## 4 Torsional Stiffness Analysis  
The dynamic torsional stiffness of the prototype was analyzed using the above dynamics model. The main parameters of the prototype are shown in Table 1. The bearing stiffness and damping were calculated according to the reference [23], while the meshing stiffness value of planetary gears was obtained from the same article. The meshing stiffness of the beveloid gear was obtained from [24] by using FE calculation. The disc spring stiffness is based on literature [25] by taking \(k_{0}\) = 4.5 \(\times\) 10\({}^{6}\) N/m. The torsional stiffness of the whole machine can be calculated from the output torque, input angle and output angle, as shown in Equation (8). \[K=\frac{iT}{\theta_{in}-i\theta_{out}} \tag{8}\]  
where  
\(i\): transmission ration;  
\(T\): output torque;  
\(\theta_{in}\): input angle;  
\(\theta_{out}\): output angle. The Crank Distribution Circle Radius Error \(E_{rep}\) of the Crankshaft vs. the Torsional Stiffness  
Under other unchanged conditions, the error of the radius of the crank distribution circle of the crankshaft is changed to \(E_{\text{rep}}\) = 0.00, 0.01, 0.02, 0.03, 0.04 mm. Through simulation analysis, the corresponding influence curve of the system's torsional stiffness is obtained in Figure 3.  
Due to damping, the torsional stiffness gradually decays with time and finally stabilizes to near a value. As can be seen from Figure 3, the dynamic torsional stiffness exhibits an approximately sinusoidal variation with time. The phase difference of different curves does not evidently change (the slope is 1 \(\times\) 10\({}^{-2}\) s/mm), and the period has changed inconspicuously (the slope is 0 s/mm). The peak-to-peak torsional stiffness of the system under different \(E_{\text{rep}}\) is 0.39, 0.40, 0.41, 0.42, and 0.43 Nm/arcmin (the slope is 1000 N/arcmin), \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameter** & **Cranks** & **Sun** & **Planetary** \\ \hline \multirow{8}{*}{Crankshaft} & Mass \(m\)/kg & 0.11 & 0.015 & 0.06 \\  & Moment of inertia \(I\)/kg-mm\({}^{2}\) & 0.84 \(\times\) 10\({}^{-2}\) & 1.24 \(\times\) 10\({}^{-3}\) & 1.01 \(\times\) 10\({}^{-2}\) \\ \cline{3-5}  & & The base diameter \(r_{b}\)/mm & — & 21.1 & 42.3 \\  & & Number of teeth \(Z\) & — & 18 & 36 \\  & & Modulus \(m_{m}\)/mm & — & 1.25 & 1.25 \\  & & Engagement angle \(\alpha\)/(\({}^{\circ}\)) & — & 20 & 20 \\ \hline \multirow{8}{*}{Crankshaft} & Mass \(m\)/kg & 2.715 & 0.537 & 2.043 \\  & & Moment of inertia \(I\)/kg-mm\({}^{2}\) & 5.052 & 1.159 & 10.19 \\ \cline{1-1}  & & The base diameter \(r_{b}\)/mm & — & 109.9 & 112.8 \\ \cline{1-1}  & & Number of teeth \(Z\) & — & 78 & 80 \\ \cline{1-1}  & & Modulus \(m_{m}\)/mm & — & 1.5 & 1.5 \\ \cline{1-1}  & & Engagement angle \(\alpha\)/(\({}^{\circ}\)) & — & 47.6 & 47.6 \\ \cline{1-1}  & & Vertebral angle \(\gamma\)/(\({}^{\circ}\)) & — & 2.5 & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Prototype parameters of PRVT. while the average values are 74.32, 74.31, 74.33, 74.30, and 74.32 Nm/arcmin (the slope is 0 N/arcmin), respectively. It can be seen from the figure that the peak-to-peak value of torsional stiffness increases with the increase of \(E_{\text{rep}}\). However, its mean value does not change significantly. The Distribution Circle Radius Error \(E_{\text{rec}}\) of the Crankshaft Bearing Hole on the Carrier vs. the Torsional Stiffness  
Under other unchanged conditions, the variation in the error of the partial circle radius of the crankshaft bearing hole was investigated for \(E_{\text{rec}}=0.00\), 0.01, 0.02, 0.03 and 0.04 mm. Through simulation analysis, the corresponding influence curve of the system's torsional stiffness is obtained in Figure 4.  
It can be seen from Figure 4 that the dynamic torsional stiffness exhibits approximately sinusoidal fluctuations with time. The phase difference of the different curves increases  
Figure 4: Dynamic torsional stiffness under different \(E_{\text{rec}}\). Figure 3: Dynamic torsional stiffness under different \(E_{\text{rep}}\).