1041
1 INTRODUCTION
The shaft line performs coupled lateral axial and
torsional vibrations under excitation effect generated
by the main engine, propeller and ship hull. In theory,
when the excitation amplitudes are much larger than
the system's dynamic stiffness, resonance effects
become dominant, leading to excessive vibrations.
Consequently, reducing vibrations and stresses can be
achieved by decreasing excitation amplitudes, altering
the dominant excitation component, or enhancing the
structural dynamic stiffness to shift resonance through
a change in the natural frequency.
Forced lateral vibrations from the propulsion
shafting system lead to fatigue failure of the bracket
and aft stern tube bearing, destruction of high-speed
shafts with universal joints, and contribute to noise and
hull vibrations [1].The continual increase in propelling
power and ship length, the use of high-strength steel,
the reduction in wear and corrosion allowances, and
the bolder use of lower safety factors lead to an
increased disparity between shaft-line stiffness and
hull flexibility, and tend to amplify the vibration
phenomenon [2]. Lateral vibration may be the cause of
dynamic bearing reactions and additional cyclic
bending stresses in propeller shaft. Nowadays, all
Classification Societies require the propulsion shafting
whirling vibration calculation, named in some Class
Rules as lateral or bending vibration [1]. Systems are
required to operate with acceptable levels of axial
bending and torsional vibrations during continuous
and transient operating, as stipulated by the makers
and pertinent exigence [3]. For all main propulsion
shafting systems, shipbuilders must ensure that the
lateral vibration characteristics remain satisfactory
throughout the entire speed range [4]. In the well-
known classification societies DNV [5], we can find
rules for vibration analysis of shaft line. These range
from the calculation of natural frequencies and forced
vibrations, including whirling and axial vibrations, to
machinery simulations such as torsional vibration
analysis. Zou et al. [6] carried out a dynamic nonlinear-
coupled longitudinal-transverse model of a propulsion
Calculation of Free and Forced Lateral Vibrations
of a Shaft Line Using Solution Coefficient Vectors
A. Hamiani, K. Boumediene & C. Kandouci
University of Sciences and Technology USTO-MB, Oran, Algeria
ABSTRACT: In this paper, the free and forced lateral vibrations of a marine propulsion shafting system on
anisotropic supports are computed using the vectors of solution coefficients method. The transverse vibrations
are considered in two orthogonal planes. The effect of the oil film in the bearings is taken into account. The stern
tube bearing is considered as an elastic support of Winkler’s type. The whirling mode shapes for free vibrations
of the system were calculated using the algebraic complements according to I.P. Natanson. The results show a
decrease in vertical static deformation due to the distributed weight of the shaft line during the operation of the
propelling system. This straightening of the propeller shaft explains the increase in the thickness of the oil film in
the stern bearing, as observed by naval engineers.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 3
September 2025
DOI: 10.12716/1001.19.03.39
1042
shafting system and studied the transverse super
harmonic resonances under the propeller blade
frequency excitation using the method of multiple
scales. Busquier et al. [7] considered only the propeller
shaft as a self-sustained whirling vibration system and
determined its critical speeds by solving the Van der
Pol equation. Kim et al. [8] compared the shaft line
flexibility and whirling vibration characteristics of
double stern tube bearings with those of a single stern
tube bearing in a 50000 DWT Oil Tanker. They
proposed a new optimum method to provide a
satisfactory shaft flexibility in a vessel with a single
stern tube bearing, thereby avoiding resonance in
whirling vibrations. Qianwen et al. [9] studied the
coupled lateral and torsional vibrations under both
conditions of idling and loading at different rotational
velocity using a finite element model of the marine
propulsion shafting system. Zhang et al. [10] studied
the dynamic characteristics of the shafting system
using a three-DOF transfer matrix-coupling model.Qin
H. & al. [11] suggested that incorporating
electromagnetic bearings to work in parallel with the
rear bearing could effectively reduce lateral vibration
transmission. The stern tube bearing sustains the
greatest load due to its extended length [12]. The stern
tube bearing should be considered as a continuous
elastic support at the nominal speed of the
propulsion system, whereas the intermediate bearings
can be represented as point supports [13]. In this paper,
the stern tube bearing is modelled as an elastic support
of Winkler’s type. The stiffness coefficients of the oil
film in the stern tube and intermediate bearings are
calculated using the Sommerfeld number. Bending
vibrations and static deformations in the horizontal
and vertical planes are computed within fixed local
coordinate systems using the relationships between the
solution coefficient vectors of the governing
differential equations. The traditional Transfer Matrix
Method (TMM) [14] involves sequentially multiplying
transfer matrices, which can lead to numerical
instability and computational inefficiency, especially
for long shaft lines. However, with the presented
approach, the product of multiplied matrices can be
represented as a single matrix, when adjacent shaft
segments have the same diameters and physical
properties, thereby enhancing computational
efficiency and numerical stability. From the available
literature, we can cite the application of this approach
to develop a mathematical model describing linear
axial and torsional vibrations of shaft line [15], and to
study the whirling vibrations of a shaft carrying three
rigid disks resting on rigid supports using the
Bernoulli–Euler beam theory [16], and the Timoshenko
beam theory [17]. More recently, Kandouci & al. [18]
extended this technique to study the whirling
vibrations of unbalanced rotor resting on viscoelastic
supports.
