Study of effects of sweep on the flutter of cantilever wings Page: 6 of 25
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REPORT 1014-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
C =C(k,)= F(k,)+ iG(k,)
is the function associated with the wake developed by
Theodorsen in reference 5; the reduced frequency parameter
k. is defined by
kn -- cos A
n v COS h
As has already been stated, the foregoing expressions were
developed and apply for steady sinusoidal oscillations,
h = hi (y') e*
The amplitude, velocity, and acceleration in each degree of
freedom are related as in the degree of freedom h; that is,
h = -Wh
Expressions for force and moment.-With the use of such
relations, equations (6) and (7) may be put into the form
P= - rp b3 (h Be + OBa)
Ala= -rp b tcO (hBa + OBe)
Be~bAa+a tean A -i (-+ As)+
B=eAca+j b tan A (Acr)+F tan' Ak,
Bea=1 Ae.+E tan A (- iA (a)+A4)+
D r ka
in which the four following coefficients:
8 2 k
wica- a a
,2F _ I F2G i [, ) 2
Aa=a 1 +a 2G (+it4)2( Q 2F)
are identical with those used in the case of the unswept wing.
A ,4 [-iF+ a)+- a) ca]
A,= iu - (++a' ) i ([a)) 4 - A]
It is of interest to note that equations (6) and (7) reduce,
for the case of the wing in steady flow (k,= 0), to
P= --2rpbv. 2[+O tan A+ rb tan A + --a)+
b- - tan2 A- b' i- tan' A
2z by? 2 by'LI 11_
a= 2rpb's2[(o+cr tan A)Q+a)+rab tanA
U - tan' A- b' + a6r tan' A)
2 by' 1) \TC by 1tll 11
per unit length of wing.
Introduction of modes.-Equations (10a) and (11a) give
the total aerodynamic force and moment on a segment of a
sweptback wing oscillating in a simplo harmonic manner.
Relations for mechanical equilibrium applicable to a wing
segment may be set up, but it is preferable to bring in directly
the three-dimensional-mode considerations. (See for example,
reference 6.) This end may be readily accomplished by
the combined use of Rayleigh type approximations and the
classical methods of Lagrange. The vibrations at flutter
are assumed to consist of a combination of fixed mode shapes,
each mode shape representing a degree of freedom associated
with a generalized coordinate. The total mechanical
energy, the potential energy, and the work done by applied
forces, aerodynamic and structural, are then obtained by the
integration of the section characteristics over the span.
The Rayleigh type approxhnation enters in the represen ta-
tion of the potential energy in terms of the uncoupled
As is customary, the modes are introduced into the problem
as varying sinusoidally with time. For the purpose of sin-
plicity of analysis, one bending degree of freedom and one
torsion degree of freedom are carried through in the present
development. Actually, any number of degrees of freedom
may be added if desired, exactly as with an unswept wing.
Let the mode shapes be represented by
where h=h,,"' is the generalized coordinate in the bending
degree of freedom, and 0= 0,e'"' is the generalized coordinate
in the torsion degree of freedom. (In a more general treat-
ment the mode shapes must be solved, but in this procedure
fA,(y') and f(y') are chosen, ordinarily as real functions of y'.
Complex functions may be used to represent twisted modes.)
The constants he, and 0o are in general complex and thus
signify the phase difference between the two degrees of
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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/m1/6/: accessed September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.