Study of effects of sweep on the flutter of cantilever wings Page: 4 of 25
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2I2REPORT 1014-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
case (for example, slope of the lift curve). As the angle of
sweep approaches 900, obviously the mechanism for the
generation of lift is different from the one postulated here;
for example, a tip condition may replace the trailing-edge
condition and c6hsiderations of very small aspect ratio arise.
Basic considerations.-Consider the configuration shown
in figure 1 where the vertical coordinate of the wing surface
is denoted by z'=Z(x',y',t) (positive downward). The
effect of the position and motion of the wing may be given
by the disturbanice-velocity distribution 'to be superposed
on the uniform stream in order to represent the condition
of tangential flow at the wing surface. This velocity
distribution normal to the surface (positive upward) is, for
( , y, t+= t +v-
where t is the coordinate in the wind-stream direction.
With the use of the relation
bZ bZ bx'+ bZ by'
= cos A+ -, sin
the vertical velocity at any point is
bZ bZ bZ
w (x', y', ) + v cos A+ v sin A
Let the wing be bending so that a segment dy' (see fig. 1) is
displaced from its equilibrium position by an incremental
distance h (positive down) and also let the wing segment be
twisting about the elastic axis through an incremental angle 0
(positive leading edge up). The position of each point of
the segment may be defined, for small deflections, by
The velocity distribution normal to the surface, equation (b),
wv= +z'x+ vO cos A+v(cr+z'r) sin A
where = is the local bending slope of the elastic axis
and is thus analogous to dihedral, and where r=y is the
local change of twist of the elastic axis.
In accordance with assumption (c) the noncirculatory-flow
velocity potentials associated with the vertical-velocity dis-
tribution are first needed. In equation (3) the terms involv-
ing K, 0, and a are constant across the chord, whereas those
involving 0 and r vary in a lulear manner. The noncircula-
tory velocity potentials as in reference 5 and the new poten-
tials associated with a and r are
, = v b 1- 2
,o= v,a tan Ab i--
-=xb 2a) 1/1 -x'
=v2r tan Ab
= ar tan Ab' b2-a) N /1 - Tz s
where v.=v cos A and x is the nondimensional chordwise
coordinate measured from the midehord as in reference 5
and related to the coordinate x' in the manner
The velocity potential for the circulatory flow associated
with the wake may be developed on the basis of assumption (c)
and the concepts for the infinite unswept wing introduced
in reference 5. (Thus the circulatory-flow pattern for a sec-
tion dy' of the finite swept wing is to be obtained from the
corresponding flow pattern for an infinite uniform yawed
wing. This infinite wing is assumed to have undergone har-
monic oscillations for a long time; the full wake is established,
remains where formed, and consequently is harmonically
distributed in space. For the infinite uniform yawed wing,
results for the circulatory flow are like those of reference 5
with v replaced by the component v, and with the addition
of terms to take care of a and r.) In particular, the strength
of the wake acting on each section is determined by the condi-
tion of smooth flow (the velocity remaining finite) at the trail-
ing edge. This condition is utilized in the form 9 (r+4.v)
is equal to a finite quantity at the trailing edge (where *r is
the velocity potential due to the vorticity in the wake, and
4w is the total noncirculatory velocity potential), and this
condition leads to a relation analogous to equation (VII) of
reference 5 involving the basic quantity
Q=h+v,0+v.ar tan A+b (-a)(+v,, tan A)
which occurs in the terms associated with the wake. The net
result of these considerations is that the circulatory-flow
velocity potential may be regarded as determined.
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Barmby, J. G.; Cunningham, H. J. & Garrick, I. E. Study of effects of sweep on the flutter of cantilever wings, report, September 9, 1948; (digital.library.unt.edu/ark:/67531/metadc60354/m1/4/: accessed November 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.