2 PHYSICAL MODEL OF THE MARINE
PROPULSION SHAFTING SYSTEM
Figure 1 provides a physical model of a marine
propulsion shafting system made up of n uniform shaft
segments [(1), (2), …, (i), (i+1), …, (n-1), (n)]. Each shaft
segment (i) is assigned a length Li and a fixed
coordinate system (xi, yi, zi) with its axes parallel to
those of the fixed reference system (X, Y, Z).
Figure 1. Physical model of the marine propulsion shafting
system.
The propeller, coupling flange and sun gear of
reduction gearbox are considered to be lumped
masses. The division is performed at the junction of
two adjacent shaft segments that differ in diameter
and/or mechanical properties, at the cross-section
passing through the center of gravity of disk (i), where
shaft segments (i) and (i+1) connect, and at the end of
the i-th shaft segment mounted on an elastic support
(i), such as intermediate bearings and gearbox
bearings.
M0 denotes the propeller inertia matrix, which
includes the added mass of seawater,
0 02 03 05 06 0
, , ,
T
f f f f f f
==
refers to the vector of
hydrodynamic excitations acting on the propeller, =2
and =3 correspond to the vertical and horizontal
forces, respectively, =5 and =6 concern the bending
moments in the horizontal and vertical planes,
respectively, Mi-1 and Mi-4 are the inertia matrices
associated with the flange couplings located at the ends
of the (i-1)th and (n-4)th shaft segments, respectively,
Mn-1 and Mn indicate the inertia matrices of the sun gear
in the reduction gearbox and the mass at the ends of
the (n-2)th and nth shaft segments, respectively, Ki is
the stiffness matrix of the oil film in the intermediate
bearing, situated at the end of the ith shaft segment, Kn-3
and Kn-1 are the stiffness matrices of the oil film in the
reduction gearbox bearings, located at the ends of the
(n-3)th and (n-1)th shaft segments, respectively, k
V
oil
and k
H
oil represent the stiffness coefficients of the oil
film in the stern tube bearing, modelled as a Winkler-
type elastic support, in the vertical and horizontal
directions, respectively.
2.1 Equations of motion
Neglecting the gyroscopic effect in the shaft, the
uncoupled differential equations describing the
vertical and horizontal transverse vibrations of the ith
rotating shaft segment are:
24
,,
,
24
( , ) ( , )
0
i y i y
i i i i z
w x t w x t
A E I
tx
+=
(2.1)
24
,,
,
24
( , ) ( , )
0
i z i z
i i i y i
w x t w x t
A E I
tx
+=
(2.2)
where wi,y(x,t) and wi,z(x,t) represent the vertical and
horizontal transverse vibrations of the cross-sectional
centroid of the ith shaft segment at the axial coordinate
xi=x and time t, respectively. Ei(n/m
2
) and
i(kg/m
3
) are
modulus of elasticity and mass density of the ith shaft
segment. Ii,y(m
4
) and Ii,z(m
4
) are diametric moments of
1043
inertia of the cross-sectional area Ai about the yi- or zi-
axis. In the case of a circular shaft, Ii,y=Ii,z=Ii
The vibrations of the shaft segment in the stern
bearing (i= 2) are described as follows
24
2, 2,
2 2 2 2 2,
24
( , ) ( , )
( , ) 0
yy
V
oil y
w x t w x t
A E I k w x t
tx
+ + =
(2.3)
24
2, 2,
2 2 2 2 2,
24
( , ) ( , )
( , ) 0
zz
H
oil z
w x t w x t
A E I k w x t
tx
+ + =
(2.4)
The vibration vector at the axial coordinate xi=x and
time t is designated as:
,
,,
,
( , ), ( , ), ( , ), ( , )
( , )
T
iz
i y i z
i i y
w x t w x t x t x t
w x t
=
The rotational vibrations, bending moment and
shear forces, as described by Bernoulli–Euler beam
theory, are expressed as follows:
,
,
,,
( , )
( , )
( , ) , ( , )
iy
iz
i y i z
w x t
w x t
x t x t
xx
= − =
(2.5)
3
3
,
,
,,
33
( , )
( , )
( , ) , ( , )
iy
iz
i y i z
i i i i
w x t
w x t
Q x t E I Q x t E I
xx
= − = −
(2.6)
2
2
,
,
,,
22
( , )
( , )
( , ) , ( , )
iy
iz
i y i i i z i i
w x t
w x t
M x t E I M x t E I
xx
= = −
(2.7)
The internal force and moment vector at axial
coordinate xi=x and time t is expressed as:
, , , ,
( , ), ( , ), ( , ), ( , )
( , )
T
i y i z i y i z
i
Q x t Q x t M x t M x t
p x t
=
The propulsion shafting system's boundary
conditions at its left and right ends, as illustrated in
Figure 1, are expressed as follows:
( )
1 0 1 1 0
(0, ) (0, ) (0, )p t E M w t w t E f− + + =
(2.8)
( , ) ( , ) 0
n n n n n
p L t EM w L t− − =
(2.9)
where M0=Mp+Mw, Mp is the inertia matrix of the
propeller, Mw denotes the inertia matrix of the added
water mass,
( , , , ), (1, 1, 1, 1)
p
pp
dd
M diag E diag
m m J J
= = − −
22 32 52 62
32 33 62 52
52 62 55 65
62 52 65 55
w
m m m m
m m m m
M
m m m m
m m m m
−−
=
−−
mp and Jd represent the mass and diametrical mass
moment of inertia of the propeller under dry
conditions.
