438
11 Vibration of Ship and Aircraft as a Beam
Equation (11.42) can be written in matrix form as
{Y } = ω
2 [K ]{Y }
(11.43)
Size of [K] is 2n×2n. Equation (11.43) is a typical eigenvalue problem, which can
be solved by using any suitable procedure. For the present case, [K] is an unsymmetric
matrix.
11.6 Numerical Examples
The flexibility matrix method presented in Sects. 11.4 and 11.5 is somewhat mathematically involved. For ships or aircrafts, when a minimum of 20 divisions of the
structure is required for a reliable result, the computations done in different stages
are to be performed with the help of a computer by preparing the necessary software.
Two results are presented in the following.
Example 11.1 The following data are given for an oil tanker. The ship is divided
into 20 stations. The length between two consecutive stations is 11.17 m.
Determine the natural frequencies for the vertical vibration of the ship.
The problem has been solved both by the stiffness method and the flexibility
method with same number of divisions of the ship. Results from both the methods
are presented in the table. They are pretty close. It may be of some interest to note that
incorporation of shear deformation results in a reduction of the natural frequency.
This effect is more pronounced at higher modes.
Section number
Ship’s weight
(kN)
Entrained water
mass (kN)
Moment of inertia
(m4)
Effective shear
area (m 2 )
1
4310
880
47.3
0.249
2
10,850
6960
145.0
0.765
3
12,340
19,740
200.0
1.054
4
11,090
34,590
200.0
1.054
5
9860
48,410
182.8
0.962
6
60,190
59,210
192.5
1.013
7
57,020
64,900
192.5
1.013
8
42,960
66,770
192.5
1.013
9
42,960
66,770
192.5
1.013
10
43,040
66,770
192.5
1.013
11
35,110
66,770
192.5
1.013
12
27,310
66,770
192.5
1.013
13
30,500
65,910
192.5
1.013
(continued)
11 Vibration of Ship and Aircraft as a Beam
Equation (11.42) can be written in matrix form as
{Y } = ω
2 [K ]{Y }
(11.43)
Size of [K] is 2n×2n. Equation (11.43) is a typical eigenvalue problem, which can
be solved by using any suitable procedure. For the present case, [K] is an unsymmetric
matrix.
11.6 Numerical Examples
The flexibility matrix method presented in Sects. 11.4 and 11.5 is somewhat mathematically involved. For ships or aircrafts, when a minimum of 20 divisions of the
structure is required for a reliable result, the computations done in different stages
are to be performed with the help of a computer by preparing the necessary software.
Two results are presented in the following.
Example 11.1 The following data are given for an oil tanker. The ship is divided
into 20 stations. The length between two consecutive stations is 11.17 m.
Determine the natural frequencies for the vertical vibration of the ship.
The problem has been solved both by the stiffness method and the flexibility
method with same number of divisions of the ship. Results from both the methods
are presented in the table. They are pretty close. It may be of some interest to note that
incorporation of shear deformation results in a reduction of the natural frequency.
This effect is more pronounced at higher modes.
Section number
Ship’s weight
(kN)
Entrained water
mass (kN)
Moment of inertia
(m4)
Effective shear
area (m 2 )
1
4310
880
47.3
0.249
2
10,850
6960
145.0
0.765
3
12,340
19,740
200.0
1.054
4
11,090
34,590
200.0
1.054
5
9860
48,410
182.8
0.962
6
60,190
59,210
192.5
1.013
7
57,020
64,900
192.5
1.013
8
42,960
66,770
192.5
1.013
9
42,960
66,770
192.5
1.013
10
43,040
66,770
192.5
1.013
11
35,110
66,770
192.5
1.013
12
27,310
66,770
192.5
1.013
13
30,500
65,910
192.5
1.013
(continued)