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Perturbation Equations
The last chapter completed the toolbox for modeling aerospace vehicle dynamics. You are now well acquainted with Newton’s and Euler’s laws as modeling tools for the equations of motion. In Chapters 8-10 we shall put them to work, simulating the dynamics of aircraft, hypersonic vehicles, missiles, and even Magnus rotors. Before pursuing that ambitious goal, I will address another important subject of modeling and simulation that deals with the linearization of the equations of motions.
Why should we, living in the computer age, still concern ourselves with the simplification of the dynamic equations? I can think of three reasons, and you may be able to add some more.
1) Stability investigations are an important part of any vehicle design. They require the linearization of the equations of motion in order to take advantage of linear stability criteria.
2) Control engineers will always need linearized representations of the plant, be they transfer functions or in state variable form.
3) For a basic understanding of the vehicle dynamics, the eigenvalues of the linear equations serve to indicate frequency and damping.
These simplifications are accomplished with perturbation techniques. There is the classical small perturbation method, developed to solve specific problems in atmospheric flight mechanics. It employs scalar perturbations and relates them, for each type of flight vehicle, to a special coordinate system. Instead of deriving the general perturbation equations first, restrictive assumptions are made, and, consequently, the perturbation equations are limited to steady flight regimes.
The objective of this chapter is to introduce the general perturbation equations of atmospheric flight mechanics that are valid even for unsteady flight regimes. To keep the derivation simple, the flight vehicles are assumed rigid bodies.
I will discuss three techniques, the scalar, total, and component perturbations and use the latter to derive the general perturbation equations of aerospace vehicles. They apply to any type of vehicle from aircraft to spinning missiles. Then I will address the expansion of the aerodynamic forces and moments into Taylor series. Taken together, they deliver the linear dynamic equations. Examples of pitch and roll linear state equations demonstrate practical applications. I will also venture into the realm of unsteady flight with nonlinear effects to challenge your imagination.
