Page 299 - Introduction to Naval Architecture

284 VIBRATION, NOISE AND SHOCK

b

C
Figure 11.4 (a) One-node; (b) Two-node; (c) Three-node

Flexing of a beam
Before proceeding to the ship it is instructive to consider the flexural
vibration of a simple beam. Take a beam of negligible mass of length Z,
supported at its ends and carrying a mass Mat its centre. Under static
conditions the deflection at mid span will be MgZ3/48EL
If the beam is deflected y from its ~quilibrium position the restoring
force will be y( 48EI) / 13. Thus 48EI/p is equivalent to the spring stiffness k
considered earlier. It follows that the frequency of vibration will be:

-(-) 48EI Om5 = 1.103($) 0.5
I
zn MP
It can be shown that if the mass M is uniformly distributed along the
beam the frequency of vibration becomes:

where n is any integer. For n = 1, the frequency is 1.57(EI/M13)0*5. The
frequency in which a beam vibrates depends upon the method of
support. For vibrations in the free-free mode the frequency becomes:
0.5

where n = 4.73,7.85 and 10.99 for the two-, three- and four-node modes
respectively.