# Blade Shape Optimization and Analysis of a Propeller for VTOL Based on an Inverse Method
## 1 Introduction  
With the rapid development of flight control technology and electric propulsion systems, vertical takeoff and landing (VTOL) aircraft have attracted a wide range of studies. Tilt-rotor and tilt-wing aircraft are the two most widely used VTOL aircraft in cargo transportation and urban flight, for example [1; 2; 3; 4; 5; 6]. The propellers applied to tilt-wing or tilt-rotor aircraft work in a wide range of operation conditions and may have two forms; a variable pitch propeller (VPP) or a fixed pitch propeller (FPP). Several VPPs participate in attitude control through a cyclic pitch mechanism, which are, in fact, helicopter rotors such as those used on the Osprey V-22 [7] and Bell Eagle Eye [8]. A few VPPs include only a collective pitch mechanism; attitude control is achieved through the difference between the collective pitch and/or the revolutionary speed of multiple propellers (greater than or equal to three) such as those used on the Boeing Vahana A3 [9] and Joby S4 [10]. FPPs are mostly used in micro VTOL UAVs or demonstrators to reduce the system complexity such as the GL-10 [11] and QTW-UAV [12]. The research of this paper does not involve cyclic pitch propellers; i.e., helicopter rotors. The distinct difference between the vertical flight mode and the level flight mode increases the difficulty of blade shape optimization. On one hand, the required thrustduring VTOL is roughly several times that of the level flight mode. On the other hand, the advance ratio during VTOL is much smaller than that of level flight [13]. [PERSON] summarized the technical challenges of a proprorotor design and claimed that the higher the efficiency of level flight, the lower the figure of merit (FM) of the rotor during vertical flight [14]. There is a natural contradiction between the two flight modes as far as the energy efficiency is concerned. A few research works on propeller blade shape optimization have been conducted for VTOL aircraft. [PERSON] et al. [4] optimized the VPP and FPP used in a micro VTOL UAV using the vortex theory and a modified parallel particle swarm optimization algorithm (PSOA). The result proved the advantages of VPP over FPP when applied to the VTOL UAV. [PERSON] et al. [15] used polynomial equations to parameterize the distribution of the chord-length and twist angle. A multi-objective optimization framework for a proprotor based on the blade element momentum theory (BEMT) was constructed. A blade shape with a higher hovering efficiency was taken as the initial design and the optimization results showed a significant improvement to the propulsive performance. [PERSON] et al. [16] carried out a multi-objective optimization design of the three-dimensional aerodynamic shape of a VPP with the help of the commercial software ModelCenter. The design variables in this paper included relevant parameters describing the blade sweep angle and the dihedral angle. [PERSON] [13] used the BEMT and genetic algorithm (GA) to optimize the chord, twist and section airfoil distribution of a VPP and then performed a second optimization on the tip shape of the VPP. In order to improve the aerodynamic efficiency of the JVX VPP, [PERSON] et al. [17] used CAMRAD II software developed from the free wake vortex lattice method (FWVLM) to perform the optimization design using a decoupling manner. The above works have enriched the research content of the blade shape optimization of VTOL propellers. However, none of the works above considered the maximum thrust-to-weight ratio at hover, which is closely related to the wind-resistance capability and maneuverability during takeoff and landing. One of the major factors affecting the takeoff and landing maneuverability is the maximum thrust-to-weight ratio at hover. The larger the maximum thrust-to-weight ratio of the propeller, the more thrust the propeller can provide for attitude and altitude control [18]. For multi-rotors, work has been conducted to improve the thrust-to-weight ratio [19; 20; 21].