## 3 The System Equations of Motion for Double Tripod Industrial Driveshaft's FTV
 \tag{19}\]  
Using the mathematical expression (18) into (10) and the mathematical expression (19) into (12), the equations of FTV for DTID's DIE1 and DOE3 are:  
\[\begin{array}{l}\bar{\varphi}_{1}+2\varOmega_{1}\sqrt{\frac{1+\varsigma_{1} \cos 2\varphi_{1}}{(1+\varsigma_{2}\cos 2\varphi_{1})(1+\varsigma_{4}\cos 2 \varphi_{1})}\varphi_{1}}+\varOmega_{1}^{2}\frac{1+\varsigma_{1}\cos 2 \varphi_{1}}{(1+\varsigma_{2}\cos 2\varphi_{1})(1+\varsigma_{4}\cos 2 \varphi_{1})}\varphi_{1}=\frac{\frac{1}{\mathrm{M}_{\mathrm{AM}}}\frac{1+ \varsigma_{2}\sin\frac{\zeta_{2}\sin\left(\frac{\varphi_{2}(I_{\mathrm{CM2},t })}{2}\right)}}{\left(1+\varsigma_{4}\sin 2\varphi_{1}\right)}}{\left(1+ \varsigma_{4}\sin 2\varphi_{1}\right)}(1-3A_{\mathrm{TJ1}}\sin 3\varphi_{1})\\ +\sum\limits_{\mathrm{k}=1}^{\infty}A_{\mathrm{k}}\Bigl{[}\frac{I_{\mathrm{ III}}}{I_{\mathrm{X_{2}DE1}}}\omega_{\mathrm{k}}^{2}\cos\delta_{1\mathrm{k}}+\frac{G_{ \mathrm{X_{2}DE1}}}{I_{\mathrm{X_{2}DE1}}}\frac{\omega_{\mathrm{k}}}{\mathrm{ c}}\sin\delta_{1\mathrm{k}}\Bigr{]}\frac{\cos(\omega_{\mathrm{k}}t-\delta_{2\mathrm{k}})}{ \left(1+\varsigma_{4}\cos 2\varphi_{1}\right)},\\ \\ \bar{\varphi}_{3}+2\varOmega_{3}\sqrt{\frac{1+\varsigma_{3}\cos 2\varphi_{3}}{(1+ \varsigma_{4}\cos 2\varphi_{3})(1+\varsigma_{4}\cos 2\varphi_{3})(1+\varsigma_{4}\cos 2 \varphi_{3})(1+\varsigma_{4}\cos 2\varphi_{3})}\varphi_{3}}+\varOmega_{3}^{2}\frac{1+ \varsigma_{1}\cos 2\varphi_{3}}{(1+\varsigma_{4}\cos 2\varphi_{3})(1+\varsigma_{4}\cos 2 \varphi_{3})(1+\varsigma_{(p. 425), one of the most important resonance regions is the PPRR for excitation frequencies \(\eta_{1}\simeq 2\Omega_{1},\eta_{3}\simeq 2\Omega_{3}\). The following paragraph will investigate the chaotic manifestation for NFTV of DIE1 and DOE3, expressed by the SEs (20) and (21), using time-history analysis in PPRR.  
Detection of Chaotic Manifestation for Nonstationary FTV of Double Tripod Industrial Driveshaft in Principal Parametric Resonance Region  
To detect chaotic manifestation for NFTV of DTID in PPRR, the Equations (20) and (21) are used to compute the phase portrait graphs \(\left(\varphi_{1},\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}\right)\), for DIE1 and DOE3 in PPRR. The DTID's elements DIE1 and DOE3 considered to have the characteristics of the DTID's material, the geometry characteristics such as the AGMIs, and their nonuniformity, are presented in Tables 1 and 2. These given geometric features are in agreement with those considered by [PERSON] [18] (p. 111).  
Based on AUTOCAD, \(\mathrm{J_{1B1},J_{2B1},J_{1B3},J_{2B3}}\) were calculated using the generic dimensions of the DTID's DIE1 and DOE3. MATLAB software determined the variation graph of the functions \(\Omega_{1}(\beta_{1}),\Omega_{3}(\beta_{2})\), representing the NPs of  
The expressions must define the excitation frequencies \(\eta_{1},\eta_{3}\) of DTID's DIE1 and DOE3 in the PPRR:  
\[\eta_{1}=\frac{2\mathrm{d}\varphi_{1}}{\mathrm{dt}}\simeq 2\Omega_{1}, \tag{27}\]  
\[\eta_{3}=\frac{2\mathrm{d}\varphi_{3}}{\mathrm{dt}}\simeq 2\Omega_{3}. \tag{28}\]  
To calculate the time-history graphs \((\varphi_{1}(t),t),(\frac{\mathrm{d}\varphi_{1}}{\mathrm{dt}}(t),t),\left( \varphi_{1}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{dt}}(t),t\right)\), \(\left(\varphi_{1}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{dt}}(t)\right)\) for DIE1 and \((\varphi_{3}(t),t),(\frac{\mathrm{d}\varphi_{3}}{\mathrm{dt}}(t),t)\), \(\left(\varphi_{3}(t),\frac{\mathrm{d}\varphi_{3}}{\mathrm{dt}}(t),t\right)\), \(\left(\varphi_{3}(t),\frac{\mathrm{d}\varphi_{3}}{\mathrm{dt}}(t)\right)\) for DOE3, the Equations (25) and (26) were numerically integrated using MATLAB software based on function ode 45 (Runge-Kutta integration method of fifth-order). The graphs \(\left(\varphi_{1}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{dt}}(t)\right)\), \(\left(\varphi_{3}(t),\frac{\mathrm{d}\varphi_{3}}{\mathrm{dt}}(t)\right)\) represent the phase portraits of NFTV for DIE1 and DOE3, being the detection method of possible chaotic dynamic behavior.