## 2 The Modified Physically Consistent Model of FTV for Double Tripod Industrial Driveshaft
The rigidity and the damping of the DTID's elements are like those presented in [9] because the DTID also has three elements, DIE1, DME2, and DOE3, as illustrated in Figure 8, linked through the joints TJ1 (tripod joint 1) and TJ2 (tripod joint 2), with dynamic torsional characteristics expressed as follows:  
Figure 8: The MPCM of FTV for DTID. Figure 7: The general scheme of DTID. 1. The first DTID element is DIE1 (see Figures 7 and 8), which has the torsion rigidity and the torsion damping coefficient given by the equations: \[\mathrm{k_{t1}}=\mathrm{G}\frac{\mathrm{J}_{\mathrm{X_{2}A1}}\mathrm{J}_{\mathrm{ X_{2}B1}}}{\mathrm{J}_{\mathrm{X_{2}A1}}\mathrm{L}_{\mathrm{B1}}+\mathrm{J}_{ \mathrm{X_{2}B1}}\mathrm{L}_{\mathrm{A1}}},\mathrm{c_{t1}}=2\zeta\sqrt{\mathrm{ k_{t1}}\mathrm{I}_{\mathrm{X_{2}DIE1}}},\] (1) where \(\mathrm{G}\) is the steel shear modulus, \(\mathrm{k_{t1}},\mathrm{c_{t1}}\) are the stiffness and the damping coefficient in torsion of DIE1, and \(\zeta\) is the damping ratio of the DIE1's material, steel, having the value range 0.0016-0.0318 [9];
2. The TJ1 is the tripod that links DIE1 and DME2 (see Figures 7 and 8), having the torsion stiffness \(\mathrm{k_{tTJ1}}\) and the coefficient of torsion damping \(\mathrm{c_{tTJ1}}\);
3. The second DTID element is DME2 (see Figures 7 and 8), which has a continuous elastic behavior in torsion and has fixed through splines at both ends of the tripods with the inertial characteristics in torsion \[\mathrm{I_{X_{2}TJ1}}=\mathrm{J}_{\mathrm{X_{2}T1}}\rho\mathrm{L_{T1}}=\left( \mathrm{J_{1T1}}+\mathrm{J}_{\mathrm{ZT1}}+\mathrm{J}_{\mathrm{X_{2}DM2}} \right)\rho\mathrm{L_{T1}},\] (2) Figure 4.0 (3) Figure 4.1 (4) where \(\mathrm{I_{X_{2}TJ1}}\), \(\mathrm{I_{X_{2}TJ2}}\) are the axial mass moment of inertia of the tripods with the DME2's axis included, \(\mathrm{I_{X_{2}DM2}}\) has the same signification for DME2, \(\mathrm{L_{T1}}\),\(\mathrm{L_{T2}}\) are the lengths of the tripods, and \(\mathrm{d_{DM2}}\) is the diameter of the DTID's element DME2;
4. The TJ2 is the tripod that links DME2 and DOE3 (see Figures 7 and 8), having the torsion stiffness \(\mathrm{k_{tTJ2}}\) and the coefficient of torsion damping \(\mathrm{c_{tTJ2}}\);
5. The third DTID element is DOE3 (see Figures 7 and 8), which has the torsion rigidity and the torsion damping coefficient given by the equations: Figure 4.2 (5) where \(\mathrm{k_{t2}},\mathrm{c_{t2}}\) are the stiffness and the damping coefficient in torsion of DOE3. The equation gives the load torque received by DTID from the asynchronous electric motor (see Figures 7 and 8) [25; 26]: Figure 4.3 (6) Figure 4.4 (7) where \(\chi_{\mathrm{AEM}}\) is the nonuniformity of the asynchronous electric motor torque, having the value in the field 0.01-0.07 [25], \(\overline{\mathrm{M}}_{\mathrm{AEM}}\) is the amplitude of the engine torque in Nm, and \(\mathrm{n_{AEM}}\) is the output velocity angle of the asynchronous electric motor in rot/min. As mentioned in [25], due to the manifested nonuniformity instantaneous electromagnetic torque components in the range of \(-0.1\)-\(0.1\) (manifestation that exists even for synchronous motor [26]), the active and reactive electric power has a nonuniformity in the range \(\pm 0.07\)[25; 26]. Therefore, the output torque has the same nonuniformity. Usually, these nonuniformities have a harmonic variation given by the equation [25]: Figure 4.5 (8)A reactive torque charges the output element DOE3 of DTID due to the milling system (see Figures 7 and 8), having a similar harmonic variation due to the variable granulation of the substance to mill as the input torque M\({}_{\rm{AEM}}\), given by the equation:  
Figure 4.6  
where \(\overline{\mathrm{M}}_{\rm{MS}}\) is the amplitude of reactive torque of the milling system and \(\chi_{\rm{MS}}\) is a moderate nonuniformity due to the granulation of the material milled in the system, having a value in the range 0-0.17 [27] (pp. 319-348).