Page 583 - Aircraft Stuctures for Engineering Student

564 Elementary aeroelasticity

Y

Fig. 13.18 Cantilever beam of Example 13.5.

we obtain

O=C+F 6)
O=AB+xD (ii)
0 = -X2BsinXL- X2CcosXL+X2DsinhXL+X2FcoshAL (iii)

0 = -X3Bcos XL + X3C sin XL + X3D cosh XL + X3F sinh AL (iv)
From Eqs (i) and (ii), C = -F and B = -D. Thus, replacing F and D in Eqs (iii) and
(iv) we obtain
B(- sin XL - sinh XL) + C( - cos XL - cosh XL) = 0 (v)
and

B(- cos XL - cosh XL) + C(sin XL - sinh XL) = 0 (vi)
Eliminating B and C from Eqs (v) and (vi) gives
(- sin XL - sinh XL) (sinh XL - sin XL) + (cos XL + cosh XL)2 = 0

Expanding this equation, and noting that sin2 XL + cos2 XL = 1 and
cosh2 XL - sinh’ XL = 1, yields the frequency equation
cosXLcoshXL+ 1 = 0 (vii)

Equation (vii) may be solved graphically or by Newton’s method. The first three roots
XI, X2 and X3 are given by
AIL = 1.875, X2L = 4.694, X3L = 7.855

from which are found the natural frequencies corresponding to the first three normal
modes of vibration. The natural frequency of the rth mode (r 2 4) is obtained from
the approximate relationship

X,L = (r - ;)7r
and its shape in terms of a single arbitrary constant K, is
V,(z) = K,[coshX,z - cos X,z - k,(sinhX,z - sin X,z)]