Perturbation Equations of Steady Flight

After this excursion into aerodynamic modeling, let us pick up the discussion from Sec. 7.2. The equations of motion, Eqs. (7.28) and (7.29), must be completed by the aerodynamic expansions of the right-hand sides. The linear terms of the Taylor expansion can be grouped according to the state variables

and

[i. e., frame A = frame 1 results in [evBBp — ev’BBp according to Eq. (7.8), and

devB d t

+ tef,]Bp + ([T]BpBr – [E])[T]Br,[fgr]1

and Euler’s equation

These fundamental stability equations of steady flight are fully coupled by the state variables [svlB]Bp and appearing in the aerodynamic terms and

in addition by [coBpBr]Bp being present on the left-hand side of Eq. (7.55). They are now linear differential equations, possibly with time-dependent coefficients through the variable Mach number. The controls [erfBp are the inhomogeneous input to the equations.

The acceleration variable [ev’BBp appears on both sides of Eq. (7.55). For a state-space representation they would have to be combined. Yet frequently in the aerodynamic expansions, [evB]Bp is expressed as d. fi and, particularly in missile dynamics, often combined with the damping effect caused by [coBpBr]Bp, thus eliminating the acceleration term on the right-hand sides.

For even greater simplicity the coupling term m[QBpBrBpvl/jr}Br is usually neglected in the development of plant dynamics for autopilot designs, and so are the thrust perturbations and the gravitational term. We will follow this practice as we derive transfer and state equations for roll, acceleration, rate, and flight-path angle autopilots.