ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 3, September 2015 : 940 – 948
942 It can be known that the displacement amplitude of the shaft is
i
Y t
and the shape of the shaft is
i
x
. As
F
is zero in equation 1 for free vibration, when putting Equation 2 into 1, 1 can be induced as follows:
4 s
i i
s si
x Y t x Y t
E I ρ
3 Equation 3 can be defined as follows:
4 4
i s
i i
s si
x Y t
a x
E I Y t
ρ
4
Where
2 4
s s
i s si
a E I
ω ρ
，
s
ω
is the natural frequency. From reference [9], the mode function
i
x
can be defined as follows:
cos sin
cosh sinh
i i
si i
si i
si i
si
x A
a x B a x C
a x D a x
5
Where
i
A
，
i
B
，
i
C
，
i
D
are the real constants of each shaft segment.
2.2. Boundary Conditions and Hull Deformation Excitations
The compatibility conditions enforce continuities of the displacement field, the slope, the bending moment and the shearing force, respectively, across each support and can be
expressed as follows:
1
, ,
L R
i i
i i
U X
t U
X t
6a
1
, ,
L R
i i
i i
U X t
U X
t x
x
6b
2 2
1 2
2
, ,
L R
i i
i i
s si
s si
U X
t U
X t
E I E I
x x
6c
3 3
1 1
3 3
, ,
,
L R
R i
i i
i s
si s
si i
i i
U X t
U X
t E I
E I K U
X t
x x
6d Where
L i
X
is the left section location of the shaft on the ith bearing and
R i
X
is the right section location.
As the propulsion is on the free-free condition, the boundary of the propulsion is as follows:
2 2
1 1
1 1
2
0, 0,
s s
U t
U t
E I j
x x
ω
7a
3 2
1 1
1 1
3
0, 0,
s s
U t
E I U
t m x
ω
7b
TELKOMNIKA ISSN: 1693-6930
Automation System Vibration Analysis Taking Environmental Factors into… Zhe Tian 943
2 1
1 2
,
n s
sn
U L t
E I x
7c
3 1
1 3
,
n s sn
U L t
E I x
7d Where the propeller is considered as a mass at the end of the shaft.
Substituting Equation 5 to compatibility condition Equations 6a, 6b, 6c and 6d and boundary condition Equations 7a, 7b, 7c and 7d, a matrix equation is obtained as
follows.
1 1
2 2
1 1
1 1
2 1
2 3
2 2
1 1
1
[ ] [
] [
] [
] [
] [
] [
] [
] [
] [
]
n n
n
L R
b L
R L
R n
n L
R n
n n
n n
n
B P
C X C X
P C X
C X P
C X
C X
C X
C X P
B
8
Where
1 2
1 1
1 1
1 1
1 1
[ ]
[ ]
T T
b n
n n
n n
P P
P P
A B
C D
A B
C D
is the
coefficient matrix.
,1 ,4
,1 ,4
4 4 ,1
,4 ,1
,4
=1,...,
i i
i i
i i
i i
i
U U
U U
C i
n U
U U
U
is the matrix form of the compatibility
conditions.
,1 , 4
2 4 ,1
, 4
=1,...,
i i
i i
i
U U
B i
n U
U
is the matrix form of the boundary conditions. For the propulsion system, hull deformations are acted on the support bearings. As a
result, Dirac delta function was introduced as the hull deformation excitations
i
F
at the support bearing
i
K
.
i i
i
F F
x X
Y t δ
9 Where
i
F
is the amplitude of the force, X
i
is the location of the bearing in the coordinate system x-o-y. The compatibility conditions 6d is rewritten as follows:
3 3
1 1
3 3
, ,
,
L R
R i
i i
i s si
i i
i i
i
U X t U X t
E I EI
SU X t F
x x
10
Let
4 1 hi
i
F F
，
2 1 hb
F
， the matrix equation of the propulsion taking the
hull deformations into account is derived as follows:
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 3, September 2015 : 940 – 948
944
1 1
2 2
1 1
1 1
2 1
1 2
3 2
2 2
1 1
1
[ ] [
] [
] [
] [
] [
] [
] [
] [
] [
] [
] [
] [
] [
] [ ]
n n
n
hb L
R b
h L
R h
L R
n n
L R
hn n
n n
n hb
n
B F
P C X
C X F
P C X
C X F
P C
X C
X F
C X
C X P
F B
11
3. Vibration Analysis of the Propulsion System
