# Blade Shape Optimization and Analysis of a Propeller for VTOL Based on an Inverse Method
## 4 Case Study
### Optimization Result  
#### 4.2.1 VPP Optimization Result  
Figure 15a shows the Pareto-optimal front. It was obvious that there was a contradiction between the mission energy and the maximum thrust-to-weight ratio at hover, which means that the more maneuverable at hover the tilt-wing is, the lower energy efficiency it will have. We transformed the multi-objective optimization problem to a single objective one by defining a new objective as:  
\[J=E_{mission}-\kappa \tag{12}\]  
where the original two objectives had the same weight factor. We then obtained a comprehensive optimal FPP with a minimum \(J\), marked as green circle 3 in Figure 15a. The maximum thrust-to-weight ratio at hover for this FPP was 1.53 and the energy consumption was 4.49 kWh. The 3D geometry of this selected optimal VPP is shown in Figure 16. The chord-length and twist distribution of this selected optimal VPP, the most energy-saving VPP and the most maneuverable VPP are shown in Figure 17. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Flight State**} & \multicolumn{3}{c}{**Mission Profile Parameters**} \\ \hline  & **Name** & **Symbol** & **Velocity (m/s)** & **Thrust Required (N)** & **Time(s)** \\ \hline  & Cruising & S\({}_{4}\) & 65.25 & 155.8 & 920 \\ \hline  & Hover & S\({}_{1}\) & 0 & 922.4 & 30 \\ \cline{2-5}  & Vertical Takeoff & S\({}_{2}\) & 3 & 922.4 & 34 \\ \cline{2-5}  & Climbing & S\({}_{3}\) & 37 & 169.4 & 124 \\ \cline{2-5}  & Vertical Landing & S\({}_{5}\) & \(-1.5\) & 922.4 & 67 \\ \cline{2-5}  & Vertical Climb & & & \\ with Maximum Velocity & S\({}_{6}\) & 10 & 922.4 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mission profile parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Given Parameters** & \multicolumn{3}{c}{**Estimated Parameters**} \\ \hline Maximum input power \(P_{limited}^{BLDC}\) (kW) & 40 & Speed constant \(K_{V}\) (rpm/V) & 8.0 \\ \hline Maximum revolutionary speed \(\Omega_{limited}\) (rpm) & 5500 & Internal resistance \(R_{0}\) (\(\Omega\)) & 0.25 \\ \hline Maximum output torque \(Q_{limited}^{BLDC}\) (Nm) & 200 & No-load current \(l_{0}\) (A) & 2.0 \\ \hline Range of input voltage \(U\) (V) & 24\(\sim\)800 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of BLDC motors used in Vahana A\({}^{3}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design Variables** & **Range** & **Design Variables** & **Range** \\ \hline \(c_{rotet}\) (m) & 0.075 \(\sim\) 0.2 & \(r_{mid}\) (m) & 0.2\(R\)\(\sim\) 0.7\(R\) \\ \hline \(c_{tip}\) (m) & 0.04 \(\sim\) 0.12 & \(\omega_{4}\) (rpm) & 1000 \(\sim\) 3300 \\ \hline \(p\) & 0 \(\sim\) 2 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Range of design variables. The performance of this selected optimal VPP with different collective pitch and advance ratios is shown in Figure 18. The collective pitch mechanism provided a wide range of high efficiency, as shown in Figure 18b. Figure 18d shows the hover efficiency of this optimal VPP with a different revolutionary speed and collective pitch angle. To investigate the twist and chord-length distribution of the optimization objectives, we selected six optimal propellers from the Pareto front, as shown in Figure 15a. Figure 15b shows the 3D geometry of these propellers. To describe the average twist amplitude, we defined a parameter denoted as \(\tau\) and its expression was:  
\[\tau=\int\limits_{R_{hub}}^{R}\frac{d\theta}{d(r/R)}dr \tag{13}\]  
Figure 16: Selected comprehensive optimal VPP. Figure 17: Twist angle and chord-length distribution of a comprehensive optimal VPP. (**a**) Twist angle distribution; (**b**) chord-length distribution. Figure 15: Pareto front and optimal VPPs. (**a**) Sample points and Pareto front; (**b**) geometries of the propeller.