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 to the influence of proportional automatic control Page: 1 of 22
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REPORT 1241
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 TO THE
INFLUENCE OF PROPORTIONAL AUTOMATIC CONTROL'
By HOWARD J. CURFMAN, JR.SUMMARY
A study has been made of the effects of two nonlinear stability
derivatives, caused by the nonlinear variations of pitching-
moment and lift coefficients with angle of attack, on the longitudi-
nal motions of an aircraft. Theoretical methods involving the
Laplace transformation 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 variations have been ap-
proximated by three linear segments. The principal means of
studying the effects 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 effects 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 with proportional control for certain
nonlinear pitching-moment variations and autopilot adjust-
ments.
INTRODUCTION
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.
The present work is presented in two independent parts.
The first part considers several analytical approaches that
may be used in the analysis and the study of the transient
t Supersedes recently declassified NACA RM LfL01, 1 1.motions of the aircraft. These methods include the use of
the operational calculus (Laplace transformation), the sta-
bility criterion of Routh, and the theories of nonlinear
mechanics. The second part presents some of the results
obtained by using electrical analog equipment. The cases
presented are for a hypothetical canard aircraft assumed to
be flying at supersonic speed.
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S
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angle of attack, positive when nose is above
relative wind vector, radians unless other-
wise noted
aircraft attitude or pitch angle, positive when
nose is above horizontal, radians
aircraft elevator deflection angle, positive
when trailing edge is down, radians unless
otherwise noted
aircraft velocity, fps
dynamic pressure, lb/sq ft
wing (reference) area, sq ft
mean aerodynamic chord, ft
mass of aircraft, slugs
pitching moment of inertia, slug-ft2
differential operator, d/dt
time
pitching-moment coefficient,
Pitching moment/gS'
lift coefficient, Lift/qS
941
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Curfman, Howard J., Jr. 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 to the influence of proportional automatic control, report, December 29, 1950; (https://digital.library.unt.edu/ark:/67531/metadc65579/m1/1/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.