Page 587 - Aircraft Stuctures for Engineering Student

568 Elementary aeroelasticity

where the origin for z is taken at the built-in end and a is a constant term which
includes the tip load and the flexural rigidity of the beam. From Eq. (i)

d2 V
V(L) = 2aL3 and - dz2 - 6a(L - '1
-
Substituting these values in Eq. (13.57) we obtain

dz
36EIa2 $(L - z)~
2
w1 = (ii)
dz
pAa2 Jk(3L - z)~z~ + 2m(2ai3)2
Evaluating Eq. (ii) and expressing pA in terms of m we obtain
w1 = 1.1584dZ EI
(iii)
which value is only 0.02 per cent higher than the true value given above. The
estimation of higher natural frequencies requires the assumption of further, more
complex, shapes for V(z).
It is clear from the previous elementary examples of normal mode and natural
frequency calculation that the estimation of such modes and frequencies for a
complete aircraft is a complex process. However, the aircraft designer is not restricted
to calculation for the solution of such problems, although the advent of the digital
computer has widened the scope and accuracy of this approach. Other possible
methods are to obtain the natural frequencies and modes by direct measurement
from the results of a resonance test on the actual aircraft or to carry out a similar
test on a simplified scale model. Details of resonance tests are discussed in Section
13.4. Usually a resonance test is impracticable since the designer requires the
information before the aircraft is built, although this type of test is carried out on
the completed aircraft as a design check. The alternative of building a scale model
has found favour for many years. Such models are usually designed to be as light
as possible and to represent the stiffness characteristics of the full-scale aircraft.
The inertia properties are simulated by a suitable distribution of added masses. A
full description of model construction, testing techniques and the estimation of
normal modes and frequencies is given in Ref. 3. The calculation of normal modes
and frequencies is also treated in Refs 3 and 4.

13.4 to "wutter'

We have previously defined flutter as the dynamic instability of an elastic body in an
airstream. It is found most frequently in aircraft structures subjected to large
aerodynamic loads such as wings, tail units and control surfaces. Flutter occurs at
a critical or flutter speed Vf which in turn is defined as the lowest airspeed at which
a given structure will oscillate with sustained simple harmonic motion. Flight at
speeds below and above the flutter speed represents conditions of stable and unstable
(that is divergent) structural oscillation respectively.
Generally, an elastic system having just one degree of freedom cannot be unstable
unless some peculiar mechanical characteristic exists such as a negative spring force or