426
11 Vibration of Ship and Aircraft as a Beam
The shifted stiffness matrix ([K ] + α[M]) will be non-singular, though [K] is
singular. It may be noted that α should be of the same order as p
2 . The eigenvalues
will be in terms of ( p
2
+ α), from where p can be evaluated. The shift approach has
the advantage that the mode shapes are not changed.
11.3 Added Mass of a Ship
Ship’s hull poses a typical problem. A ship’s hull vibrates in a liquid medium with an
air–water interface. The natural frequency of the hull, when calculated on the basis
of the vessel’s weight and elastic properties, results in a much higher value of the
calculated frequency. In order to reconcile this difference, it is necessary to include
the energy imparted to the surrounding water by the vibrating ship. The total mass
thus considered is known as the virtual mass. The addition of mass over the hull
weight due to the surrounding fluid is known as the added virtual mass or simply
added mass.
The ship is divided into a number of sections. We consider the section as vibrating,
by considering the motion of the fluid as two-dimensional. However, it is to be borne
in mind that the actual motion of the fluid is three-dimensional. Therefore, the value
of the added mass that has been obtained on the basis of two-dimensional motion is
to be corrected for actual three-dimensional motion.
Considerable work has been done for evaluating added mass of ship sections. We
shall not go into any detailed hydrodynamic theory, but will directly use the results.
Let us first consider the vertical vibration of ship’s hull. If the ship is divided into
n stations with the distance between each station being L, the added mass of water
per unit length per section is given by
M i =
1
2
πρ B
2
i C i L J i K i
(11.3)
where
ρ is the density of water,
B i is the half-breadth of the section at water line,
C i is the shape factor depending on the relation between A i /B i and B i /d i
(Fig. 11.1),
A i is the half-sectional area up to water line
d i is the section draft,
J i is the three-dimensional correction factor, which varies with each mode and is
dependent of L wl /2B i (Fig. 11.2).
L wl is the length of the ship at the waterline and
K i is a local correction factor (Fig. 11.3).
An interesting observation may be made at this stage. As the three-dimensional
correction factor is dependent on the mode of vibration, Eq. (11.3) reveals that the
11 Vibration of Ship and Aircraft as a Beam
The shifted stiffness matrix ([K ] + α[M]) will be non-singular, though [K] is
singular. It may be noted that α should be of the same order as p
2 . The eigenvalues
will be in terms of ( p
2
+ α), from where p can be evaluated. The shift approach has
the advantage that the mode shapes are not changed.
11.3 Added Mass of a Ship
Ship’s hull poses a typical problem. A ship’s hull vibrates in a liquid medium with an
air–water interface. The natural frequency of the hull, when calculated on the basis
of the vessel’s weight and elastic properties, results in a much higher value of the
calculated frequency. In order to reconcile this difference, it is necessary to include
the energy imparted to the surrounding water by the vibrating ship. The total mass
thus considered is known as the virtual mass. The addition of mass over the hull
weight due to the surrounding fluid is known as the added virtual mass or simply
added mass.
The ship is divided into a number of sections. We consider the section as vibrating,
by considering the motion of the fluid as two-dimensional. However, it is to be borne
in mind that the actual motion of the fluid is three-dimensional. Therefore, the value
of the added mass that has been obtained on the basis of two-dimensional motion is
to be corrected for actual three-dimensional motion.
Considerable work has been done for evaluating added mass of ship sections. We
shall not go into any detailed hydrodynamic theory, but will directly use the results.
Let us first consider the vertical vibration of ship’s hull. If the ship is divided into
n stations with the distance between each station being L, the added mass of water
per unit length per section is given by
M i =
1
2
πρ B
2
i C i L J i K i
(11.3)
where
ρ is the density of water,
B i is the half-breadth of the section at water line,
C i is the shape factor depending on the relation between A i /B i and B i /d i
(Fig. 11.1),
A i is the half-sectional area up to water line
d i is the section draft,
J i is the three-dimensional correction factor, which varies with each mode and is
dependent of L wl /2B i (Fig. 11.2).
L wl is the length of the ship at the waterline and
K i is a local correction factor (Fig. 11.3).
An interesting observation may be made at this stage. As the three-dimensional
correction factor is dependent on the mode of vibration, Eq. (11.3) reveals that the