Page 300 - Introduction to Naval Architecture

VIBRATION, NOISE AND SHOCK 285

Formulae for ship vibration

The formulae for uniform beams suggests that for the ship an
approximation will be given by a formula of the type:

Assuming the hull is made of various grades of steel, for which E is
effectively constant, this can be represented by:

where 7 a is the moment of inertia of the midships section.
2
This formulation was first proposed by Schlick . Suggestions for the
3
value of the constant for different ship types have been made but these
can only be very approximate because of the many variables involved in
ships. The most important are:

(1) Mass and stiffness distribution along the length.
(2) Departure from ordinary simple theory due to shear deflection
and structural discontinuities.
(3) Added mass.
(4) Rotary inertia.

Shear deflection
Simple bending theory assumes that when a beam is bent sections
which were plane and perpendicular to the neutral axis before bending
remain so after. For this condition there can be no shear stress in the
beam. The influence of shear stresses is two-fold:

(1) The stress is no longer simply proportional to the distance from
the neutral axis.
(2) Additional deflection of the beam occurs, the shear deflection.

To illustrate the effects of shear consider again the massless beam
carrying a mass Mat its centre. If the beam has a solid rectangular cross
section the deflection due to shear alone is given by BMgl/WbdG, where
b and d are the breadth and depth of the beam and G is the Modulus of
rigidity.