## 3 Performance Analysis
### Velocity Stability  
The aircraft possessing velocity stability have an edge on landing risks. The normal landing aircraft velocity stability criterion is given by Equation (6). \[\frac{dT_{r}}{d\overline{q}}>0 \tag{6}\]  
where \(T_{r}\) is the required thrust, and \(\overline{q}\) is the dynamic pressure. Nevertheless, the velocity stability criterion on SRVL is still not published. Thus, this part derives the velocity stability criterion of the SRVL from that of the normal landing. Equation (7) is satisfied when the aircraft is in a steady straight-level flight. \[\left\{ \begin{array}{l}T_{r}=D=C_{\overline{D}}\overline{q}S\\ W=L=C_{\overline{L}}\overline{q}S\end{array} \right. \tag{7}\]  
Thus, Equation (8) is given:  
\[\frac{dT_{r}}{d\overline{q}}>0 \tag{8}\]  
Figure 8: Trajectory stability analysis. Therefore, the velocity stability criterion can be rewritten as:  
\[\frac{dT_{r}}{d\overline{q}}=\frac{d(\frac{C_{D}}{C_{L}})}{d(\frac{1}{C_{L}})} \tag{9}\]  
For the steady straight gliding, the force analysis function is:  
\[\left\{ \begin{array}{l}T_{r}=D+W\sin\gamma\\ L=W\cos\gamma\end{array} \right. \tag{10}\]  
Therefore, compared to Equation (8), the variables used in Equation (9) become:  
\[\left\{ \begin{array}{l}\frac{C_{D}}{C_{L}}=\frac{T_{r}-W\sin\gamma}{W\cos \gamma}=\frac{T}{W\cos\gamma}-\tan\gamma\\ \frac{S}{C_{L}}=\frac{T}{W\cos\gamma}\end{array} \right. \tag{11}\]  
Since the gliding angle, \(\gamma\), is a fixed value set by the users and is usually a small number, \(\cos\gamma\) equals approximately one and \(\tan\gamma\) equals approximately zero, that is,\(\cos\gamma\approx 1\) and \(\tan\gamma\approx 0\). Therefore, \(\frac{dC_{D}}{dC_{L}}\) derives from Equation (11) and is almost the same as that derived from Equation (8), as is \(\frac{1}{C_{L}}\). This means that the velocity stability criterion is still applicable in the steady straight-gliding case. For the SRVL case, the propulsion system provides direct lift. Thus, Equation (10) should be adjusted to Equation (12). \[\left\{ \begin{array}{l}T_{r}=D+W\sin\gamma\\ L=W\cos\gamma-L_{T}\end{array} \right. \tag{12}\]  
where \(L_{T}\) is the direct lift generated by the propulsion system. Now, Equation (11) converts to:  
\[\left\{ \begin{array}{l}\frac{C_{D}}{C_{L}}=\frac{T_{r}}{W\cos\gamma}-\tan \gamma\\ \frac{1}{C_{L}}=\frac{T}{W\cos\gamma-L_{T}}\end{array} \right. \tag{13}\]  
In the steady straight-gliding case, the gliding angle is a constant. When discussing stability issues, the control inputs are all given fixed values, which means the control inputs are constants too. Therefore, the direct lift provided by the propulsion system is also a constant. Then, the derivative of (\(\frac{1}{C_{L}}\)) versus time is still a constant that multiplies the derivative of \(\overline{q}\) versus time. To make it clear, \(d(\frac{1}{C_{L}})=\frac{S}{W\cos\gamma-L_{T}}\cdot d\overline{q}\). Thus, Equation (9) in the SRVL case turns into:  
\[\frac{dT_{r}}{d\overline{q}}=\frac{W\cos\gamma-L_{T}}{S}\frac{d(\frac{C_{D}}{ C_{L}})}{d(\frac{1}{C_{L}})} \tag{14}\]  
This means the derivative of the required thrust versus dynamic pressure in the SRVL landing case is what it is in the normal landing case multiplying \(\frac{W\cos\gamma-L_{T}}{S}\). Usually, the direct lift provided by the propulsion system is no more than \(W\cos\gamma\) unless the aerodynamic lift is a negative value. Therefore, the velocity stability criterion of the SRVL case is still Equation (6). Then, the velocity stability area can be calculated by Equation (6) for each pitch angle. Take the pitch angle of 3 degrees as an example. For a set forward flight airspeed and descending rate, the gliding angle is determined, as are the angle of attack and the aerodynamic coefficients, \(C_{L}\) and \(C_{D}\). With these parameters, the velocity stability was calculated by the equations deduced above. The results are given in Figure 9a, where the green points are the stable points and the red points are the unstable points.