Page 584 - Aircraft Stuctures for Engineering Student

13.3 Structural vibration 565

l-1 4- --- - .

.

'IT 1.758E
1.758E
-
fl = - --
=
fl
L'
L'
'IT
(a)

(b) (C)
Fig. 13.19 The first three normal modes of vibration of the cantilever beam of Example 13.5.
where
cos X,L + cosh X,L
k, = r= 1,2,3,..
sin X,L + sinh X,L '
Figure 13.19 shows the first three normal mode shapes of the cantilever and their
associated natural frequencies.

13.3.1 Approximate methods for determinina natural freauencies

The determination of natural frequencies and normal mode shapes for beams of non-
uniform section involves the solution of Eq. (13.46) and fulfilment of the appropriate
boundary conditions. However, with the exception of a few special cases, such
solutions do not exist and the natural frequencies are obtained by approximate
methods such as the Rayleigh and Rayleigh-Ritz methods which are presented
here. (A review of several methods is given in Ref. 3.) Rayleigh's method is
discussed first.
A beam vibrating in a normal or combination of normal modes possesses kinetic
energy by virtue of its motion and strain energy as a result of its displacement from
an initial unstrained condition. From the principle of conservation of energy the
sum of the kinetic and strain energies is constant with time. In computing the
strain energy U of the beam we assume that displacements are due to bending strains
only so that