428
11 Vibration of Ship and Aircraft as a Beam
added mass of water varies with each mode of vibration. Therefore, for ship’s hull,
one natural frequency is to be considered at a time. In vibration problems of other
structures, once the stiffness matrix and mass matrix are formed, all the natural
frequencies and mode shapes required can be obtained from that directly. However,
in the case of ship’s hull, the mass matrix is to be modified every time, after evaluating
each natural frequency.
In order to calculate added mass in horizontal vibration, the same curves given
by Figs. 11.1, 11.2 and 11.3 can be used, with B/d to be changed to d/2B, and L/2B
to L/d for reading C i from the graph for a particular section and for obtaining J i .
Not much work has been done on added mass moment of inertia for torsional
vibrations. Kumai [8] has made two-dimensional calculations of the added mass
moment of inertia of ship sections and has proposed the basic equation as
I = C I ρπd
4
(11.4)
where
I = added mass moment of inertia per unit length,
ρ = density of the fluid,
d = draft of the section,
and C I = coefficient, dependent on the shape of the section and on the
location of centre of rotation.
Kumai found that the coefficient for added mass moment of inertia C I depended
upon the shape of the section, the ratio of the beam-to-draft and the distance between
the centre of rotation and the waterline (y o ). For different ship sections, C I values
have been presented in Fig. 11.4 [6, 8].
11.4 The Flexibility Matrix Method for Determining
Natural Frequencies of a Free-Free Beam in Vertical
Vibration
The flexibility matrix method can be applied for determining natural frequencies
and mode shapes for a non-uniform beam having both ends free. The rigid body
displacements, which creep into the equations, are suitably eliminated with the help
of dynamic equilibrium equations. The free vibration analysis of a free-free nonuniform beam is presented in the following.
Consider a beam of variable mass and section properties, which is representative
of a ship or an aircraft (Fig. 11.5). Both ends are considered to be free. The beam is
divided into a number of stations at equal intervals. The left-hand station has been
designated at 0 and succeeding stations to the right as 1, 2, 3, . . . , n, respectively. y 0
and θ 0 are the deflection and the angular rotation for the station 0.
11 Vibration of Ship and Aircraft as a Beam
added mass of water varies with each mode of vibration. Therefore, for ship’s hull,
one natural frequency is to be considered at a time. In vibration problems of other
structures, once the stiffness matrix and mass matrix are formed, all the natural
frequencies and mode shapes required can be obtained from that directly. However,
in the case of ship’s hull, the mass matrix is to be modified every time, after evaluating
each natural frequency.
In order to calculate added mass in horizontal vibration, the same curves given
by Figs. 11.1, 11.2 and 11.3 can be used, with B/d to be changed to d/2B, and L/2B
to L/d for reading C i from the graph for a particular section and for obtaining J i .
Not much work has been done on added mass moment of inertia for torsional
vibrations. Kumai [8] has made two-dimensional calculations of the added mass
moment of inertia of ship sections and has proposed the basic equation as
I = C I ρπd
4
(11.4)
where
I = added mass moment of inertia per unit length,
ρ = density of the fluid,
d = draft of the section,
and C I = coefficient, dependent on the shape of the section and on the
location of centre of rotation.
Kumai found that the coefficient for added mass moment of inertia C I depended
upon the shape of the section, the ratio of the beam-to-draft and the distance between
the centre of rotation and the waterline (y o ). For different ship sections, C I values
have been presented in Fig. 11.4 [6, 8].
11.4 The Flexibility Matrix Method for Determining
Natural Frequencies of a Free-Free Beam in Vertical
Vibration
The flexibility matrix method can be applied for determining natural frequencies
and mode shapes for a non-uniform beam having both ends free. The rigid body
displacements, which creep into the equations, are suitably eliminated with the help
of dynamic equilibrium equations. The free vibration analysis of a free-free nonuniform beam is presented in the following.
Consider a beam of variable mass and section properties, which is representative
of a ship or an aircraft (Fig. 11.5). Both ends are considered to be free. The beam is
divided into a number of stations at equal intervals. The left-hand station has been
designated at 0 and succeeding stations to the right as 1, 2, 3, . . . , n, respectively. y 0
and θ 0 are the deflection and the angular rotation for the station 0.