General equations of unsteady motion
The basis for analysis, computation; or simulation of the unsteady motions of flight vehicles is the mathematical model of the vehicle and of its subsidiary systems, i. e. their general equations of motion. Although a useful first step is to treat the vehicle as a single rigid body, and many important results can be derived from this model, we cannot in general avoid facing up to the reality of the situation, which is that vehicles are deformable and contain articulated or rotating subsystems such as control surfaces and engines. Furthermore the external forces and couples that act on aircraft and spacecraft are in general complicated functions of shape and of motion. This is especially true of the aerodynamic forces in atmospheric flight which are known only approximately. The attention that must be devoted to their representation dominates the formulation of the mathematical model. The forces and couples provided by the space environment (gravitational, magnetic, radiation pressure) are generally not so uncertain, and the problem of deriving an adequate mathematical model is consequently less difficult for spacecraft during extra-atmospheric operation.
In the following sections, we first treat the general motion of a particle over the rotating Earth, then derive the dynamical and kinematical equations for an arbitrary deformable vehicle in flight. Finally the equations for small disturbance from steady flight are presented in both dimensional and nondimensional form.