gw
= +
22 32 52 62
32 33 62 52
52 62 55 65
62 52 65 55
0 0 0 0
0 0 0 0
,
0 0 0
0 0 0
gw
xx
xx
c c c c
c c c c
J c c c c
J c c c c
−−
= =
− − −
g and w denote the matrices of gyroscopic moments
induced in the propeller and viscous damping
coefficients, respectively. Jxx is the propeller polar
moment of inertia about the shaft axis. The added mass
and damping matrices, (Mw and w), are calculated
using the relationships given by Carlton [19].
The hydrodynamic excitations acting on the
propeller rotating at speed
are expressed in terms of
a Fourier series as follows
( ) ( ) ( )
( )
( ) ( )
0
0 0 0 0
cos , 1,2,... , =2,3,5,6f f f t
= + + =
(2.10)
where
( )
0
f
is the complex amplitude defined as
( ) ( )
()
0
( ) ( )
00
ˆ
, 1,2,... , =2,3,5,6
i
f f e
==
(2.11)
In the above equation,
( )
0
ˆ
f
and
()
0
represent the
real amplitude and initial phase of the
th component
of the hydrodynamic excitation in the direction, (=2,
3, 5, 6) at propeller angular velocity.
The vibrations can be found as the real parts of the
following expressions
( ) ( )
, , , ,
( , ) ( ) , ( , ) ( )
j t j t
i y i y i z i z
w x t w x e w x t w x e
==
(2.12)
( ) ( )
, , , ,
( , ) ( ) , ( , ) ( )
j t j t
i y i y i z i z
x t x e x t x e
==
(2.13)
Substituting Eq. (2.12) into differential Eqs. (2.1) and
(2.2) yields
()
( , ) ( ) , 0,1,2,...
jt
ii
w x t w x e
==
(2.14)
( )
( , ) ( ) 0,1,2,., ..
jt
ii
p x t p x e
==
(2.15)
( ) ( )
()
( ) ( )
i i i
u x C x a
=
(2.16)
( ) ( ) ( )
( ) ( )
i i i
p x D x a
=
(2.17)
In Eqs. (2.16) and (2.17),
()
1, 2,...,8
()
i iq
a = a , q =
is
the vector of solution coefficients for the ith shaft
segment, and it can be determined using the continuity
of displacements, equilibrium of forces (and moments)
as well as boundary conditions (2.8) and (2.9). While:
− for
=0 and i2, we get
( )
( )
32
32
0
2
2
0
0 0 0 0
1
1
0 0 0 0
()
3 2 1 0
0 0 0 0
0 0 0 0
3 2 1 0
6 0 0 0 0 0 0 0
0 0 0 0 6 0 0 0
()
0 0 0 0 6 2 0 0
6 2 0 0 0 0 0 0
i
i
x x x
x x x
Cx
xx
xx
D x EI
x
x
=
− − −
−
−
=
−−
1044
− for
0 and i2, we obtain
( ) ( )
( )
( )
( )
12
( ) ( ) : ( )
i i i
C x C x C x
=
and
( ) ( )
( )
( )
( )
12
( ) ( ) : ( ) ,
i i i
D x D x D x
=
where
( )
( )
( )
( )
( )
( )
1
2
3 3 3 3
1
22
cos sin c
0 0 0 0
( ) ,
0 0 0 0
sin cos
0 0 0 0
cos sin c
()
sin cos
0 0 0 0
sin cos
0 0 0 0
()
0 0 0 0
cos sin
i
i
i
x x h x sh x
Cx
x x sh x ch x
x x h x sh x
Cx
x x sh x ch x
x x sh x ch x
D x EI
x
=
−
=
− − −
− − −
=
( )
( )
22
3 3 3 3
2 2 2 2
2
,
0 0 0 0
- sin cos - -
()
- cos - sin
0 0 0 0
i
x ch x sh x
x x sh x ch x
D x EI
x x ch x sh x
−−
=
with
For the shaft segment in the stern tube bearing (i =2),
the substitution of Eq. (2.12) into differential Eqs. (2.3)
and (2.4) leads to
( )
22
( , ) ( ) , 0,1,2,...
