## 7 Results and Discussion
It can also be remarked that the increase in the damping ratio in the range of 0.0016-0.0318 increased the minimum normalized stationary amplitude from 0.13 to 0.42 for one branch, and for the second branch, the increase in the minimum normalized stationary amplitude was from 0.32 to 0.425. This can be explained by the existence of dynamic instability in the region of principal parametric resonance, as mentioned in the literature [28] (pp. 132-160). Figures 13 and 14 present the variation of the stationary phase angle for the FBV in the PPR region for the tulip of an AD. \begin{table}
\begin{tabular}{c c c c} \hline
**ENF (Hz)** & **NSNE (Hz)** & \(\mathbf{\
u_{n1}\) (Hz)} & \(\mathbf{\
u_{n2}\) (Hz)} \\ \hline
683.59 & 686.77 & 683.56 & 1343.3 \\ \hline - & 748.01 & 748.1 & 1470.1 \\ \hline
1406.25 & 1457.8 & 1406.2 & 2763.4 \\ \hline
1464.84 & 1466.6 & 1464.9 & 2878.7 \\ \hline - & 2207.4 & 2207.2 & 4337.3 \\ \hline  & 2235.6 & 2235.5 & 4393 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of the experimental natural frequencies (ENFs) of the driveshaft and numerical simulations using the finite element analysis (FEA) program Ansys of the natural frequencies (NSNFs) in [13] with those obtained for the midshaft based on Equations (57) and (83). When analyzing the stationary parametric phase angle presented in the graphs in Figures 13 and 14 for \(\zeta=0.0016\)-\(0.0318\) and \(\chi_{\rm nT}=0.25\), it can be seen that for the cases, we had a manifestation of a \"soft spring\" with one branch for \(\eta_{1}\leq 2\Omega_{1}\), while for \(\eta_{1}\geq 2\Omega_{1}\), two branches of a \"hard spring\" exist, but with the increase in the damping ratio passing through the principal parametric resonance for \(\eta_{1}>2\Omega_{1}\), and in the region of a \"hard\" spring, the two branches are approaching one against the other. In addition, 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). Through experiments, [PERSON] demonstrated that the nonlinear parametric dynamic behavior of automotive driveshafts is like that of geared systems [4] (p. 117), [32] (pp. 132-160), and therefore, we observed similar pitting phenomena as well as micro-cracks inside the tulip and the bowl of the CVJ tulip-tripod and bowl-inner race joints [6] (pp. 88-94), which was also mentioned in the literature [13]. These aspects will \"conduct\" dynamic behavior through a chaotic dynamic that has, as a practical effect, an accelerating effect on pitting phenomena and cracking, as mentioned by [PERSON] [4]  
Figure 11: Normalized stationary parametric amplitude W of the tulip for \(\chi_{\rm nT}=0.25\), where \(\mu=0.623\times 10^{-4}\). (a) \(\zeta=0.0016\). (b) \(\zeta=0.0116\). Figure 12: Normalized stationary parametric amplitude W of the tulip for \(\chi_{\rm nT}=0.25\), where \(\mu=0.623\times 10^{-4}\). (a) \(\zeta=0.0216\). (b) \(\zeta=0.0318\). (pp. 88-94) or, in the worst case, results in the manifestation of cracks followed by failure (breaking) of the tulip [4] (p. 89). Figures 15 and 16 present the variation of the normalized stationary amplitude for the FBV in the PPR region for the bowl of an AD (around 6306.6 Hz). The fundamental natural frequency in bending for the bowl (3153.3 Hz) was computed with Equation (57). As can be seen, the natural frequency for the bowl was much bigger than the natural frequency of the tulip, and the PPR region for the bowl was in the PPR region for the forced torsional vibrations of the tulip, as mentioned in the literature [29]. When analyzing the normalized stationary parametric amplitude of the bowl presented in the graphs in Figures 15 and 16 for \(\zeta\) = 0.0016-0.0318 and \(\chi_{\text{HB}}\) = 0.10, the same manifestation in the PPR region for the bowl as for the tulip can be seen but in a different range frequency.