## 4 Aerodynamic Analysis Based on the Multi-Plane Simulation Method
Therefore, the multi-plane method can be employed to more accurately simulate the wing's motion and deformation. By dividing the wing into multiple planes and assuming that the angular velocities of these rigid planes are the same as the leading-edge angular velocity of the wing while considering the differences in the angles of attack and their derivatives for each plane, the aerodynamic forces of different sections of the wing can be calculated more precisely. Figure 14 presents a quasi-steady aerodynamic simulation method incorporating the multi-plane approach. The figure depicts a schematic view of the wing observed from the opposite direction of the Y-axis in the wing coordinate system after determining the wing's shape in three-dimensional space. Similar to the analysis of a single rigid plane, the wing is divided into multiple blade segments along the span direction, each with a width of \(dr\); however, in contrast, these blade segments reside on planes in three-dimensional space. This implies that the angle of attack \(\alpha\) varies for each blade segment on different planes, necessitating a separate analysis for each plane. When calculating the aerodynamic forces for each blade segment, it is essential to consider the current wing shape, as the morphological parameter \(c(r)\) varies with time at different positions along the wing. Therefore, the analysis needs to be conducted in conjunction with the wing's present morphology. Additionally, as depicted in Figure 14, the number of intersections between the sampled blade segments (which serve as the reference in the wing coordinate system's \(YZ\) plane) and the planes differs due to the varying values of \(r\). The quasi-steady aerodynamic analysis using the multi-plane approach can be approximated as the superposition of quasi-steady aerodynamic analyses conducted on multiple planes. However, an issue arises when analyzing the additional aerodynamic forces caused by the overlapping of partial air volumes between different blade segments, as illustrated in Figure 13:9 It is challenging to determine whether it is necessary to treat this overlap as an equivalent longer blade segment for the calculation of additional mass-induced aerodynamic forces. Nevertheless, previous studies [40; 42; 43] have shown that the aerodynamic forces caused by additional mass are relatively small compared to the translational and rotational terms, suggesting that they can be approximated so as to have no significant impact on the results. Additionally, it is worth noting that we have omitted the potential interplay between the translational aerodynamic forces and rotational aerodynamic forces across different planes. Whether this simplification is permissible in actual physical analysis requires further verification in subsequent research. In summary, under the assumption of three planes, the quasi-steady unsteady aerodynamic forces using the multi-plane approach can be expressed as follows:  
\[F_{tr}^{{}^{\prime}}=F_{Ltr}^{{}^{\prime}}+F_{Dtr}^{{}^{\prime}}, \tag{20}\]0  
\[F_{tr}^{{}^{\prime}}=F_{Ltr}^{{}^{\prime}}+F_{Dtr}^{{}^{\prime}}, \tag{20}\]1  
\[F_{tr}^{{}^{\prime}}=F_{Ltr}^{{}^{\prime}}+F_{Dtr}^{{}^{\prime}}, \tag{20}\]2 Schematic diagram of blade segmentation in the multi-plane method from a three-dimensional perspective. The instantaneous lift and drag obtained through the multi-plane approach are denoted as \(F_{Lall}\) and \(F_{Dall}\), respectively, while \(F_{Lp1}\) and \(F_{Dp1}\) respectively represent the instantaneous lift and drag of plane1 obtained using the quasi-steady aerodynamic model. The definitions for the remaining variables are similar.