THE p DERIVATIVES (Cv Cv C„s, C7la/ Chr)

When an airplane rolls with angular velocity p about its x axis in the reference state (the flight direction for wind axes), its motion is instantaneously like that of a screw. This motion affects the airflow (local angle of attack) at all stations of the wing and tail surfaces. This is illustrated in Pig. 8.12 for two points: a wing tip and the fin tip. It should be noted that the non-dimensional rate of roll, p = p&/2F is, for small p, the angle (in radians) of the helix traced by the wing tip. These angle-of-attack changes bring about alterations in the aerodynamic load distribution over the surfaces, and thereby introduce perturbations in the forces and moments. The change in the wing load distribution also causes a modification to the trailing vortex sheet. The vorticity distribution in it is no longer symmetrical about the x axis, and a sidewash (positive, i. e. to the right) is induced at a vertical tail conventionally placed. This further modifies the angle-of-attack distribution on the vertical – tail surface. This sidewash due to rolling is characterized hy the derivative
да/dp. It has been studied theoretically and experimentally by Michael (ref. 8.1), who has shown its importance in relation to correct estimation of the tail contributions to the rolling derivatives. Finally, the helical motion of the wing produces a trailing vortex sheet which is not flat, but helical. For the small rates of roll admissible in a linear theory, this effect may be neglected with respect to both wing and tail forces.