Page 577 - Aircraft Stuctures for Engineering Student

558 Elementary aeroelasticity

Hence

dz

jo EI
612 = 62, = (21 - z)(Z - z) dz
from which we obtain
61 13 51
611=E, 622=-, slz=621=-
3EI 6EI
Writing X=mZ3/6EI and solving Eqs (i) and (ii) in an identical manner to the
solution of Eqs (i) and (ii) in Example 13.1 results in a quadratic in Xu2, namely

188(X~~)~ (vii)
- 44Xw2 + 1 = 0
Solving Eq. (vii) we obtain
44fd442-4x188x1
Xl3 =
376
which gives
xW2 = 0.21 or 0.027
The lowest natural frequency therefore corresponds to Xu2 = 0.027 and is

27r
Example 13.3
Determine the natural frequencies of the system shown in Fig. 13.13 and sketch the
normal modes. The flexural rigidity EI of the weightless beam is 1.44 x lo6 N m2,
1 = 0.76 m, the radius of gyration r of the mass m is 0.152 m and its weight is 1435 N.

Fig. 13.13 Massheam system for Example 13.3.