## 7 Results and Discussion
Figure 25 illustrates the nonstationary velocity amplitude spectrum \(\frac{\mathrm{d}W}{\mathrm{d}t}\) of the FBVM for the bowl during the transition through the PPR region. When analyzing the graphs in Figure 25, the moment of passing in close vicinity of the PPR was marked by a sudden increase in the peak of velocity amplitude, while the manifestation of beatings was like that of the nonstationary velocity amplitude spectrum for the tulip (see Figure 22). The manifestation for the bowl was different than that for the tulip (see Figure 22) because from the beginning, in the whole range of the damping ratio (0.0016-0.0318), the increase in the damping ratio had the effect of decreasing the maximum peak of the nonstationary velocity amplitude by approximately 5 times in the range of 0.0016-0.0116, while in the range of 0.0216-0.0318, the increase in the damping ratio had the effect of decreasing the maximum peak by more than 8 times. It can be concluded that the growth of the damping ratio had the beneficial impact of reducing the nonstationary velocity amplitude in the whole range (0.0016-0.0318) of the damping ratio. Figure 26 illustrates the graphs of phase space \(\left(\mathrm{W},\frac{\mathrm{dW}}{\mathrm{d}t}\right)\) for the nonstationary FBVM of the bowl during the transition through the PPR region. When analyzing the graphs in Figure 26, the same manifestation for the bowl as for the tulip (see Figure 23) can be seen, that being a transition to chaotic behavior for the FBVM of the bowl due to the presence of limit cycles or even strange attractors. However, this last conclusion needs to be certified by detailed analysis using the methods of chaotic movements. This manifestation is valid only in the damping ratio range of 0.0016-0.0096. One aspect that is evident is how the transition through the PPR region for the bowl was also an unstable one. Unfortunately, there are no published studies which analyze in detail the dynamic behavior of each element of the AD for the FBVM (the tulip, bowl, and midshaft) apart from [4], as all the studies analyzed the global dynamic behavior of the automotive driveshaft. As can be seen from this paragraph, the use of AMA coupled with [PERSON]'s principle allowed the investigation of the stationary motion for the FBV movements of an AD's tulip and bowl in the PPR region by computing the amplitude (see Figures 10, 11, 14 and 15, the phase angle (see Figures 7, 12, 13 and 16), and the dynamic instability frontiers (see Figures 18 and 19) in the PPR region. In the meantime, the use of the AMA coupled with [PERSON]'s principle allowed the investigation of the nonstationary motion for the FBV movements of an AD's tulip and bowl in the transition through a PPR region by computing the amplitude spectrum (see Figures 21 and 24), the velocity amplitude spectrum (see Figures 22 and 25), and the velocity amplitude versus the amplitude in the phase space \(\left(W,\frac{dW}{dt}\right)\) (see Figures 23 and 26). Figures 23 and 26 allowed the investigation of the dynamic instability in the transition through the PPR region. As is noted in Figures 21-26, the transition to the PPR region had an aspect of chaotic manifestation due to the manifestation of beating effects. To check whether this nonstationary dynamic behavior is deterministic chaos or a stochastic process, it would be necessary to use [PERSON]'s exponents method coupled with the Poincare map method. All these aspects represent novelties in the dynamic instability investigation for the AD. Based on the phenomena presented above, the direction for future research involves the study of chaotic motion generated by the FBVM of the AD elements. Another future research direction is to investigate the dynamic instability in the proximity of each possible resonance type for the stationary and nonstationary FBVM of the AD elements, such as subharmonic resonance, superharmonic resonance, and combination and internal resonance.