## 4 Results and Discussion
### Frequency Response Curves (FRC)  
The response amplitude is influenced by both the detuning parameter and the excitation amplitude \(f\). Equations (35) and (42) are solved numerically and graphically to derive the solution for the amplitudes of all the main systems and the PPF controller at similar times. Figures 4-14 are structured in two columns. The left column illustrates the capacity of the main system (\(a_{1}\)) on the vertical axis against the tuning parameter (\(\sigma_{1}\)) on the horizontal axis. The right column similarly depicts the capacity of the control unit (\(a_{2}\)) on the vertical axis, also plotted against the tuning parameter (\(\sigma_{1}\)) on the horizontal axis. The graphical solution is obtained concerning the detuning parameter, offering a visual depiction of how the amplitudes of these systems fluctuate as the detuning parameter \(\sigma_{1},\sigma_{2}\) changes two beaks are used to indicate this. Figure 3: The amplitude of the main system before and after adding a positive position feedback (PPF) controller. Figure 4: Frequency resonance curves of (**a**) main system and (**b**) PPF controller. Figure 5: The effectiveness of the external excitation force \(f\) on (**a**) the main system (**b**) the PPF controller. Figure 6: The effectiveness of control signal gain \(\lambda_{1}\) on (**a**) the main system (**b**) the PPF controller. Figure 7: The effectiveness of feedback signal \(\lambda_{2}\) on (**a**) the main system (**b**) the PPF controller. Figure 8: The effectiveness of damping parameter \(\alpha_{1}\) on (**a**) the main system (**b**) the PPF controller. Figure 10: The effectiveness of nonlinear parameter \(\beta\) on (**a**) the main system (**b**) the PPF controller. Figure 9: The effectiveness of damping parameter \(\alpha_{2}\) on (**a**) the main system (**b**) the PPF controller. The frequency response curves depicted in Figure 4 exhibit the characteristics of the main system and the PPF controller. The frequency response of the controlled system is represented by a stable solution drawn as solid lines. Figure 4 depicts the frequency response curves of the main system, whereas Figure 4 shows the frequency response curves of the controller. These figures specify a visual comparison of how the systems perform at various frequencies and highlight the stability regions represented by the solid lines. We can conclude from this graph that the minimal value of the main system amplitude occurs at \(\sigma_{1}=0\). This has implied that the PPF controller is intelligent in reducing main system vibrations in the super-harmonic resonance. The amplitudes of the main system and the PPF controller increase when the value of a harmonic excitation force grows, the jump phenomena happen, and the minimum value of the main system amplitude occurs at \(\sigma_{1}=0\) as illustrated in Figure 5,b. The control signal gain \(\lambda_{1}\) is represented in Figure 6,b. It seems like a description of a series of results and observations related to the effects of the PPF controller and its  
Figure 14: Effect between the different types of controllers (PPF-NVC-PD). Figure 12: Effect of varying the natural frequency (\(\omega_{1}=\omega_{2}\)) for \(\sigma_{2}=0\) on the frequency response curves of (**a**) the main system (**b**) the PPF controller. Figure 4:0 Effect of varying \(\sigma_{2}\) on the frequency response curves of (**a**) the main system (**b**) the PPF controller. parameters on the system behavior. The PPF controller's amplitude shift to the right implies an expansion in the control signal's bandwidth, as shown in Figure 6a. Figure 6b demonstrates that the PPF controller's amplitude decreases monotonically, aligning with the intended purpose of the control signal gain. Figure 7a,b depict the impact of the feedback signal \(\lambda_{2}\); in Figure 7a, augmenting the feedback signal results in the vibration reduction frequency shift towards higher values for the controller. In Figure 7b, higher rates of the feedback signal have led to increased controller amplitudes.