## 2 Integrated Dynamics Model of the Helicopter Engine/Transmission System/Rotor
### Interaction Effects Analysis of Multivariate Time Series in RSM Models
In Figure 8, when \(t_{hc}\) = 9 s, the changes in \(t_{ha}\) and \(t_{hb}\) have no significant impact on the engine output speed; when \(t_{hb}\) = 12.5 s, the earlier \(t_{ha}\) and \(t_{hc}\) can reduce the overshoot of the engine output speed; when \(t_{hu}\) = 9 s, the earlier \(t_{hb}\) and later \(t_{hc}\) can reduce the overshoot of the engine output speed. Figure 8: Response surface of engine speed overshoot during upshift. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Response Target** & **F Value** & \(p\)**-Value** & **R\({}^{2}\)** & **R\({}^{2}\)adj** \\ \hline \(\Delta\Omega\mathrm{e}\) & 28.61 & 0.0001 & 0.9735 & 0.9395 \\ \(\Delta T_{in}\) & 69.28 & \textless{}0.0001 & 0.9889 & 0.9746 \\ \(\Delta\omega_{MR}\) & 86.58 & \textless{}0.0001 & 0.9911 & 0.9797 \\ \(\Delta T_{out}\) & 52.46 & \textless{}0.0001 & 0.9854 & 0.9666 \\ \(\Delta\tau_{f}\) & 71.21 & \textless{}0.0001 & 0.9829 & 0.9753 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Significance analysis results of the upshift regression model. In Figure 12, when \(t_{hc}=10.5\) s, earlier \(t_{ha}\) and \(t_{hc}\) can reduce clutch torque overshoot; when \(t_{hb}=12.5\) s, later \(t_{ha}\) and earlier \(t_{hc}\) can reduce the clutch torque overshoot; when \(t_{hu}=16.5\) s, later \(t_{hb}\) and earlier \(t_{hc}\) can reduce the clutch torque overshoot. Based on the downshift strategy detailed in Table 7, simulations are executed. The response targets for the downshift process include overshoot values for engine speed \(\Delta\Omega_{e}\), transmission system input torque \(\Delta T_{in}\), rotor speed \(\Delta\omega_{MR}\), and transmission system output torque \(\Delta T_{out}\). Given that the one-way clutch is engaged passively to convey torque during the downshift, the torque overshoot of the one-way clutch \(\Delta\tau_{o}\) is also identified as a response target. Table 10 delineates the downshift simulation strategy and its associated response targets. Regression model between the different response targets during downshift and starting points (\(t_{lat}\), \(t_{lb}\), \(t_{lc}\)) of fuel flow \(Fue_{2}\), hydraulic \(Hyd\), and pitch angle \(Pit_{2}\). \[\Delta\Omega e=-468.61+24.13t_{lb}-10.16t_{lc} \tag{31}\]  
\[\Delta T_{in}=70489.29-1512.67t_{la}-3095.85t_{lb}+1388.96t_{lc} \tag{32}\]  
\[-49.26t_{lb}t_{lc}+16.27t_{la}^{2}+61.77t_{lb}^{2}+8.37t_{lc}^{2}\]  
\[\Delta\omega_{MR}=-30.83+1.71t_{lb}-0.7t_{lc} \tag{33}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Strategy**} & \multirow{2}{*}{\(\Delta\Omega_{e}\) **(rad/s)**} & \multirow{2}{*}{\(\Delta\Omega_{in}\) **(Nm)**} & \multicolumn{2}{c}{\(\Delta\omega_{MR}\)} & \multirow{2}{*}{\(\Delta\Omega_{out}\) **(Nm)**} & \multirow{2}{*}{\(\Delta\tau_{o}\) **(Nm)**} \\  & & & & & **(rad/s)** & \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 10: Downshift strategy and response objectives. Figure 12: Response surface of friction clutch torque overshoot during upshift. \[\Delta T_{out}=3044.54-100.92t_{lb}+41.14t_{lc} \tag{34}\]  
\[\Delta\tau_{o}=1219.09-37.15t_{lb}+15.061t_{lc} \tag{35}\]  
The significance analysis results of the downshift regression model are shown in Table 11, with \(p\) values less than 0.05, indicating that the models are statistically significant. The multiple correlation coefficient R\({}^{2}\) is greater than 0.52, and the correction determination coefficient R\({}_{\text{adj}}\)\({}^{2}\) is greater than 0.46, indicating a high degree of similarity between the calculated results of the response surface model and the simulated results of the existing mathematical model. Therefore, the downshift regression model can be used to analyze and predict the response characteristics of downshift strategies in different time series. Based on the regression model analysis results, the interaction effect between the downshifting time points and the relationship between multiple response targets was determined.