jt
u x t u x e
==
(2.18)
( )
22
( , ) ( ) 0,1,2,...,
jt
p x t p x e
==
(2.19)
( ) ( )
()
2 2 2
( ) ( )u x C x a
=
(2.20)
( ) ( ) ( )
2 2 2
( ) ( )p x D x a
=
(2.21)
where
− for
=0, we get
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( )C x C x C x
=
and
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( ) ,D x D x D x
=
where
( )
( )
0
2
1
0 0 0 0
( ) ,
0 0 0 0
( ) ( ) ( ) ( )
V V V V
V V V V V V V V V V V V
Q N S T
Cx
S N Q T Q T N S
=
− − + +
( )
( )
0
2
2
0 0 0 0
( ) ,
( ) ( ) ( ) ( )
0 0 0 0
H H H H
H H H H H H H H H H H H
Q N S T
Cx
S N T Q Q T N S
=
− − − + − +
( )
( )
3 3 3 3
0
2
1
2 2 2 2
( ) ( ) ( ) ( )
0 0 0 0
2,
0 0 0 0
V V V V V V V V V V V V
x
V V V V V V V V
N S Q T T Q S N
D EI
T S N Q
+ + − −
=
−−
( )
( )
3 3 3 3
0
2
2 2 2 2
2
0 0 0 0
( ) ( ) ( ) ( )
2,
0 0 0 0
H H H H H H H H H H H H
x
H H H H H H H H
N S Q T T Q S N
D EI
T S N Q
+ + − −
=
−−
with
( )
22
cos( ) ( ), cos( ) ( ),
sin( ) ( ), sin( ) ( ).
cos( ) ( ), cos( ) ( ),
sin( ) ( ), sin( ) ( ),
,(
V V V V V V
V V V V V V
H H H H H H
H H H H H H
1/ 4
VH
V oil 2 H oil 2
Q x ch x N x sh x
S x ch x T x sh x
Q x ch x N x sh x
S x ch x T x sh x
= k / 4E I and = k / 4E I
==
==
==
==
).
1/ 4
For
0, four cases that can be encountered for the
shaft segment in the stern tube bearing, (i =2):
A- if
()
22
22
( ) 0
V
oil
kA
−
and
()
22
22
( ) 0
H
oil
kA
−
,
we obtain
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( )C x C x C x
=
and
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( ) ,D x D x D x
=
where
( )
( )
( )
( )
( )
( )
2
1
2
2
2
1
cos sin
0 0 0 0
( ) ,
0 0 0 0
sin cos
0 0 0 0
cos sin
( ) ,
sin cos
0 0 0 0
()
V V V V
V V V V V V V V
H H H H
H H H H H H H H
x x ch x sh x
Cx
x x sh x ch x
x x ch x sh x
Cx
x x sh x ch x
Dx
− − − −
− − − − − − − −
− − − −
− − − − − − − −
=
−
=
− − −
=
( )
( )
3 3 3 3
2 2 2 2
3 3 3 3
2
22
2
sin cos
0 0 0 0
,
0 0 0 0
cos sin
0 0 0 0
sin cos
()
cos sin
V V V V V V V V
V V V V V V V V
H H H H H H H H
H H H H H
x x sh x ch x
EI
x x ch x sh x
x x sh x ch x
D x EI
xx
− − − − − − − −
− − − − − − − −
− − − − − − − −
− − − − −
− − −
−−
− − −
=
−−
22
,
0 0 0 0
H H H
ch x sh x
− − −
with
1/4
( ) 2 2
2 2 2 2
( ) / ( )
V
oil
VV
A k E I
−−
= = −
and
1/4
( ) 2 2
2 2 2 2
( ) / ( ) .
H
oil
HH
A k E I
−−
= = −
B- If
()
22
22
( ) 0
V
oil
kA
−
and
()
22
22
( ) 0
H
oil
kA
−
, we
get
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( )C x C x C x
=
and
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( ) ,D x D x D x
=
where
( )
( )
( )
( )
( )
( )
2
1
2
2
3
2
1
0 0 0 0
( ) ,
0 0 0 0
( ) ( ) ( ) ( )
0 0 0 0
( ) ,
( ) ( ) ( ) ( )
0 0 0 0
2
V V V V
V V V V V V V V V V V V
H H H H
H H H H H H H H H H H H
VV
x
Q N S T
Cx
N S Q T Q T N S
Q N S T
Cx
S N T Q Q T N S
NS
D EI
+ + + +
+ + + + + + + + + + + +
+ + + +
+ + + + + + + + + + + +
++
=
− − + +
=
− − − + − +
+
=
( ) ( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )
3 3 3
2 2 2 2
3 3 3 3
2
2 2 2 2
2
0 0 0 0
,
0 0 0 0
0 0 0 0
2,
0 0 0 0
V V V V V V V V V V
V V V V V V V V
H H H H H H H H H H H H
x
H H H H H H H H
Q T T Q S N
T S N Q
N S Q T T Q N S
D EI
T S N Q
+ + + + + + + + + +
+ + + + + + + +
+ + + + + + + + + + + +
+ + + + + + + +
+ − −
−−
+ + − −
=
−−
with
1045
()
cos( ) ( ), cos( ) ( ),
sin( ) ( ), sin( ) ( ),
cos( ) ( ), cos( ) ( ),
sin( ) ( ), sin( ) ( ),
(
V V V V V V
V V V V V V
H H H H H H
H H H H H H
O
VV
Q x ch x N x sh x
S x ch x T x sh x
Q x ch x N x sh x
S x ch x T x sh x
k
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
++
==
==
==
==
==
1/4
22
2 2 2 2
1/4
( ) 2 2
2 2 2 2
) / (4 ) , and
( ) / (4 )
V
il
H
oil
HH
A E I
k A E I
++
−
= = −
C- If
()
22
22
( ) 0
V
oil
kA
−
and
()
22
22
( ) 0
H
oil
kA
−
, we
obtain
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( )C x C x C x
=
and
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( ) ,D x D x D x
=
with
( )
( )
2
1
()Cx
and
( )
( )
2
1
()Dx
as in case A, while
( )
( )
2
2
()Cx
and
( )
( )
2
2
()Dx
as in case B,
D- If
()
22
22
( ) 0
V
oil
kA
−
and
()
22
22
( ) 0
H
oil
kA
−
,
we get
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( )C x C x C x
=
and
( ) ( )
( )
( )
( )
2 2 2
12
( ) ( ) : ( ) ,D x D x D x
=
with
( )
( )
2
1
()Cx
and
( )
( )
2
1
()Dx
as in case B, while,
( )
( )
2
2
()Cx
and
( )
( )
2
2
()Dx
as in case A.
