# Study of effects of sweep on the flutter of cantilever wings Page: 3 of 25

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STUDY OF EFFECTS OF SWEEP ON THE FLU'flER OF CANTILEVER WINGS

ra nondimensional radius of gyration of wing

about elastic axis ( mj

El bending stiffness, pound-inches' in tables,

pound-feet2 in analysis

GJ torsional stiffness, pound-inche-s2 in tables,

pound-feet in analysis

g structural damping coefficient for bending

vibration

g structural damping coefficient for torsional

vibration

P oscillator lift per unit length. positive down-

ward (defined in equation (6))

10 oscillatory moment about elastic axis, positive

leading edge up (defined in equation (7))

[ ] a special bracket used to identify terms which

are due solely to inclusion of the last term

in equation (5b)

In order to preserve continuity and to facilitate comparison

%with previous work on the unswept wing, the subscript a

rather than 0 is retained with certain quantities to refer to

.te torsional degree of freedom.

ANALYTICAL INVESTIGATION

GENERAL

Assumptions.--An attempt is first made to point out the

main assumptions which seem to be applicable for swept

wings of moderate taper and of high or moderate length-

chord ratios.

(a) The assumptions, such as small disturbances and poten-

tial flow, commonly employed in linearized treatment of

unswept wings in an ideal incompressible fluid are made.

(b) The structural behavior is such that over the main part

of the wing the elastic axis may be considered straight. The

wing is also considered sufficiently stiff at the root so that it

behaves as if it were clamped normal to the elastic axis.

An effective length 1' needed for integration reasons may be

defined (for example, as in fig. 1). The angle of sweepback

is measured in the plane of the wing from the direction nor-

mal to the air stream to the elastic axis. All section param-

eters such as semichord, locations of elastic axis and center

of gravity, radius of gyration, and so forth, are based on

sections normal to the elastic axis.

Actfu root 3''

Effective root >,a

X 'I . sectkrns normal to

be eratsc oxis-- +1<r A

h

a x

SecCtion B-B

PER;Rn 1.-Nonunitfrn mwept wing treated in the present analyst.(c) The aerodynamic behavior is such that any section dy'

of the wing normal to the elastic axis, taken in the direction

of the component r cos A of the main-stream velocity, gener-

ates a velocity potential associated with a uniform infinite

swept wing having the same instantaneous distribution over

the chord of velocity normal to the wing surface as does the

actual section.

Additional remarks on these assumptions are appropriate.

With regard to assumption (a), in accordance with lineariza-

tion of the problem, the boundary conditions are stated and

treated with respect to a reference surface, in this case a

plane, containing the mean equilibrium position of the wing

and the main-stream velocity. Furthermore, incompressible

flow is assumed in order to avoid complexity of the analysis,

although modifications due to Mach number effects can be

added. Such modifications may be based, for example, for

wings having large length-chord ratios, on existing theoretical

calculations of aerodynamic coefficients for subsonic or super-

sonic two-dimensional flow appropriate to the component

v cos A. On the other hand the modifications may be partly

empirical, especially for "transonic" conditions and for small

length-chord ratios. The transonic conditions and the gen-

eral aerodynamic behavior of swept wings may depend, for

large length chord ratios, on the component v cos A, bu the

dependence may shift to the stream velocity v for small length-.

chord ratios.

With respect to assumption (b), results of analyses of and

experiment on unswept wings having low ratios of bending

frequency to torsion frequency show that small variations of

position of the elastic axis are not important. The assump-

tion of a straight elastic axis over the main part of a swept

wing, similarly, is not critical for many cases. This assump-

tion is made for convenience, however, and modifications for

a curved elastic axis can be made when necessary, for example,

for plate-like wings. Small differences in the angle of sweep-

back of the leading edge, quarter-chord line, elastic axis, and

so forth, are neglected. The analysis could be further modi-

fied to take into account variation of the angle of sweepback

along the length of the wing.

Assumption (c) implies that associated with the action.

of the wing in pushing air downward there is a noncirculatory

potential-type flow similar to that around sections of an

infinite flat-plate wing. Furthermore, as in the case of the

unswept airfoil, a circulatory potential-type flow is generated

in which for the swept airfoil the component c cos A is

decisive in fixing the circulation. (This assumption differs

from that made in the strip theory of references 3 and 4

which employs the main-stream velocity together with

sections of the wings parallel to the stream direction.)

Effects of the floating of the wake in the stream direction

rather than in the direction of v cos A and induced effects

of variation of the strength of the wake in the wing-length

direction are neglected, as are three-dimensional tip effects.

For large values of the reduced frequency k,, a given segment

of the wing might be influenced chiefly by the nearby wakeand the correction would be small. On the other hand, for

small values of k. a given segment might be influenced by a

more widespread portion of the wake; corrections for this

condition may possibly be based on knowledge of the static231

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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/m1/3/: accessed July 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.