## 3 The Equations of ADFTV and the Mathematical Solutions of FOAM
After mathematical calculus yielding the ADFTV's equations,  
\[\ddot{\varphi}_{1}+2\zeta\overline{\Omega}_{1}\sqrt{\frac{1+{\rm a }_{1}\cos 2\varphi_{1}}{(1+{\rm a}_{2}\cos 2\varphi_{1})(1+\chi_{\text{GTn}}\cos 2 \varphi_{1})}}\dot{\varphi}_{1}+\overline{\Omega}_{1}^{2}\frac{1+{\rm a}_{1} \cos 2\varphi_{1}}{(1+{\rm a}_{2}\cos 2\varphi_{1})(1+\chi_{\text{GTn}}\cos 2 \varphi_{1})}\varphi_{1}=\] \[=\frac{\overline{\text{M}}_{\text{e}}}{\tilde{\text{I}}_{\text{ X}_{2}\text{GT}}}\frac{1+{\rm x}_{\text{e}}\cos(n\Omega_{\text{e}}t)}{(1+\chi_{ \text{GTn}}\cos 2\varphi_{1})}(1-3\text{A}_{\text{TTr}}\sin 3\varphi_{1})+ \tag{4}\] \[+\sum\limits_{\text{n}}\Phi_{2n}\Bigg{[}\frac{\tilde{\text{I}}_{ \text{I}0\text{Ms}}}{\tilde{\text{I}}_{\text{X}_{2}\text{GT}}}\omega_{\text{ n}}^{2}\cos\Theta_{1n}+\frac{\text{G}\text{J}_{\text{X}_{2}\text{Ms}}}{\tilde{ \text{I}}_{\text{X}_{2}\text{GT}}}\frac{\omega_{\text{n}}}{\text{c}}\sin \Theta_{1n}\Bigg{]}\frac{\cos(\omega_{\text{n}}t-\Theta_{2n})}{(1+\chi_{\text{ GTn}}\cos 2\varphi_{1})},n=1,2,3,\ldots,\]  
\[\ddot{\varphi}_{3}+2\zeta\overline{\Omega}_{3}\sqrt{\frac{1+{\rm a }_{3}\cos 2\varphi_{3}}{(1+{\rm a}_{4}\cos 2\varphi_{3})(1+\chi_{\text{GBn}}\cos 2 \varphi_{3})}}\dot{\varphi}_{3}+\overline{\Omega}_{3}^{2}\frac{1+{\rm a}_{3} \cos 2\varphi_{3}}{(1+{\rm a}_{4}\cos 2\varphi_{3})(1+\chi_{\text{GBn}}\cos 2 \varphi_{3})}\varphi_{3}=\] \[=-\frac{\text{M}_{\text{H}}}{\tilde{\text{I}}_{\text{X}_{2}\text{ GB}}}\frac{\left[1+{\rm a}_{3}^{\text{H}1\text{e}}-{\rm q}_{2}^{2}\right]}{(1+\chi_{ \text{GBn}}\cos 2\varphi_{3})}(1-3\text{A}_{\text{BIr}}\sin 3\varphi_{3})+ \tag{5}\] \[+\sum\limits_{\text{n}}(-1)^{\text{n}}\Phi_{2n}\Bigg{[}\frac{ \tilde{\text{I}}_{\text{Q}\text{Ms}}}{\tilde{\text{I}}_{\text{X}_{2}\text{GB} }}\omega_{\text{n}}^{2}\cos\Theta_{1n}+\frac{\text{G}\text{J}_{\text{X}_{2} \text{Ms}}}{\tilde{\text{I}}_{\text{X}_{2}\text{GB}}}\frac{\omega_{\text{n}}} {\text{c}}\sin\Theta_{1n}\Bigg{]}\frac{\cos(\omega_{\text{n}}t-\Theta_{2n})}{( 1+\chi_{\text{GBn}}\cos 2\varphi_{3})},n=1,2,3,\ldots,\]  
where \(\tilde{\text{I}}_{\text{X}_{2}\text{GT}}\) is the GT's AIMM concerning the axis \(\text{X}_{2}\) for \(\beta_{1}=0^{\circ}\), \(\chi_{\text{GTn}}\) is the GT nonuniformity (as in Appendix B), \(\tilde{\text{I}}_{\text{X}_{2}\text{GB}}\) is the GB's AIMM concerning the axis \(\text{X}_{2}\) for \(\beta_{2}=0^{\circ}\), \(\chi_{\text{GBn}}\) is the GB nonuniformity (as in Appendix B), and \(\overline{\Omega}_{1}\) and \(\overline{\Omega}_{3}\) are the circular eigenfrequencies in torsion of the GT and GB as functions of \(\beta_{1},\beta_{2}\), having the mathematical expressions  
\[\overline{\Omega}_{1} =\frac{\zeta}{\Gamma_{1}}\Bigg{[}\frac{1-\frac{\tilde{\text{I}}_{ \text{X}_{2}\text{AT}}\text{P}_{\text{AT}}}{\tilde{\text{I}}_{\text{X}_{2} \text{GT}}}}{1+\left(\frac{\tilde{\text{I}}_{\text{X}_{2}\text{AT}}}{\tilde{ \text{I}}_{\text{X}_{2}\text{GT}}}\right)^{2}\frac{\tilde{\text{I}}_{\text{X}_{ 2}\text{GT}}}{\tilde{\text{I}}_{\text{X}_{2}\text{AB}}\text{P}_{\text{AB}}} \Bigg{]}^{1/2}, \tag{6}\] \[\overline{\Omega}_{3} =\frac{\zeta}{\Gamma_{\text{B}}}\Bigg{[}\frac{1-\frac{\tilde{\text{I }}_{\text{X}_{2}\text{AB}}\text{P}_{\text{AB}}}{\tilde{\text{I}}_{\text{X}_{2} \text{GB}}}}{1+\left(\frac{\tilde{\text{I}}_{\text{AB}}}{\tilde{\text{I}}_{ \text{B}}}\right)^{2}\left(\frac{\tilde{\text{I}}_{\text{X}_{2}\text{GB}}}{ \tilde{\text{I}}_{\text{X}_{2}\text{AB}}\text{P}_{\text{AB}}}-1\right)}\Bigg{]}^ {1/2}.\]  
For the dynamic instability analysis for ADFTV, different approaches can be used, such as the harmonic balance method (HBM), asymptotic method (AM) [31] (pp. 299-393), and the method of multiple scales (MMS) [32] (pp. 424-427). The HBM is very efficient but allows only the computation of the stationary amplitude of vibrations; the AM permits the calculation of the amplitude and phase angle for both stationary and nonstationary oscillations, while MMS raises computational difficulties due to zeroing the secular term and supplementary analysis for the detuning parameter [31] (pp. 424-427).