2.2 Transfer matrices related to the solution coefficient
vectors for different adjacent shaft segments
A junction between two adjacent shaft segments, (i)
and (i+1), is shown in Figure 2. The axes xi, yi, zi of the
ith shaft segment are parallel to the axes X, Y, Z of the
fixed local coordinate system, as illustrated in Figure 1.
The length of the ith shaft segment is denoted by
Figure 2. Schematic illustration of displacements, slopes,
shear forces and bending moments acting at junction of two
adjacent shaft segments (i) and (i+1), with different diameters
and/or physical properties.
2.2.1 Case of shaft segments (i) and (i+1) with different
mechanical properties and/or diameters
The continuity of displacements and the
equilibrium of shearing forces and bending moments,
at a junction of two shaft segments, (i) and (i+1), with
different mechanical properties and/or diameters
require that
( ) ( )
1
( ) (0)
i i i
w L w
+
=
(2.22)
( ) ( )
1
( ) (0)
i i i
p L p
+
=
(2.23)
Taking into account Eqs. (2.14-2.17), the conditions
(2.22) and (2.23) can be expressed in the following form
( ) ( ) ( ) ( )
11
( ) (0)
i i i i i
C L a C a
++
=
(2.24)
( ) ( ) ( ) ( )
11
( ) (0)
i i i i i
D L a D a
++
=
(2.25)
( ) ( ) ( )
1i i i
a B a
+
=
(2.26)
( )
( )
( )
( )
( )
1
1
1
(0) ( )
(0) ( )
i i i
i
i i i
C C L
B
D D L
−
+
+
=
(2.27)
where
( )
i
B
is a block diagonal matrix,
( )
( )
( )
( ) ( )
11
00
2
23
1 1 1 1
3
2
32
0 0 0
03
00
,
0
3 2 1 0
1
ii
ii
i i i i i
i i i
i i i i
i
ii
i i i
EI
EI
B E I E I
B B B
E I E I
B
LL
L L L
++
+ + + +
= = =
(2.28)
+
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
1 1 2 2
3 3 4 4
12
2 2 1 1
4 4 3 3
()
cos sin
sin cos
cos sin
sin cos
ii
i
l l ch l sh l
l l sh l ch l
BB
l l ch l sh l
e l e l sh l ch l
−
==
−
(2.29)
( )
( )
( )
( )
( )
( )
( )
( )
22
12
22
1 1 1 1 1 1
33
34
33
1 1 1 1 1 1 1 1
11
, ,
2 2 2 2
,
2 2 2 2
i i i i i i
ii
i i i i i i
i i i i i i i i
ii
i i i i i i i i
E I E I
E I E I
E I E I
E I E I
+ + + + + +
+ + + + + + + +
= + = −
= + = −
(2.30)
Thus, the relationships between the vectors of
solution coefficients
( )
i
a
and
( )
1i
a
+
are written in
matrix form. For the (r-1) following shaft segments, we
have the relationship:
( ) ( ) ( ) ( ) ( )
12
....
i r i r i r i i
a B B B a
+ + − + −
=
(2.31)
For long shaft line vibration, the number of
multiplied matrices in Eq. (2.31) may be large,
however, compared with the classical transfer matrix
method (TMM), the matrices
( )
i
B
present an
advantageous property for shaft segments with the
same mechanical and geometric properties, that
reduces the number of multiplied matrix product in Eq.
(2.31) to a single matrix as follows
( ) ( ) ( )
i s i i
aa
+
=
(2.32)
where the matrix
( )
i
is obtained from the matrix
( )
i
B
by setting
( )
( )
( )
( )
13
1
ii
==
,
( )
( )
( )
( )
24
0
ii
==
and
replacing Li by
1p i r
ip
pi
LL
= + −
=
=
(2.33)
The matrix
( )
i
can be understood as a transfer
matrix associated with the solution coefficient vectors
( )
i
a
and
( )
1i
a
+
. However, it should not be confused with
the transfer matrix commonly used in the well-known
Transfer Matrix Method (TMM).
