# Study of effects of sweep on the flutter of cantilever wings Page: 7 of 25

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STUDY OF EFFECTS OF SWEEP ON THE FLUTTER OF CA NTILEVER WINGS

In the subsequent treatment the reader will notice that

in some expressions, namely for force and moment, h and 0

can conveniently and logically be retained in their complex

form. In other expressions, notably for energy, one is

forced to utilize A and 0 as real quantities. Appropriate

statements will be made where necessary.

For each degree of freedom an equation of equilibrium

may be obtained from Lagrange's equationd (bT 6T bU=Q

dt 4ia _ qg-, 6 q,(13)

where q, is a generalized coordinate and Q, is the correspond-

ing generalized force. The kinetic energy of the mechanical

system is

T=4 2, ,j mLf,(y 'dy a [y fd

22 mLftJy lfo ojv f( dy' +o mXab [fL(y')] Lf,(y')1 dy'

(14)

where h and 0 here and in the subsequent equations (15) are

to be interpreted as real in order that the energy be always

positive (or zero), and for definiteness can be regarded as the

real parts of ,heol* and Oe'," respectively; and where

m mass of wing per unit length, slugs per foot

I, mass moment of inertia of wing about its elastic axis

per unit length, slug-feet' per foot

X,,b distance of sectional center of gravity from the elastic

axis, positive rearward, feet

The potential energy of the mechanical system may be

expressed in a form not involving bending-torsion cross-

stiffness terms:u1 2 'E f d + I d'G'aj fd"uJ

U E dy', dy'+ dy'"(15a)

where

El bending stiffness, pound-feet2

GJ torsional stiffness, pound-feet'

If Rayleigh type approximations are used to introduce

frequency, the expression for the potential energy may be

written in a more convenient form:U 1 Wfo mfdy'+ fa I,fldy'

Another expression for the potential energy is

U = f CJddy'+ 1 Of' C.f,dy'

a 2-(15b)

(1i5c)

The effective spring constants Ca and C, correspond to unit

length of wing and thus conform to their use in references

5 to 7. The constants are effectively defined byC2 f'dy'

f mfldy',I.fy'dy'

These effective spring constants are related to the frequencies

associated with the chosen modes. For so-called uncoupledmodes the frequencies appropriate to pure modes (obtained

by proper constraints) are often used. On the other hand,

employment of the normal or natural modes and frequencies

appropriate to them, which might be obtained by proper

ground test or by calculation, may be preferred. In either

case the convenience of not having cross-stiffness terms in

the potential-energy expression is noted.

Application is now made to obtain the equation of equilib-

rium in the bending degree of freedom. Equation (13)

becomes(16)

The term (}Q represents all the bending forces not derivable

from the potential-energy function and consists of the aero-

dynamic forces together with the structural damping

forces. The virtual work aTVt done on the wing by these

forces as the wing moves through the virtual displacements

mh and 50 is

a=f" P- C )ah+ l- C. ~ 80]dy'

_f(P-- m w 2 fdy'<(.t- I2c f f, dy' a46= QA 5-Q- Q#

(17)

where

gA structural damping coefficient for bending vibration

g. structural damping coefficient for torsional vibration

In this expression the aerodynamic forces appropriate to

sinusoidal oscillations are used. The application of the

structural damping as in equation (17) (proportional to

deflection and in phase with velocity) corresponds to the

manner in which it is introduced in reference 7. In accord-

ance with the preceding development, the aerodynamic and

structural damping forces and moments in equation (17)

are regarded as compIex, but the virtual displacements 5h

and 50 should be considered real. Thus, the physically

significant part of the resulting expression for virtual work is

the real part. Since the subsequent analysis reverts to ex-

pressions for forces and moments, no further qualifications on

the use of h and 0 in their convenient complex forms are

needed.

For the half-wing

Q=f '(P- m&2&LfA)fady'

= Fp b A

Aa) (tan .))fd f+ A, b (tan'A) f4 +

dfo

OA.,fe f+0Acb (tan A fA +

a, Pd'f 1 (18)

kP dy bK C7235

+ b= QA

diit bh A b

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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/m1/7/: accessed December 11, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.