## 5 Results and Discussion
### Forward Flight Validation  
For verifying the reliability of the established nonlinear model in high-speed forward flights, the simulation results from NMM are compared with the trim data. As shown in Figure 11, the solid points and hollow points represent the trim results in the stable speed sections. One data point represents one flight. The NMM predictions are presented by lines. In Configuration D, the maximum speed from experiments is 32 m/s, while in Configuration WD, the maximum speed is 30.7 m/s. Figure 11a shows the only one compound control signal \(\delta_{duct}\). \(\delta_{duct}\) reached its highest value in Configuration D 32 m/s. Due to the fact that Configuration WD had greater drag then Configuration D, \(\delta_{duct}\) in Configuration WD reached the highest point at 27 m/s. Subsequently, above 27 m/s, the pitch angle (Figure 11e) presented greater negative value and the helicopter showed worse overall performance relative to the optimal. In the pitch angle simulation (Figure 11e), the nonlinear model gives very good agreement with data for both conditions. In Figure 11b,h there are some underestimates on the collective pitch and the total power in Configuration WD above 27 m/s. It is because a greater interference from the rotor wake under a large negative pitch angle cannot be captured accurately by NMM. In Figure 11c,d the trim values of \(\delta_{lat}\) and \(\delta_{lon}\) all focused around zero, which represents a balanced moment condition. In the roll angle calculation (Figure 11f), the nonlinear model fits the trim results of Configuration D closely. In Configuration WD, the simulation of the roll angle shows a lower estimation. The flight data of roll angle in Configuration WD was more scattered. This is because the Configuration WD carried a large wing component, which was more sensitive to the influence of lateral wind. Figure 11g,h show the overall characteristics of the compound helicopter. \(\zeta\) is a defined ratio, named the forward thrust ratio:  
\[\zeta=\frac{X_{duct}/mg}{X_{duct}/mg+\sin(-\theta)} \tag{45}\]  
where \(\sin(-\theta)\) represents the projection of main rotor lift on the x-axis, \(X_{duct}/mg\) represents the contribution of forward thrust by the ducted fan. This ratio is simple measure of the contribution of the ducted fan to net forward propulsive force. In the designing process, the expectation of the ratio in forward speed above 25 m/s is 80-90%. The real \(\zeta\) (Figure 11g) of Configuration D was exactly in 80-90%, which meets the expectation. In \(\zeta\) calculations, the nonlinear model predicts closely to the real data in both conditions. Figure 11h shows the total power recorded by the main ESC. In Configuration D, the nonlinear model predicts the power in high accuracy and in Configuration WD below 27 m/s, the power calculation shows credible results around the flight results. Additionally, in Figure 12, groups of time-domain data were extracted for simulation. The data contain part of the accelerate section and the stable speed section. Two sets of data are presented in the figure: one is Configuration D under reference speed of 30.5 m/s, the other is Configuration WD under reference speed of 30 m/s. The blue dash lines show the NMM responses following the real control signals. The calculated \(v\) focuses around zero, similar to the flight results. In \(u\) calculations, the acceleration rate and the target speed value are captured accurately. That is to say the aerodynamic drags in these high-speed periods are well evaluated. \(\theta\) and \(\phi\) smoothly fluctuated following the control signals. Unlike the sweep signals in the hover identification, the fluctuations of \(\theta\) and \(\phi\) in Figure 12 were well controlled within 2\({}^{\rm o}\). The calculation of NMM has high quality and VAF are close to 100% from the results. Figure 12: Time-domain validation in forward flights under Configuration D and Configuration WD.