# 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

small disturbances,bZ bZ

( , y, t+= t +v-(la)

where t is the coordinate in the wind-stream direction.

With the use of the relation

bZ bZ bx'+ bZ by'

bZ bZ

= cos A+ -, sin

the vertical velocity at any point isbZ bZ bZ

w (x', y', ) + v cos A+ v sin A(lb)

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, byZ=h+x'O

(2)

The velocity distribution normal to the surface, equation (b),

consequently becomeswv= +z'x+ vO cos A+v(cr+z'r) sin A

(3)

bh

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 thoseinvolving 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(4)

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.232

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### Reference the current page of this Report.

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 August 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.