Theory of Wings in Nonstationary Flow Page: 3 of 22
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NACA M No. 1154
Then the equation of motion of the material point will be
d + 2 -- + 2x 0
The integral of this equation, if p is small, is of the form
z = ae1R sin (qk2 -p2t + c)
where a and are arbitrary constants. We see that if the
quantity p is positive the material point will be in harmonic o
vibratory motion with damping. We shall now assume that itself
depends on a certain parameter w so that p - p(w), where for
w = wcr p = 0 and for w>Wvor p is negative, that is, equal
to - X (where 7>0). Then for w = wor the above equation assumes
x - a sin (kt + )
that is, we shall have the usual harmonic vibration of a poift, but
for w>Wcr we obtain
x= aet sin ( A 2t + )
that is, the amplitude of the vibration begins to increase exponen-
tially. Thus, whereas in the case of resonance the amplitude of
vibration has, as i known, the form at, in the case of flutter
it has the form ae". Table I shows the increase of the amplitude
with time in resonance and in flutter for different values of the
t at ae,It ae0,5t
0 0 a a a
1 a 1,105a l,649a 2,71Ca
2 2a 1,221a 2,718a 7,389a
3 a 1,351a 4,482a 20,086a
4 4a 1,492a 7,389a 54,598a
5 5a 1,649a 12,183a 148,413a
We note that whatever the value of the positive number A the
ratio e t/t approaches infinity with increasing t. For the air-
plane the parameter w is the velocity of flight where the velocity
w = wer, the value at which the coefficient p becomes zero, is
called the critical flatter velocity.
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Nekrasov, A. I. Theory of Wings in Nonstationary Flow, report, June 1947; (digital.library.unt.edu/ark:/67531/metadc63973/m1/3/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.