## 7 Results and Discussion
Here, m = 8 \(\times\) 10\({}^{-5}\) octaves/min. Figure 20: Stationary dynamic instability surface frontier for the bowl. (**a**) \(\mu\) \(>\) 0. (**b**) \(\mu\) \(<\) 0. It may be noted that while the excitation frequency was varied logarithmically through the resonant regime, the load parameter \(\mu\) was kept constant. A vast number of numerical tests was performed to arrive at a reasonable value for the sweep rate of the excitation frequency m = 0.00008 octaves/min. When analyzing the nonstationary parametric amplitude spectrum presented in Figure 20 for \(\zeta\) = (1.6-31.8) \(\times\) 10\({}^{-3}\) and \(\chi_{\text{nT}}\) = 0.25, the manifestation of beating effects as well as a significant decrease in the amplitude peaks with the increase in the damping ratio value in the range mentioned above can be seen. The maximum peak of the amplitude was obtained when the excitation frequency was in the close vicinity of PPR 1038 Hz. Each time the damping ratio increased at a rate of 0.01, the maximum peak amplitude decreased by more than five times. Figure 22 illustrates the nonstationary velocity amplitude spectrum \(\frac{\text{dW}}{\text{dt}}\) of the forced bending vibrating movement (FBVM) for the tulip during transition through the PPR region. When analyzing the graphs in Figure 22, the moment of passing in close vicinity of the PPR by a sudden increase in the peak velocity amplitude can be seen, while the manifestation of beatings was like the nonstationary amplitude spectrum (see Figure 20). In the range of 0.0016-0.0116, the increase in the damping ratio had the effect of increasing the maximum peak of the nonstationary velocity amplitude by approximately 15%, while in the range of 0.0116-0.0318, the increase in the damping ratio had the beneficial effect of decreasing the maximum peak by 72%. It can be concluded that the growth of the damping ratio had a beneficial impact through a reduction of the nonstationary velocity amplitude only in the range of 0.0116-0.0318. Figure 23 illustrates the graphs of the phase space \(\left(\mathrm{W},\frac{\mathrm{d}\mathrm{W}}{\mathrm{d}t}\right)\) for the nonstationary FBVM of the tulip in the PPR region. By analyzing the graphs in Figure 23, a transition to chaotic behavior for the FBVM of the tulip due to the presence of limit cycles or even of strange attractors can be seen, but this last conclusion needs to be certified by detailed analysis using the methods of chaotic movements. One aspect that is evident is that the transition through the PPR region for the tulip was an unstable one. Figure 24 illustrates the nonstationary amplitude spectrum W of the FBVM for the bowl during the transition through the PPR region. As for the tulip, a vast number of numerical tests was performed to arrive at a reasonable value for the sweep rate of the excitation frequency m = 1.4 \(\times\) 10\({}^{-5}\) octaves/min. By analyzing the nonstationary parametric amplitude spectrum presented in Figure 24 for \(\zeta\) = (1.6-31.8) \(\times\) 10\({}^{-3}\) and \(\chi_{\rm nB}\)= 0.10, the manifestation of beating effects as well as a significant decrease in the amplitude peaks with the increase in the damping ratio value  
Figure 24: Nonstationary parametric amplitude spectrum W of the bowl for \(\chi_{\rm nB}\) = 0.10, where \(\mu\) = 0.754 \(\times\) 10\({}^{-4}\). (**a**) \(\zeta\) = 0.0016. (**b**) \(\zeta\) = 0.0036. (**c**) \(\zeta\) = 0.0076. (**d**) \(\zeta\) = 0.0096. Here, m = 1.4 \(\times\) 10\({}^{-5}\) octaves/min. in the range of (1.6-11.6) \(\times\) 10\({}^{-3}\) can be found, while in the range of (11.6-31.8) \(\times\) 10\({}^{-3}\), the decrease was more significant. The maximum peak of the amplitude was obtained when the excitation frequency was in close vicinity of the PPR for the bowl (6306 Hz). Each time the damping ratio increased at a rate of 0.01, the maximum peak amplitude decreased by more than 10 times (5 times for the tulip), because the bowl had a more considerable rigidity than the tulip.