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13.3 Structural vibration 563
1 L Li
I- -1
__----- --
I,-- -.
f,="E
\
2
L2
f2 = $6 f3= 9a E
2 L2
(b) (C)
Fig. 13.17 First three normal modes of vibration of the beam of Example 13.4.
The only non-trivial solution (XL # 0) of Eqs (iii) and (iv) is D = 0 and sin XL = 0.
It follows that
XL=nr n= 112131...
Therefore
IZ= 11213!...
and the normal modes of vibration are given by
nrz
v(z, t) = B, sin-sin(w,t + E,)
L
with natural frequencies
(vii)
The first three normal modes of vibration are shown in Fig. 13.17.
Example 13.5
Find the first three normal modes and corresponding natural frequencies of the
uniform cantilever beam shown in Fig. 13.18.
The boundary conditions in this problem are V = 0, dV/dz = 0 at z = 0 and
d2V/dz2 = 0, d3V/dz3 = 0 at z= L. Substituting these in turn in Eq. (13.50)
