ON THE APPLICATION OF SPECIAL FUNCTIONS TO NON-LINEAR AND UNSTEADY STABILITY: (PART II) SOLUTION IN TERMS OF BESSEL AND HYPERGEOMETRIC FUNCTIONS
Title:
ON THE APPLICATION OF SPECIAL FUNCTIONS TO NON-LINEAR AND UNSTEADY STABILITY: (PART II) SOLUTION IN TERMS OF BESSEL AND HYPERGEOMETRIC FUNCTIONS
Author:
Campos, L. M. B. C. Azinheira, J. R. C.
Appeared in:
Integral transforms and special functions
Paging:
Volume 14 (2003) nr. 2 pages 167-180
Year:
2003-04
Contents:
In Part I, an iterative method of solution of a coupled non-linear system of ordinary differential equations was presented, consisting of successive cycles of solutions of three linear ordinary differential equations. The application chosen was to the longitudinal stability of an airplane, and the three coupled equations specify the airspeed, angle-of-attack and flight path angle, as a function of time. The first cycle leads to solutions in terms of Bessel or hypergeometric functions, depending on whether in the zeroth-order iteration the airspeed is a linear or exponential function of distance, and on whether damping is taken into account or not. The cycles of iteration can be stopped (§1) when the flight path deviation becomes small (§2.1). If this is the case after the first cycle, then the solutions in terms of Bessel and hypergeometric functions are used to plot the angle-of-attack response, as a function of distance along flight path (§2) for: (§2.2) undamped motion with constant airspeed or linear airspeed variation (Fig. 2); (§2.3) damped motion in the same cases (Fig. 3); (§2.4) exponential airspeed variation, without, and with damping, for several values of airspeed change and aerodynamic lengthscale (Figs. 3-6). It is shown (§3) that different airspeed profiles have (i) little effect on the spatial periodicity of angle-of-attack oscillations, but can have a significant effect on (ii) the amplitude of oscillations and (iii) their decay. The latter (iii) decay of oscillation due to airspeed changes is algebraic, rather than exponential, as in the case of damping. For simplicity the equations, references and Figures are numbered sequentially between Parts I and II of the present paper.
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