The essence of this two-part paper is the analytical, aerodynamic modelling of insectlike flapping wings in the hover for microair vehicle applications. A key feature of such flappingwing flows is their unsteadiness and the formation of a leading-edge vortex in addition to the conventional wake shed from the trailing edge. What ensues is a complex interaction between the shed wakes which, in part, determines the forces and moments on the wing. In an attempt to describe such a flow, two-novel coupled, non-linear, wake-integral equations are developed in this first part of the paper, and these form the foundation upon which the rest of the work stands. The circulation-based model thus developed is unsteady and inviscid in nature and essentially two-dimensional. It is converted to a 'quasi-three-dimensional' model using a blade-element-type method, but with radial chords. The main results from the model are force and moment data for the flapping wing and are derived as part of this article using the method of impulses. These forces and moments have been decomposed into constituent elements. The governing equations developed in the study are exact, but do not have a closed analytic form. Therefore, solutions are found by numerical methods. These are described in the second part of this paper.

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The essence of this two-part paper is the analytical, aerodynamic modelling of insect-like flapping wings in the hover for micro-air-vehicle applications. A key feature of such flapping-wing flows is their unsteadiness and the formation of a leading-edge vortex in addition to the conventional wake shed from the trailing edge. What ensues is a complex interaction between the shed wakes, which, in part, determines the forces and moments on the wing. In an attempt to describe such a flow, two novel coupled, non-linear, wake integral equations were developed in the first part of the paper. The governing equations derived were exact, but did not have a closed analytical form. Solutions were, therefore, to be found by numerical methods and implemented in Fortran. This is the theme of the second part of the paper. The problem is implemented by means of vortex methods, whereby discrete point vortices are used to represent the wing and its wake. A number of numerical experiments are run to determine the best values for numerical parameters. The calculation is performed using a time-marching algorithm and the evolution of the wakes is tracked. In this way, both flow field and force data are generated. The model is then validated against existing experimental data and very good agreement is found both in terms of flow field representation and force prediction. The temporal accuracy of the simulations is also noteworthy, implying that the underlying flow features are well captured, especially the unsteadiness. The model also shows the similarity between two-dimensional and three-dimensional flows for insect-like flapping wings at low Reynolds numbers of the order of Re 200.