Page 302 - Introduction to Naval Architecture

VIBRATION, NOISE AND SHOCK 287

Added mass
When a body vibrates in any fluid its motion causes a movement of the
fluid. The total kinetic energy of the system is the kinetic energies of
the body and the fluid it has set in motion. If the fluid is air the kinetic
energy in it is low because of its very low density. The effect on
frequency can be safely ignored. When the fluid is water, as in the ship
case, the water density is much larger and the kinetic energy of the
water cannot be neglected.
As a section of the ship moves into the water it pushes fluid aside and
as it moves out of the water the fluid returns to fill the void. The water
is thus in a constant state of oscillation and this effect is transmitted out
to the surrounding water. The phenomena is often termed the entrained
water effect and is equivalent to an increase in mass of the system. This
increase is referred to as the added virtual mass or simply added mass. It
can be shown that for a horizontal circular cylinder, immersed to half
its diameter, and executing small vertical oscillations, the virtual added
mass is equal to the mass of the cylinder. That is, the effect of the added
mass is to double the mass of the system. An effect of possibly this
magnitude must be allowed for in calculating vibratory behaviour.

Virtual added mass for ship forms
The theory used for the oscillating cylinder can be adapted to a typical
ship section. It is reasonable to assume that the added mass will depend
upon the slenderness of the section in the direction of motion, that is
the breadth to draught ratio. If it assumed the section has an area A, a
waterline breadth of b and a depth t then the cylinder case suggests that
the added mass for that section will be approximated by pA(b/2,t). It
gives the correct result for a circular cylinder and allows for the section
shape. For the complete ship this gives:

integrated over the ship length.
It would be reasonable, following this general train of thought, to say
that the total virtual mass of a ship should be able to be expressed in
terms of the ship's beam and draught. That is

where M is the mass of the ship.
3
k is a constant which Burrill suggested should be 0.5. Todd on the
other hand suggested: