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National Advisory Committee for Aeronautics, Report - Comparison of Several Methods for Obtaining the Time Response of Linear Systems to Either a Unit Impulse or Arbitrary

Several methods of obtaining the time response of linear
systems to either a unit impulse or an arbitrary input from
frequency-response data are described and compared.

Comparisons indicate that all the methods give good accuracy
when applied to a second-order system; the main difierence is
the required computing time. Several of the methods when
applied to higher order systems require excessive computing
time in order to obtain the same degree of accuracy. The
methods generally classified as inverse Laplace transform
methods were found to be most reflective in determining the
response to a unit impulse from frequency—response data of
higher order systems.

Some discussion and examples are given of the use of the
methods as fiight—data—analysis techniques in predicting loads
and motions of a flexible aircraft on the basis of simple calcu-
lations when the aircraft frequency response is known.

The frequency-response type of analysis used on linear
systems has found extensive application in the field of air-
craft stability and in the determination of overall dynamic
characteristics of an aircraft. In fact the current trend
to perform analysis on flight test data in frequency-response
form appears to be gaining favor especially in the case of
flexible aircraft. Considerable emphasis has, therefore, been
given to methods of determining the frequency response of
a system from transient responses. A concise résumé and
comparison of methods for obtaining the frequency response
from transient responses are presented in reference 1.

This report briefly considers the next step in the process—
the converting of the information contained in a frequency
response to the time plane in the form of the response to
a unit impulse. For certain purposes this conversion gives
data in more useful form. The response of a linear system
to a unit impulse may be used in conjunction with Duhamel's
(superposition) integral to determine (1) the aircraft tran-
sient response to any type of input or (2) the input required
to cause any required aircraft transient response.