• Version
• 146 Downloads
• 1.18 MB File Size
• 1 File Count
• December 13, 2016 Create Date
• December 13, 2016 Last Updated

National Advisory Committee for Aeronautics, Technical Notes - Application of the Laplace Transformation to the Solution of the Lateral and Longitudinal Stability Equations

The application of the Laplace transformation to the solution of
the lateral and longitudinal stability equations is presented. The
expressions for the time history of the motion in response to a
sinusoidal control motion are derived for the general case in which
the initial conditions, initial displacements and initial velocities,
are assumed different from zero. Some illustrative examples of the
application of the Laplace transform to ordinary linear differential
equations with constant. coefficients and a numerical example of a
specific problem are presented in appendixes.

Recent developments in piloted and pilotless aircraft, equipped
with automatic devices, have directed the attention of engineers to
the theoretical investigation of dynamic longitudinal and lateral
stability problems of aircraft designed for high—speed and high—
altitude flight. In the past, the dynamic stability investigations
were usually limited to the determination of Routh‘s condition for
stability and for the calculation of the roots of the characteristic
stability equation to determine the damping of the modes of motion
and the period of the oscillation. A more complete analysis of the
problem requires the calculation of a time history of the-airplane
motion in response to a gust disturbance or in response to the
application of the control surfaces. As the methods of classical
analysis (references 1 and 2) proved to be inadequate for this
purpose, new methods of operational mathematics, representing a
more powerful tool, were used. These methods are known today as the
Heaviside operational calculus and the Laplace transformation.

The application of the Heaviside operational calculus to the calculation
of airplane motions is discussed in references 3, 11-, and 5. However,
the Laplace transformation is considered a more powerful method than
the Heaviside operational calculus because the initial conditions of ,"
the problem, initial displacements and initial velocities, are inher—
ently taken into account by the Laplace transformation, whereas in
the Heaviside operational calculus, all initial conditions are zero.

In this paper, the Laplace transformation is applied to both the
longitudinal and lateral stability equations for the general case where
the initial.displacements and initial velocities were assumed different
from zero.