The derivation, solution, and analysis of airplane spin equations modeled in an inertial coordinate system

Abstract

The general equations of motion for a rigid body are derived in cylindrical coordinates by Lagrangian dynamics and used to model the motion of an airplane in a steady spin. After simplification, the equations are cast into a form utilizing conventional aerodynamic derivatives along with other derivatives which may be significant in spins. An iterative numerical solution procedure is outlined which should simplify the problem of solving the nonlinear differential equations, and relationships between the Euler Angles used in the equations and the more familiar ordered set of pitch, roll, and yaw are derived to permit computer input and output of orientation to be more easily visualized. The inverse problem is also considered , and equations are derived to translate spin test data into the six Lagrange coordinates used so that computer calculations may be checked against test results and actual aerodynamic forces can be compared with computed values. The paper concludes with a discussion of the implications of this spin analysis along with applications and extensions.