Class of shockfree airfoils producing the same surface pressure

Abstract

Writing the loads in this fashion emphasizes that it is not the airfoil section pitch and heave motions (a and h) that must be identified, but rather the mean and linear components of the normal velocity distribution over the airfoil chord. Observe that the a terms actually arise from two sources: from the time derivative of (h + Ua) as well as from the pitch rate. It is worthwhile noting that in this form the result is applicable as well to the case of time-varying freestream velocity £7 (see Ref. 2), except that the Theodorsen lift deficiency function must be generalized to account for the expansion and compression of the shed wake vorticity. The result is also applicable for arbitrary motion, again except for the lift deficiency function; indeed, the derivation of Ref. 1 does not find it necessary to introduce the assumption of harmonic motion until the final step of evaluating C( k) [Eq. (5-308) of Ref. 1]. The most common approach in helicopter analyses has been to identify h as the normal velocity up at the rotor blades, and a as the pitch rate 0 + Q/3 (or sometimes even just 0). The numerical results for the lift will not be greatly influenced by this error, since the unsteady lift is small compared to the steady component. The steady moment component is normally small however, so the calculation of the moment will be seriously affected if the unsteady terms are incorrect. Another consequence of the incorrect identification of h and a, important for the lift as well as for the moment, is that the equivalence of flapping and feathering motion for an articulated rotor blade will be violated. As an example, consider the rigid flap and rigid pitch motion of an articulated rotor blade in forward flight. Unsteady airfoil theory requires (h + Ua), which is the air velocity normal to the airfoil section, at the pitch axis; and d, which is the equivalent pitch rate or camber of the airfoil. Hence for the present case h + Ua = u T6 - u p = ( Qr + QRfjLsin\l/ ) d