## 3 Validation and Numerical Analysis
### Time-Varying Effect of the Original Helix Angle  
As can be seen in Figure 6, the difference value between the time-varying and the constant conditions is around 0.793 degrees for the helix angle, around 0.671 degrees for the transverse pressure angle, around 0.2243 mm for the gear pair center distance, and around 0.01696 for the total contact ratio. In this section, the difference values between the time-varying and the constant conditions under five different original helix angles are discussed. In the real helical gear systems, the range of helix angle is about 15-35 degrees. The results of the difference values are listed in Table 5. With the original helix angle raised, the difference values between the time-varying and the constant conditions of the helix angle, the transverse pressure angle, the gear pair center distance, and the total contact ratio are enhanced, especially the helix angle and total contact ratio. The time-varying variables have a greater effect on the higher initial helix angle and lower effect on the lower initial helix angle. This demonstrates that the time-varying variables of the higher initial helix angle are not negligible.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mode No.**} & \multicolumn{2}{c}{**Natural Frequency (rad/s)**} & \multirow{2}{*}{**Relative Errors (\%)**} & \multirow{2}{*}{**Mode Description**} \\ \cline{2-2} \cline{4-5}  & **Present** & & & \\ \hline
1 & 3574 & 3583 & \(-\)0.26 & coupled lateral-torsional \\
2 & 4225 & 4237 & \(-\)0.29 & 1 st lateral, Y-direction, driving shaft \\
3 & 4237 & 4245 & \(-\)0.20 & coupled lateral-torsional \\
4 & 4250 & 4246 & 0.09 & 1 st lateral, Z-direction, driven shaft \\
5 & 15,804 & 15,816 & \(-\)0.08 & coupled lateral-torsional \\
6 & 20,643 & 20,796 & \(-\)0.74 & 2 nd lateral, Y-direction, driving shaft \\
7 & 20,643 & 20,796 & \(-\)0.74 & 2 nd lateral, Z-direction, driving shaft \\
8 & 20,978 & 21,084 & \(-\)0.50 & 2 nd lateral, Y-direction, driven shaft \\
9 & 20,978 & 21,084 & \(-\)0.50 & 2 nd lateral, Z-direction, driven shaft \\
10 & 38,093 & 38,336 & \(-\)0.64 & 3 rd lateral, Y-direction, driving shaft \\
11 & 38,143 & 38,374 & \(-\)0.60 & coupled lateral-torsional \\
12 & 38,235 & 38,432 & \(-\)0.51 & 3 rd lateral, Y-direction, driven shaft \\
13 & 38,406 & 38,614 & \(-\)0.54 & coupled lateral-torsional \\ \hline \hline \end{tabular}
\end{table}
Table 3: Natural frequencies and corresponding mode shapes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Shaft**} \\ \hline
**Parameters** & **Driving Shaft** & **Driven Shaft** \\ \hline Shaft length (mm) & 254 & 254 \\ Outer diameter (mm) & 37 & 37 \\ Inner diameter (mm) & 0 & 0 \\ Modulus of elasticity (Cpa) & 207.8 & \\ Density of material (kg/m\({}^{3}\)) & 7806 & \\ Poisson’s ratio & 0.3 & \\ \hline \multicolumn{1}{c}{**Gear**} \\ \hline
**Parameters** & **Driving Gear** & **Driven Gear** \\ \hline Tooth number & 28 & 28 \\ Base circle radius (mm) & 24.4 & 24.4 \\ Addendum circle radius (mm) & 28.4 & 28.4 \\ Mass (kg) & 1.84 & 1.84 \\ Mass moment of inertia (kg-m\({}^{2}\)) & 0.0009 & 0.0009 \\ Polar mass moment of inertia (kg-m\({}^{2}\)) & 0.0018 & 0.0018 \\ Face width (mm) & 17.4 & 17.4 \\ Pitch diameter (mm) & 52.07 & 52.07 \\ Initial transverse pressure angle (degree) & 20 \\ Initial helix angle (degree) & 20 \\ Mesh stiffness during two teeth contact (N/m) & \(1.9\times 10^{8}\) \\ Mesh stiffness during three teeth contact (N/m) & \(2.4\times 10^{8}\) \\ \hline \multicolumn{1}{c}{**Bearing Parameters**} \\ \hline All radial bearing stiffness (N/m) & \(2\times 10^{7}\) \\ \hline All axial bearing stiffness (N/m) & \(1\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Helical-geared rotor-bearing system parameters.  
Figure 6: Time-varying and constant (**a**) helix angle, (**b**) transverse pressure angle, (**c**) gear pair center distance, and (**d**) total contact ratio.