## 3 SCH Nonlinear Mathematical Model (NMM)
### Modeling According to First Physical Principles
#### 3.1.6 Wing Model  
In the previous research such as in Speedhawk compound helicopter, for reducing the cost of drag under hover condition, a large area of flap was partitioned on the wing. Apart from the drag in hover, problems arise during a transition flight when the angle of attack (AoA) is influenced by the airframe pitch angle. To solve these problems, a free-rotate wing was designed and installed on the test model. The wings on both sides of the airframe can rotate around its pitch axis controlled by servo actuators. The wing angle can change from \(-20^{\circ}\) to \(90^{\circ}\), which meets the demand of the VTOL inflow condition. Different input signals in each half of the wing also made the aileron control possible. The innovative wing solved the interference problem to a simple one. When in hover, the wing rotated to vertical orientation and in forward flight, the wing rotated to level orientation. During forward flight, the influence of the airframe pitch angle on the wings was balanced automatically.  
\[\text{Mode 1}\qquad\text{Hover}:\ \theta_{w}=\theta_{best} \tag{17}\]  
\[\text{Mode 2}\qquad\text{Forward flight }:\ \theta_{w}=\theta_{best}-\theta(1-e^{- st_{w}})\pm\theta_{a}\ \,\ \ \ \ \ (\theta_{w}+\theta)\in[-20^{\circ},90^{\circ}] \tag{18}\]  
Figure 5: Ducted fan thrust and power curves.  
\[\theta_{best}=\left\{\begin{array}{ccc}90^{\circ}&,&\text{while}&\text{Mode1}\\ 5^{\circ}&,&\text{while}&\text{Mode2}\end{array}\right. \tag{19}\]  
In the mode illustrations, \(\theta\) is the pitch angle of helicopter. \(\theta_{a}\) is the maneuver signals of aileron differential angles on each half of the wing. \(t_{w}\) is the delay of the pitch control. \(\theta_{w}\) is the pitch angle of the wing with respect to the helicopter's body-fixed axes. During hover, \(\theta_{best}\) was set to \(90^{\circ}\) and the wing was in vertical position for reducing drag. In forward flight, \(\theta_{best}\) was selected for a greater ratio of lift-to-drag coefficient with consideration of a suitable moment. In the experiments, \(\theta_{best}\) was set to \(5^{\circ}\) in mode 2. The angles mentioned above are shown in Figure 6.  
In the modeling, to reduce the complexity, the interference from the main rotor to the wing was simplified by a linear interference assumption: First, calculate \(\chi\), the wake skew angle of main rotor.  
\[\tan\chi=\frac{\mu}{\lambda}=\frac{u\cos\theta}{v_{intr}-u\sin\theta} \tag{20}\]  
With the aid of the CATIA helicopter model, it is known that when \(\chi<70^{\circ}\), the wing was located inside the wake of the rotor. In this condition, \(v_{intr}\) should be considered. When \(\chi>70^{\circ}\), the wing was clear from the rotor wake and thus \(v_{intr}\) should not be a factor. Then a linear change of \(v_{intr}\) between \(\chi\in[0^{\circ},70^{\circ}]\) was assumed. The forces of the wings can be decided and the lift and drag in mode 2 can be written as:  
\[dL=\frac{1}{2}\rho v_{y}^{2}c_{y}C_{t}dy \tag{21}\]  
\[dD=\frac{1}{2}\rho v_{y}^{2}c_{y}C_{d}dy \tag{22}\]  
\[Z=\sin\alpha_{w}\int dD-\cos\alpha_{w}\int dL \tag{23}\]  
\[X=-\cos\alpha_{w}\int dD-\cos\alpha_{w}\int dL \tag{24}\]  
\[C_{l}=f_{l}(\theta_{w})\ \,\ \ \ \ \ C_{d}=f_{d}(\theta_{w}) \tag{25}\]  
where \(y\) is the distance to the rotor shaft. The subscript \(y\) represents the value of variables at the position \(y\). \(c\) is the wing chord. \(C_{l}\) and \(C_{d}\) are the lift and drag coefficients, respectively. \(\alpha_{w}\) represents the inflow angle.