# General Equations of. Unsteady Motion

4.1 General Remarks

The basis for analysis, computation, or simulation of the unsteady motions of a flight vehicle is the mathematical model of the vehicle and its subsystems. An airplane in flight is a very complicated dynamic system. It consists of an aggregate of elastic bodies so connected that both rigid and elastic relative motions can occur. For exam­ple, the propeller or jet-engine rotor rotates, the control surfaces move about their hinges, and bending and twisting of the various aerodynamic surfaces occur. The ex­ternal forces that act on the airplane are also complicated functions of its shape and its motion. It seems clear that realistic analyses of engineering precision are not likely to be accomplished with a very simple mathematical model. The model that is developed in the following has been found by aeronautical engineers and researchers to be very useful in practise. We begin by first treating the vehicle as a single rigid body with six degrees of freedom. This body is free to move in the atmosphere under the actions of gravity and aerodynamic forces—it is primarily the nature and com­plexity of aerodynamic forces that distinguish flight vehicles from other dynamic systems. Next we add the gyroscopic effects of spinning rotors and then continue with a discussion of structural distortion (aeroelastic effects).

As was noted in Chap. 1, the Earth is treated as flat and stationary in inertial space. These assumptions simplify the model enormously, and are quite acceptable for dealing with most problems of airplane flight. The effects of a round rotating Earth are treated at some length in Etkin (1972).

Extensive use is made in the developments that follow of linear algebra, with which the reader is assumed to be familiar. Appendix A. 1 contains a brief review of some pertinent material.