system when setting \(\eta_{2}=\eta_{4}=\eta_{7}=\eta_{8}=0\). Figure 4 shows the oscillation amplitudes of both the rotor system (i.e., \(a_{1}\) and \(a_{2}\)) and the connected PPF-controller (i.e., \(a_{3}\) and \(a_{4}\))  
Figure 3: Vibration amplitudes of the rotor system without the LIPPF-controller using \(\sigma\) as a bifurcation parameter when \(f=0.001,\ 0.004,\ 0.007,\ 0.01\), and \(0.013\): (a) vibration amplitude in \(X_{1}\) direction and (b) vibration amplitude in \(X_{2}\) direction. along the \(\sigma\)-axis at different levels of the rotor eccentricity when \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\) and \(\eta_{2}=\eta_{4}=\eta_{7}=\eta_{8}=0\). Comparing Figures 3 and 4, it is clear that the coupling of the conventional PPF-controller to the considered system eliminated the rotor vibrations in the vicinity of \(\sigma=0\). However, the vibration amplitudes of the rotor system were amplified on both the right and left sides of \(\sigma=0\), which is one of the drawbacks of the conventional PPF-controller. To eliminate this undesired phenomenon of the PPF-controller, the LIPPF-controller was introduced as a new control strategy that possesses the advantages of both PPF and linear integral resonant controllers. Figure 5 shows the oscillation amplitudes of both the  
Figure 4: Vibration amplitudes of the rotor system and PPF-controller using \(\sigma\) as a bifurcation parameter when \(f=0.005,\ 0.01,\ 0.015,\ 0.02\), and \(0.025\): (**a**) vibration amplitude in \(X_{1}\) direction, (**b**) vibration amplitude in \(X_{2}\) direction, and (**c**,**d**) vibration amplitudes of the PPF-controller. rotor system (i.e., \(a_{1}\) and \(a_{2}\)) and the connected LIPPF-controller (i.e., \(a_{3}\) and \(a_{4}\)) when \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\) and \(\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\). It is clear from Figure 5 that the proposed LIPPF-controller eliminated the rotor vibrations in the vicinity of \(\sigma=0\) and suppressed the vibration amplitudes to a very small value on both sides of \(\sigma=0\). The vibration suppression mechanism of the introduced LIPPF-controller can be explained simply based on the working principle of both the conventional PPF-controller and the IRC-controller. It is well known that the PPF-controller absorbs the non-linear vibrations of the connected oscillatory system with high efficiency via establishing an energy bridge when the controller's natural frequency has the same value as the excitation frequency of the targeted system [25; 26; 27; 28; 29]. On the other hand, the IRC-controller can mitigate the non-linear oscillation via increasing the linear damping of the targeted system [30; 31; 32; 33; 34; 35; 36; 37; 38]. Figure 5: Vibration amplitudes of the rotor system and LIPPF-controller using \(\sigma\) as a bifurcation parameter when \(f=0.005,\ 0.01,\ 0.015,\ 0.02,\) and \(0.025\): (**a**) vibration amplitude in \(X_{1}\) direction, (**b**) vibration amplitude in \(X_{2}\) direction, and (**c,d**) vibration amplitudes of the LIPPF-controller.