## 7 Results and Discussion
The manifestation of a \"soft spring\" with one branch for \(\eta_{3}\leq 2\Omega_{3}\) can be noted, indicating the presence of interaction between the principal parametric resonance and the primary resonance of the bowl, while for \(\eta_{3}\geq 2\Omega_{3}\), two branches of a \"hard spring\" existed, which indicates the manifestation of interaction with the combination resonance or with the internal resonance for the bowl [28] (pp. 132-160). This aspect agrees with the fact that the two branches of the bowl are not identical, and the same behavior is observed for the two branches of the bowl. Figure 16: Normalized stationary parametric amplitude W of the bowl for \(\chi_{\text{HB}}\) = 0.10, where \(\mu\) = 0.754 \(\times\) 10\({}^{-4}\), (**a**) \(\zeta\) = 0.0216 (**b**) \(\zeta\) = 0.0318. Figure 15: Normalized stationary parametric amplitude W of the bowl for \(\chi_{\text{HB}}\) = 0.10, where \(\mu\) = 0.754 \(\times\) 10\({}^{-4}\). (**a**) \(\zeta\) = 0.00016 (**b**) \(\zeta\) = 0.00016. When analyzing the stationary parametric phase angle for the bowl presented in the graphs in Figures 17 and 18 for \(\zeta=0.0016\)-\(0.0318\) and \(\chi_{\rm HB}=0.10\), the same kind of manifestation as that for the tulip, but in a different range frequency, can be seen, and for the cases, we had a manifestation of a \"soft spring\" with one branch for \(\eta_{3}\leq 2\Omega_{3}\), while for \(\eta_{3}\geq 2\Omega_{3}\), two branches of a \"hard spring\" existed, but with the increase in the damping ratio, the passing through principal parametric resonance was found for \(\eta_{3}>2\Omega_{3}\), and in the region of the \"hard\" spring, the two branches were approaching one against the other. For the bowl, this aspect of the two branches closing one against the other was more accentuated. Additionally, the manifestation of increasing the value of the minimum of the second branch was sensitive to the increase in the value of the damping ratio in the range mentioned above (0.0016-0.0318). Figure 19a,b illustrates the stationary dynamic instability region of the tulip in the space \((\eta_{1},\xi,\mu)\) using MATLAB software and Equation (81). When analyzing Figure 19a,b, it can be noticed that the two folded surfaces obtained were symmetrical concerning the plan given by \((\eta_{1},\xi)\) for the excitation frequency and the damping ratio. In contrast, the excitation coefficient \(\mu\) could be positive or negative. The folded surface of the dynamic instability \"kept\" inside the two branches the region where it manifested the stationary instability. It can also be seen that increasing the damping ratio had a stabilizing effect on the dynamics of the tulip, as expected. Figure 20a,b illustrates the stationary dynamic instability region of the bowl in the space \((\eta_{3},\xi,\mu)\) using MATLAB software and Equation (81). By analyzing Figure 20a,b, it can be noticed that the two folded surfaces obtained were symmetrical concerning the plan given by \((\eta_{3},\xi)\) for the excitation frequency and the damping ratio, while the excitation coefficient \(\mu\) could be positive or negative. The folded surface of the dynamic instability \"kept\" inside the two branches the region where it manifested the stationary instability for the bowl. It can also be seen that increasing the damping ratio had a stabilizing effect on the dynamic of the bowl, as expected. The only difference for these phenomena for the bowl was that the manifestation was in another range frequency than that of the tulip: the range frequency given by the natural frequency of the bowl in bending. Figure 21 illustrates the nonstationary amplitude spectrum W of the FBVM (forced bending vibrating movement) for the tulip during transition through the PPR region. Figure 21: Nonstationary parametric amplitude spectrum W of the tulip for \(\chi_{\rm nT}\) = 0.25, where \(\mu\) = 0.623 \(\times\) 10\({}^{-4}\). (**a**) \(\zeta\) = 0.0016. (**b**) \(\zeta\) = 0.0116. (**c**) \(\zeta\) = 0.0216. (**d**) \(\zeta\) = 0.0318.