According to the classical Prandtl's lifting-line theory, the bound vortex is located along the quarter chord, and the trailing vortex is parallel to the x-axis along the free-stream velocity, as shown in Figure 1.
With the traditional aerodynamic theory on a finite wing with no sweep, the induced velocity generated by its own bound vortex at an arbitrary location along the lifting line can be ignored.
In addition, the bound vortex as well as the trailing vortex generated by the other lifting surface can both produce an induced velocity at an arbitrary location along the lifting line.
where [phi] is the angle between the induced velocity and the z-axis direction as shown in Figure 2, cos ([phi]) = [S'.sub.t]/([S'.sup.2.sub.t] + [G'.sup.2]), [phi] is the angle between the segment of the bound vortex and the line through the point and the vortex segment as shown in Figure 3, sin ([phi]) = [([S'.sup.2.sub.t] + [G'.sup.2]).sup.1/2]/r, r is the distance from the point to the vortex segment, and r = [([([y.sub.2]-[y.sub.1]).sup.2] + [S'.sup.2.sub.t] + [G'.sup.2]).sup.1/2], [S'.sub.t] is the longitudinal separation along the x-axis of the two wings and G' is the vertical distance along the z-axis between the two wings, as shown in Figure 2, and can be calculated with the coordinate transformation as
Applying the Biot-Savart law into the segment of the bound vortex along the lifting line of the other lifting surface and integral along the spanwise, the x-velocity component induced by the bound vortex at an arbitrary point along the lifting line is
Assume that the UAV helicopter has k-blade rotor and the circulation of each blade is [GAMMA]; the circulation of a bound vortex corresponding to d[theta] is (k[GAMMA]/2[pi])d[theta].
The relationship between the bound vortex and cylindrical vortex circulation is given by
Refer to the flapping motion equation combined with blade-element theory and Kutta-Joukowski equation; the components of the bound vortex in first-order Fourier series can be expressed as
The aerodynamic force necessary to stay aloft is created solely because of the so-called bound vortex (or circulation), which is complementary to the starting vortex and constitutes a measure of difference in flow speeds over and under the wing.
Therefore, the ring rotation speed, the corresponding bound vortex strength, and their contribution to useful force generation have been increased for the duration of the stroke.