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National Advisory Committee for Aeronautics, Report - Matrix Method of Determining the Longitudinal Stability Coefficients and Frequency Response of an Aircraft from Transient Flight Data

A matrix method is presentedfor determining the longitudinal—
stability coefiicie-nts and frequency response of an aircraft
from arbitrary maneuvers. The method is devised so that it
can be applied to time-history measurements of combinations
of such simple quantities as angle of attack, pitching relocity,
load factor, elecator angle, and hinge moment to obtain the
over—all coqficients. Although the method has been devised
primarily for the evaluation of stability coeficients which are
(9’ primary interest in most aircraft loads and stability studies,
it can be used also, with a simple additional computation, to
determine the frequency—response characteristics. The entire
procedure can be applied or extended to other problems which.
can be expressed by linear difierential equations.

The longitudinal characteristics of an aircraft are often
related by a second-order linear differential equation in
which the aircraft is assumed to have freedom in pitch and
in vertical motion; changes in forward velocity are so small
that they can be neglected. In the evaluation of tail loads,
the coefficients of the differential equation and the elevator
forcing function are generally assumed to be known and the
response is to be determined. In the evaluation of gust
problems the response and the coefficients are assumed to
be known and the forcing function is to be determined.

By analogy in stability and control work, it is desirable to
determine the restoring—force and‘ damping-force coefficients
from known forcing functions and responses. In case the
damping is small enough to obtain the rate of decay (or
logarithmic decrement) and period from the oscillation, the
required damping and restoring coefficients are easily com—
puted. Models employed in rocket-powered and drop tests
can be and usually are so ballasted that such well-defined
oscillations are obtained; however, the longitudinal oscilla-
tions of piloted airplanes ordinarily are nearly critically
damped and this analysis procedure cannot be applied. In
any case. additional data and analysis are required to
evaluate the control-effectiveness coefficients.