## 3 The Dynamic Model for FBV Movements of an AD's Elements
bowl-inner race joint with the bowl (see Figure 7), with the inertial characteristics given by the relations below:  
\[J_{Y_{2}\rm Tr}=0.5(J_{\rm 1 Tr}+J_{\rm 2 Tr})[1+\chi_{\rm nTr}\cos(2\varphi_{2})],J_{Z_{2}\rm Tr}=0.5(J_{\rm 1 Tr}+J_{\rm 2 Tr})[1-\chi_{\rm nTr}\cos(2\varphi_{2})], \tag{21}\]  
\[J_{Y_{2}\rm Ir}=0.5(J_{\rm 1 Ir}+J_{\rm 2 Ir})[1+\chi_{\rm nIr}\cos(2\varphi_{2})],J_{Z_{2}\rm Ir}=0.5(J_{\rm 1 Ir}+J_{\rm 2 Ir})[1-\chi_{\rm nIr}\cos(2 \varphi_{2})], \tag{22}\]  
where \(J_{\rm 1 Tr},J_{\rm 2 Tr}\) are the principal geometric moments of inertia for the tripod, \(J_{\rm 1 Ir},J_{\rm 2 Ir}\) are the principal geometric moments of inertia for the inner race, \(\chi_{\rm nTr}\) and \(\chi_{\rm nIr}\) are the geometric nonuniformities of the tripod and inner race, and \(J_{Y_{2}\rm Tr},J_{Z_{2}\rm Tr},J_{Y_{2}\rm Ir},J_{Z_{2}\rm Ir}\) are the geometric moments of inertia of the tripod and inner race concerning the axes \(Y_{2},Z_{2}\);  
3. The bowl has a stiffness \(k_{2}\), given by the serial springs \(k_{21},k_{22}\) (see Figure 7), and damping \(c_{2}\), given by the serial dampers \(c_{21},c_{22}\), for the bending vibration rigid movement of the bowl regarding the axis \(Z_{2}\) and a stiffness \(k_{2}\), given by the serial springs \(k_{t_{21}},k_{t_{22}}\), as well as damping \(c_{2}\), given by the serial dampers \(c_{t_{21}},c_{t_{22}}\), for the angular bending vibration rigid movement of the bowl regarding the axis \(Y_{2}\), given by the following relations:  
\[k_{21}=\frac{3E_{\rm 2 BBA}}{L_{\rm BA}^{3}},k_{22}=\frac{3E_{\rm 2B}k_{ 22}}{L_{\rm BB}^{3}},k_{2}=\frac{k_{21}k_{22}}{k_{21}+k_{t_{22}}},c_{2}=\frac{ 2\Delta}{\sqrt{4\pi^{2}+\Delta^{2}}}\sqrt{k_{2}m_{\rm B}}, \tag{23}\]  
\[k_{t_{21}}=\frac{\rm GJ_{Y_{2}BA}}{L_{\rm BA}},k_{t_{22}}=\frac{G_{Y_{2}\rm BB }}{L_{\rm BB}},k_{t2}=\frac{k_{t_{21}}k_{t_{22}}}{k_{t_{21}}+k_{t_{22}}},c_{2}= \frac{2\Delta}{\sqrt{4\pi^{2}+\Delta^{2}}}\sqrt{k_{t2}I_{Y_{2}}B}, \tag{24}\]  
where \(m_{\rm B}\) is the bowl's mass. 4. The tulip-tripod joint in FBV movement (see Figures 3 and 6) realizes the elastic link between the tulip and the midshaft through the stiffness \(k_{\rm TIT}\) and the damping \(c_{\rm TITr}\) for the bending vibrating movements concerning the \(Z_{2}\) axis and the angular stiffness \(k_{\rm TITr}\) and angular damping \(c_{\rm TITr}\) for the bending vibrating movements concerning \(Y_{2}\) axis. 5. The bowl-inner race joint in FBV movement (see Figures 4 and 7) realizes the link between the bowl and the midshaft through the stiffness \(k_{\rm rIB}\) and the damping \(c_{\rm IrB}\) for the bending vibrating movements concerning the \(Z_{2}\) axis and the angular stiffness \(k_{\rm tHB}\) and the angular damping \(c_{\rm UHB}\) for the bending vibrating movements concerning the \(Y_{2}\) axis. The wheel induces excitations as a moderate impulsive shock force \(F_{\rm s}\) acting in the \(Z_{2}\) direction, and the excitation load can be expressed as  
\[F_{\rm S}=\overline{F}_{\rm S}\big{[}1+q_{3}t^{q_{1}}e^{-q_{2}t}\big{]}, \tag{25}\]  
where \(\overline{F}_{\rm S}\) is the amplitude of the shock on the bowl's longitudinal axis \(X_{3}\) transmitted from the wheel axis and \(q_{i},i=\overline{1,3}\), are experimental constants, depending on the type of shock applied at the wheel by the road excitation [4] (pp. 142-172). All the computations were performed considering that the tulip was a cantilever beam fixed in the gearbox with simple elastic supports in the tulip-tripod joint. The bowl was a cantilever beam fixed in the steering wheel by simple elastic supports in the bowl-inner race joint. As mentioned in the literature [4] (pp. 142-172), the shock excitation loads produce huge automotive stress solicitations in the car suspension system and the rim-tire system. These two systems can absorb 90% of the shock energy.