Formulation of second order differential equation and its analysis for aircraft flying qualities

Since many physical systems can be modeled by second order differential equation, airplane can also be thought of as an equivalent spring-mass-damper system. This makes the analysis of aircraft's damping and longitudinal static stability much simpler. For that matter, a model was proposed in which only pure pitching motion was considered and dependence of pitching moment on various factors was sought. A second order differential equation was then formulated in terms of stability derivatives in one variable only. From its characteristic equation, two roots can be obtained whose nature decide the stability of aircraft. The variation of different system parameters such as damping ratio, Z and natural frequency, ran as a function of various flight conditions such as flight velocity and altitude was also studied. Comparing these graphs with the pilot opinion contours, a judgment is passed about the flying qualities of aircraft at the specified conditions. If the aircraft is not safe and easy to handle, a substantial amount of damping should be added to it. Since the stability derivatives are a function of geometric and aerodynamic characteristics of the airplane, designer has some control over longitudinal dynamics by varying different structural parameters, if only that does not cause a subsequent loss in its flight performance. Hence, just by the virtue of second order differential equation, an optimum solution for the performance and handling of aircraft is estimated at different flight conditions.