Page 592 - Aircraft Stuctures for Engineering Student

13.4 Introduction to 'flutter' 573
1

Centre of gravity Mean position

-\

Flexural axis

Fig. 13.23 Flutter of a wing section.

Myy = 1ypc4y/c
2
2
M,a = m,pc V a/c
3
M&ci = m&pc Vcilc
= m6pc4ii/c
~ ~ i i
in which m, etc. re analogous to the steady motion local pitching moment
coefficients.
Now consider the wing section shown in Fig. 13.23. The wing section is oscillating
about a mean position and its flexural and torsional stiffnesses are represented by
springs of stiffness k and ke respectively. Suppose that its instantaneous displacement
from the mean position is y, which is now taken as positive downwards. In additior, to
the aerodynamic lift and moment forces of Eqs (13.60) and (13.61) the wing section
experiences inertial and elastic forces and moments. Thus, if the mass of the wing
section is m and Io is its moment of inertia about 0, instantaneous equations of
vertical force and moment equilibrium may be written as follows. For vertical force
equilibrium
L - my i- mgcii - ky = 0 (1 3.62)
and for moment equilibrium about 0
M - I0&+mgcy - kea = 0 (13.63)
Substituting for L and M from Eqs (13.60) and (13.61) we obtain
(m - Ly)j - LFj + (k - L,)y - (mgc + Lh)2i! - L&& - L,a = 0 ( 1 3.64)
(-mgc - Mj)j - Mjj - M,y + (Io - M&)ii - M&ci + (ke - M,)a = 0 (13.65)

The terms involving y in the force equation and a in the moment equation are known
as direct terms, while those containing a in the force equation and y in the moment
equation are known as coupling terms.
The critical flutter speed Vf is contained in Eqs (13.64) and (13.65) within the terms
L,, Lj, L,, L&, My, My, M, and Mb. Its value corresponds to the condition that these