## 3 The System Equations of Motion for Double Tripod Industrial Driveshaft's FTV  
For the MPCM of FTV for DTID (see Figure 8), considering the same mathematical procedure as in [9], Hamilton's Principle, it yields the system equations of motion:  
\[\mathrm{I_{\chi_{\rm{DDE1}}}(\varphi_{1})\ddot{\varphi}_{1}+[\epsilon_{\rm{t1}} +3c_{\rm{tTJ1}}A_{\rm{TJ1}}\sin(3\varphi_{1})]\varphi_{1}+[k_{\rm{t1}}-k_{\rm{ tTJ1}}A_{\rm{TJ1}}\cos(3\varphi_{1})]\varphi_{1}=\overline{\mathrm{M}}_{\rm{AEM}} \left[1+\chi_{\rm{AEM}}{\sum\limits_{i=0}^{5}\frac{\cos\left(2^{i}\Omega_{\rm{ AEM}}t\right)}{2^{i+1}}}\right](1-3A_{\rm{TJ1}}\sin(3\varphi_{1})), \tag{10}\]  
\[\frac{\partial^{2}\varphi_{2}}{\partial t^{2}}=c^{2}\frac{\partial^{2}\varphi_ {2}}{\partial x^{2}},c=\sqrt{\frac{G}{\rho}}, \tag{11}\]  
\[\mathrm{I_{\chi_{\rm{DDE3}}}(\varphi_{3})\ddot{\varphi}_{3}+[\alpha-3c_{\rm{tTJ 2}}A_{\rm{TJ2}}\sin(3\varphi_{3})]\dot{\varphi}_{3}+[k_{\rm{t2}}+k_{\rm{tTJ2}} A_{\rm{TJ2}}\cos(3\varphi_{3})]\varphi_{3}=-\overline{\mathrm{M}}_{\rm{MS}} \left[1+\chi_{\rm{MS}}{\sum\limits_{j=1}^{7}\frac{\cos\left(2^{j-1}\varphi_{3} (t)\right)}{2^{j}}}\right](1-3A_{\rm{TJ2}}\sin(3\varphi_{3})), \tag{12}\]  
\[\mathrm{where\ A_{TJ1}}\mathrm{,\ A_{TJ2}}\mathrm{ are [1] (p. 78), [9]:}\]  
\[\mathrm{A_{TJ1}}=0.5\frac{\mathrm{R_{TJ1}}}{\mathrm{L_{DME}}}\tan\beta_{1}\tan^ {2}\frac{\beta_{1}}{2}, \tag{13}\]  
\[\mathrm{A_{TJ2}}=0.5\frac{\mathrm{R_{TJ2}}}{\mathrm{L_{DME}}}\tan\beta_{2}\tan^ {2}\frac{\beta_{2}}{2}, \tag{14}\]  
where \(\mathrm{L_{DME}}\) is the length of DME2 measured between the centroids of TJ1 and TJ2, \(\mathrm{R_{TJ1}}\), \(\mathrm{R_{TJ2}}\) are the geometrical radii of the tripods joints TJ1 and TJ2, and the boundary conditions of DTID are expressed by the equations:  
\[\mathrm{I_{\chi_{\rm{TJ1}}}\frac{\partial^{2}\varphi_{2}(0,t)}{\partial t^{2} }-3c_{\rm{tTJ1}}A_{\rm{TJ1}}\sin(3\varphi_{1})\varphi_{1}+k_{\rm{tTJ1}}A_{\rm{ TJ1}}\cos(3\varphi_{1})\varphi_{1}-G\mathrm{J_{\chi_{\rm{D}M2}}}\frac{\partial \varphi_{2}(0,t)}{\partial x}=0,\mathrm{atx}=0, \tag{15}\]  
\[\mathrm{I_{\chi_{\rm{TJ2}}}\frac{\partial^{2}\varphi_{2}(L_{\rm{DME}},t)}{ \partial t^{2}}+3c_{\rm{tTJ2}}A_{\rm{TJ2}}\sin(3\varphi_{3})\dot{\varphi}_{3}- k_{\rm{tTJ2}}A_{\rm{TJ2}}\cos(3\varphi_{3})\varphi_{3}+G\mathrm{J_{\chi_{\rm{2} DM2}}}\frac{\partial\varphi_{2}(L_{\rm{DME}},t)}{\partial x}=0,\mathrm{atx}=L_{\rm{DM2}}. \tag{16}\]  
The system of Equations (SEs) (10)-(12) coupled with the Equations (13) and (14) and the boundary conditions (15) and (16) expressed the nonlinear dynamic behavior of FTV for DTID excited by an input harmonic torque generated by an asynchronous electric motor and having an output harmonic excitation caused by the granulation of materials neededto be milled. Equations (13) and (14) represent the geometric and kinematic nonuniformity of DTID generated by the effect that the tripod's joints are quasi-homokinetic [1; 2; 3]. In close regard, it is evident that Equations (10) and (12) are the equations of forced torsional parametric vibrations of DIE1 and DOE3 representing modified Mathieu-Hill equations; in contrast, (11) is a differential equation for the vibrations of a uniform shaft in torsion. The DME2 has fixed-fixed boundary conditions in torsion, and therefore, the solution of Equation (11) yields:  
\[\varphi_{2}(\mathrm{x},t)=\sum\limits_{\mathrm{k}=1}^{\infty}A_{\mathrm{k}} \cos\Bigl{(}\frac{\omega_{\mathrm{k}}\mathrm{x}}{\mathrm{c}}-\delta_{1 \mathrm{k}}\Bigr{)}\cos(\omega_{\mathrm{k}}t-\delta_{2\mathrm{k}}),\omega_{ \mathrm{k}}=\frac{\mathrm{k}\pi\mathrm{c}}{\mathrm{L}_{\mathrm{DM2}}},\mathrm{ k}=1,2,3,\ldots \tag{17}\]  
where \(A_{\mathrm{k}}\) represents the amplitude in torsion of DME2, \(\delta_{1\mathrm{k}},\delta_{2\mathrm{k}}\) are the phase's angles, and \(\omega_{\mathrm{k}}\) is the natural pulsation (NP) of the free vibrations in torsion of the DME2. Injecting the Equation (17) into the boundary conditions (15) and (16) yields:  
\[