A mathematical model for the thrust force generated by a flapping elastic wing

Abstract

The physical nature of the thrust force generated by flapping wings is of a long-time interest of many researchers. The idea of the thrust effect came from the observation of birds’ flight. Apparently, Leonardo da Vinci was first who tried to explain the mechanism of the flapping wing trust, for possible engineering applications. Nevertheless, the fundamental basics of a theoretical study of wing oscillations were laid only near the beginning of the 20th century. The thrust effect of the flapping wing was explained by Knoller in 1909 and Betz in 1912, independently. The principal problem in this theory is to define an optimal deformation law which provides the flapping wing to work with highest efficiency. In the present paper we study a rectangular elastic wing of finite span as a propulsion device. We propose an analytical approach, to study harmonic oscillations of a thin elastic rectangular wing at zero attack angle in a flow of inviscid incompressible fluid. The problem is reduced to an integro-differential equation, in frames of the “plane sections” hypothesis.