Page 573 - Aircraft Stuctures for Engineering Student

554 Elementary aeroelasticity

m2, .... mi, .... m, may be connected by any form of elastic weightless member. Thus,
if mi is the mass at a point i where the displacement is xi and 6, is the displacement at
the point i due to a unit load at a point j (note from the reciprocal theorem 6, = S,),
mlxlS,l + m2x2Sn2 + ... + mixiSni + ... + mnxnS,, + x,, = 0 1 (1 3.39)
the n equations of motion for the system are ...
...
mlxlS11 + m2x2S12 + + mixiSli + + rnnjt,S1, + x1 = 0
...
...
mlxlS21 + m2x2622 + + mixi62i + + mnxnS2, + x2 = 0
.............................................................................................
mljilSil + m2x2Siz + ... + mixiSii + ... + mnxn,4 + xi = 0

or .............................................................................................
n
mjxjS, + xi = 0 (i = 1,2, .... n) (13.40)
j= 1
Since each normal mode of the system oscillates with simple harmonic motion,
then the solution for the ith mode takes the form x = xf sin(wt + E) so that jii =
-Jxf sin(wt + E) = -w2xi. Equation (13.40) may therefore be written as
n
-JCmjsijxj +xi = o (i = 1,2,. .. ,n) (1 3.41)
j= 1
For a non-trivial solution, that is xi # 0, the determinant of Eqs (13.41) must be zero.
Hence

(Jm1611 - 1) dm2612 ... w2mi sli ... w2mn s~,,
w2mlS21 (w2m2S22 - 1) ... w2miS2i ... w2mn 6%
..................................................................................................................
w2ml ail w2m2si2 ... (w2miSii - 1) ... w m,S,
2
I .................................................................................................................. I=O
I w2mlsn1 Jm26n2 ... w miSni ... (w2wI,,~,,,, - 1) I
I
2
(13.42)
The solution of Eqs (13.42) gives the normal frequencies of vibration of the system.
The corresponding modes may then be deduced as we shall see in the following
examples.

Example 13.1
Determine the normal modes and frequencies of vibration of a weightless cantilever
supporting masses m/3 and m at points 1 and 2 as shown in Fig. 13.9. The flexural
rigidity of the cantilever is EI.

The equations of motion of the system are
(m/3)ij1611 + mij2Sl2 + w1 = 0 (9
(rn/3)ijlS2, + mij2S22 + wz = 0 (ii)