Lateral dynamic flight stability of a model hoverfly in normal and inclined stroke-plane hovering

Many insects hover with their wings beating in a horizontal plane ('normal hovering'), while some insects, e.g., hoverflies and dragonflies, hover with inclined stroke-planes. Here, we investigate the lateral dynamic flight stability of a hovering model hoverfly. The aerodynamic derivatives are computed using the method of computational fluid dynamics, and the equations of motion are solved by the techniques of eigenvalue and eigenvector analysis. The following is shown: The flight of the insect is unstable at normal hovering (stroke-plane angle equals 0) and the instability becomes weaker as the stroke-plane angle increases; the flight becomes stable at a relatively large stroke-plane angle (larger than about 24°). As previously shown, the instability at normal hovering is due to a positive roll-moment/side-velocity derivative produced by the 'changing-LEV-axial-velocity' effect. When the stroke-plane angle increases, the wings bend toward the back of the body, and the 'changing-LEV-axial-velocity' effect decreases; in addition, another effect, called the 'changing-relative-velocity' effect (the 'lateral wind', which is due to the side motion of the insect, changes the relative velocity of its wings), becomes increasingly stronger. This causes the roll-moment/side-velocity derivative to first decrease and then become negative, resulting in the above change in stability as a function of the stroke-plane angle.