In this paper, analytical expressions for cycle-averaged aerodynamic forces generated by flapping wings are derived using a force model and flapping kinematics suitable for the forward flight of avian creatures. A strip theory-based formulation is proposed and the analytical expressions are found as functions of the amplitude of twist profile, mean twist angle, the flow separation point on the upper surfaces of the wings, and Strouhal number. Numerical results are obtained for a rectangular planform as well as for a representative avian wing planform. Utilizing these results, it is shown that there exists a narrow Strouhal number range where cycle-averaged net thrust, lift, and lift to drag ratio are optimal for a given flow pattern over the upper surfaces of the wings. This narrow Strouhal number range, found to be between 0.1 and 0.3, corresponds to the cruising range for a large number of avian creatures, as documented in current literature. An explanation, based on force constraints and local optimization in aerodynamic force generation, is provided for the unique range of Strouhal numbers utilized in avian cruising flight. The results and the approach outlined in the paper can be utilized to design efficient bio-inspired flapping vehicles.

在本文中，使用力模型和适用于禽类动物向前飞行的拍打运动学，推导了拍打翅膀产生的周期平均空气动力的解析表达式。提出了一种基于条形理论的公式，并根据扭转轮廓的幅度，平均扭转角，机翼上表面的流分离点和斯托洛哈尔数来找到解析表达式。对于矩形平面形式以及代表性的禽翼平面形式，获得了数值结果。利用这些结果表明，存在一个狭窄的Strouhal数范围，对于给定的流型，机翼上表面，周期平均净推力，升力和升阻比最佳。狭窄的Strouhal数范围介于0.1和0.3之间，对应于当前文献中记载的大量鸟类生物的巡航范围。基于力约束和空气动力学力产生的局部优化的解释为鸟类巡航飞行中使用的斯特劳哈尔数的独特范围提供了解释。本文概述的结果和方法可用于设计高效的仿生扑扑车辆。