Linearization

In their basic form, the equations of airplane motion are a set of nine simultaneous nonlinear differential equations. One of the most far-reaching steps taken by Bryan was the development of a perturbation, linearized, form of these equations. The perturbation motion of a simple mechanical object, such as a pendulum, about a state of rest is a familiar concept. In his Mecanique Analytique of 1788, J. L. Lagrange developed the theory of small perturbation motions of systems having many degrees of freedom about a position of stable equilibrium. Bryan extended Lagrange’s work by replacing the position of stable equilibrium by a steady equilibrium motion.

The utility of Bryan’s linearization arises from the nature of airplane perturbed motions. Under normal operating conditions, such as personal-airplane and airliner climbs, cruises, and landing approaches, airplanes are among the most linear dynamic systems known. Aerodynamic force and moment are quite closely proportional to airplane perturbed motions, without any equivalent to coulomb friction. Small-perturbation or linearized equa­tions are perfectly suitable to describe the motions experienced by the crew and passengers, and for the design of stability augmenters and automatic pilots.

Bryan analyzed small perturbations about steady, symmetric, rectilinear flight, either level, climbing, or diving. Most of the subsequent literature on airplane dynamics is based on the same model. Equations of perturbed airplane motion about steady turning and steady sideslipping flight came soon after Bryan, in an important 1914 report by Leonard Bairstow. A further extension to general curvilinear flight was made using earth-referred coordinates (Frazer, Duncan, and Collar, 1938). Still later investigators (Abzug, 1954; Billion, 1956) used the more useful body-fixed coordinates. Then, in a series of NASA papers dating from 1981 to 1983, Robert T N. Chen applied linearization to the case of perturbations from uncoordinated turns. Chen’s immediate goal was to represent perturbation motions of single-rotor helicopters in low-airspeed, steep turns, in which appreciable amounts of sideslip are quite normal.

The 1914 linearization work by Bairstow suffered the fate of theory that was too far ahead of its time. The later investigators mentioned above seemed to have been unaware that Bairstow had already extended the original Bryan equations.

Bryan’s linearization of the equations of airplane motion reduces them to two sets of three simultaneous linear differential equations, each set of fourth order. The linearized equations shown in Figure 18.4 illustrate three typical features of these equations. Differentiation is indicated by the Laplace variable s, operating on the small-perturbation quantities such as u, w, в, and в. Aerodynamic variations with small-perturbation quantities, called stability derivatives, are in the “dimensional” form, suitable for closed-loop system studies and for simulation.

Finally, the derivatives are the primed form such as Lp’ rather than Lp for the rolling moment due to rolling velocity. Primed derivatives combine inertial terms with aerody­namic terms, simplifying the lateral set and putting these equations into state-variable form (Sec. 11).

The fact that the linearized equations of motion separate into two independent sets is of enormous significance. Engineers can treat airplane dynamics as two individual problems:

longitudinal stability and control, arising from the symmetric equation set, and lateral stability and control, arising from the asymmetric set. However, separation into independent longitudinal and lateral sets fails for perturbations from curvilinear or sideslipping flight. Coupled lateral-longitudinal equations of up to eighth order result. Bairstow (1920) treated perturbations from circling flight.