###### Abstract  
In the design of flapping-wing micro-size air vehicles capable of hovering, wings serve as the primary source of hovering power, making the analysis of aerodynamics and aerodynamic efficiency crucial. Traditional quasi-steady models treat the wings as single rigid plane, neglecting the deformable characteristics of flexible wings. This paper proposes a multi-plane method that, in conjunction with various design parameters of flexible wings in a two-dimensional plane, analyzes their deformation characteristics under the assumption of multiple planes in three-dimensional space, and describes the deformation of wings during flapping. By combining the quasi-steady aerodynamic model, aerodynamic analysis of the deformed wings can be conducted. The relationship between the slack angle, wing flapping position, and wing deformation are analyzed, along with their effects on aerodynamics and aerodynamic efficiency. Experiments validate the deformation patterns of wings during flapping and compare the simulated aerodynamic forces with measured ones. The results indicate that wing deformation can be accurately described by adjusting the parameters in the multi-plane method and that the aerodynamic analysis using this method closely approximates the average lift results. Additionally, the multi-plane method establishes a connection between wing morphology and aerodynamic forces and efficiency, providing valuable insights for aerodynamic analysis. F 2024 14 4258 19 February 2024 19 February 2024 19 May 2024 17 May 2024 122024 14 4258 19 February 2024 14 4258 19 February 2024 19 February 2  
and engineering guidance for the design of hovering flapping-wing aircraft. In recent decades, a variety of micro-sized hovering flapping-wing aircraft have emerged, such as those from Harvard University [14; 15], Shanghai Jiao Tong University [16; 17], Purdue University [18; 19], Universite libre de Bruxelles [20], University of Maryland [21], and Konkuk University [22; 23], as well as the nano-hummingbird from Aero Vironment [24]. These prototypes, regardless of their flapping structures or propulsion methods, have achieved varying durations of hovering flight, proving from an engineering perspective the feasibility of in-depth research on flapping-wing aircraft for hovering flight. Hovering organisms generate aerodynamic forces by flapping their wings, which has always been a focal point in the research of flapping-wing aircraft. In the study of flapping-wing aerodynamics, the most common methods currently include experimental methods, CFD-FSI (Computational Fluid Dynamics and Fluid-Structure Interaction) computational methods [25; 26; 27; 28; 29; 30; 31; 32; 33], and blade element theory based on quasi-steady models [5; 34; 35]. Among these, experimental methods are undoubtedly the most accurate; however, they require testing of the designed and manufactured wings, and lack theoretical guidance prior to the design phase. Thus, relying solely on experimental methods for wing design and selection can waste a lot of effort. On the other hand, the computational fluid dynamics method combined with fluid-structure interaction [28; 29; 30; 31; 32; 33] can provide a discrete quantitative description of the flow field by solving fluid dynamics equations numerically, predicting the movement patterns of the fluid, and offering more precise aerodynamic simulations for flapping wings. However, the accuracy of aerodynamic calculations using this method is highly dependent on the precision of the grid model, leading to high computational complexity, significant impacts from different models, and challenges in parameter selection such as grid division and time step [25; 26; 27]. When designing wings, considering factors such as efficiency and accuracy, academia often tends to favor the blade element theory based on quasi-steady models. The quasi-steady model has a history of nearly fifty years in the aerodynamic analysis of flapping wings. In 1973, [PERSON] [5] first proposed the quasi-steady flapping model.