## 2 The Analysis of DMADFTV  
Figure 4 shows a simplified picture of the AD, where each AD's element has attached a Cartesian reference system such as  
* \(X_{1}Y_{1}Z_{1}\) attached to the GT (global tulip: TB and TA), having a rigid rotation with \(\varphi_{1}\) around axis \(X_{1}\),
* \(X_{2}Y_{2}Z_{2}\) attached to the MA, having a rigid rotation with \(\varphi_{2}\) around axis \(X_{2}\),
* \(X_{3}Y_{3}Z_{3}\) attached to the GB (global bowl: BB and BA), having a rigid rotation with \(\varphi_{3}\) around axis \(X_{3}\), while \(\beta_{1}\) is the angle between the longitudinal directions of the GT and the MA regarding the axis \(Z_{1}\), \(\beta_{1}\in[0^{\circ},15^{\circ}]\), and \(\beta_{2}\) is the angle between the longitudinal directions of the GB and the MA regarding the axis \(Z_{3}\), \(\beta_{2}\in[0^{\circ},47^{\circ}]\). The angle \(\beta_{1}\) between the longitudinal directions of the GT and the MA regarding the axis \(Z_{1}\), is given by the internal length of the trucks inside the TB, the external diameter of the MA, and the internal diameter of the TB (see Figure 5a), conforming to the designer calculus [27] (pp. 214-230). The angle \(\beta_{2}\) between the longitudinal directions of the GB and the MA regarding the axis \(Z_{3}\) is given by the internal length of the trucks inside the IR, the external diameter of the MA, and the internal diameter of the BB (see Figure 5b), conforming to the designer calculus [27] (pp. 153-208).  
Figure 4: Schematical representation of movements for an AD.  
Figure 5: The geometry of the CVJ automotive driveshaft.  
An AD is a multibody driveline transmission from the automotive geared box to the wheel with the geometrical details presented in Figure 5.  
To determine the differential equations of the ADFTV based on the variational approach of [PERSON]'s principle (VAHP), it is mandatory to realize the reduction of the AIMM (axial inertial mass moment) and the AIGM (axial inertial geometric moment) for the GT and the GB to the longitudinal axis of the MA, respectively \(X_{2}\), as illustrated in Figure 5. The AIMM and the AIGM of the GT and the GB consider the angles \(\beta_{1}\) and \(\beta_{2}\) as well as the specific geometry dimensions of each GT's components: TB, TA reduced in the tripod's CM (center mass); and GB's components: BB, BA reduced in the inner race's CM, as expressed by the Equations (A1)-(A4) in Appendix A. In Appendix A, Equation (A5) expressed the AIMM and the AIGM for the AD's MA, while Equation (A6) expressed the stiffnesses and the dampings for the GT and the GB.  
Figure 6 illustrates the DMADFTV (dynamic model of the ADFTV).  
The DMADFTV comprises the following elements: the GT, the MA, and the GB, the GT being linked in torsion with the MA through TTJ-CVJ while the GB is linked in torsion with the MA through BIRJ-CVJ. The geared box injects in the AD a moment produced by the internal combustion engine, yielding [3] (p. 361)  
\[M_{Gb}=\overline{M}_{e}[1+\chi_{e}cos(n\Omega_{e}t)],n\in N,\Omega_{e}=\frac{ \pi n_{e}}{30}, \tag{1}\]  
where \(\chi_{e}\) is the engine moment nonuniformity, having values 0.10-0.12 [3] (p. 363), \(\overline{M}_{e}\) is the moment's amplitude, and \(n_{e}\) is the engine crankshaft rpm. The moment induced at the wheel is a moderately impulsive kind and has the expression  
\[M_{w}=M_{H}\big{[}1+q_{3}t^{q_{1}}e^{-q_{2}t}\big{]}, \tag{2}\]  
where \(M_{H}\) is the moment of adhesion [27] (p. 130), where \(q_{i}\), \(i=\overline{1,3},q_{1}\gg q_{2},q_{3}\geq 1.1\) are experimental constants depending on the shock applied to the wheel by the excitation [28].