## 3 The Equations of ADFTV and the Mathematical Solutions of FOAM
For the above reasons, the present work uses the asymptotic method approach (AMA) for the solutions of Equations (4) and (5). The use of first-order approximation of AM makes the investigation of dynamic instability of ADFTV in both stationary and nonstationary torsional vibrations in the APPR possible. A small positive parameter must be introduced for both stationary and nonstationary torsional vibrations; the slowing time \(\tau=\varepsilon t\), \(\varepsilon\) being a positive small number, as mentioned in [30] (p. 299). These equations must be modified to apply AM before raising \(\tau\) in Equations (4) and (5). For the AM's first-order approximation (FOAM), the last terms in Equations (4) and (5) are neglected because the excitation frequency in APPR for GT and the GB is not in the range of free natural frequencies for torsional vibrations of the midshaft. The development gives the approximations used in the mathematical procedure of FOAM in [PERSON]'s series of the following functions:  
\[ \begin{array}{l}\frac{1+\mathrm{a}_{3}\cos 2\varphi_{1}}{(1+\mathrm{a}_{2} \cos 2\varphi_{1})(1+\mathrm{X}_{\mathrm{C}\mathrm{C}\mathrm{I}}\cos 2 \varphi_{1})}\simeq 1+2\mu_{1}\cos 2\varphi_{1}\Rightarrow\mu_{1}\simeq(1+ \alpha_{1})\alpha_{2},\\ \frac{\mathrm{M}_{\mathrm{e}}}{\mathrm{I}_{\mathrm{X}_{2}\mathrm{GT}}}\frac{1 +\mathrm{X}_{\mathrm{C}\mathrm{I}}\cos(\mathrm{n}\mathrm{f}\mathrm{c}\mathrm{c }\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c }\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}})(1-3\mathrm{A}\mathrm{T}_{\mathrm{T} \mathrm{T}}\sin 3\varphi_{1})\simeq\Gamma_{1}\varphi_{1}^{3}\Rightarrow\Gamma_{1}=9.112 5\frac{\mathrm{A}\mathrm{T}_{\mathrm{T}\mathrm{T}\mathrm{T}}}{\mathrm{X}_{ \mathrm{C}\mathrm{I}}\mathrm{T}}\frac{\mathrm{M}_{\mathrm{e}}}{\mathrm{I}_{ \mathrm{X}_{2}\mathrm{GT}}},\\ \\ \frac{1+\mathrm{a}_{3}\cos 2\varphi_{3}}{(1+\mathrm{a}_{4}\cos 2\varphi_{3})(1+ \mathrm{X}_{\mathrm{C}\mathrm{I}}\cos 2\varphi_{3})}\simeq 1+2\mu_{3}\cos 2 \varphi_{3}\Rightarrow\mu_{3}\simeq(1+\alpha_{3})\alpha_{4},\\ -\frac{\mathrm{M}_{\mathrm{H}_{\mathrm{I}}}}{\mathrm{X}_{\mathrm{C}\mathrm{G} \mathrm{R}}}(\frac{1+\mathrm{q}\mathrm{g}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c }\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c} \mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c }\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}\mathrm{c}})(1-3 \mathrm{A}\mathrm{B}\mathrm{I}_{\mathrm{T}}\sin 3\varphi_{3})=\Gamma_{3}\varphi_{3}^{3}+ \Gamma_{5}\varphi_{5}^{5},\\ \\ \Rightarrow\Gamma_{3}=-9.1125\frac{\mathrm{A}\mathrm{B}\mathrm{I}_{\mathrm{T} \mathrm{T}}}{\mathrm{X}_{\mathrm{C}\mathrm{G}\mathrm{R}}}\frac{\mathrm{M}_{ \mathrm{H}}}{\mathrm{I}_{\mathrm{X}_{2}\mathrm{C}\mathrm{G}\mathrm{R}}},\Gamma _{5}=-18.225\frac{\mathrm{A}\mathrm{B}\mathrm{I}_{\mathrm{T}\mathrm{A}\mathrm{ B}\mathrm{I}}}{\mathrm{X}_{\mathrm{C}\mathrm{G}\mathrm{R}}}\frac{\mathrm{M}_{ \mathrm{H}}}{\mathrm{I}_{\mathrm{X}_{2}\mathrm{C}\mathrm{G}\mathrm{R}}},\end{array}  \tag{7}\]  
whose coefficients \(\alpha_{i},i=1,2,3,4\) are presented in Appendix B.