Page 306 - Introduction to Naval Architecture

VIBRATION, NOISE AND SHOCK 291

is also important. A correction is applied based on the ratio of the
rotational energy to translational energy, r r. The correction to the
05
frequency of vibration calculated ignoring rotation, is 1/(1 + r r ) ,

Direct calculation of vibration
Empirical formulae enable a first shot to be made at the frequency of
vibration. The accuracy will depend upon the amount of data available
from ships on which to base the coefficients. It is desirable to be able to
calculate values directly taking account of the specific ship character-
istics and loading. These days a full finite element analysis could be
carried out to give the vibration frequencies, including the higher
order modes. Before such methods became available there were two
methods used for calculating the two-node frequency:

(1) The deflection method or full integral method.
(2) The energy method.

The deflection method
In this method the ship is represented as a beam vibrating in simple
harmonic motion in which, at any moment, the deflection at any
position along the length is y = f(x) sin pt The function f(x) for non-
uniform mass and stiffness distribution is unknown but it can be
approximated by the curve for the free-free vibration of a uniform
beam.
Differentiating y twice with respect to time gives the acceleration at
any point as proportional to y and the square of the frequency. This
leads to the dynamic loading. Integrating again gives the shear force
and another integration gives the bending moment. A double
integration of the bending moment curve gives the deflection curve. At
each stage the constants of integration can be evaluated from the end
conditions. The deflection curve now obtained can be compared with
that originally assumed for f(x). If they differ significantly a second
approximation can be obtained by using the derived curve as the new
input to the calculation.
In using the deflection profile of a uniform beam it must be
remembered that the ship's mass is not uniformly distributed, nor is it
generally symmetrically distributed about amidships. This means that
in carrying out the integrations for shear force and bending moment
the curves produced will not close at the ends of the ship. In practice
there can be no force or moment at the ends so corrections are
needed. A bodily shift of the base line for the shear force curve and a
tilt of the bending moment curve are used.