# The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance
## 5 Non-Linear Analysis of Stability  
The main aim of this section is to investigate the stability analysis of the considered system using the non-linear stability approach of Routh-Hurwitz. The examined system is exposed to various forces and moments, such as the excited harmonic forces \(F_{1},F_{3}\) and the moment \(M_{\phi}\). It is found that the natural frequencies \(\overline{\omega_{j}}\left(j=1,2,3\right)\), damping coefficients \(c_{j}\), and the parameters of detuning perform a crucial destabilizing role in the criteria's stability. To plot the stability diagrams of the system of Equation (46), the previous values of the different parameters are used. The modified amplitudes \(h_{j}\ (j=1,2,3)\) are shown with the detuning parameter \(\sigma_{1}\) in Figures 11-14 for different values of \(\overline{\omega_{j}}\) when \(\sigma_{2}=\sigma_{3}=0\). The outlined curves of these figures contain the possible fixed points. Resonance curves are plotted in Figures 11-14 according to the previous data of parameters. Each of the resonance curves has different stable and unstable regions with the various values of the natural frequencies \(\overline{\omega}_{1}\) and \(\overline{\omega}_{2}\) (see Figures 11-14). More precisely, the inspection of parts of Figure 11 shows that the instability and stability ranges are \(-0.1<\sigma_{1}<0.009\) and \(0.009\leq\sigma_{1}<0.1\), respectively, as indicated in Figure 11 in blue. On the other hand, when \(\overline{\omega}_{1}=3.43\), the red curves represent the instability and stability ranges of \(-0.1<\sigma_{1}<0.012\) and \(0.012\leq\sigma_{1}<0.1\), respectively, while the black curves represent the areas of instability and stability when \(\overline{\omega}_{1}=3.02\), which are \(-0.1<\sigma_{1}<0.014\) and \(0.014\leq\sigma_{1}<0.1\), respectively. The included curves of each part have different fixed points associated with the different values of \(\overline{\omega}_{1}\), as indicated in Figure 11. Moreover, the instability regions decrease with an increase in the value of \(\overline{\omega}_{1}\), due to the impact of the natural frequency \(\overline{\omega}_{1}\). Figure 12 illustrates the different ranges of stability criterion, where the instability and stability areas, indicated in blue, are determined according to the ranges \(-0.1<\sigma_{1}<0.009\) and \(0.009\leq\sigma_{1}<0.1\), respectively. At \(\overline{\omega}_{2}=5.72\), the red curves show the instability and stability areas with ranges \(-0.1<\sigma_{1}<0.021\) and \(0.021\leq\sigma_{1}<0.1\), respectively, while the black curves show the instability and stability areas at \(\overline{\omega}_{2}=6.68\), which are \(-0.1<\sigma_{1}<0.033\) and \(0.033\leq\sigma_{1}<0.1\), respectively. According to the calculations of the graphed curves in Figures 13 and 14, we can state that three different peak fixed points are produced when \(\overline{\omega}_{2}\) has different values for the resonance curves of \(h_{2}\) and \(h_{3}\), which are varied with \(\sigma_{1}\). The instability and the stability areas become \(-0.1<\sigma_{1}<0.009\) and \(0.009\leq\sigma_{1}<0.1\), respectively, denoted in blue (see Figures 13 and 14), while at \(\overline{\omega}_{2}=5.72\), the red curves indicate the instability and stability areas \(-0.1<\sigma_{1}<0.021\) and \(0.021\leq\sigma_{1}<0.1\), respectively. On the other hand, the black curves represent the areas of instability and stability when \(\overline{\omega}_{2}=6.68\), which are \(-0.1<\sigma_{1}<0.033\) and \(0.033\leq\sigma_{1}<0.1\), respectively. Figure 12: The variation of the frequency response \(h_{1}\) with \(\sigma_{1}\) at \(\sigma_{2}=0\), \(\sigma_{3}=0\) when \(\overline{\omega}_{2}=(4.43)\), \(5.72\),\(6.68)\).