On the application of special functions to non-linear and unsteady stability: (Part 1) - Method of iterative solution of a coupled non-linear system

A new approach to longitudinal airplane stability is presented (1), which is more general than the method of linear, constant stability derivatives, and allows oscillations which are non-sinusoidal and non-periodic, and amplitude growth or decay which is non-exponential. This approach to the solution of the non-linear, coupled equations of longitudinal motion of an airplane (2.1), uses an iterative, cyclic method (2.2), as follows: (i) a constant flight path angle is assumed, and elimination between the two force balance equations, gives airspeed as a function of distance along the flight path (2.3); (ii) this airspeed variation is fed into the pitching moment equation, to specify an angle-of-attack oscillation with changing frequency (2.4); (iii) the flight path angle is calculated from the airspeed (i) and angle-of-attack (ii), to determine the deviation from the preceding iteration (4.1), and stop the cyclic process if the change is small (as discussed in Part II). The airspeed variation is (a) an exponential function of distance for small deviation from initial airspeed (2.3), and simplifies further to a (b) a linear function of distance, for distance short compared to aerodynamic lengthscale (3.1). The angle-of attack response (3) is specified by: (3.1) Bessel functions, for linear airspeed variation without damping: (3.2) confluent hypergeometric functions for linear airspeed variation with damping; (3.3-3.4) Gaussian hypergeometric functions, for exponential airspeed variation without (3.3) or with (3.4) damping. These results are further developed in Part II.