1046
2.2.2 Case of adjacent shaft segments (i) and (i+1) joined
by a discrete mass mi
When the section connecting two adjacent shaft
segments, (i) and (i+1), passes through a gravity center
of discrete mass mi with a diametrical mass moment of
inertia Ji (e.g. flange coupling, sun gear of reduction
gear box), the continuity of displacements and the
equilibrium of forces (and moments) require that
( ) ( )
1
( ) (0)
i i i
w L w
+
=
(2.34)
( ) ( ) 2 ( )
1
( ) (0) ( )
i i i i i i
p L p M w L
+
=+
(2.35)
where Mi is the diagonal inertia matrix of the discrete
mass mi located at the end of the ith shaft segment
( , , , )
i
ii
ii
M diag
m m J J
=
Equations (2.34) and (2.35) can be expressed as
( ) ( )
( ) ( )
11
( ) (0)
i i i i i
C L a C a
++
=
(2.36)
( ) ( ) ( ) ( ) ( ) ( )
2
11
( ) (0) ( )
i i i i i i i i i
D L a D a M C L a
++
=+
(2.37)
Thus, the relationships between the solution
coefficient vectors
( )
i
a
and
( )
1i
a
+
, for two shaft
segments joined at a discrete mass, can be formulated
in a matrix form as
( ) ( ) ( )
1i i i
a R a
+
=
(2.38)
where
( )
( )
( )
( )
( ) ( )
1
1
2
1
(0) ( )
(0) ( ) ( )
i i i
i
i i i i i i
C C L
R
D D L M C L
−
+
+
=
−
(2.39)
( )
i
R
serves as the transfer matrix that relates the
solution coefficient vectors
( )
i
a
and
( )
1i
a
+
via mass (i).
2.2.3 Case of intermediary anisotropic supporting bearing
In such a case, the continuity of deformations and
equilibrium of internal forces and moments require
that
( ) ( )
1
( ) (0)
i i i
w L w
+
=
(2.40)
( ) ( ) ( )
1
( ) (0) ( )
i i i i i i
p L p EK u L
+
=+
(2.41)
where Ki is the stiffness matrix of the oil film in the ith
intermediate bearing, defined as follows
22 23
32 33
00
00
0 0 0 0
0 0 0 0
i
kk
kk
K
=
The equations (2.40) and (2.41) can be expressed in
the following form
( ) ( )
( ) ( )
11
( ) (0)
i i i i i
C L a C a
++
=
(2.42)
( ) ( ) ( ) ( ) ( ) ( )
11
( ) (0) ( )
i i i i i i i i i
D L a D a EK C L a
++
=−
(2.43)
Writing Eqs. (2.42) and (2.43) in matrix form
produces
( ) ( ) ( )
1i i i
a S a
+
=
(2.44)
where
( )
( )
( )
( )
( ) ( )
1
1
1
(0) ( )
(0) ( ) ( )
i i i
i
i i i i i i
C C L
S
D D L EK C L
−
+
+
=
+
(2.45)
where
( )
i
S
is the transfer matrix through the
intermediate bearing (i).
Using the above relationships between different
adjacent shaft segments, the vectors of solution
coefficient
( ) ( ) ( )
23
, ,...,
n
a a a
for the vibrations from the
second to the nth shaft segment can be determined, as
function of
()
1
a
, which represents the coefficient vector
for lateral vibrations of the first shaft segment. For the
system as shown in Figure 1, we get
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
()
2 1 1
( ) ( )
3 2 1 1
()
4 3 2 1 1
( ) ( )
5 4 3 2 1 1
( ) ( )
1 4 3 2 1 1
,
,
,
,
... .
nn
a B a
a B B a
a R B B a
a S R B B a
a B S R B B a
−
=
=
=
=
=
(2.46)
3 FREE VIBRATION ANALYSIS
Similarly, the boundary conditions (2.8) and (2.9) can
be written in the following matrix
( )
( )
0
1
0
Ef
a
=
(3.1)
where
( )
( )
( )
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
0
11 12 18
2
()
1,0 0 1,0
21
2
1 5 4 3 2 1
81 88
==
( ) ( ) .....
n n n n n n
D j c EM C
D L EM C L B B S R B B
−
− + −
−+
...
...
...
...
(3.2)
with
( ) ( ) ( ) ( ) ( )
1 1,1 1,2 1,3 1,8
, , ...
T
a a a a a
=
.
3.1 Natural frequencies and associated mode shapes
For
=1, free vibrations occur when
( )
1
0
0f =
, and the
characteristic equation is given by
( )
1
1
.0a=
(3.3)
with
(1) (1) (1) (1) (1)
1 1,1 1,2 1,3 1,8
, , ...
T
a a a a a
=
.
