## 2 Integrated Dynamics Model of the Helicopter Engine/Transmission System/Rotor
### Interaction Effects Analysis of Multivariate Time Series in RSM Models  
Following the upshift strategy outlined in Table 6, simulations are conducted. During the upshift, the response targets include the engine speed overshoot value \(\Delta\Omega_{\rm e}\), transmission system input torque overshoot value \(\Delta T_{in}\), rotor speed overshoot value \(\Delta\omega_{MR}\), and transmission system output torque overshoot value \(\Delta T_{out}\). Given that the friction clutch serves as an active control component during the upshift process, the torque overshoot of the friction clutch \(\Delta T_{f}\) is also established as a response target. The upshift simulation strategy and its associated response targets are listed in Table 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Strategy**} & \multicolumn{2}{c}{_Fue\({}_{2}\)_} & \(Hyd\) & \(Pit_{2}\) \\ \cline{2-4}  & \(t_{la}\) **(s)** & \(t_{lb}\) **(s)** & \(t_{lc}\) **(s)** \\ \hline
1 & 48 & 40.5 & 42 \\
2 & 45 & 40.5 & 39 \\
3 & 45 & 40 (39) & 40.5 \\
4 & 48 & 40.5 & 39 \\
5 & 46.5 & 42 & 42 \\
6 & 46.5 & 40.5 & 40.5 \\
7 & 48 & 42 & 40.5 \\
8 & 46.5 & 40.5 & 40.5 \\
9 & 46.5 & 39 & 39 \\
10 & 46.5 & 40.5 & 40.5 \\
11 & 46.5 & 41 (39) & 42 \\
12 & 45 & 42 & 40.5 \\
13 & 46.5 & 42 & 39 \\
14 & 45 & 40.5 & 42 \\
15 & 48 & 40 (39) & 40.5 \\
16 & 46.5 & 40.5 & 40.5 \\
17 & 46.5 & 40.5 & 40.5 \\ \hline \hline \end{tabular} Note: Downshift strategies 3, 11, and 15 do not converge in the simulation; therefore, the hydraulic pressure relief time is selected, and the original design value in “\(0\)” is replaced.  
\end{table} 
Table 7: The starting time point under downshift strategy BBD design (_Fue_\(1\): 45 s, _Pit_\(1\): 42 s). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Strategy**} & \multirow{2}{*}{\(\Delta\Omega_{\rm e}\) **(rad/s)**} & \multirow{2}{*}{\(\Delta T_{in}\) **(Nm)**} & \(\Delta\omega_{MR}\) & \multirow{2}{*}{\(\Delta T_{out}\) **(Nm)**} & \multirow{2}{*}{\(\Delta\tau_{f}\) **(Nm)**} \\  & & & & **(rad/s)** & \\ \hline
1 & 92.30 & 342.88 & 8. The data in Table 8 is processed using Design-Expert v.13. A regression model is established between the various response targets observed during the upshift and starting points (\(t_{har}\), \(t_{hb}\), \(t_{hc}\)) of fuel flow \(Fue_{2}\), hydraulic \(Hyd\), and pitch angle \(Pit_{2}\). \[\Delta\Omega\mathrm{e}=2826.91-76.88t_{ha}-336.74t_{hb}+28.44t_{hc}+9.34t_{ha} t_{hb}+0.05t_{ha}t_{hc} \tag{26}\] \[-15.01t_{hb}t_{hc}-1.30t_{ha}^{2}+12.35t_{hb}^{2}+8.17t_{hc}^{2}\]  
\[ \begin{array}{l}\Delta T_{in}=-1811.35-235.59t_{ha}+1063.21t_{hb}+313.59_{hc} +36.34t_{ha}t_{hb}\\ -22.69t_{hb}t_{hc}-6.90t_{ha}^{2}-64.26t_{hb}^{2}+31.26t_{hc}^{2}\end{array}  \tag{27}\]  
\[ \begin{array}{l}\Delta\omega_{MR}=53.65+3.89t_{ha}-16.48t_{hb}+2.82t_{hc}-0. 513t_{ha}t_{hb}\\ +0.29t_{hb}t_{hc}+0.08t_{ha}^{2}+0.98t_{hb}^{2}-0.36t_{hc}^{2}\end{array}  \tag{28}\]  
\[ \begin{array}{l}\Delta T_{out}=6313.72-651.8t_{ha}+833.43t_{hb}-637.87t_{hc} +89.75t_{ha}t_{hb}\\ -71.95t_{hb}t_{hc}-14.92t_{ha}^{2}-78.87t_{hb}^{2}+79.25t_{hc}^{2}\end{array}  \tag{29}\]  
\[ \begin{array}{l}\Delta\tau_{f}=-1739.84-171.09t_{ha}+846.96t_{hb}-233.04t_{ hc}+26.77t_{ha}t_{hb}\\ +0.004t_{ha}t_{hc}-16.41t_{hb}t_{hc}-5.16t_{ha}^{2}-50.07t_{hb}^{2}+22.98t_{hc} ^{2}\end{array}  \tag{30}\]  
Table 9 presents the significance analysis results of the upshift regression model. With \(p\) values below 0.0001, these results highlight the model's statistical significance. Both the multiple correlation coefficient R\({}^{2}\) and adjusted determination coefficient R\({}_{\text{adj}}\)\({}^{2}\) are close to 1, illustrating a strong correspondence between the response surface model's calculations and the existing mathematical model's simulation results. Consequently, the upshift regression model is suitable for analyzing and forecasting the response traits of various sequential upshifting strategies in variable-speed power systems.