Study of effects of sweep on the flutter of cantilever wings Page: 5 of 25
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STUDY OF EFFECTS OF SWEEP ON THE FLUTTER OF CANTILEVER WINGS
The pressure difference between upper and lower surfaces of the wing at a point x is (positive downward)
p=-2p ( t
2p ( +0 ~4cos A+v 4sinA (Sa)
where 4 is in general the total potential (the sum of circulatory-flow and noncirculatory-flow potentials). The last term in
equation (Sa) is the product of the component of main-stream velocity taken along the wing and the lengthwise change
in the velocity potential and is often neglected even in steady-flow work. The question of the retention or neglect of this
last term seems partly dependent on the order in which the approximations are introduced-specifically, whether velocity
potentials for the whole flow pattern are found and then the integrated forces are determined or whether section forces are
first determined and then integrated. It seems appropriate to retain at least the noncirculatory part 4~ of 0 in the last
term of equation (5a). In view, however, of the nature of the approximate treatment of the circulatory potential and of the
inherent shortcomings of a strip analysis, in particular the neglect of lengthwise variations in wake vortex strength, compli-
eating the results by also including Or in this term does not appear worth while. (This neglect of Or and retention of 4) is
realized to involve some inconsistencies in that account may not be taken of other higher order terms associated with length-
wise variation of the wing wake, which may be of the same order as terms retained.) Thus equation (5a) becomes
p=-2( btp cosA+v - sin A) (Sb)
For harmonic motion in each degree of freedom, relations for the pressure may be integrated over the chord to yield
expressions for the air forces and moments. For the sake of separating and identifying the terms in force and moment ex-
pressions which are due solely to the inclusion of the last term in equation (5b), a special bracket L is employed. Thus
these terms may be readily omitted. Numerical checks among the calculations made for the present report showed the
effect of inclusion of the last term in equation (5b) on the calculated results to be quite small, even for 600 of sweepback
within the range of other parameters investigated.
The expressions for the aerodynamic lift (positive down) and for the moment about the elastic axis (positive leading
edge up), each perlunit length of the wing, are as follows:
P=-2rp sbC hA+.+ r. tan A+by-a (0+rr tanA) -
rpb A+. +a.o,& tan A+(E.& tan A+r, r tan A+-v~, tan'A}+
prPbla[+ vs tan A+j.r tan A+r5 r., tan2 A (6)
M = 2rp b t b fa C + 0 v0+vctan+b (4-a) (O+vr tan A)1-
rpvab a --a O+ vr tan A ]+rpba [h+v. tan A+
vIf. tan A+ v2r tan A+ ~ tan' A7 J7 rp br ( +
ver tan A+. u+ tan A+ r,' bC tan'4 (7)
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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/m1/5/: accessed November 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.