Page 580 - Aircraft Stuctures for Engineering Student

13.3 Structural vibration 561

aZv Inertia
p~sz dt2 (force

(a)
Fig. 13.15 Vibration of a beam possessing mass.

element gives

from which, neglecting second-order terms, we obtain

aMx
s,. = - (13.43)
dZ
Considering the vertical equilibrium of the element
(Sy +2&) pA6z- d2V = 0
S,
-
-
at2
so that
( 13.44)

From basic bending theory (see Eqs (9.20))

(13.45)

It follows from Eqs (13.43), (13.44) and (13.45) that
a2
(1 3.46)

Equation (13.46) is applicable to both uniform and non-uniform beams. In the latter
case the flexural rigidity, EI, and the mass per unit length, pA, are functions of 3. For
a beam of uniform section, Eq. (13.46) reduces to
6% a2v
+
EI - PA- - 0 (13.47)
az4 at2 -
In the normal modes of vibration each element of the beam describes simple harmonic
motion; thus

v(z: t) = V(z) sin(wr + E) (13.48)