Nontrivial solution for the column vector
( )
1
1
a
,
requires that
0=
(3.4)
The natural frequencies for the non-rotating shaft
line may be obtained by solving the above eigenvalue
equation (3.4). For each value of natural frequency, the
1047
matrix in equation (3.4) is singular. In such cases,
Natanson's technique [20], which use the algebraic
complements, can be applied to determine the
eigenvector for the first shaft segment, up to a
multiplicative constant
:
(1) (1) (1)
1,1 1,1 1,2 1,2 1,8 1,8
. , . , ... , .a a a
= = =
(3.5)
22 23 28 21 23 28 21 22 27
32 22 22
11 12 18
82 88 81 88 81 87
= , = - ,..., = -
... ... ...
... ... ...
... ... ...
... ... ...
Thus, the mode shapes of vertical and horizontal
bending vibrations of the first shaft segment, (i=1), can
be determined as
( ) ( ) ( ) ( )
1 1 1 1
1, 11 12 13 14
( ) cos( ) sin( ) ( ) ( )
y
w x x x ch x sh x
= + + +
(3.6)
( ) ( ) ( ) ( )
1 1 1 1
1, 15 16 17 18
( ) cos( ) sin( ) ( ) ( )
z
w x x x ch x sh x
= + + +
(3.7)
with
1 11 12 18
, , ... ,
T
=
For the remaining of shaft segments, (i=2,3...n), the
eigenvectors,
2 3 8
( , , ... , )
, can be determined
using the relationships given by Eq. (2.46), as follows
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1
2 1 1
11
3 2 1 1
111
4 3 2 1 1
1111
5 4 3 2 1 1
1 1 1 1 1
1 4 3 2 1 1
,
,
,
,
... .
nn
B
BB
RBB
S R B B
B S R B B
−
=
=
=
=
=
(3.8)
3.2 Forced bending vibrations analysis
The coefficient vector for forced whirling vibration of
the first shaft segment can be determined from Eqs.
(3.1) as follows
( )
( )
( )
( )
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
0
1
2
()
1 0 1
0
1
2
1 5 4 3 2 1
(0) (0)
, 1,2,...
0
( ) ( ) .....
n n n n n n
D j c EM C
Ef
a
D L EM C L B B S R B B
−
−
− + −
==
−+
(3.9)
Using the relationships given by equation (2.46), the
vectors of constant coefficients
( ) ( ) ( )
23
, ,...,
n
a a a
can be
easily calculated.
4 APPLICATION TO A TYPICAL MARINE
PROPELLER SHAFTING SYSTEM
A marine propeller shafting system of a real cargo ship
(see figure 3) is considered to calculate the static
deformations and the natural frequencies with their
associated mode shapes of lateral vibrations. The
responses of the system under the first and second
blade frequency excitations are also examined. The
analysed system consists of a propeller, stepped
propeller shaft composed of three shaft segments ( (1),
(2), and (3)) supported by an anisotropic continuous
bearing of Winkler’s type, intermediate shaft
composed of two shaft segments ( (4) and (5))
supported by an anisotropic bearing, a reduction gear
supported by two anisotropic bearings and composed
of four shaft segments ( (6), (7), (8) and (9)), and a thrust
bearing. The given data for Figure 5 are as follows:
Young's modulus
11 2
2.1 10 /
i
E N m=
, and mass
density
3
7850 /
i
kg m
=
. The other parameters of the
shafts and propeller are provided in Table 1 and Table
2, respectively.
Figure 3. Analysed marine propeller shafting system.
Table 1. The details of marine shafting system.
Length
L1
L2
L3
L4
L5
L6
L7
L8
L9
0.840m
3.560m
1.550m
1.650m
4.800m
0.850m
0.970m
0.975m
0.922m
Diameter
d1
d2
d3
d4
d5
d6
d7
d8
d9
0.56m
0.555m
0.565m
0.405m
0.405m
0.420m
0.420m
0.420m
0.420m
Mass
m3
m5
1363.6kg
1363.6kg
Diametrical mass
moment of
inertia
J3
J5
82.4109kgm
2
82.4109kgm
2
Intermediate
bearing
k22
k23
k32
k33
2.56667 x 10
7
N/m
1.64815 x 10
7
N/m
0.32963 x 10
7
N/m
0.89 x 10
7
N/m
Reduction gear
bearing
(propeller side)
kyy
kzz
1.2613 x 10
7
N/m
6.8869 x 10
7
N/m
Reduction gear
bearing (motor
side)
kyy
kzz
1.2223 x 10
7
N/m
6.5366 x 10
7
N/m
Stern tube
bearing
k
V
oil
k
H
oil
9.37537 x 10
6
N/m
2
4.49802 x 10
7
N/m
2
1048
Table 3. Propeller hydrodynamic excitations.
Harmonic of
blades
Amplitude and phase of variables hydrodynamic excitations
()
02
()fN
()
02
()
03
()fN
()
03
()
05
( . )f N m
()
05
()
06
( . )f N m
()
06
=1 x z=4
10220
153°
9744
56°
46887
149°
47431
58°
=2 x z=8
1173
17°
1917
-66°
7773
2°
12555
-83°
=0
Value of constant hydrodynamic excitations
(0)
02
()fN
(0)
03
()fN
(0)
05
( . )f N m
(0)
06
( . )f N m
288
12713
1662
119175
According to pressure measurements on the
submerged part of the ship's stern and calculations
using lifting surface theory conducted by the
shipbuilder, the first and second blade harmonics are
considered in this analysis. The propeller
hydrodynamic excitations are expressed in the
following form
( ) ( ) ( )
( )
( ) ( )
0
0 0 0 0
cos , 4,8 , =2, 3, 5, 6f f f t
= + + =
Table 2. Parameters of the propeller.
