Page 593 - Aircraft Stuctures for Engineering Student

574 Elementary aeroelasticity

equations represent simple harmonic motion. Above this critical value the equations
represent divergent oscillatory motion, while at lower speeds they represent damped
oscillatory motion. For simple harmonic motion
y = yo e'"', a = a. e iwt
Substituting in Eqs (13.64) and (13.65) and rewriting in matrix form we obtain

-w (m - L?) - idj, + k - Ly J(mgc + Lii) - iwL& - La
['
w2(mgc + Mj) - iwMl - M), -J(Io - M6) - iwM& + ke - M, I{ E} =O
(13.66)
The solution of Eq. (13.66) is most readily obtained by computer4 for which several
methods are available. One method represents the motion of the system at a general
speed V by
(6+w)z (6+iw)t
Y'Yoe , a=ao
in which S + iw is one of the complex roots of the determinant of Eq. (13.66). For any
speed V the imaginary part w gives the frequency .of the oscillating system while S
represents the exponential growth rate. At low speeds the oscillation decays (6 is
negative) and at high speeds it diverges (6 is positive). Zero growth rate corresponds
to the critical flutter speed V,, which may therefore be obtained by calculating 6 for a
range of speeds and determining the value of Vf for S = 0.

13.4.3 Prevention of flutter

We have previously seen that flutter can be prevented by eliminating inertial,
aerodynamic and elastic coupling by arranging for the centre of gravity, the centre
of independence and the flexural axis of the wing section to coincide. The means by
which this may be achieved are indicated in the coupling terms in Eqs (13.64) and
(1 3.65).
In Eq. (13.65) the inertial coupling term is mgc + My in which My is usually very
much smaller than mgc. Thus, inertial coupling may be virtually eliminated by
adjusting the position of the centre of gravity of the wing section through mass
balancing so that it coincides with the flexural axis, i.e. gc = 0. The aerodynamic
coupling term Mjj vanishes, as we have seen, when the centre of independence
coincides with the flexural axis. Further, the terms M,J and L&d! are very small
and may be neglected so that Eqs (13.64) and (13.65) now reduce to
(m - Lji)ji - Lpj + (k - Ly)y - L,a = 0 (13.67)
and
(Io - Me)& - Mbd! + (ke - Ma)a = 0 (1 3.68)
The remaining coupling term L,a cannot be eliminated since the vertical force
required to maintain flight is produced by wing incidence.
Equation (13.68) governs the torsional motion of the wing section and contains no
coupling terms so that, since all the coefficients are positive at speeds below the wing