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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Barmby, J. G.; Cunningham, H. J. & Garrick, I. E. Study of effects of sweep on the flutter of cantilever wings, report, January 1, 1951; (digital.library.unt.edu/ark:/67531/metadc60354/m1/7/: accessed March 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.