Parameters
Value
Number of blades
z=4
Mass
mp=13500kg
Diametrical mass moment of
inertia
Jyy=9996kgm
2
Polar mass moment of inertia
Jp=19992kgm
2
Nominal speed
Ω =112 rpm
Diameter
D=5.5m
The propeller hydrodynamic excitations are shown
in Table 3.
5 NUMERICAL RESULTS AND DISCUSSION
The natural frequencies and associated mode shapes as
well as global responses for lateral vibrations of the
propulsion shafting system were calculated using a
developed computer programs. The natural
frequencies and their associated mode shapes are
presented in Table 5 and Figures 4-11, respectively.
The major propeller excitations at MPP (Maximum
Propulsion Power) are listed in table 4. The rotation
speed at MPP is Ω =112 rpm.
Table 4. The major propeller excitations at MPP.
Order
Excitation frequency [rpm]
4. Ω
448
8. Ω
896
Table 5. Natural frequencies of the lateral vibration mode
shapes.
Mode shape
Frequency [rpm]
Separation margin
(Excitation order)
1
st
238
-46,87% (4.Ω)
2
nd
365
-18,52%\ (4.Ω)
3
rd
561
-37,38% (8.Ω)
4
th
758
-15,40% (8.Ω)
Figure 4. First mode shapes of vertical and horizontal
vibrations.
Figure 5. Second mode shapes of vertical and horizontal
vibrations.
Figure 6. Third mode shapes of vertical and horizontal
vibrations.
1049
Figure 7. Fourth mode shapes of vertical and horizontal
vibrations.
Figure 8. The first whirling mode shape of lateral vibrations.
Figure 9. The second whirling mode shape of lateral
vibrations.
Figure 10. The third whirling mode shape of lateral
vibrations.
Figure 11. The fourth whirling mode shape of lateral
vibrations.
Equations (2.1) and (2.4) were also employed to
analyse the deformations caused by the constant
components of the propeller excitations (i.e., for
=0),
discarding the (
2
,
2
( , )
iy
w x t
t
and
2
2,
2
( , )
y
w x t
t
) terms and
adding the distributed load qi from the weight of the ith
shaft segment, to the right-hand side of these
equations.
Figure 12 illustrates the static vertical deflection
Y1(x) caused by the distributed weight of the shaft line,
including the propeller and flange couplings. In
contrast, Figure 13 depicts the static vertical deflection
Y(x) induced by the constant components of the
propeller excitations. The superposition of both effects
is illustrated in Figure 14.
It is concluded that vertical deflection decreases
during propulsion system operation due to the
constant components of propeller excitations (moment
and force).
The decrease at the propeller level (from 6 mm to
5 mm) is crucial for the stern tube bearing's
performance, as it decreases unit pressure and
influences oil film thickness.
41
Figure 12. Vertical deflection Y1(x) due to the distributed
weight of the shaft line.
1050
Figure 13. Vertical deflection Y(x) due to constant component
of propeller excitations.
Figure 14. Total vertical deflection Y2(x)=Y1(x)+Y(x)
Figure 15. Horizontal deflection Z(x) due to the constant
component of propeller excitations.
Figure 15 depicts the horizontal deflection of the
propulsion shafting system resulting from the constant
components of propeller excitations
Since the separation margins for the second and
fourth modes were within 20% of the (4Ω) and (8Ω)
excitation frequencies at MPP, an additional harmonic
excitation response analysis was conducted to
investigate the effect of propeller excitations on the
magnitude of the lateral displacement of the shafting
system.
The vertical and horizontal vibration amplitudes
are plotted in Figures 16 and 17, respectively. The
vertical and horizontal displacements at the propeller
do not exceed 0.3 mm. Therefore, lateral vibration
issues are not expected in the propulsion shafting
system under normal operating conditions.
Figure 16. Amplitudes of vertical vibrations.
Figure 17. Amplitudes of horizontal vibrations.
6 CONCLUSION
In this work, the matrix formulation of relationships
derived from equilibrium and compatibility equations
at each junction of adjacent shaft segments enables
efficient computation of natural frequencies and lateral
whirling vibration amplitudes, considering the
propeller's gyroscopic effect, as well as the added mass
and damping of seawater.
The constant deflection due to the weight of the
shafts, propeller, and flange couplings is significantly
greater than the transverse vibration amplitudes in the
analysed shaft line.
The constant components of the bending force and
moment from the operating propeller can contribute to
the straightening of the propeller shaft, reducing the
weight-induced deflection of the propeller shaft end by
approximately 1 mm in the analysed system, thereby
improving the operating conditions of the stern
bearing. However, this does not reduce the importance
of static calculations in ensuring proper shaft line
alignment and uniform load distribution on the radial
bearings.
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