## 2 Computation of the Inertial Characteristics of the AD Elements for the FBV Movements
The geometric and mass moments of inertia for the tulip \(J_{Y_{2}T},J_{Z_{2}T}\), \(I_{Y_{2}T},I_{Z_{2}T}\) reduced to the \(Y_{2},Z_{2}\) axes of the midshaft are given by the following equations:  
\[J_{Y_{2}T}=J_{Y_{2}TB}+J_{Y_{2}TA}, \tag{1}\]  
\[J_{Z_{2}T}=J_{Z_{2}TB}+J_{Z_{2}TA}, \tag{2}\]  
\[I_{Y_{2}T}=J_{Y_{2}TB}\rho L_{TB}+J_{Y_{2}TA}\rho L_{TA}, \tag{3}\]  
\[I_{Z_{2}T}=J_{Z_{2}TB}\rho L_{TB}+J_{Z_{2}TA}\rho L_{TA}, \tag{4}\]  
\[J_{Y_{2}TB}=0.5(J_{1T}+J_{2T})\Big{[}1+\sin^{2}\beta_{1}+\chi_{nT}\cos(2\varphi _{1})\cos^{2}\beta_{1}\Big{]}+\frac{S_{T}L_{TB}^{2}}{12}\cos^{2}\beta_{1}+S_{T }(d_{CT})^{2}, \tag{5}\]  
\[\chi_{nT}=\frac{J_{1T}-J_{2T}}{J_{1T}+J_{2T}}, \tag{6}\]  
\[J_{Y_{2}TA}=\frac{\pi d_{TA}^{4}}{64}\Big{(}1+\sin^{2}\beta_{1}\Big{)}+\frac{ \pi d_{TA}^{2}}{4}\frac{L_{TA}^{2}}{12}\cos^{2}\beta_{1}+\frac{\pi d_{TA}^{2}} {4}(L_{TB}+0.5L_{TA})^{2}, \tag{7}\]  
\[J_{Z_{2}TB}=0.5(J_{1T}+J_{2T})[1-\chi_{nT}\cos(2\varphi_{1})]+\frac{S_{T}L_{ TB}^{2}}{12}+S_{T}(d_{CT})^{2}, \tag{8}\]  
Figure 5:0  
where \(J_{1T},J_{2T}\) are the principal geometric moments of inertia for the tulip's bell, \(J_{Y_{2}TB},J_{Y_{2}TA}\), \(J_{Z_{2}TB},J_{Z_{2}TA}\) are the geometric moments of inertia for the tulip's elements reduced to the axes \(Y_{2},Z_{2}\), \(\rho\) is the mass density of the AD elements, \(d_{CT}\) is the distance between the center mass of the tulip and the tripod's center mass, \(S_{T}\) is the surface of the cross-section for the tulip's bell, \(\chi_{nT}\) is the nonuniformity of the geometric moments of inertia for the tulip (see Figure 3), \(L_{TB}\) is the length of the tulip's bell, \(L_{TA}\) is the length of the tulip ax, and \(d_{TA}\) is the diameter of the tulip ax. The geometric and mass moments of inertia for the bowl \(J_{Y_{2}B},J_{Z_{2}B}\), \(I_{Y_{2}B},I_{Z_{2}B}\) reduced to the \(Y_{2},Z_{2}\) axes of the midshaft are given by the following equations:  
Figure 5:1  
Figure 5:2  
Figure 5:3  
Figure 5:4Figure 5:5  
Figure 5:6  
Figure 5:7  
Figure 5:8  
Figure 5:9  
where \(\mathrm{J}_{1\mathrm{B}},\mathrm{J}_{2\mathrm{B}}\) are the principal geometric moments of inertia for the bowl's bell, \(\mathrm{J}_{Y_{2}\mathrm{BB}},\mathrm{J}_{Y_{2}\mathrm{BA}}\), \(\mathrm{J}_{Z_{2}\mathrm{BB}},\mathrm{J}_{Z_{2}\mathrm{BA}}\) are the geometric moments of inertia for the bowl's elements reduced to the axes \(\mathrm{Y}_{2},Z_{2}\), \(\mathrm{d}_{\mathrm{CB}}\) is the distance between the center mass of the bowl and the inner race's center mass, \(S_{\mathrm{B}}\) is the surface of the cross-section for the bowl's bell, \(\chi_{\mathrm{nB}}\) is the nonuniformity of the geometric moments of inertia for the bowl (see Figure 4), \(\mathrm{L}_{\mathrm{BB}}\) is the length of the bowl's bell, \(\mathrm{L}_{\mathrm{BA}}\) is the length of the bowl ax, and \(\mathrm{d}_{\mathrm{BA}}\) is the diameter of the bowl ax. By analyzing Equations (1)-(18), it can be remarked that the geometric and mass moments of inertia of the tulip and the bowl, reduced to the referential of the midshaft, are functions of the angles \(\varphi_{1},\varphi_{3},\Phi_{1},\Phi_{3},\beta_{1},\beta_{2}\), meaning that they depend on the dynamic behavior of the tulip and the bowl, with this aspect being a novelty in designing dynamic models for the AD elements' movements.