Page 581 - Aircraft Stuctures for Engineering Student

562 Elementary aeroelasticity

where V(z) is the amplitude of the vibration at any section z. Substituting for TJ from
Eq. (13.48) in Eq. (13.47) yields
d4 V PAW’ V=O ( 13.49)
d# EI
Equation (13.49) is a fourth-order differential equation of standard form having the
general solution
V = Bsin Xz + Ccos XZ + D sinh XZ + F cosh Xz (1 3 .SO)
where

PAJ
4 =-
EI
and B, C, D and F are unknown constants which are determined from the boundary
conditions of the beam. The ends of the beam may be:

(1) simply supported or pinned, in which case the displacement and bending
moment are zero, and therefore in terms of the function V(z) we have V = 0 and
d2V/& = 0;
(2) fixed, giving zero displacement and slope, that is V = 0 and dV/dz = 0;
(3) free, for which the bending moment and shear force are zero, hence
d2V/d2 = 0 and, from Eq. (13.43), d3V/dz3 = 0.

Example 13.4
Determine the first three normal modes of vibration and the corresponding natural
frequencies of the uniform, simply supported beam shown in Fig. 13.16.

Since both ends of the beam are simply supported, V = 0 and d2 V/d2 = 0 at z = 0
and z = L. From the first of these conditions and Eq. (13.50) we have
O=C+F 6)
and from the second
0 = -X2C + X2F (ii)
Hence C = F = 0. Applying the above boundary conditions at z = L gives

0 = BsinXL+DsinhXL (iii)
and
0 = -X2B sin XL + X2D sinh XL 6.1

Fig. 13.16 Beam of Example 13.4.