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National Advisory Committee for Aeronautics, Report - Theoretical and Analog Studies of the Effects of Nonlinear Stability Derivatives on the Longitudinal Motions of an Aircraft in Response to Step Control Deflections and the Influence of Proportional Automatic Control

A study has been made of the efiects of two nonlinear stability
derivatives, caused by the nonlinear variations of pitching-
moment and lift coefiicients with angle of attack, on the longitudi-
nal motions of an aircraft. Theoretical methods involving the
Laplace transformatwn and the procedures of nonlinear
mechanics have been presented along with electrical-analog re-
sults for the responses of a canard aircraft to step control deflec-
tions and to the influence of two types of proportional automatic
control.

In all cases except the example illustrating the procedures of
nonlinear mechanics, the nonlinear tariations have been ap-
proximated by three linear segments. The principal means of
studying the efiects of the nonlinearities were through the charac-
teristics of the time responses in angle of attack and, in some
cases, pitching velocity. The changes in the period and damping
of the oscillations due to the nonlinearities have been discussed.
The efi'ects of the autopilot proportionality constant on system
stability were also investigated. The occurrence of continuous
hunting oscillations was predicted and demonstrated for the at—
titude stabilization system un'th proportional control for certain
nonlinear pitching-moment variations and autopilot adjust-
ments.

In the classical treatments of the study of aircraft dynamics
an assumption is usually made that the forces and moments
resulting from small disturbances in position, velocity, and
acceleration are linear variations with these quantities.
However, some aircraft configurations do not exhibit linear
force and moment characteristics in some flight conditions
and, therefore, the linearized methods cannot be a satisfac-
tory approximation over an appreciable range of the param-
eter involved. This report is concerned with the problem
of aircraft dynamics as affected by nonlinear stability de-
rivatives and, in particular, with the short-period longitudi-
nal mode of motion of an aircraft flying at constant velocity.
Consideration is also given to the problem of applying simple
forms of automatic control to such an aircraft.