Theoretical calculation of the power spectra of the rolling and yawing moments on a wing in random turbulence Page: 1 of 20
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THEORETICAL CALCULATION OF THE POWER SPECTRA OF THE ROLLING AND YAWING
MOMENTS ON A WING IN RANDOM TURBULENCE
By Jom M. EGGLESTON and FRAN-A N W. DIEDERI H
The correlation functions and power spectra of the rolling
and yawing moments on an airplane wing due to the three
components of continuous random turbulence are calculated.
The rolling moments due to the longitudinal (horizontal) and
normal (vertical) components depend on the spanwise distribu-
tions of instantaneous gust intensity, which are taken into
account by using the inherent properties of symmetry of iso-
tropic turbulence. The results consist of expressions for
correlation functions or spectra of the rolling moment in terms
of the point correlation functions of the two components of
Specific numerical calculations are made for a pair of correla-
tion functions given by simple analytic expressions which fit
available experimental data quite well. Calculations are made
for four lift ditributions. Comparison is made with the
results of previous analyses which assumed random turbulence
along the flight path and linear variations of gust velocity across
The rolling moment due to lateral (side) gusts, which is small,
is expressed in terms of the instantaneous value of the gust near
the center line of the fuselage, so that the effect of spanwise
variation in gust intensity is ignored. The yawing moments
are considered to be proportional to the rolling moments with the
constants of proportionality given by simple aerodynamic
The gust velocities acting on an airplane flying through
turbulent air are functions of position or time known only
in a statistical sense. Consequently, aerodynamic forces and
moments produced by the lifting surfaces of the airplane
can be known only in a statistical sense. If the statistical
characteristics of the turbulence are assumed to be invariant
with position along the flight path, flight through turbulent
air may be considered to be a stationary random process
and the mathematical techniques developed for such pro-
cesses (see ref. 1, for instance) may then be used in this
This approach has been adopted in many papers on this
subject, among them references 2 and 3. Inasmuch as in
these papers the motions and forces associated with the
longitudinal degrees of freedom were of primary interest, the
assumption was made, implicitly, that the gust intensity is
uniform along the span at any instant. However, for the
problem of analyzing the motions and forces associated
with the lateral degrees of freedom, this assumption is
inadequate, inasmuch as it implies that the vertical and
horizontal gusts produce zero rolling and yawing moments
on the wing. This problem has been treated in references 4
and 5 and elsewhere on the basis of the assumption that at
any instant the gust intensity varies linearly across the span.
A fundamental method of accounting for the lift on a wing
due to random variations of the gust velocities in both the
flight-path and the spanwise directions is given in reference 6
for the longitudinal response of an airplane in atmospheric
turbulence. The approach is based on the assumption that -
the turbulence is axisymmetric (according to ref. 7), so that,
at any arbitrary time or position in the turbulence, the
statistical characteristics of the turbulence encountered by an
airplane do not depend on the heading of the airplane. On
the basis of this assumption, the variation of gust intensity
across the span can be related to the variation of the gust
intensity along the flight path.
In the present report the approach of reference 6 is extended
to the calculation of the rolling and yawing moments on a
wing due directly to vertical gusts, longitudinal gusts (herein-
after referred to as horizontal gusts), and lateral or side
gusts. These moments are required as a first step in cal-
culating the motions of a complete airplane in atmospheric
turbulence; the moments due to the motions caused by these
input moments can be calculated by conventional methods
and will not be considered herein.
In the first part of the report, a theoretical analysis is made
defining the power spectra of the rolling and yawing moments
of a wing in terms of the statistical characteristics of the
atmospheric gust velocities. By using an analytical expres-
sion to define these characteristics, a numerical solution of
the lateral moments is presented in the last part of the report.
wing mean aerodynamic chord
complete elliptic integrals of the second and
first kind, respectively, of modulus k
longitudinal correlation function for iso-
Fourier transform of f
SSupersedes NACA Technical Note 3864 by John f. Eggleston and Franklin W. Diederoch, 196.
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Eggleston, John M. & Diederich, Franklin W. Theoretical calculation of the power spectra of the rolling and yawing moments on a wing in random turbulence, report, September 6, 1956; (https://digital.library.unt.edu/ark:/67531/metadc60721/m1/1/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.