## 4 Results and Discussion  
Figure 7 illustrates the variation of the AD's GT and GB natural frequencies based on Equation (6). Comparing the obtained results with the published ones in [5] (pp. 119-139), the conclusion is clear that the DMADFTV-computed circular eigenfrequencies agree with the data in the literature. The potential sources of discrepancies between the theoretical predictions and experimental measurements mentioned before might be manufacturing tolerances, material inconsistencies, or assembly imperfections. These practical technical aspects could explain the differences between the simulated results and the experimental data published in [5] (pp. 119-139), which are less than 3% for the eigenfrequency of the GT and less than 4% for the GB. The free natural frequencies of the GT and the GB that are computed based on Equation (6) are not sensitive to the logarithmic decrement \(\xi\), that participates in Equations (14) and (15) in the RHS as an attenuation term of the excitations. The nonuniformities of the AIMM and AIGM for the GB and the GT are expressed in the terms \(\chi_{\text{mB}}\), \(\chi_{\text{nT}}\) (see Equations (A1)-(A3) in Appendix A) and \(\chi_{\text{GBn}}\), \(\chi_{\text{GTn}}\) (see Equations (A13) and (A14) in Appendix B), and the angles \(\beta_{1},\beta_{2}\) influence the free eigenfrequencies computed with Equation (6). Based on Equation (23), MATLAB software was developed to calculate the dynamic instability boundary frontiers (DIBF) for the stationary GT's FTV and GB's FTV; the graphs are presented in Figures 8 and 9. Analyzing the graphs presented in Figures 8 and 9, it can be remarked that the folded surface of the DIBF manifests symmetry concerning the plane (excitation frequency, damping ratio), having the same manifestation as for the global tulip and the global bowl. Because the damping ratio \(\xi\), is an isotropic physical property of the material, analyzing the needed limit values to obtain DIBF for the global tulip and the global bowl, the condition is that the steel composition must satisfy the condition \(\xi\geq 0.0044\) if it is used for the fabrication of  
Figure 8: Stationary DIBF for the global tulip \(\beta_{1}=15^{\circ}\). (**a**) \(\mu_{1}>0\) (**b**) \(\mu_{1}<0\). Figure 7: Natural frequency in torsion of (**a**) global tulip, (**b**) global bowl. the driveline of the same steel. It can also be remarked that the folded surfaces illustrated in Figures 8 and 9 close, between their two branches, the instability volume in the space \((\eta_{1},\xi,\mu_{1})\) for the global tulip and the space \((\eta_{3},\xi,\mu_{3})\) for the global bowl. Figures 10-12 illustrate the graphs for investigating the dynamic instability of NFTV for the global tulip in transition through APPR \(\eta_{1}\simeq 6131.6\) Hz, considering the worst case for the global tulip \(\beta_{1}=15^{\circ}\), \(\mu_{1}=0.1001\), \(\overline{\Omega}_{1}=19263\) rad/s, \(\chi_{\rm GTM}=0.0073\), \(\Gamma_{1}=1.9298\cdot 10^{4}\) s\({}^{-2}\)rad\({}^{-2}\), \(\mathrm{n}=0.000008\) [octaves/min], and the damping ratio \(\xi=0.0016\ldots 0.0043\). MATLAB software was developed based on DSOE (22) for NFTV of the global tulip to calculate \(\Phi_{1}(\eta_{1})\), \(\frac{\Phi_{1}}{\mathrm{dt}}(\eta_{1})\). For the graphs illustrated in Figures 10-12, the excitation frequency has a logarithmic balayage for NFTV of the GT in transition through APPR. In contrast, the characteristics of internal and external excitations \(\mu_{1},\chi_{\text{GTn}},\Gamma_{1}\) are constants. In Figures 10 and 11, it is remarked that the nonstationary SA and SVA \(\Phi_{1}(\eta_{1})\), \(\frac{\Phi\Phi_{1}}{\text{dt}}(\eta_{1})\) for NFTV of the GT manifest the \"beatings\" and a rapid decrease of the extreme values with the damping's increases in the range \((1.6\dots\dots\,4.3)\times 10^{-3}\). The greatest value of the nonstationary SA and SVA for NFTV of the GT is reached when \(\eta_{1}=6136\) Hz. Figure 12 shows that the \"look\" is similar to the strange attractor's representations.